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Trick or Truth Essay Contest (2015)
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How Mathematics Meets the World by Tim Maudlin
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Author Tim Maudlin wrote on Jan. 26, 2015 @ 21:49 GMT
Essay AbstractThe most obvious explanation for the power of mathematics as the language of physics is that the physical world has the right sort of structure to be represented mathematically. But what this in turn means depends on the mathematical language being used. I first briefly review some of the physical characteristics required in order to unambiguously describe a physical situation using integers, and then take up the much more difficult question of what characteristics are required to describe a situation using geometrical concepts. In the case of geometry, and particularly for the most basic form of geometry— topology—this is not clear. I discuss a new mathematical language for describing geometrical structure called the Theory of Linear Structures. This mathematical language is founded on a different primitive concept than standard topology, on the line rather than the open set. I explain how some other geometrical concepts can be defined in terms of lines, and how in a Relativistic setting time can be understood as the feature of physical reality that generates all geometrical facts. Whereas it is often said that Relativity spatializes time, from the perspective of the Theory of Linear Structures we can see instead that Relativity temporalizes space: all of the geometry flows from temporal structure. The Theory of Linear Structures also provides a mathematical language in which the fact that time is a fundamentally directed structure can be easily represented.
Author BioTim Maudlin is Professor of Philosophy at NYU. He received his B.A. in physics and philosophy from Yale and his Ph.D. in History and Philosophy of Science from the University of Pittsburgh. His books include Quantum Non-Localtiy and Relativity (Blackwell), The Metaphysics Within Physics (Oxford), Philosophy of Physics: Space and Time (Princeton), and New Foundations for Physical Geometry: The Theory of Linear Structures (Oxford). He has been a Guggenheim Fellow.
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Edwin Eugene Klingman wrote on Jan. 27, 2015 @ 19:14 GMT
Dear Tim Maudlin,
You emphasize that associating a mathematical structure with physical items is
not the same as postulating that they
are mathematical entities. I fully agree with this.
As you note, boundaries and structural integrity through time is sufficient for enumeration, and, per Kronecker, given integers, the rest of math follows.
You then ask, what...
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Dear Tim Maudlin,
You emphasize that associating a mathematical structure with physical items is
not the same as postulating that they
are mathematical entities. I fully agree with this.
As you note, boundaries and structural integrity through time is sufficient for enumeration, and, per Kronecker, given integers, the rest of math follows.
You then ask, what features must a physical entity have in order to display a geometrical structure? Perhaps this question has been less frequently asked because there is no obvious answer. You discuss topology as the standard basis of continuity based on open sets. I find open sets highly abstract and artificial, and difficult to relate to physical reality in the natural way that integers (counting) relate to discrete physical entities. It is therefore of some interest that you switch tools to your
Theory of Linear Structures and ordered, i.e., directed lines. As this is, in my opinion, a preferred approach to geometry, it then suggests the question, "What physical feature of the universe might be responsible for creating lines?" Your suggestion that it is
time that underlies geometry is both novel and fascinating.
Thank you for your essay presenting these ideas to the FQXi community. My own ideas seem compatible with your ideas. In my
Automatic Theory of Physics (1979) I essentially identified directed lines as time. Once lines are associated with physical time as the basis of linear ordered events underlying geometry, one can then go to second-order and employ
geometry to construct sequential switching (see my 'End Notes') to construct automata, which can then be mapped into any axiom-based physical theory, answering Wigner.
I invite you to read and comment on my current essay. I believe you'll find it compatible with your essay, while conflicting with your work on Bell. I'm interested in your comments.
Best,
Edwin Eugene Klingman
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Author Tim Maudlin replied on Jan. 27, 2015 @ 21:13 GMT
Dear Dr. Klingman,
Thanks for your remarks. I will be reading other essays once the semester here settles down a bit and look forward to yours.
Regards.
Tim Maudlin
Demond Adams wrote on Jan. 27, 2015 @ 21:02 GMT
Tim,
Thank you for contributing an interesting essay. Perhaps I am misinterpreting your ideas, but for clarification, are you suggesting in an effort to understand the fundamentals of mathematics in physics, we use a different language or include more "descriptive fundamental concepts or words"? Furthermore, is it your argument that the physics we describe in our universe is somewhat...
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Tim,
Thank you for contributing an interesting essay. Perhaps I am misinterpreting your ideas, but for clarification, are you suggesting in an effort to understand the fundamentals of mathematics in physics, we use a different language or include more "descriptive fundamental concepts or words"? Furthermore, is it your argument that the physics we describe in our universe is somewhat coincidentally predisposed to an application of fundamental mathematics and if so, why is it? Perhaps I am a bit confused with my interpretation and will look again.
Nevertheless, the universe is dynamic which is why relativity needed a serious upgrade from Einstein introducing GR. I find the definition of any abstract concept, event, or object leads to ambiguity (fuzziness), therefore the interpretation of these observations described in physics leads to relative descriptions in which we use mathematics to explain them quantitatively.
Finally I am curious about your definition of time. (Excuse the pun, but only if you have time and don't mind?)
Thanks for your essay - I look forward to your reply.
Best Regards,
D.C. Adams
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Author Tim Maudlin replied on Jan. 27, 2015 @ 21:25 GMT
Dear Demond,
The issue is not so much words (although of course we have to use words to convey what we mean) but rather what the fundamental mathematical concepts are in a particular mathematical theory. The application of those concepts typically requires that certain axioms be satisfied, so the clearest way to understand why the mathematical concepts would apply usefully to the physical...
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Dear Demond,
The issue is not so much words (although of course we have to use words to convey what we mean) but rather what the fundamental mathematical concepts are in a particular mathematical theory. The application of those concepts typically requires that certain axioms be satisfied, so the clearest way to understand why the mathematical concepts would apply usefully to the physical world is to postulate that the physical world contains physical structures that satisfy the axioms. Since different branches of mathematics use different basic concepts explicated by different axioms, the choice of one or another of these mathematical languages makes a difference to how we might postulate the sort of physical structure there is. None of this is controversial, I think, but we often use a particular set of mathematical tools just because they are the only ones around. By constructing different mathematical structures we have a broader set of choices, and also become more aware of the tacit decisions we make when employing one or another mathematical tool.
The way to get ambiguity and fuzziness out of a description is to have the foundational concepts precisely defined via clear axioms. The particular set of mathematical concepts I define is just as exact and unambiguous as the standard concepts, but because they rest on different axioms they have different application to systems.
I do not try to "define" time: it seems like a very good choice of a physically primitive feature of the world that is not analyzed in terms of anything else. The point is rather than if time orders events, and if furthermore the way it does is well described by Relativity, then the particular geometrical language I have developed has clear application to the physical world.
Regards,
Tim
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Koorosh Shahdaei wrote on Jan. 29, 2015 @ 19:58 GMT
Hi Tim,
Thanks for the enjoyable essay. Although i share your view that "The physics ... physical world has the right sort of structure to be represented mathematically", but I think it is not end of the story, It is to say, in same physical world we have e.g. rise of self-consciousness in particles, that can't be represented mathematically. The other words, there are structures that can't fit into math either partially or entirely.
The essential fact is that whatever thing that has a quantity can fit into math, for example the true mechanism behind the quantum entanglement can't be observed and hence doesn't have a quantity and can't be fitted into.math. These are what I had to say in my article.
Kind regards
Koorosh
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Author Tim Maudlin replied on Jan. 29, 2015 @ 21:18 GMT
Dear Koorosh,
I agree that consciousness raises a very difficult problem, which I do not pretend to be able to solve. Since physics has gotten along so well dealing with non-conscious systems, there should be a way to understand that which is independent of that issue. That is all I was trying to discuss.
Regards,
Tim
Florin Moldoveanu wrote on Jan. 30, 2015 @ 03:21 GMT
Dear Professor Maudlin,
You make excellent point in your essay. I want to comment on three ideas. First, on “The
Unreasonable Relevance of Some Branches of Mathematics to Other Branches”, I would recommend Connes’ essay “A view of mathematics: ”http://www.alainconnes.org/docs/maths.pdf (see page 3): “there is just “one” mathematical world, whose exploration is the...
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Dear Professor Maudlin,
You make excellent point in your essay. I want to comment on three ideas. First, on “The
Unreasonable Relevance of Some Branches of Mathematics to Other Branches”, I would recommend Connes’ essay “A view of mathematics: ”http://www.alainconnes.org/docs/maths.pdf (see page 3): “there is just “one” mathematical world, whose exploration is the task of all mathematicians and they are all in the same boat somehow.” “Where things get really interesting is when unexpected bridges emerge between parts of the mathematical world that were previously believed to be very far remote from each other in the natural mental picture that a generation had elaborated. At that point one gets the feeling that a sudden wind has blown out the fog that was hiding parts of a beautiful landscape.”
Then on “Why should any mathematical concepts that do not fall into the class of naturally suited ones nonetheless be of use in physics?” I can offer a modest suggestion: all mathematical structures are “unique” but only some of them are “distinguished”, meaning they are used by nature. For example, SO(3,1) is the group that describes special theory of relativity, and this is distinguished from say SO(345,2411) which has no role in nature. Why are some mathematical structures distinguished? Because they are consequences of physical principles (see my essay on this http://www.fqxi.org/community/essay/winners/2009.1#Moldovean
u ).
Last, on continuous and discrete spaces, the proper formalism is that of non-commutative geometry (see the large table in http://fmoldove.blogspot.com/2014/01/solving-hilberts-sixth-
problem_17.html from http://arxiv.org/pdf/math/0408416v1.pdf which shows how non-commutative mathematical structures generalize the commutative ones). This has extremely deep implications for the Standard Model:
http://fmoldove.blogspot.com/2014/11/clothes-for-standard-mo
del-beggar.html where instead of quantizing gravity Connes is geometrizing the theory from a strange space: a Cartesian product of a manifold with a set of two points. See also http://www.mth.kcl.ac.uk/~streater/lostcauses.html#XXII of why the other way around does not work.
Sincerely,
Florin Moldoveanu
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Author Tim Maudlin replied on Jan. 30, 2015 @ 04:55 GMT
Dear Prof. Moldoveneanu,
Thanks for the message. The comment by Connes is on target, of course. I recently was discussing my work with Shahn Majid, who also works on non-commutative geometry, and he did find commonalities and points of contact between the two. But I doubt that the two approaches will yield exactly the same formalism or results, so there is more to explore about the...
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Dear Prof. Moldoveneanu,
Thanks for the message. The comment by Connes is on target, of course. I recently was discussing my work with Shahn Majid, who also works on non-commutative geometry, and he did find commonalities and points of contact between the two. But I doubt that the two approaches will yield exactly the same formalism or results, so there is more to explore about the similarities and differences. In the most radical version of the non-commutative approach you do not recover an underlying point set, but in my approach there always is one. Whether one regards that as an advantage or a disadvantage would require a long discussion.
Regards,
Tim Maudlin
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Florin Moldoveanu replied on Feb. 2, 2015 @ 04:12 GMT
I was wondering in relationship to your question: "If it is correct, then we might see how the time itself creates the geometry of space-time" if you are aware of 1963 Zeeman's result: causality implies the Lorentz group: https://download.wpsoftware.net/causality-lorentz-group-zeem
an.pdf
Time demands causality and Zeeman's result (which was independently discovered by 2 more people) is...
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I was wondering in relationship to your question: "If it is correct, then we might see how the time itself creates the geometry of space-time" if you are aware of 1963 Zeeman's result: causality implies the Lorentz group: https://download.wpsoftware.net/causality-lorentz-group-zeem
an.pdf
Time demands causality and Zeeman's result (which was independently discovered by 2 more people) is quite powerful and I think it might represent the end goal of your program. I also see natural links with Sorkin's causal set program and with efforts to derive special relativity in the framework of first order logic.
Florin
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Author Tim Maudlin replied on Feb. 2, 2015 @ 04:31 GMT
Dear Florin,
Yes, the Zeeman result is familiar to me, although it is actually much too restrictive for my purposes, as it presupposes a Minkowski metric, and we want to have a theory that can describe all of the the solutions to the General Relativistic Field Equations, and even other possible geometries (e.g. discrete geometries) as well. What is easy to show in my setting is that the...
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Dear Florin,
Yes, the Zeeman result is familiar to me, although it is actually much too restrictive for my purposes, as it presupposes a Minkowski metric, and we want to have a theory that can describe all of the the solutions to the General Relativistic Field Equations, and even other possible geometries (e.g. discrete geometries) as well. What is easy to show in my setting is that the Linear Structure of a General Relativistic space-time already contains the complete conformal structure. Contrast this with the standard approach, in which the topology is just a four-dimensional manifold, that contains no information about the light-cone structure. To recover the entire Relativistic metric, in the continuum case, one needs to add a temporal metric. All of this is uncontroversial.
However, if one uses these resources to build a discrete space-time structure, then one can use the counting measure to complete the metric...but you have to be careful to count the right thing! It is natural to see similarities to Causal Sets here, but the actual implementation is quite different. The way I do it, the fundamental lines that constitute the geometry of a discrete space-time are all light-like. But in Causal Sets, the chance of any of the event being light-like related is zero (using the usual sprinkling method). I did not have space to go into all this in the essay.
Cheers,
Tim
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Florin Moldoveanu replied on Feb. 3, 2015 @ 00:35 GMT
Here is another reference, a work by Jochen Rau http://arxiv.org/pdf/1009.5523.pdf on reconstructing general relativity (see Fig 1 for his line of argument). I do have a question about your framework: how do you get to curvature? Are your “lines” really straight lines, or geodesics?
Can I also ask you a question in a different area? If one introduces a new quantum mechanics interpretation, what questions must one address? For reference here is a draft of a paper on a new QM interpretation I am working on: http://fmoldove.blogspot.com/2015/01/the-composability-inter
pretation-before.html
Florin
PS: I don't know if you associate the face with the name, I go every year to the New Directions in the Foundations of Physics conference in DC.
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Lawrence B Crowell wrote on Feb. 4, 2015 @ 11:50 GMT
Tim,
I thought your essay was interesting, though I have somewhat different ideas about things. I think spacetime is emergent from quantum entanglement. The emergence of time occurs in the Wheeler De-Witt equation HΨ[g] = 0. The wave functional is defined on an entire spatial manifold, but in general spatial slices only have diffeomorphisms with each other that define time on a local chart or patch. We may then consider the projection onto Hilbert subspaces H_i ⊂ H, P_i so that
P_i|Ψ[g]> = e^{θ_i}|ψ_i>,
which may be accomplished with a sum over other states
P_i = sum_{j=!i}|ψ_j>
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Lawrence B Crowell replied on Feb. 4, 2015 @ 11:53 GMT
I forgot that this editor does not like carrots, so I use parentheses instead.
P_i = sum_{j=!i}|ψ_j)(ψ_j|. It is simplest of course to consider i = 1 or 2 for two different regions. The relative phase θ_i = ω_it, and defines a local time.
A simple case of this is the de Sitter spacetime with two patches in static coordinates or standard dS slicing, such as seen in the diagram below. In this case there are two patches with different time “arrows,” or better put two sets of diffeomorphisms that correspond to local spacetime.
Cheers LC
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Author Tim Maudlin replied on Feb. 4, 2015 @ 12:44 GMT
Dear Lawrence,
Your proposal is a bit compressed here, but one obvious question, apart from others, is where the ω you mention comes from. Without it, you do not define a t. In the usual case, ω is E, the eigenvalue in an energy eigenstate, and the energy eigenstates are the eigenstates of H. But of course, in Wheeler-deWitt H annihilates the state, so the eigenvalue is zero. That is the classic problem of time in Wheeler-deWitt.
Also, if you are somehow dealing with deSitter, the you already have a space-time structure (since the Ψ[g] is defined over the entire space of metrics, how did we get to de Sitter?), so the "emergence" problem must already have been solved to make sense of the rest of the construction. How, in that case, does time "emerge"?
Regards,
Tim
Lawrence B Crowell replied on Feb. 4, 2015 @ 19:20 GMT
The WDW equation does not have any time, for spacetimes in general relativity most often do not have a single set of diffeomorphisms that include the entire manifold. The conjugate meaning of this is that with E conjugate to t there is no manner in most manifolds by which one can form a Gaussian surface to define mass-energy. Mass-energy is not localizable. This only happen for stationary spacetimes with some asymptotic flatness.
What I mentioned above is a situation where the Hilbert space for the WDWE is partitioned into two parts so that locally in the manifold corresponding to either of them time can be maybe defined. The states for the two Hilbert space subsets are entangled and this entanglement is exchanged with the occurrence of horizons separating the two regions. The horizons, such as in a black hole, are entangled with states in the exterior. This is in effect a sort of coarse graining, and spacetime is a coarse graining of entangled states. This coarse graining reflects a lack of information about the nature of the entanglement.
Cheers LC
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Gary D. Simpson wrote on Feb. 6, 2015 @ 10:20 GMT
Tim,
This is a very interesting essay. I have one minor criticism. The first four pages are very philosophical. That's ok, no problem. You lay out some history, some limitations, etc ... Basically counting and geometry. Then at the end of page four you introduce the true objective of your essay. A lazy reader would be denied this new knowledge:-)
Regarding your new topology, is it necessary to use two of the same thing to create the linear structure? By this I mean can you only form a linear structure using geometric points? The reason for my question is that it appears to me that you are applying topology with its vast legacy knowledge to Hamilton's quaternions.
All in all, very well done. Thank you for sharing these new ideas.
Best Regards and Good Luck,
Gary Simpson
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Author Tim Maudlin replied on Feb. 7, 2015 @ 01:26 GMT
Hi Gary,
Since I am proposing the Linear Structure as the most fundamental geometrical structure that exists, the objects over which it is defined are automatically geometrical points, by which I mean they have no intrinsic geometrical structure. I get that for free. But of course, they could differ in all kinds of other ways. They could have different intrinsic properties (as an illustrative example, think of them as having different electrical charges), but these would not be geometrical properties. The structure of these other properties would also be unconstrained by the Linear Structure over the elements. One could, I imagine, use quaternions to represent these intrinsic properties, but then the theory is going beyond geometry.
Thanks, and best wishes,
Tim
Sylvain Poirier wrote on Feb. 7, 2015 @ 17:09 GMT
Hello. I have several remarks about your work, first about general topology, then about physics.
I also considered this problem, of how else topology might be formalized so as to be more convenient for different purposes, may it be easiness of use, restrictions or generalizations, or modified kinds of structures (usually more rigid than some topology, since topology is usually the lightest...
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Hello. I have several remarks about your work, first about general topology, then about physics.
I also considered this problem, of how else topology might be formalized so as to be more convenient for different purposes, may it be easiness of use, restrictions or generalizations, or modified kinds of structures (usually more rigid than some topology, since topology is usually the lightest non-trivial interesting structure that may be found in a given space), and I came to consider other options, I will share with you.
First, I agree with you about the importance of linear orders: it is a remarkable property of lines that their topology is equivalently expressible in that very simple formalism (linear order), unlike other kinds of topological spaces, so we may consider to use this fact to define general topological spaces by somehow "reducing" them to this 1-dimensional case, explicitly involving lines somewhere in the general definition of topology. However there are different possible ways of doing so, and we need to see the larger picture of possible methods and how they may differ, in order to find out which method may be more relevant for given purposes, instead of picking up a fixed method just because it is the first method we happen to think of.
Let us consider this problem in its abstract generality : if there is a general kind of "spaces" we have intuitive ideas of, but we do not know how to formalize (by which kind of operation or relation, as none seems to fit), then can we still find a method to "anyway simply formalize it no matter what a subtle kind of system it is", that will be general enough to include topology ?
As surprising as it may seem, the answer is yes.
Let us explain this amazingly general and still simple toolbox.
First step is to look at the intended kind of "spaces" as forming a category: no matter how topology may be formalized, we expect the concept of "continuous function" between spaces to make sense. We have a clear intuition what it means for a function to be continuous, so there should exist a definite set of all continuous functions from a space to another, the only problem is that we cannot write a definition for it since we did not formalize what is the topological structure we want these functions to preserve. Never mind, we will define this later. At first, let us just admit this concept of continuous function as primitive, forming a category.
Now the problem is: starting with an arbitrary category, is there any general, systematic way to interpret its objects as systems with a kind of structures, so that the morphisms in this category will happen to preserve these structures, and that will still be interesting even if (as in usual topological spaces) there does not exist any non-trivial invariant algebraic operation in these spaces ? We can make it, and here is how.
First we need to pick up a fixed object K in the category, with the quality of being "flexible enough" to serve as the prototype from which the structure of other objects will be formalized. For the needs of expressing topology in physics, we can take K=R (the line of real numbers).
For even more general cases where one fixed object (space) is not "flexible enough", we can take a series of different spaces that would hopefully "complete each other" in their exhaustion of the different kinds of possible shapes.
Now with a fixed K, here is how to interpret any object M in the category as a system with structures. In fact there are 2 structures we can automatically define on all objects M, that will be preserved by all morphisms in the category:
1) The set Mor(K,M) of all morphisms from K to M;
2) The set Mor(M,K) of all morphisms from M to K.
Indeed, for any objects M and N, any morphism from M to N maps Mor(K,M) into Mor(K,N), and also Mor(N,K) into Mor(M,K). I wrote
a page with a representation theorem that goes deeper than what I found usually done elsewhere on the topic.
Now as compared to this, your approach can be described as follows:
- You take as K not just the real line, but all possible linearly ordered sets;
- You only consider the embeddings of such K into all possible M, instead of any continuous functions. You refuse those mapping 2 non-neighbor points into neighbor points (in the case of finite graphs).
Some questions:
- It may look good to handle continuous lines as well as discrete lines in similar ways, however what is the use of "mixing" them by not formally assuming K as fixed, so as to admit the case of hybrid systems ? I imagine that sometimes we need to formalize continuous spaces, other times discrete graphs, but I hardly see the interest of hybrid systems. Anyway, there is no interesting continuous map in either way between a discrete and a continuous line.
- There is a diversity of non-isomorphic linear orders, beyond finite ones and that of R. Your formalism automatically admits them. Are you interested in such generalizations, despite the fact that only the case of R comes in usual theories of physics, namely in general relativity which seems to be what you focus on ? Examples : the set of rationals ; the set of irrationals ; the long line ; lines where all countable monotone sequences converge but that are incomplete since gaps have uncountable cofinality; Suslin lines, whose existence is undecidable in ZF.
- By only accepting embeddings of lines into your topological spaces, you make it straightforward to define embeddings between them, however it leaves the concept of other continuous functions less straightforward to define. Do you see the notion of embedding as a more important correspondence between topological spaces than continuous functions ?
I will write more comments later.
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Author Tim Maudlin replied on Feb. 7, 2015 @ 19:17 GMT
Dear Sylvain Poirier,
Thanks for the careful reading of the paper (which is, of course, very brief) and the comments. Let me try to advance the conversation.
There are a few different threads of thought here. Let's separate them.
One is how the Theory of Linear Structures interacts with category theory. It is an interesting question—when I taught a seminar on this, there was...
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Dear Sylvain Poirier,
Thanks for the careful reading of the paper (which is, of course, very brief) and the comments. Let me try to advance the conversation.
There are a few different threads of thought here. Let's separate them.
One is how the Theory of Linear Structures interacts with category theory. It is an interesting question—when I taught a seminar on this, there was a math graduate student who wanted to look into the question. My own understanding of category theory is not at all strong, so at the moment I do not feel qualified to say much in detail, and everything here should be taken with a grain of salt. It does seem clear that there are enough similarities between the approaches to explore how they agree and differ. One obviously does not need to decide between them in any final way: the more tools in the toolbox the better, as long as one has a good understanding of them.
As far as the structure of the conceptual analysis goes, I follow standard topology in this: I define the notion of a continuous function in terms of the function and the geometries of the domain and the range. If I follow, you want to take the notion of a continuous function as a primitive, not in need of further analysis. I don't think that there is any general method for evaluating whether a particular concept ought to be taken as primitive (and associated with some structural axioms, as happens with "open set" in the standard approach and "line" in my approach) or taken as definable in terms of some more conceptually primitive concept. So all I can ask is that one look at the actual definition of a continuous function that I give (in the book, not in the paper) and see whether it does a good job of capturing one's intuitions about continuous functions. I can say this: my definition of a continuous function is not the same as the standard definition in topology, and where the two approaches disagree about particular functions (whether they are continuous or not) mine better accords with intuition. But to discuss that we would have to get into the details of the definitions. I think that the ability to provide an explicit definition that works well does argue in favor of taking a concept as non-primitive. I could not tell from your description of your approach whether there are any structural features the set of continuous functions are required to have, or any axioms they must satisfy. If there are, one could look and see how intuitive they are.
Of course, given a domain and a range whose geometries are otherwise undefined and set of functions that are to count as "continuous", one ought to be able to recover (or define) something about the geometries of the spaces. I guess I find this procedure a bit odd since the notion of a function obviously already requires the domain and the range, and in the standard approach (and mine) one specifies the geometries of the domain and range as spaces, completely independently of any considerations of functions between them. So I think there is a naturalness of this standard order of definition here. But again, one has to look at entire approaches to see what they accomplish.
The advantage of taking the notion of a "line" as primitive is that the axioms governing lines make use only of the notion of a linear order, and that in turn is easily and cleanly defined.
Let me answer your questions:
1) It is true that my system handles "hybrid" geometries (single spaces with some continuous and some discrete lines, and even spaces with some lines whose linear orders are complete and some that are not). That is just a consequence of relying on the notion of a linear order, and the fact that there are linear orders that are dense, orders that are not, dense, orders that are complete, orders that are not complete, etc. It is not that I expect the hybrid spaces to come up in physics. It is easy to define a subclass of Linear Structures all of whose lines have the same sort of linear order—e.g. all of them are dense, or none are dense, or all are complete, or none are complete. I call these "uniform" Linear Structures, and my expectation is that the physical universe should be a uniform Linear Structure. But letting the definition range too widely is not a problem: if you want to restrict to some subclass, you just restrict. It also would handle, e.g., "infinitessimals" without any trouble, since they also should have a linear order. It is not beyond the realm of possibility that those have physical application. It is also obviously possible that physical space-time is discrete.
2) I think our mind should be open to all these possibilities. The reason that R is used so much is that R is familiar and R has nice algebraic properties. I think that a lot of the algebraic properties (i.e. ones that depend on the fact that the elements of R can be added and multiplied) evidently have nothing to do with geometrical structure as such. R may keep coming up in physics just because it is the tool we always pull out of the toolbox! In the book, I discuss the "rational line" in great detail, and the set of irrationals (with the usual linear order) as well. Your other examples are more sophisticated, and I have not considered them. But again, having cast the net too wide (allowing one to define geometries that will not come up in physics) is not a sin: having cast too narrowly (so you miss the right structure) is.
3) I provide an explicit definition of continuity that covers all functions from one space to another. I also define various weaker notions (continuous at a point, continuous at a point in a direction, etc.) I also define both weaker and stronger properties of functions. For example, I define what it is for a function to be "convergent" (which actually corresponds most closely to the standard topological definition of continuity), and is a weaker notion than continuity. I also define what it is for a function to be "lineal", which is a stronger notion than continuity. All of these are defined using just the resources provided by the theory. I would have to refer you to the book to see how straightforward the definitions are.
By the way, the definition does provide for continuous functions from a continuum to a discrete space. For example, the floor and ceiling functions from R to Z are continuous by my definition. So when you say that there is no interesting continuous function between a continuous line and a discrete space I disagree! Not every such function, by my definition, is continuous, but those are. The only continuous functions from a discrete space to a continuum, though are constant functions. So in that direction it is trivial.
Regards,
Tim Maudlin
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Sylvain Poirier replied on Feb. 7, 2015 @ 23:39 GMT
When I mentioned to consider the sets of morphisms as primitive, I meant only : in a first draft of consideration, until fixing a definition of the structures that the morphisms will preserve. Also, while not strictly necessary, I generally admit that the objects in the category are given as sets (with structures to be introduced later), and the morphisms are maps between these...
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When I mentioned to consider the sets of morphisms as primitive, I meant only : in a first draft of consideration, until fixing a definition of the structures that the morphisms will preserve. Also, while not strictly necessary, I generally admit that the objects in the category are given as sets (with structures to be introduced later), and the morphisms are maps between these sets.
Finally, the only sets of morphisms I take as primitive are the sets M
0=Mor(K,M) and/or M*=Mor(M,K) with fixed K, which I take to play the roles of structures for M.
The condition of "preserving the structure" is thus defined as follows:
Putting on each M the structure M
0, A function f from M to N is said to preserve that structure if ∀x ∈ M
0, fox∈N
0.
Example : M is an affine space and M
0 is the set of all affine maps from R to M.
This condition on f means that f preserves every operation of barycenter between 2 points with any coefficients (because the barycenter of points x and y with coefficients (1-u) and u is the image of u by the unique affine map from R to M which sends 0 to x and 1 to y), so that it is an affine map, mapping every straight line into a straight line.
Now if we put on each M the dual structure M*, that is the notion of affine function with real values, the preservation condition for this structure by a map f from M to N says : ∀y ∈ N*, yof∈M*, that is, the pullback of an affine form is an affine form. Then, since the preimage of a singleton by an affine form is either an hyperplane or the empty set or the whole space, this directly implies that the preimage of an hyperplane by such a preserving function, is either an hyperplane, the empty set or the whole space. The advantage here is that morphisms so defined between infinite dimensional topological affine spaces are automatically continuous, a condition which the algebraic definition by barycenters does not ensure (while both concepts of morphism are equivalent in finite dimensional spaces).
The very same tools can as well define the structure of vector space (with fewer axioms than usual by taking the dual space as primitive), topological spaces with the particular case of topological manifolds, then Lipschitz structures on topological manifolds, and also differential manifolds with whatever degree of smoothness you choose.
It is very simple to introduce the notion of measure on a topological manifold M : take M* the vector space of continuous functions with real values, then the space of measures on M is the vector space of linear forms on M* that is "generated by M", i.e. the set of limits of sequences of linear combinations of elements of M in the dual of M*. Now if you take as M a differential manifold and M* the set of smooth functions on M, then what you get in this construction (closed vector space generated by M) is the space of distributions on M.
I do not need to check your book to know that you define continuous functions as functions f such that the image of any line with endpoints x and y either contains a line with endpoints f(x) and f(y), or f(x)=f(y); and that f is continuous at a point x if for any line with endpoint x, either f is constant near x on this line or the image by f of this line contains a line with endpoint f(x). So it is indeed less straightforward than with the general tool I told you, where (in the direct version) the condition implies that the direct image of any figure of some kind in M (conceived as the direct image of K by some morphism from K to M) is already (instead of : contains) that kind of figure, while, in the dual version, the condition implies that the preimage of any figure of some kind is a figure of that kind.
I see some differences with usual topological concepts, which you did not specify in your essay.
For example, in the set of (x,y)∈R
2 such that (y=0≤x) or (0 < y ≤ x and 1/x is an integer), the line (y=0) is a neighborhood of (0,0) according to your definition but not in the standard topological definition, where any neighborhood of (0,0) in this set must contain the whole subset of points with x smaller than some positive value. Does your definition of "neighborhood" fit your intuitive idea of this concept in this case ? Note that if, instead of only admitting lines as subsets, you worked with the tools I gave you, allowing squeezed lines, defined as continuous maps from a totally ordered set into the space but not required to be an embedding, then the resulting concept of neighborhood would coincide with the traditional one in this particular case.
I do see a specific problem with your concept of neighborhood : according to the classical definitions, if f is continuous in x then the preimage of a neighborhood of f(x) by f, is a neighborhood of x. I can imagine a continuous idempotent function that squeezes {(x,y) | 0 ≤ y ≤ x} onto the above set, however the preimage of the neighborhood (y=0) of (0,0) by this map is not a neighborhood of (0,0).
Another difference, is that your topology does not admit any Cantor space, unless you give it a very different topology that makes it... connected, and thus no more feeling like a Cantor space in the usual topological sense.
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Author Tim Maudlin replied on Feb. 8, 2015 @ 00:39 GMT
Well, I won't try to comment on your approach, given that there is obviously not room to explicate it properly, and in any case it is not directly relevant to my paper.
There is quite a bit I did not specify in my paper, as it is a less-than-9-page introduction to something that takes hundreds of pages to present!
But to your questions: Yes, by my definitions, y=0 is indeed a neighborhood of (0,0) in that set, and that does completely align with my intuitions. Any continuous line that arrives a (0,0) has a segment in y = 0. In that sense, which is the only obvious one, the set of points y=0 completely surrounds (0,0) in that set, which is what we want a neighborhood to do. You seem to think that it is a problem that my definitions yield different results than the standard ones. In this case, mine in the more intuitive. As for Cantor sets, neither I nor anyone else has any real intuitions about them, so it is hard to argue that any result about them is either good or bad.
I should also note that you simply repeat certain properties of the standard definition as if they are desirable but without any argument. This is particularly the case with pre-images. The most natural thing is to define properties of functions by the geometrical characteristics that they preserve under their action: that is, the geometrical characteristics of the domain that are preserved in the range. The whole idea of looking at what is preserved backwards, i.e. by the pre-images. is just strange. You are used to it because it is what has been done, but it is just peculiar.
There are lots of differences between my definitions and the standard ones. You seem to think that these differences are per se objectionable. But there is no argument to that effect.
I am also a bit mystified by your assertion that you do not need to check my book to see how I have defined things. Are you psychic?
Sylvain Poirier replied on Feb. 8, 2015 @ 10:39 GMT
No, I'm not psychic :) I believe psychic abilities exist but I'm not sure if it would work for this purpose. What I meant is that I find only one possible definition that mathematically fits. Now I understand that your continuous functions from continuous spaces into finite graphs are those that make only one step at a time.
Well if you like your definitions... like tastes these cannot be...
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No, I'm not psychic :) I believe psychic abilities exist but I'm not sure if it would work for this purpose. What I meant is that I find only one possible definition that mathematically fits. Now I understand that your continuous functions from continuous spaces into finite graphs are those that make only one step at a time.
Well if you like your definitions... like tastes these cannot be discussed.
For example according to your definitions, the identity map (inclusion) of the rationals (or the irrationals) into the reals is discontinuous ; this seems odd to me.
If your ideas take hundreds of pages to present, I'm afraid this means they are not as simple and intuitive as you are trying to advertise.
You may fail to have an intuition about Cantor sets, however please do not project your failure on others. I have been quite interested in the Mandelbrot set and Julia sets, and this provides
direct visual images of Cantor sets.
Now about the physics. What physics ? The only physics clearly related with your topology, is that of General Relativity (as for quantum gravity, a main candidate is Loop Quantum Gravity, that, if I understood well, does not fit with your topological concepts with time orientation). But for this, what you do is that you start with a time order taken as primitive to define oriented lines, and then you say that this concept of oriented lines can be taken as a basis for topology and thus geometry. But if in this case, the linear structure is equivalent with the time order (i.e. each is definable from the other), then what is the interest of developing the concept of linear structure, rather than just looking at the time order, which is formally simpler ?
Then, this linear structure, or time order, defines the conformal structure of space-time... though, even using the linear structure, it seems complicated to me to define the tangent space at each point, to be able to express that it indeed forms a vector space and the light cone near a point is actually a quadratic cone. But what is the use to point out that conformal structure, intermediate between those of topological manifold and pseudo-Riemannian manifold ? I may have seen that it was considered in some works, however I fail to figure out any physical context where that structure naturally remains fixed while the pseudo-Riemannian structure may arbitrarily vary (multiplied by arbitrary scalar fields).
You may point out that particles or planets, whose mass affect the space-time curvature, follow time-like lines in space-time. However they do not follow any such lines but only geodesics if they are isolated, which involves the whole metric and not just the conformal structure ; moreover, the electromagnetic field contributes to the space-time curvature without being contained in any line.
Finally, the main equation of General Relativity (the Einstein field equations) is a tensor formula on the tangent space at each point, that involves the fields and the metric in a way formally independent of the particular signature of the metric; the same equation may be written, keeping much of its properties, in a manifold with any signature, thus making the particular time-oriented linear structure irrelevant to the understanding of this equation.
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Sophia Magnusdottir wrote on Feb. 8, 2015 @ 09:49 GMT
Hi Tim,
Clearly one of the best essays yet :) I have one issue to pick. You state that the physical world has the right structure to be describable as math as if that was a fact, but forget to question whether that is indeed so. It is arguably true that some of it is describable as math. But is all of it describable by math? What if not? That's what I've addressed in my essay.
-- Sophia
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Author Tim Maudlin replied on Feb. 8, 2015 @ 14:15 GMT
Hi Sophia,
So what we know for sure is that certain aspects of the physical world have structures that can be described to very high accuracy by mathematically formulated theories: everyone uses the anomalous magnetic moment of the electron as the example of precision, and the entire standard model is tremendously successful, as is the General Theory of Relativity. It does not follow that everything can be captured mathematically, but whatever cannot must not play a very noticeable role in producing all of the phenomena that physics has been able to predict so accurately. If some aspect of the world cannot be represented mathematically, then no mathematically formulated theory will succeed predictively for it.
In any case, I did not claim that it is all describable by math (probably no mathematical theory will account for consciousness, for example) but that the geometry of space-time is. That's a hard enough problem all by itself!
Regards,
Tiim
Akinbo Ojo wrote on Feb. 11, 2015 @ 19:08 GMT
Dear Tim,
I just read your interesting essay, and also your comments elsewhere on the Heraclitean and Parmenidean views of reality. Your essay touches on fundamental questions in geometry and how to correct the perceived wrong conceptual foundation. I am sure others will have other questions on your interesting contribution. For me since I discuss similar interest in my essay, I will have...
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Dear Tim,
I just read your interesting essay, and also your comments elsewhere on the Heraclitean and Parmenidean views of reality. Your essay touches on fundamental questions in geometry and how to correct the perceived wrong conceptual foundation. I am sure others will have other questions on your interesting contribution. For me since I discuss similar interest in my essay, I will have two questions for you:
Question 1. You state and I quote:
"The fundamental structural characteristic of an open line is this: given the points in an open line, there is a linear order among its points such that all and only the intervals of that linear order are themselves open lines. This basic structural characteristic of the open line holds for lines with infinitely many points (such as lines in Euclidean space) and lines with only finitely many elements… So the Theory of Linear Structures is capable of describing the geometry of continua and of discrete spaces (such as lattices) using the same conceptual and definitional resources."How can either of the two varieties of lines be cut? In the first variety, there will always be a point at the incidence of cutting, and a point is uncuttable, so how does cutting proceed? In the second, the elements, whatever they may be are uncuttable, being fundamental and if the interval between them is a distance, distance also consists of points, so where can you cut successfully?
Question 2: You seem to dismiss the Parmenidean view. I believe it should not be dismissed but be reviewed appropriately as I do in my essay. My question: If the universe itself can perish,
can your Linear Structures similarly perish or are they eternally existing objects immune to perishing? If in a region, the 'Linear Structures' therein perish and in some other region, 'Linear Structures' appear spontaneously or a mixture of the two is occurring in a rhythmic pattern in some region, what could be physically manifest?
Best regards and good luck in the competition,
Akinbo
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Author Tim Maudlin replied on Feb. 11, 2015 @ 19:24 GMT
Dear Akinbo,
Let me try to answer your questions. If "cutting" a line just means partitioning it into two parts, each of which are themselves lines, that is partitioning a line into two segments, then this can be done for any linear order: it is exactly what Dedekind called a "Schnitt". Now Dedekind wanted something more in order to define a continuum: for every Schnitt, there should be either one or two points that correspond to the Schnitt. Think of this as either the greatest or least element of one of the two parts of the partition. Some linear orders can be cut in this way without there being such an element. For example, the set of positive rational numbers can be partitioned in two groups: those whose squares are greater than 2 and those whose squares are less than two. That is a perfectly good Schnitt, but there is no greatest element of the one part or least element of the other. So, by Dedekind's definition, the set of rational numbers is not a continuum.
I have no problem with Dedekind's definition. It just shows that lines can be defined—and cut—even if the space is not a continuum.
As for perishing: the physical lines I have in mind are sets of events, ordered by a temporal order. The universe could have a maximal element in time—a last event. That is a claim about the overall geometry of the universe. If you mean by "perishing" that any object in the past has "perished", then lines do indeed perish: lines made of events in my past have, from my present perspective, perished. That is just the same sense in which we generally talk about things in time perishing: no longer existing.
Regards,
Tim Maudin
Akinbo Ojo replied on Feb. 12, 2015 @ 09:22 GMT
Dear Tim,
Thanks for finding the time to reply. Following your response, I checked on
'Schnitt', which is German for 'cut'. So as not to confuse issues, by cutting of 'a line', I do not mean mathematical cutting of the number line in Dedekind's sense. By line, I mean extension in Euclid's sense. A point cannot be cut by definition, and unlike the number line where an irrational number can be invented as a 'trick' to provide a "gap" in order for cutting to take place, on an extended line "gap" itself will connote either an extension, distance or space and therefore consist of points, all of which similarly cannot be cut. Probably, if you later read my essay you may get my meaning.
If the physical lines you have in mind are sets of events, rather than extension that I mean, then of course events cannot be cut in two.
Then on "perishing" and the possibility of your own type of line perishing, please give a thought of the implication of this in resolving Zeno's Dichotomy paradox, even though Calculus is mathematically used to find a solution to it. However, the 'infinitesimal' of calculus or "ghost of departed quantities" as is famously called challenges aspects of physical reality. Calculus does not tell us the size of the last dx in the race. Calculus cannot also explain how to cut a line of the extended type. Thanks for the exchange.
All the best,
Akinbo
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basudeba mishra wrote on Feb. 11, 2015 @ 19:15 GMT
Dear Sir,
We have discussed Wigner’s paper in our essay to show that the puzzle is the result of unreasonable manipulation to present an un-decidable proposition and impose the unreasonableness on mathematics. We have specifically discussed complex numbers (since he has given that example) and other examples. You are welcome to read and comment on it.
Your statement: “A physical...
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Dear Sir,
We have discussed Wigner’s paper in our essay to show that the puzzle is the result of unreasonable manipulation to present an un-decidable proposition and impose the unreasonableness on mathematics. We have specifically discussed complex numbers (since he has given that example) and other examples. You are welcome to read and comment on it.
Your statement: “A physical world completely described by fluid mechanics would contain no such objects, so the physics does make a crucial contribution” ignores bosons, which also behave like fluids and are “uncuttable”. The problem with your example of the child with square tiles and Fermat’s last theorem are put in an un-decidable format by equating integers with area (tiles) and volume (cubes). The integers are scalar quantities that are related to differentiation between similars as repeat perception of ‘one’s. The “similars” can have various units. While the value of the integer; say 3, remains same, 3 apples or 3 square meters is not the same as 3 cubic meters. While apples are discrete, areas or volumes are analog. There is no puzzle here. We have discussed these in detail in our essay.
Regarding number of atoms in DNA and number of mountains in Europe, there is no puzzle in principle. It is only a matter of interest. If we want, we can easily count all. However, if we fail to define something precisely, as is done in most branches of physics (including space, time, dimension, wave-function, etc. so that there is scope for manipulation), then we cannot say it is puzzling. In our essay, we have defined each term precisely to avoid ambiguity. Due to conservation laws, cell number does not become indeterminate during cell division – it is our inability to count precisely that creates the problem. Further, name dropping is regarded as a sign of superiority and views are presented piecemeal to suit one’s own requirements – to prove that particular point anyhow. We have given one example from one of the essays here in Dr. Lee Smolin’s thread. There is a need to reassess and rewrite physics.
Regards,
basudeba
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Anonymous wrote on Feb. 14, 2015 @ 16:28 GMT
"Our understanding of the structure of time has been revolutionized by the Theory of Relativity. Intriguingly, the change from a classical to a Relativistic account of temporal structure is of exactly the right sort to promote time into the sole creator of physical geometry."
Even if the speed of light were constant, as Einstein postulated in 1905, it can be shown that no new temporal structure arises - rather, the concept of time becomes "not even wrong". But that is an obsolete argument because, as a recent experiment showed, the speed of light is not constant:
"The work demonstrates that, after passing the light beam through a mask, photons move more slowly through space."Pentcho Valev
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Author Tim Maudlin replied on Feb. 14, 2015 @ 22:09 GMT
The question of the "speed" of light does not even arise at this level of geometrical analysis. What gets built into the Linear Structure of a Relativistic space-time is just the conformal (light cone) structure. That structure has no classical analog at all, and so constitutes something completely new in relation to classical absolute time and absolute simultaneity.
Pentcho Valev replied on Feb. 14, 2015 @ 23:17 GMT
Strange argument. What could
"The question of the "speed" of light does not even arise at this level"
mean? You don't want to think of it? The postulate of the constancy of the speed of light is essential, even if all heads are in the sand, and since it is false, the return to "classical absolute time and absolute simultaneity" is unavoidable.
Pentcho Valev
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Author Tim Maudlin replied on Feb. 15, 2015 @ 05:12 GMT
The notion of the speed of anything is a metrical notion. Topology describes geometrical features of a space that are not tied so closely the metrical features: a topological feature is invariant under transformations that change metrical relations. So no speed is definable at this level of description.
Consider just the conformal structure of a Relativistic space-time. This does not contain the sort of metrical information one would need to define a speed, but does define a notion of temporal precedence and a by that a light-cone structure. Those structures are not classical. Even more, they are inconsistent with the classical account of temporal structure.
It is not a matter of what I want to think about, but of what is formally definable at a certain level of geometrical description.
Tim Maudlin
Sujatha Jagannathan wrote on Feb. 16, 2015 @ 07:57 GMT
Topology of Linear Structures is in all you've generalised, it would be more explainable if you diverse from geometrically conceived prospects, which would bring more light to the subject.
Great job & luck!
Sincerely,
Miss. Sujatha Jagannathan
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Akinbo Ojo wrote on Feb. 18, 2015 @ 09:23 GMT
Dear Tim,
I posted this elsewhere in conversation and I thought I would share this with you to add to our previous conversation. Also as something you can confirm directly from Roger Penrose being a fellow FQXi member...
Here is what Roger Penrose has to say in his book,
The Emperor's New Mind, p.113… "The system of real numbers has the property for example, that
between any two of them, no matter how close, there lies a third. It is not at all clear that physical distances or times can realistically be said to have this property. If we continue to
divide up the physical distance between two points, we should eventually reach scales so small that the very concept of distance, in the ordinary sense, could cease to have meaning. It is anticipated that at the 'quantum gravity' scale (…10
-35m), this would indeed be the case".
Hence, my asking assuming, without conceding that the system of real numbers applies to distance, how can a distance be divided if there is always a third element between two elements and going by geometrical considerations these elements are
uncuttable into parts?
Regards,
Akinbo
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Author Tim Maudlin replied on Feb. 18, 2015 @ 13:57 GMT
Dear Akinbo,
Geometrical points have no geometrical parts, by definition. Hence, a single point cannot be further divided. That is true whether the points on a line are dense (there is always a point between any other two) or not dense. Roger Penrose (and George Ellis, for example) think that in this sense physical space or space-time is not dense. My own mathematical language can handle spaces that are dense and spaces that are not dense.
A distance can be divided, as Dedekind shows, by partitioning a line into two sets of points in certain way. This does not require dividing any individual point in two.
Regards,
Tim
Richard Lewis wrote on Feb. 18, 2015 @ 12:22 GMT
Dear Tim,
I do like your theory of linear structures and it does look as if it could have application in four dimensional spacetime in which a series of points on a line in four dimensional spacetime can include variability in the space and time dimensions.
Can you describe any applications or experiments in which this theory has been used?
Regards
Richard
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Author Tim Maudlin replied on Feb. 18, 2015 @ 14:01 GMT
Dear Richard,
What I have developed is not itself a physical theory but rather a mathematical language in which physical theories can be written. The language provides clues about, for example, how to describe a discrete space-time that has Relativistic characteristics. But no complete novel theories have as yet been formulated in this mathematical language, since it has just been developed. Nonetheless, one can see how this language could be well-adapted to describing the physical world on account of its temporal structure.
Regards,
Tim
Richard Lewis replied on Feb. 20, 2015 @ 06:43 GMT
Dear Tim,
I had a further thought about applications of the Theory of Linear Structures. Can you generalize the theory so that it can progress from dealing with sequential points on a line to points on a plane or curved surface and then on to points in three dimensions and the four dimensions of spacetime.
The idea is to try to show a mathematical (or topological?) equivalence between String theory which models oscillating strings of one dimension in a hypothetical 12 dimensional spacetime (11 space plus one time) with the Spacetime Wave theory which proposes oscillations in spacetime (wave propagation of changes in spacetime curvature at speed c) as the description of photons and fundamental particles.
This objective is referenced briefly in my essay on Solving the Mystery.
Regards
Richard
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Author Tim Maudlin replied on Feb. 20, 2015 @ 16:53 GMT
Dear Richard,
The Theory as it is deals with as high-dimensional spaces as you like. One specifies the Linear Structure of, say, a n-dimensional space-time by specifying those sets of events that constitute continuous lines. If the theory only worked for one-dimensional spaces it would not be worth much.
Maybe you mean this: the 2-D world-sheet of an open string can be thought of as a sequence of lines. Can one extend the machinery used to describe sequences of points that constitute lines to cover sequences of lines that constitute worlds sheets? I have not thought about that. One problem, of course, is that the worlds sheet can be partitioned into sequences of lines in different ways, and none is "the right" war to do it. There is no similar ambiguity when resolving a line into a sequence of points.
Regards,
Tim
Efthimios Harokopos wrote on Feb. 19, 2015 @ 18:33 GMT
" The physical world is as it is, and will not change at our command. But we can change the mathematical language used to formulate physics, "
I'm not sure about either one of these two hypotheses. It is a realist view that does not conform with the standard model of quantum mechanics. It complies with the tensless view of relativity but these are different theories and definitively not fully united other than in QED. Can we be sure that the laws of physics will not change one million years from now? Obviously, if we say they will not, this is an axiom and far from a common sense truth.
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Author Tim Maudlin wrote on Feb. 19, 2015 @ 23:04 GMT
I can't imagine what you have in mid by this comment. If you think you can change fundamental physics by your command, you are welcome to try. And if you are unable to change the mathematical language you use to write physical theories, then you are oddly constrained. I did not say that the laws won't change (I do not mention laws anywhere here), but in any case they will not change at our command. so if we want our mathematical theories to describe the sort of structure the world has, we have to change the theories to fit the world rather than the other way around.
Efthimios Harokopos replied on Feb. 20, 2015 @ 21:33 GMT
"If you think you can change fundamental physics by your command, you are welcome to try"
What I tried to say and maybe I did not say it correctly is that your comment " The physical world is as it is, and will not change at our command." involves a hypothesis that there is something we can call fundamental physics. You or anyone else have not proved that. I can assume that we live in a virtual reality in which the laws were changed by its creators. There is an infinite regression of fundamental physics founded on fundamental physics. Except if you are talking about fundamental assumptions like particles. I think Einstein changed fundamental physics by his command, meaning the fundamental physics that people thought they were. Speaking about absolute fundamental physics makes no sense imo. It's like speaking about unicorns. Because it is true that: Two unicorns imply 1+1 = 2
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Author Tim Maudlin replied on Feb. 20, 2015 @ 21:52 GMT
Ah. By "the physical world" I mean the physical world, not our theories about the physical world (however widely accepted they may be). In this sense, Einstein did not change the physical world at all: he developed new theories about that world. Even if we live in a "virtual reality", then it is a physical fact about us that we do. Probably, we could never figure that out.
If we do not separate sharply between the physical world as it is and our theories about it, our discussion will be very confused. Perhaps we will never develop a completely accurate theory of the physical world. But if we do, if will be framed in a mathematical language, so we should think about what languages are available and create more if needed.
Efthimios Harokopos wrote on Feb. 21, 2015 @ 09:00 GMT
Thanks. I have a problem with this below you may want to give another shot at it although I understand that communicating through message boards is difficult especially in this area:
"If we do not separate sharply between the physical world as it is and our theories about it... "
Is it necessary that the physical world should be in a particular way or this is just an assumption to make our life easier?
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Author Tim Maudlin replied on Feb. 21, 2015 @ 13:15 GMT
I do think we are having some communication problems....Even if the laws of the universe change through time, for example, there is still some way it is (i.e. changing, and changing in a particular way), and some description of it. So I can;t see any substantial assumption is saying that there is some way the universe is, not subject to our control (in the relevant sense) that we are trying to describe.
Efthimios Harokopos replied on Feb. 21, 2015 @ 17:41 GMT
"Even if the laws of the universe change through time, for example, there is still some way it is (i.e. changing, and changing in a particular way), and some description of it."
The hidden assumption here is determinism. If this premise is true, then your statement is true otherwise it is false. I have trouble with the assumption of a "particular way" and of determinism. I think they both reflect some type of wish rather than a fact. Regardless of that, thank you.
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Author Tim Maudlin replied on Feb. 21, 2015 @ 19:47 GMT
Think of an indeterministic random walk. It is both the case that the particular details of any walk admit of a mathematical description (2 steps right, then one left, then three right, then four left...) and, in many cases, that some statistical characteristics of the walk can be predicted with high reliability. So indeterminism is not incompatible with mathematical description. Quantum theory is generally considered to be indeterministic, but still amenable to precise mathematical description.
Vladimir Rogozhin wrote on Feb. 21, 2015 @ 11:38 GMT
Dear Tim,
I read your essay with great interest. I totally agree with you: "But we can change the mathematical language used to formulate physics, and we can even seek to construct new mathematical languages that are better suited to represent the physical structure of the world." My high score.
I think only that we must first to consider the proto-structure of the Universum (matter) from the point of view of eternity ("sub specie aeternitatis"), that is, to carry out the ontological structure of matter in the proto-era, "time before times began". When we "grab" (understand) the primordial (ontological) structure of space, then we will understand the nature of time. Therefore, the fundamental physics we must move from the concept of "space-time", to the onto-topological concept
"space-matter-time". The primordial structure of matter determines the structure of the language in which Nature speaks to us, single language for mathematicians, physicists and
poets , ie, language that contains all the meanings of the "LifeWorld"(E.Husserl). I invite you to read my
essay .
Kind regards,
Vladimir
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Author Tim Maudlin replied on Feb. 21, 2015 @ 13:21 GMT
Dear Vladimir,
Certainly physics must deal with matter—a nice simple characterization of physics is the theory of matter in motion. What I have proposed here does not touch on that. Rather, it deals only with the "motion" part. Motion can be understood as the trajectory of an object through space-time, so then one question is how to characterize the structure (geometry) of space-time. That is what I have been working on. Putting the matter into the space-time arena is yet another problem (one would have to confront quantum theory). But I am just taking one step at a time—the step I have something new to offer.
Thanks for the comments.
TIm
Alan M. Kadin wrote on Feb. 21, 2015 @ 16:40 GMT
Dear Prof. Maudlin,
You make some interesting arguments about the nature of physics and mathematics, but it seems to me that the entire question of the “mysterious connection between physics and math” is misplaced. There is rather a simple explanation. Physics deals with how simple rules for relating real objects lead to more complex objects. Mathematics deals with how simple rules for relating abstract structures lead to more complex structures. So a common theme of underlying simplicity can guide them both. But there is no reason to assume that a given elegant mathematical model must ipso facto be represented in the real world.
My own essay addresses a somewhat different issue: (
"Remove the Blinders: How Mathematics Distorted the Development of Quantum Theory"I argue that premature adoption of an abstract mathematical framework prevented consideration of a simple, consistent, realistic model of quantum mechanics, avoiding paradoxes of indeterminacy, entanglement, and non-locality. What’s more, this realistic model is directly testable using little more than Stern-Gerlach magnets.
Alan Kadin
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Author Tim Maudlin wrote on Feb. 21, 2015 @ 19:39 GMT
Dear Alan Kadin,
Thanks for your remarks. Of course, even if both physics and mathematics are concerned with situations where one wants to derive complex conclusions from relatively simple rules (and at one level of abstraction that is correct), it would not follow that the actual physical world behaves in a way well-desribed by a mathematical formalism. We would certainly like, for example, simple rules from which we could derive the weather a year from now, but it seems that the physics of weather just does not admit of such rules at all. So there is a question of which physical conditions must obtain for an effective mathematical description to be possible.
I am a bit puzzled by your description of your theory. If your theory is local, then, by Bell's theorem, it cannot predict violations of Bell inequality for experiments done at space-like separation. But these experiments have been done, and Bell's inequality is violated. So if the theory is as you have described it, we do not need to do further experiments with Stern-Gerlach magnets to check: relevant experiments already exist.
Regards,
Tim Maudlin
Ed Unverricht wrote on Feb. 26, 2015 @ 20:52 GMT
Dear Prof Maudlin,
From Wignar "How do we know that, if we made a theory which focuses its attention on phenomena we disregard and disregards some of the phenomena now commanding our attention, that we could not build another theory which has little in common with the present one but which, nevertheless, explains just as many phenomena as the present theory?"
His answer: It has to be admitted that we have no definite evidence that there is no such theory.
I am not sure of the detail regarding "Theory of Linear Structures", but I think there is room for multiple models, where details will either pass of fail the test of real physical experiments.
Regards and wondering if you have a internet link for more detail?
Ed
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Author Tim Maudlin replied on Feb. 26, 2015 @ 22:30 GMT
Dear Ed,
The mathematical detail is spelled out in the book of mine I have cited, but is not online. Application of the mathematical to physical theories is the subject of a second volume that is being written now.
Regards,
Tim
Member Matthew Saul Leifer wrote on Mar. 1, 2015 @ 02:25 GMT
Tim,
Thanks for your interesting and thought-provoking essay. I was wondering if you have applied your theory of linear structures to any of the discrete structures that have been proposed as candidate theories of quantum gravity, such as causal sets, spin foams, or causal dynamical triangulations?
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Author Tim Maudlin replied on Mar. 1, 2015 @ 02:50 GMT
Hi Matt,
I have not tried to get the quantum-mechanical aspect of it, but I have done some work on using this to describe discrete Relativistic structures. So think of this as in the spirit of causal sets. I can get a simple discrete approximation to a 2-D Minkowski space-time and to a 3-D inflating space-time with horizons, and this is just from trying a few simple constructive rules for the Linear Structure and then analyzing the results. I have an idea for a general scheme for writing down constructive rules (both deterministic and stochastic) for generating Relativistic discrete Linear Structures, but there is a lot of work to do.
Just to give a taste of how this differs from causal sets, using the usual way that causal sets are generated no pair of events will be null related. But doing it my way, the entire space-time structure is built from null related events: it is all light-like in the foundations, as it were. I can also easily put in place constraints on the constructive rules that avoid some of the issues that come up for causal sets, which basically arise from the fact that the kind of graph they want to get is very much not a random graph.
The analytical advantage of a discrete space is that it comes already equipped with a natural measure—counting measure—but in the Relativistic case you have to be careful about what to count. I know that sounds cryptic, but it would take to long to explain properly...maybe we can talk about it sometime.
It may be that just being able to generate good discrete approximations to classical solutions in GR would yield clues about how to implement a fully quantum treatment, but that prospect is too far away now.
Cheers,
Tim
Member Matthew Saul Leifer replied on Mar. 26, 2015 @ 22:29 GMT
Sounds interesting. I will be at New Directions. Maybe we can discuss it there.
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Alex Newman wrote on Mar. 1, 2015 @ 08:02 GMT
This was a good essay with some interesting ideas such as the temporalization of sapce but ity is an idea that cannto be tested in a laboratory and abstract as it is it is pure speculation and increases the complexity of physical models and introduces more ambiguity. I think a theory like that presented in usenet years ago would be fircly attacked and the creator woudl be called names. What are any new predictions this temporalization ofefrs? Therefore, although I through the essay was good I think it makes undjustifuable claims that should not be made at the level of professional physics.
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Anonymous wrote on Mar. 1, 2015 @ 10:01 GMT
"Whereas it is often said that Relativity spatializes time, from the perspective of the Theory of Linear Structures we can see instead that Relativity temporalizes space"
Most theoreticians now believe that the spatialization of time (a consequence of Einstein's 1905 false constant-speed-of-light postulate) was wrong:
"And by making the clock's tick relative - what happens simultaneously for one observer might seem sequential to another - Einstein's theory of special relativity not only destroyed any notion of absolute time but made time equivalent to a dimension in space: the future is already out there waiting for us; we just can't see it until we get there. This view is a logical and metaphysical dead end, says Smolin."
"Was Einstein wrong? At least in his understanding of time, Smolin argues, the great theorist of relativity was dead wrong. What is worse, by firmly enshrining his error in scientific orthodoxy, Einstein trapped his successors in insoluble dilemmas..."
WHAT SCIENTIFIC IDEA IS READY FOR RETIREMENT? Steve Giddings: "Spacetime. Physics has always been regarded as playing out on an underlying stage of space and time. Special relativity joined these into spacetime... (...) The apparent need to retire classical spacetime as a fundamental concept is profound..."
Nima Arkani-Hamed 06:11 : "Almost all of us believe that space-time doesn't really exist, space-time is doomed and has to be replaced by some more primitive building blocks."
Pentcho Valev
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Pentcho Valev replied on Mar. 7, 2015 @ 08:24 GMT
Tim Maudlin: "The Relativistic account of space-time geometry makes the light-cone structure of space-time a fundamental part of its geometry. This, rather than the "constancy of the speed of light" lies at the heart of the theory."
This is wrong, Tim Maudlin - the constancy of the speed of light does lie at the heart of the theory, even if you want to hide it behind the light-cone structure of space-time. And since the speed of light is not constant (you know that don't you?), you will have to join Steve Giddings, Nima Arkani-Hamed and Lee Smolin in their rejection of Einstein's space-time.
Pentcho Valev
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Branko L Zivlak wrote on Mar. 1, 2015 @ 11:28 GMT
Dear Prof Maudlin,
For your subquestion 1) „Which mathematical concepts seem naturally suited to describe features of the physical world, and what does their suitability Imply about the physical world?“
I suggest three main candidates for the mathematical concept:
bit (it was the subject of the competition FQXi 2013);
exp(x) (You know the unique features of this function);
Euler's identity. There are other useful functions, but less importance.
Suitable use of pervious can to describe features of the physical World.
What are your main candidates? If you agree with me, part of the solution can be found in my essay.
Best Regards,
Branko Zivlak
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George Rajna wrote on Mar. 2, 2015 @ 08:49 GMT
Congratulation for such a brilliant essay. You deserve the best.
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Thomas Howard Ray wrote on Mar. 2, 2015 @ 17:43 GMT
Tim,
I read your essay with a mixture of exhilaration and misgiving. Exhilaration for the sheer audacious brilliance of it, and misgiving that I have not yet introduced myself to your work. I plan to start correcting the latter in short order.
One comment: "This (metric - ed.) distance is just the minimal 1 length of a continuous path between the points. It can have an affine structure, which sorts continuous paths into straight and curved. It can have a differentiable structure, which distinguishes smooth curves from bent curves. But beneath all these, already presupposed by all of these, is the most basic geometrical structure: topological structure."
The straight line being a special case for the curve, an analytical "twoity" (LEJ Brouwer's word) guarantees curved structure of metric properties. In a
2006 conference paper I identified the complex plane structure that guarantees a counting function without appealing to the axiom of choice, with a physical definition of "time: n-dimension infinitely orientable metric on random, self-avoiding walk."
Looking forward to immersing myself in your research.
Tom
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Author Tim Maudlin replied on Mar. 2, 2015 @ 18:05 GMT
Dear Tom,
Thanks for the comments. I was not aware of the Brouwer, and a quick look at some discussions shows that it will not be an easy thing to really understand. It is, of course, possible to describe the geometry of a space with enough structure to define the affine structure but not enough for a full metric. The so-called "Galilean" or "Neo-Newtonian" space-time is like this (if you try to use a standard full metric is it degenerate). This is particularly nice if one is trying to translate physical laws into a purely geometrical vocabulary. Newton's First Law, for example, becomes "The trajectory of a body is a straight line through space-time unless a force is put on it". The fundamental distinction between the affine and metrical structure also shows up when one demands, in General Relativity, that the metric be compatible with the connection on the tangent bundle.
Regards,
Tim
Anonymous replied on Mar. 3, 2015 @ 14:53 GMT
Hi Tim,
I think a full metric description is, just as you imply -- native to point set topology, and not to affine space. There is an arithmetic theorem that any point maps simultaneously to any set of points provided it is far enough away. In reverse, this gives us the degenerate result of Galilean or Newtonian space. There is no time parameter.
Einstein, by introducing time by way of Minkowski space, may have hoped the point "far enough away" would avoid the singularity and instead found that expanding 4 dimension spacetime (by Hubble's result) places the singularity at every arbitrarily chosen point of 3-space. There will be singularities in general relativity. No point is far enough away to overcome the Poincare-Hopf theorem.
Best,
Tom
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Cristinel Stoica wrote on Mar. 5, 2015 @ 12:57 GMT
Dear Tim,
I was very happy to find your essay here. I read it with pleasure and I like it so much. I am a mathematical physicist working in general relativity (singularities). Also, I started recently teaching a master class at the Faculty of Philosophy, together with the philosopher Iulian Toader, and we are using as main resource your book Philosophy of Physics. Space and Time, which we...
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Dear Tim,
I was very happy to find your essay here. I read it with pleasure and I like it so much. I am a mathematical physicist working in general relativity (singularities). Also, I started recently teaching a master class at the Faculty of Philosophy, together with the philosopher Iulian Toader, and we are using as main resource your book
Philosophy of Physics. Space and Time, which we both consider great.
In addition to the part containing the general discussion about the effectiveness of mathematics in physics, I enjoyed very much the part about your theory of Linear Structures. I like the idea that, once you have the linear structure, with directed (causal) lines, you can recover not only the topological and the causal structures of relativistic spacetime, but also the conformal structure, that is, the metric up to a scaling factor. I have some comments and questions.
You said somewhere "Whether those axioms could be modified in a natural way to treat of pointless spaces is a question best left for another time." I think the answer to this is positive, and that in the same way linear structures reconstruct topology, a sort of linear structures can reconstruct the generalization of topology which may or may not have points, named
"locales".
The linear structure able to lead to the recovery of an Euclidean or relativistic spacetime has to be very special. In other words, it has to be subject of some constraints, which lead to the topology, the affine structure, and the metric of the Euclidean space.
In the case of relativity, without an underlying structure similar to the usual topological structure of spacetime, the directed lines can be distributed in so many ways. Consider first that we have a point and a local homeomorphism to R
4 around that point. The directions of the directed lines in R
4 at that point can be any subset of the sphere from R
4 centered around the origin of the lines. This means that, in order to get the causal cones in relativity, one needs to ensure that at least the topology of the directions is that of a cone. Otherwise, we can obtain various kinds of spacetimes, in particular the Galilei spacetime is of this form.
But how to recover the topology of R
4 from directed lines in relativity? The problem seems to me to be that future oriented timelike vectors of length smaller than the unit form a noncompact set, and they can't be used to reconstruct the topology. In addition, the spacelike directions, which have to be undirected if we want to talk about them, are disconnected from the timelike ones, so it is difficult to use them even together to reconstruct open sets. I have some ideas how to do this, but I am still not sure if this reconstruction can be made simpler than the standard one.
The condition of homeomorphism with R
4 is quite natural in the standard notion of topology, but you can object that this is because we are in the old paradigm. However, the condition that the directed lines give causal cones is more natural assuming a (4D) tangent bundle (which can only be defined if we assume a 4 dimensional topological manifold endowed with a differential structure), on which a Lorentz metric is defined, or at least the corresponding conformal structure. Both these conditions seem to me to make the case for 4D open sets rather than lines, because it seems difficult to recover the differential structure from lines rather than open sets.
Did you find additional axioms to those of a directed linear structure, which would make it in a 4D manifold with differential structure and causal cones just as in general relativity, in a more natural way? Because at this point it seems to me that adding such axioms would lead to a much more complicated definition of relativistic spacetime, and by this would make the advantages of the simplicity and concision of the linear structure vanish.
Maybe some of my questions are already answered in your second volume or other works. Or in your future results, since it is natural to think that such a theory takes some time to answer to the most important questions.
Anyway, thinking at this led me to some ideas of simple constraints to supplement your causal structures to recover relativistic spacetime. If you are interested, I can try to detail them.
Reconstructing relativistic spacetime in a natural way, based on more intuitive and physical principles, can be a good starting point for generalizing the structure to include matter and quantization.
I want to make clear that the fact that in order to recover relativity one has to add to the linear structure less natural axioms than the standard is due to the richness and generality of the linear structures, and it happens the same even if we start from open sets topology. And is not necessarily a disadvantage. By contrary, it may be an advantage, because we have the freedom to use other axioms, that give something different than general relativity. For instance, consider the possibility that the linear structure behaves differently at different scales, maybe this would lead to the dimensional reduction which would be useful to perturbatively quantize gravity (
my own approach to quantum gravity is based on singularities, which are naturally accompanied by dimensional reduction).
Another advantage of linear structures approach over standard general relativity is that it is much richer. Maybe this richness can be used somehow to describe matter on spacetime, although I don't have a clue how to do this. Also, I think that Sorkin's causal sets can be seen as a particular case of your linear structure approach to general relativity.
Another feature I liked at your theory of Linear Structures is that it equally works for discrete geometries. This made me think at the following. In 2008
I proposed a mathematical structure, based on sheaf theory, which allows to construct all sorts of spacetimes and fields on them - a general framework which contains as a particular case any theory in physics (but it is not a TOE). This sheaf theoretical approach works equally for discrete approaches like causal sets, spin networks, CDT etc., but I did not develop it beyond what is in that paper, for lack of time. My sheaf theory approach works with both discrete and continuum theories and captures some essential features in a simpler and more general structure, otherwise there is no parallel with your theory.
Best wishes,
Cristi Stoica (
link to my essay)
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Author Tim Maudlin replied on Mar. 5, 2015 @ 22:38 GMT
Dear Crista,
Thanks so much for the comments. I am familiar with some work on pointless spaces (some of it older that the things coming from category theory) and I can only say that there seemed to be no obvious way to adapt this approach there. The basic problem is that the points in a 1-D open manifold are linearly ordered automatically, but while regions in a pointless space can include one another, it is hard to define the same sort of linear order if the basic structures are not 1-D (and hence constructed from things are are 0-D, i.e. points). This is not so say it is impossible, but that it is at least not obvious.
The language of Linear Structures is, as you appreciate, very flexible. That is good in one way (lots of possible geometries) and bad in another (you need strong constraints to narrow down to what you want). One interesting place to look is discrete spaces, when one can consider various constructive rules for generating a geometry and then analyze the character of the geometries that result. I have quickly found rules that give good discrete approximations to a 2-D flat space-time and the 3-D expanding space-time with a horizon structure. It will take more research to figure out exactly what features of the constructing rule control the outcome.
The idea is a bit like Causal sets, but the actual implementation is quite different. The most fundamental structure is light-like rather than time-like, and the Causal sets it is time-like.
Cheers,
Tim
Cristinel Stoica wrote on Mar. 6, 2015 @ 22:09 GMT
Dear Tim,
I am happy to announce you that I found a natural way to supplement linear structures (directed null lines) with simple constraints to recover relativistic spacetime in n dimensions, without directly imposing to be locally IR
n. If you think it would be interesting, I can post it here, I don't think it will take me more than a couple of pages to explain it in detail.
Best wishes,
Cristi Stoica
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Author Tim Maudlin replied on Mar. 6, 2015 @ 23:56 GMT
Dear Cristi,
Of course I would be very interested! Maybe you should send it directly in an e-mail so I can keep it more easily. My e-mail is twm3@nyu.edu.
Thanks!
Tim
Jose P. Koshy wrote on Mar. 7, 2015 @ 10:02 GMT
Dear Tim Maudlin,
I really enjoyed the thread of your argument: simple arithmetic and geometry, based on our acquaintance with the physical world, leading to complex mathematical concepts, and these in-turn having some role in physics, and thus the Wigner's puzzle being solved. The whole thing would have been a smooth curve, but for your sudden jump: in explaining geometry, you jumped from...
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Dear Tim Maudlin,
I really enjoyed the thread of your argument: simple arithmetic and geometry, based on our acquaintance with the physical world, leading to complex mathematical concepts, and these in-turn having some role in physics, and thus the Wigner's puzzle being solved. The whole thing would have been a smooth curve, but for your sudden jump: in explaining geometry, you jumped from the 'geometry of the bodies' to the 'geometry of the space'.
“And as soon as items are countable, other mathematical concepts can be brought to bear: ratios and proportions for example.” 'Countable' is a factor; but, imagine the universe has a finite number of atoms, but does not change with time, what structure will it have? For any change, these countable entities should move, or 'motion' is the most fundamental factor. Motion is a space- time relation that obeys mathematical laws, and so the changing world obeys the laws of mathematics. That provides the simplest solution for Wigner's puzzle. Please go through my essay:
A physicalist interpretation of the relation between Physics and Mathematics.
“We can even seek to construct new mathematical languages that are better suited to represent the physical structure of the world”. As you have pointed out, any new mathematics depends on the basic mathematics, and cannot be 'genuinely new'. The theory of linear structures, in fact, is not new mathematics; the 'newness' is in the 'directional aspect' which is a property, a property which you assume. You have assumed the conceptual primitive, regarding space, to be lines instead of points. The mathematical rules remain the same, but the resultant structures are different; so there is justification in calling it a new mathematical language used to describe space-time.
The physicalist point that I propose is that the properties we assume, whether about space or about atoms, should be in conformity with what we observe directly. As pointed out by you, everything started from the simple observations in the nature. The concept can become more and more complex, but should not go against that simple facts. The essence of your arguments in the first part of the essay is also the same.
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Author Tim Maudlin replied on Mar. 7, 2015 @ 15:54 GMT
Dear Jose Koshy,
I agree that the motion of matter is described by some mathematical law, but the formulation of that law is beyond the scope of my present project. If we start with the general idea that physics is the theory of matter in motion, this presents two targets for an exact theory: matter and motion. The "motion" of a localized object seems to be best understood (and this is not how Newton would have conceptualized it) as a trajectory through space-time. So that leaves us with the problem of describing the geometry of the space-time, and the problem of understanding how the trajectories of objects are produced. You have to solve the first in order to even approach the second. That is, you need to understand the geometry of the space-time before you can begin to write down a dynamics. This project is then even more basic: not yet "what is the geometry of space-time?" but rather "what is the best mathematical language to use to describe the geometry of space-time?". If the fundamentals of Relativity are correct, this looks like an especially promising sort of language. The language is new, in the sense that (as far as I know) no one has written it down before. But at a basic level, I am a Platonist about mathematics, and in that sense nothing is really new.
Regards,
Tim
Jose P. Koshy replied on Mar. 8, 2015 @ 11:53 GMT
Dear Tim Maudlin,
Your stand is clear: if the fundamentals of (General) Relativity are correct, then there is four-dimensional space-time, and you are proposing a new language to describe the geometry of the space-time. I have just downloaded "New foundations for physical geometry" to know exactly what you referred as the project.
My argument is that physicists have not explored all the possibilities classical three-dimensional space offers. For example, the model proposed by me (refer
finitenesstheory.com) views that motion at speed 'c' is the fundamental property of matter, and the reaction to this motion creates gravity. Consequently, the path of a body, in a classical 3-dimensional space, is bent by its own gravity. This can give similar results like that of GR.
From a cursory glance at the material downloaded, I get the impression that you consider going back to classical three-dimensional space is a retrograde step. However, I think that you will ultimately arrive at that conclusion.
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Author Tim Maudlin replied on Mar. 8, 2015 @ 14:28 GMT
Dear Jose Koshy,
I would not at all consider quite different space-tmes from GR as "retrograde": I am quite open to all sorts of possibilities. Here are two considerations. One is that if one rejects the idea that space-time is a continuum and uses a discrete space-time instead, the most natural way to implement that idea using this formalism results in the basic geometrical structure of space-time being all light like (null) paths. It would then follow automatically that any matter in continuous motion (i.e. following a continuous path) is always "traveling at the speed of light", and massive particles, such as electrons, must really engage in Zitterbewegung: rapid vibration. The other consequence of going to a discrete space-time is that the geometrical structure has a natural unique foliation, which can be considered a "backward" step to a more classical structure. (Not full Newtonian picture, with a single space persisting through time, but a structure with an intrinsic "simultaneity" built in, together with a light-cone structure.) Is that bad? It is actually the most straightforward way to be able to implement the non-locality implied by violations of Bell's inequality. So I am very open to these possibilities. As I said, in the first place this is just a new mathematical language. There are many things that can be described using it.
Reagrds,
Tim Maudlin
Jose P. Koshy replied on Mar. 9, 2015 @ 05:44 GMT
Dear Tim Maudlin,
One last question: Devoid of physics, can the Theory of Linear Structures be regarded as a new branch of mathematics related to topology? Or is there already any related branch in mathematics, and you are only trying to use that in physics?
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Author Tim Maudlin replied on Mar. 9, 2015 @ 16:34 GMT
The aim of the Theory of Linear Structures is to provide a precise a formalized mathematical language (axioms for the basic structure and strict definitions) that make precise the same informal notions addressed by standard topology (continuity, connectedness, closure, limit points, etc.). These new definitions are different from those of standard topology...for example some functions that are continuous according to the standard definition are not continuous according to my definitions. Conceptually, I think my definitions are better: they match better what our informal judgments are. But more importantly, in the case of physical space-time, one can see from a physical perspective why the geometry should be well described in the this language, and it is not clear why it should be well described in the standard language. So: it is a new branch related to topology and, in some sense, competing with standard topology, which I am trying to use to do physics. It also provides a common mathematical language for describing both continua and discrete spaces. Standard topology does not provide this. So insofar as we are unsure whether physical space-time is continuous or discrete, we can can still use the basic conceptual tools provided here knowing they will work in either case. This also provides clues about how to formulate discrete approximations to continuous structures.
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Michael James Goodband wrote on Mar. 9, 2015 @ 16:55 GMT
Dear Tim Mauldin
An enjoyable essay to read. I agree that “one could easily write a companion paper to Wigner’s called “The Unreasonable Relevance of Some Branches of Mathematics to Other Branches””. For me there are 2 places where this particular unreasonableness of maths transfers over to physics in a sort of pincer movement that constrains physics and potentially your proposed...
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Dear Tim Mauldin
An enjoyable essay to read. I agree that “one could easily write a companion paper to Wigner’s called “The Unreasonable Relevance of Some Branches of Mathematics to Other Branches””. For me there are 2 places where this particular unreasonableness of maths transfers over to physics in a sort of pincer movement that constrains physics and potentially your proposed descriptive language.
The first is causation for any continuous physics over a space with a metrical structure: these conditions specify the maths description must be in terms of the norm-division algebras. Both General Relativity and Standard Model are in terms of NDA valued fields, which constrains every attempt to unify them to be able to reproduce this NDA description. But the physics conditions also constrains all alternative descriptions to be capable of reproducing the NDA based description.
The second side of the pincer comes from requiring any theory to reproduce quantum theory results. My hidden propagator dynamics analysis came from having a particular theory in mind: one with discrete topological defects in a space with compactified dimensions. This is a discrete theory with a potentially discrete space. However, I was surprised to discover that my HPD analysis revealed that in order to connect with experimental results the details of the discrete nature of the theory must necessarily be erased, leaving only the same NDA-based description of experimental results as quantum theory. This is a general result for any theory with discrete particles: to connect with experimental results all discrete elements of the theory are erased, leaving only an NDA based description. This means that the entire class of HPD theories are experimentally indistinguishable from each other, as they must all reproduce the same descriptive form for experimental results.
This pincer movement would seem to include your new descriptive language of the Theory of Linear Structures. Any new descriptive language must reproduce the descriptions of existing theory. In this bigger picture context of connecting with experimental results that are already successfully described by NDA valued fields – won’t the new description be lost in the process of experimental prediction?
Michael Goodband
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Author Tim Maudlin replied on Mar. 10, 2015 @ 17:13 GMT
Dear Michael,
I do not feel confident that I am in command of the technical details of your analysis, but let me say at least this. The sense in which any new theory must "reproduce the descriptions of existing theory" is obviously a matter of approximation, not exact derivation. That is, all experimental data come with error bars, and so recovering the predictive success of present theories (such as quantum theory or the General Theory of Relativity) requires matching their predictions where they have actually been tested to within the tolerance of the errors. If one were to demand a higher degree of match than this, then the new description would of course be "lost". But the expectation is that the new description will deviate in its predictions from the present theory, but only to a small degree and only in certain circumstances. What that degree is, and which circumstances are relevant depends on the theory itself.
Regards,
Tim Maudlin
Member David Garfinkle wrote on Mar. 9, 2015 @ 18:03 GMT
Dear Professor Maudlin,
this is a very interesting essay and a very interesting program of research. It reminds me very much of the causal set research program of Rafael Sorkin. In particular, like causal sets it is a beautiful and austere way of looking at spacetime structure, but perhaps a bit too austere for my taste. Below I append the sort of questions that I usually ask about the causal set program which I think also apply to your program. If you have time to reply to any of these questions, that would be very much appreciated.
--David Garfinkle
(1) Your method gives a conformal structure, which in the case of a spacetime is equivalent to the conformal class of the metric. There is then a standard result that also giving the volume element will determine the metric. Is there a simple and natural way of specifying a volume element within your formalism?
(2) Presumably a model for a spacetime within your formalism would be a set with your structure which also (exactly or approximately) admits a differential structure (thus making it a manifold) and a metric. But most of the sets with your structure will not be manifolds, even in an approximate sense. Is their a way within your formalism of figuring out which sets are manifolds? (or approximately manifolds).
(3) I admire the austere beauty of your approach. But I'm pretty happy with the standard notion of spacetime as a manifold with a metric and with the usual definition of a manifold as a topological space with an atlas. What do I lose by not adopting your approach?
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Author Tim Maudlin replied on Mar. 10, 2015 @ 17:47 GMT
Dear Professor Garfinkle,
I completely understand seeing a similarity between this approach and causal sets, but at the level of implementation and exact detail they are quite different. So, for example, in the causal set approach there is essentially zero chance for any pair of events to be null related, and in this approach to a discrete structure all the fundamental space-time geometry...
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Dear Professor Garfinkle,
I completely understand seeing a similarity between this approach and causal sets, but at the level of implementation and exact detail they are quite different. So, for example, in the causal set approach there is essentially zero chance for any pair of events to be null related, and in this approach to a discrete structure all the fundamental space-time geometry is null. Also, as you note, in the causal set approach the idea is to get the conformal structure from the causal graph and then fill out the rest of the geometry by a volume measure. But in this approach to a discrete Relativistic space-time (which is not described here) one does not add a volume measure but rather a measure corresponding to the Interval measure on continuous paths. In a word, the basic metrical notion is length rather than volume.
Let me answer your questions.
Well, actually what I have said above answers 1) for the discrete case. The discrete case is nice because (as Riemann observed) the discrete case comes equipped with a measure: counting measure. But you have to be careful about what you count! In this case, one does not count nodes (which is what the causal set people do to recover a volume measure), and what you are quantifying is not volume but length. This may be a bit cryptic, but in short in the discrete case one can define a "corner" in a continuous path through the space-time as opposed to an "unbent segment", and when quantifying the length of a path one counts corners rather than nodes in the path. The result is that the number of corners on a path that lies on the light cone is zero (even though the number of nodes may be unboundedly large). The connection to the Interval should be obvious.
In a continuum, the measure must be imported from outside (just as Riemann said). I have not tried yet to somehow analyze that measure in terms of anything else in the continuum case: it is just an additional piece of space-time geometry supplementing the conformal structure, just as in GTR.
To the second question:
Investing a space with a Linear Structure automatically invests it with a standard topology, via the definitions I give in the appendix. So it is easy to tell if you have a manifold. But in fact, even familiar continuous space-times (e.g. Minkowski) turn out not to be manifolds once one takes account of the directionality of time. One of my points is that the whole idea of a manifold arose in the context of purely spatial (Riemannian) geometry, and the use of those mathematical tools to deal with space-time (Lorentzian) structures is a mistake. So I am not even aiming at recovering manifolds: I am aiming to recover geometries with an intrinsic light-cone structure, which a manifold does not have.
To the third:
One of the main advantages of my approach is that one can deal with contiinua and discrete spaces using the very same analytical tools and definitions. No one knows whether at (say) Planck scale space-time is discrete or not. So it would be nice to articulate theories in a mathematical form that can be adapted to each possibility. As things are, one uses manifolds and differential geometry for continua and graph theory for discrete spaces. Graph theory (including infinite graphs) is a special case of the Theory of Linear Structures, but the Theory of Linear Structures can be used to analyze continua as well. It also allows you to take ideas developed in the context of continua and see how they play out in a discrete setting. So I think that is one thing that is an advantage.
One other advantage, which I do touch on, is that the Theory of Linear Structures allows for a distinction between intrinsically directed and intrinsically undirected geometries, which is relevant to the description of time. Indeed (as I mention) it even allows one to make a distinction between intrinsically directed and intrinsically undirected topologies, which (to my knowledge) no one working in standard topology has made, and maybe cannot even be drawn without using the resources of the Theory of Linear Structures. So if space-time is discrete (which it may be) and if time is intrinsically directional (which I think is obvious!) then the analytical tools available are better than standard topology and hence manifolds.
Cheers,
Tim Maudlin
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Edward Michael MacKinnon wrote on Mar. 12, 2015 @ 03:49 GMT
the central point here is :(Q) what properties mus physical reality have for math to be applicable? As the author knows the brings in some traditional problems of scientific realism. In the simple cases treated first we know reality and then apply integers or geometric forms. In the difficult cases we know physical reality through the way successful physical theories represent reality. Here Q i still applicable but more difficult to answer. You eventually sharpen Q to: What physical features space or space-time must have to be represented by the topology of an open set? This leads into your development of a theory of linear structures. I don't feel qualified to comment on that, though it looks good. I tend t think of open sets as a mathematical trick with no direct physical significance. Maybe the development of your theory will chang that.
Ed MacKinnon
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Pankaj Mani wrote on Mar. 14, 2015 @ 16:58 GMT
Dear Tim Maudlin,
You have mentioned that for mathematics to be used as the language of physics, physical world has that sort of structure to be represented mathematically? That depends on the mathematical language being used Physical characteristics are required for mathematical structures to describe a physical situation.
Yes I agree with you and thats why I have propounded...
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Dear Tim Maudlin,
You have mentioned that for mathematics to be used as the language of physics, physical world has that sort of structure to be represented mathematically? That depends on the mathematical language being used Physical characteristics are required for mathematical structures to describe a physical situation.
Yes I agree with you and thats why I have propounded Mathematical Structure Hypothesis to explain their origin in the same line.
Question is - Who decides the symphonic structure of that language? For any mathematical structure to be compatible to explain the physical structure, we need to match their intrinsic "laws of invariance" otherwise their applications would be wrong.
This is why in context of Skolem's paradox: "A particular model fails to accurately capture every feature of the reality of which it is a model. A mathematical model of a physical theory, for instance, may contain only real numbers and sets of real numbers, even though the theory itself concerns, say, subatomic particles and regions of space-time. Similarly, a tabletop model of the solar system will get some things right about the solar system while getting other things quite wrong."
You have classified the mathematical structures into two categories based on Wigner's essay
1) One which are naturally suited to physical world e.g. Integers and what does their suitability imply about the physical world?
2) Others which are not e.g. advanced concepts e.g. complex numbers should have use in physics.
I have explained on the basis of Mathematical Structure Hypothesis that whether it falls in any category, its basically physical characteristics behind the development mathematical language which describes the physical characteristics of the physical world i.e. whether Integer or Complex numbers.
Wigner talks about Complex numbers as advanced concept but what decides the structure of complex numbers and why they are so effective in Quantum Mechanics.
Eugene Merzbacher in his book on QM has explained by deriving that for certain physical characteristic to be satisfied( for quantum waves,any displacement in the space & time dimension should not alter the physical characteristics of waves) and to satisfy these criteria, the mathematical parameters turns out to be "i"(complex number).
Here is the reason the structure of mathematical language has been matched/molded to suit the physical characteristic of quantum waves(physical world).Infact, its not the mathematics describing physics here rather their corresponding law of invariance. So, what is the law of invariance behind complex number. Its answer lies in the definition of why negative multiplied by negative turns out to be positive? Why not positive multiplied by positive also become negative? Here is hidden laws of physics behind the definition of mathematical operators structure and vice versa.
This is because mathematical structures abstractness and physical reality both are creations of the same thing Vibration, which my Mathematical Structure Hypothesis has propounded.
Anyway, your essay is indeed great.
Thanks & Regards,
Pankaj Mani
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Author Tim Maudlin replied on Mar. 14, 2015 @ 17:08 GMT
Dear Pankaj Mani,
Thank you for the comments. The use of complex numbers in quantum theory is a very interesting case, which needs a lot of discussion. My own work here is just on space-time structure, so does not touch on quantum theory directly. But I think it may help to recall that time-revesal is implemented in quantum formalism by taking the complex conjugate of the wave function. This immediately suggests a connection between the use of complex numbers and the temporal structure, indeed a connection with the direction of time.
It is harder to deal with quantum theory because there is no agreement at all about just what physical entities the theory is committed to, particularly what Bell called the "local beables" of the theory. The observable behavior of laboratory apparatus should be determined by the behavior of these local beables at microscopic scale. If you don't even know what these are, then interpreting the significance of the mathematical apparatus becomes essentially impossible. Pure space-time theory is a bit more straightforward.
Regards,
Tim Maudlin
Jayakar Johnson Joseph wrote on Mar. 14, 2015 @ 19:15 GMT
Dear Tim,
Much fascinated by your work, ‘The Theory of Linear Structures’. I think you may find some of the interesting applications of your work at a paradigm used for the comparative analysis in my essay, ‘
Before the Primordial Geometric origin: The Mysterious connection between Physics and Mathematics’. Hope you will enjoy in reading.
With best wishes,
Jayakar
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Christian Corda wrote on Mar. 14, 2015 @ 21:33 GMT
Dear Tim,
Concerning you criticisms in my Essay page, some clarifications could be needed. For rotating frame in my Essay I mean the frame in which the observer sees the detector at rest (the absorber orbits around the source). Clearly, in that frame photons propagate in the radial direction. You are of course correct in highlighting that Equivalence Principle has local behavior. On the other hand, rotating frames generate the centrifuge acceleration in the radial direction cited above, which, in turn, defines a locally accelerated frame. Thus, it seems to me that the application of Equivalence Principle is completely legitimate. I also stress that the use of the Equivalence Principle in rotating frames in general and in the Mössbauer rotor experiment in particular has a long, more than fifty-year-old, history. In the paper of Kündig, i.e. ref. [3] in my Essay, which is dated 1963, one reads verbatim: "when the experiment is analyzed in a reference frame K attached to the accelerate observer, the problem could be treated [7] by the principle of equivalence of the general theory of relativity". Reference [7] in the paper of Kündig is the historical book of Pauli on the theory of relativity dated 1958. Thus, it seems that you were wrong in those criticisms. Here the key point is not the viability of the Equivalence Principle in treating this problem, but the issue that previous literature did not take into due account clock synchronization.
I will read, comment and score your Essay soon. I wish you best luck in the Contest.
Cheers, Ch.
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Jacek Safuta wrote on Mar. 15, 2015 @ 21:19 GMT
Dear Tim,
I was reading your excellent essay with growing interest as you were getting closer and closer to the geometry of spacetime. You argue that standard geometry has been built on the wrong conceptual foundation to apply optimally to spacetime and you propose an alternative geometrical language. That is really promising as I am just looking for such languages. You finally claim that...
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Dear Tim,
I was reading your excellent essay with growing interest as you were getting closer and closer to the geometry of spacetime. You argue that standard geometry has been built on the wrong conceptual foundation to apply optimally to spacetime and you propose an alternative geometrical language. That is really promising as I am just looking for such languages. You finally claim that the Theory of Linear Structures (TLS) is capable of describing the geometry of continua [...]. Is TLS designed exclusively for a specific spacetime description or is it possible to describe e.g. Thurston geometries (the geometrization conjecture, proved by Perelman)? This is double-dealing question because the Thurston geometries, in my opinion, we can treat as a space-like, totally geodesic submanifolds of a 3+1 dimensional spacetime. In three dimensions, it is not always possible to assign a single geometry to a whole space. Instead, the conjecture states that every closed 3-manifold can be decomposed into pieces that each have one of eight types of geometric structure, resulting in an emergence of some attributes that we can observe. 3+1, in turn, means that the constant curvature geometries (S3, H3, E3) can arise as steady states of the Ricci flow, the other five homogeneous geometries arise naturally where the dynamics of the Ricci flow is more complicated and where topological changes (neck pinching or surgery) happen. That way the space flows with time and becomes a dynamical medium - spacetime. Then if TLS can really suit, what is the profit from that approach in comparison to the standard?
I agree that “Physicists seeking such a mesh between mathematics and physics can only alter one side of the equation.” However if we use a proper correspondence rule with the empirical domain than it is really sufficient to discover the set of geometric structures isomorphic to physical reality. Details in my
essay.
I would appreciate your comments.
Best regards,
Jacek
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Author Tim Maudlin replied on Mar. 15, 2015 @ 21:51 GMT
Dear Jacek,
As I mentioned in the appendix, there are topologies that cannot be recovered from any underlying Directed Linear Structure, but I am morally certain that these have no bearing at all on Thurston's conjecture, which deals with manifolds. I see no reason at all to doubt that every one of the manifolds that the conjecture deals with is generated by at least one (and in fact infinitely many) distinct Directed Linear Structures. (Indeed, by infinitely many Linear Structures, with no direction). And the way to implement the analog of surgery theory in the Theory of Linear Structures is completely straightforward: one specifies how two Linear Structures are to be combined by specifying which lines in one are continuations of lines in the other.
As for the profit: the Theory of Linear Structures describes geometrical structure in finer detail than standard topology: that is why many distinct Linear Structures typically can generate the same standard topology.
I'm not sure what you mean when you say that one cannot always assign a geometry to the whole space. I think you must be using that term in a non-obvious way. As you say, the theorem is about decomposing closed 3-manifolds into a set of pieces, each with a specified geometry. But there is obviously a geometry (in the sense of a topological and differentiable structure) assigned to all of these objects: that's why they count as 3-manifolds.
Regards,
Tim
Jacek Safuta replied on Mar. 16, 2015 @ 07:52 GMT
Thank you Tim for inspiration and clarifying as the TLS is new for me. I do not feel it yet and I have to catch up.
When I say that one cannot always assign a (single) geometry to the whole space I mean this is not possible in three dimensions. This is specific feature of three dimensions pointed out in the geometrization conjecture. Differently, for two-dimensional surfaces you can freely assign a single geometry to a whole space. It was really not clearly stated.
Best regards,
Jacek
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Member David Hestenes wrote on Mar. 17, 2015 @ 21:20 GMT
Nicely argued essay, Tim!
I believe your idea that time (motion) imposes linear ordering on space is fundamental. I suppose you know that the idea was fundamental to Newton’s fluxions. And you have convinced me to look at your book on “The Theory of Linear Structures.”
In regard to geometry, I submit that your arguments may benefit from the Geometric Algebra mentioned in my essay. It may help you with the notion of areas defined by linear ordering of line segments, and volumes defined by linear ordering of areas.
Respectfully….David Hestenes
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Author Tim Maudlin replied on Mar. 17, 2015 @ 23:10 GMT
Thanks David! I have always been an admirer of Geometric Algebra...one of my students (Doug Kutach) became much more adept at it than I did. I hope to be able to understand it more deeply. And on a different note, I have recently been looking into Zitter theory. It looks like if one uses Linear Structures to model discrete Relativistic space-times something like Zitter theory must be the right picture of matter. (The fundamental structure of the discrete models is all light-like, so all particles have to follow light-like paths at micro scale.)
Cheers,
Tim
Tommaso Bolognesi wrote on Mar. 18, 2015 @ 10:31 GMT
Dear Tim,
just a quick first reaction to your enjoyable text. We are all very pleased to live in a physical world not completely described by fluid mechanics, but in one where chairs, tables, even living bodies are possible. The step from the physics of this type of world to the mathematics of natural numbers is short and reasonable. But in this reasoning (which I am not questioning) one is actually going form physics to math, from object to description, from territory to map. The reversed scenario - from math to physics - is also interesting, and more challenging. For example, the Mathematical Universe Hypothesis (MUH) does that: it puts math at the roots of the physical world, which would neatly explain why math is
also good at
describing the physics: math—(is)-->physics--(described-by)—>math. However, MUH does not specifically address the key question of why there are objects (thus natural numbers) rather than just fluids, a circumstance that scientists should also put in the box of ‘unreasonable’ facts about the universe (nice topic for next year Contest...).
In my opinion, the most convincing explanation for the emergence of distinguishable ‘objects’ in our universe, thus of natural numbers for their mathematical description, is, currently, the one that attributes a fundamentally
algorithmic nature to the dynamics of spacetime, or of whichever discrete structure one figures sitting at the bottom of the universal architecture. We have today plenty of experimental evidence for the ‘miraculous’ emergence of distinguishable-denumerable ‘objects’ - object/background patterns - from the computations of even the simplest programs.
Best regards
Tommaso
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Jonathan J. Dickau replied on Mar. 18, 2015 @ 15:44 GMT
Hi Tim,
After reading Tommaso's comment (but before I've read your essay), I wanted to chime in to say that the piece he describes as missing is precisely what I found almost 30 years ago, and touch on in my essay. The attached JPEG shows explicitly the quantum hydrodynamic analog within the Mandelbrot Set, showing where objectified forms appear just past the primary Misiurewicz points. This image is my Mandelbrot Butterfly figure, everted about (-1, 0i) such that concentric circles are laid flat.
When I sent this image to John Bush, he replied that he found it 'quite interesting,' which I suspect is because the analogy with quantum hydrodynamics is pretty obvious. This research is still a work in progress. But if it turns out there is a robust connection, where M reveals the process of pinch-off and nucleation by which fluids form droplets - this raises the issue that Tommaso raises above. How could this be, unless the physical reality flows directly from the Math - rather than the other way around?
All the Best,
Jonathan
attachments:
Plateau.jpg
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Tommaso Bolognesi replied on Mar. 18, 2015 @ 15:46 GMT
... and another point.
You are against attributing an abstract mathematical status to democritean atoms: for being mathematically tractable (e.g. counted), they need some physical properties, that european mountains to not possess. You conclude that attributing a mathematical structure to physical items is not the same as postulating that they
are mathematical entities.
I do not know which precise physical properties can be attributed to democritean atoms (probably not color, volume, spin…), but let us consider an atomic event, or atom of spacetime, as conceived in a causal set, a model mentioned higher up in this blog. These are points, with no other attributes than those you assign to the ideal mathematical/geometrical point. They are countable - crucially, for recovering volume information - and yet totally featureless. I would say that their nature (their ontological status?) is mathematical, not physical. And yet, when collected in a causal set, or a superposition of these, they are conjectured to originate, by emergence, just about Everything - I mean the physical Everything.
I find irresistible the argument that the deeper you go in magnifying the 'fabric of the cosmos’ the more the familiar physical properties we are used to recognize tend to vanish. Brought to the limit, this means that physical reality pulverises into mathematics, or ‘baggage-free’, purely abstract objects.
I guess you disagree with this, If I understand your points correctly…
Sorry for the length
Tommaso
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Author Tim Maudlin replied on Mar. 18, 2015 @ 15:54 GMT
Dear Tommaso,
I am a Platonist about mathematics, in the sense that I think there are objective facts about abstract mathematical objects, including facts that lie beyond our abilities to prove. (For example, Goldbach's conjecture is certainly either true or false, and if it is true we may never be able to prove it.) So there are all of these non-material mathematical structures, whose existence is independent of the physical world. (The physical world could not come out one way that makes Goldbach's conjecture true and another way that makes it false.) As I understand Tegmark's hypothesis, every single one of these abstract mathematical structures is a concrete physical world. I think that there are very, very severe problems of different sorts with that hypothesis. One is that the vast majority of possible mathematical structures are not regular enough to be described simply (think of all possible sequences of integers: for most there is no compact way to specify it). So if all mathematical structures are physical, most physical worlds are not compactly describable. And it would be almost a miracle that ours is.
That is a completely different matter than the one about algorithmic dynamics. Here I think we agree: indeed, relatively few structures can be generated by a compact algorithm, just a relatively few sequences can be generated by simple rules. The search for such an algorithm is a form of the search for simple laws. And the simplicity of the laws should explain the comprehensibility and predictability of the physical world.
Regards,
Tim
Author Tim Maudlin replied on Mar. 18, 2015 @ 20:10 GMT
Dear Tommaso,
I missed the second post, so a comment. I am perfectly happy with the sort of ideas in causal sets, or with point-particles or point-events. Mathematically, an infinitude of different causal graphs exist as abstract objects. If one of these accurately describes the physical universe, it is because the physical universe is composed of physical point-like entities. If one denies the difference between the abstract mathematical objects and the physical objects, then I suppose one ends up with Tegmark's view, which has insuperable difficulties. There are all sorts of mathematical objects (e.g. operators) that can be used to describe physical things (e.g. time evolutions) but are not themselves physical things.
Physical point-event are certainly stripped down: they have little in the way of intrinsic physical characteristics. But that does not make them abstract in the sense that mathematical objects are. It just makes them have few physical characteristics.
Regards,
Tim
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Than Tin wrote on Mar. 18, 2015 @ 19:31 GMT
Dear Dr. Maudlin:
I have downloaded a fair number of FQXi-2015 Contest Essays, and tried to read through as many as I can manage. Needless to say that my understanding of the essays is based on the framework I used to view them, and that framework is described in my essay http://fqxi.org/community/forum/topic/2456 .
Simply put, I view the world “analogically,” as contextually sensitive set of duals: i.e. I frame Wigner’s Refrain of mathematics and physics as freedom and determinism (among others I can choose) and then try to understand Dr. Maudlin’s:
(1) “Wigner’s question is this: why is the language of mathematics so well suited to describe the physical world? A proper answer to this question must approach it from both directions: the direction of the mathematical language and the direction of the structure of the physical world being represented. In order for the language to fit the object in a useful way the two sides have to mesh.”
(2) “Physicists seeking such a mesh between mathematics and physics can only alter one side of the equation. The physical world is as it is, and will not change at our command. But we can change the mathematical language used to formulate physics, and we can even seek to construct new mathematical languages that are better suited to represent the physical structure of the world.”
Given my formulation of Wigner’s Thesis, there is nothing that I can disagree with Dr. Maudlin’s views as expressed in the two paragraphs above, but I like to know how the “meshing” might be accomplished in the project.
Regards,
Than Tin
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Member Ian Durham wrote on Mar. 21, 2015 @ 19:59 GMT
Hi Tim,
Nice essay. I will have to read more about your Theory of Linear Structures at some point. It seems to have some similarity to what Kevin Knuth has done with posets, and with some of the things I'm doing right now with information orders on domains (building on the domain work of Keye Martin -- not sure if you are familiar with any of it).
I think you also hit on something interesting in regard to this idea of counting and how it relates to a physical ontology. One could argue that, even if the universe is entirely continuous, our ability to measure it to arbitrary accuracy is necessarily discrete and thus the integers match up well with that discreteness (which interestingly links back to a previous FQXi essay contest). Just a thought.
Anyway, I nevertheless must admit that I didn't find your argument convincing in general. It seems to miss some subtleties. Perhaps these subtleties are addressed in your larger work on the topic, however. For instance, I disagree with you on a key point: I do think that how different branches of mathematics relate to one another, has a direct bearing on how mathematics relates to the physical universe. How could it not? If you are familiar with category theory or topos theory, think about how such theories describe both mathematics and physics and their inter-relationships.
I had a few minor quibbles as well. In the example you gave of a universe describable entirely via fluid mechanics and dynamics, you would still be faced with the distinction of "something" versus "nothing" which maps quite naturally to 1 and 0 respectively. Integers are an elementary extrapolation from there.
I also am not particularly awed by the fact that results in semi-stable elliptical curves were used to prove Fermat's Last Theorem. While I am not deeply familiar with the details of Wiles' proof, in some sense both elliptic curves and Fermat's Last Theorem deal, on some level, with polynomials. Certainly the connection is not obvious, but neither is it all that shocking, at least to me.
Cheers,
Ian
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Author Tim Maudlin replied on Mar. 21, 2015 @ 20:51 GMT
Dear Ian,
Thanks for the comments. Let me try to address some of them.
There are surface similarities to how one treats discrete spaces or space-time using this formalism and what is done in Causal Set theory (which also uses posets), but the actual details turn out to be quite different. Of course, I did not have the space to go into that here, and it is not even in the book that is out, which deals solely with the math. The second volume will apply the math to physics, and it will be done there.
I friend of mine pointed out that for another reason there are countable things even in a fluid mechanics continuum: there can be discrete vortices. (There will be problems counting when they merge, but still they can be stable and discrete over long periods.) So the claims about fluid mechanics is too strong.
I think you misinterpreted the claim about the bearing of different branches of math on one another. Of course that has implications for the connection between mathematics and the world! My point was that if one branch has unproblematic relation to the physics, then any other mathematical structure which connects to the unproblematic one will inherit a comprehensible bearing on physics. In this case, I said that Wigner's problem is solved without remainder. I just wanted to separate puzzlement about why one branch of pure math bears on another from the question of why any math bears on physics.
The example of Fermat was just illustrative: maybe the connection is not so obscure. Like you, I do not know the details. Take the Moonshine conjectures then: certainly mathematicians were surprised about the connections between group theory and the Fourier expansion there. But if the physics were using the group theory in some obvious way, the purely mathematical connection would make the Fourier expansion relevant to study.
Cheers,
Tim
Jairo Jose da Silva wrote on Mar. 23, 2015 @ 01:16 GMT
Dear Prof. Maudlin.
Despite some interesting ideas, you paper pressuposes what it was supposed to explain. As I see it, there is nothing, absolutely nothing intrinsically mathematical in brute nature. Take for instance number. Given any, any!, amount of objects, no matter how sharply distinct one from another, from a certain perspective, there's no number naturally attached to it independently of a unit determination, or, which is the same, a concept which tells us what is it that we are numbering. So, numbering is a conceptual operation and concepts are creatures of ours. In my paper ("Mathematics, the Oracle of Physics") I approached the question of the applicability of mathematics in science from a transcendental perspective. Since nature "out there" has nothing intrinsically mathematical about it, how come that mathematics has anything to do with our theory of nature? From my point of view, the answer to this question requires showing how by a series of constituting acts a suitable mathematical representation of nature is constituted from brute sensorial data. Once this is done the applicability of mathematics in physics is, as I've argued, just an instance of the applicability of mathematics in mathematics itself (in your paper you explicitly reject this identification). In short, I don't think your perspective is radical enough from a truly philosophical perspective. You take too much for granted and embrace too many idees recues. Thank you! Best! Jairo Jose da Silva
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Author Tim Maudlin replied on Mar. 23, 2015 @ 01:48 GMT
Dear Jairo Jose de Silva,
The main contribution in the essay does not concern the application of numbers to physical states, but rather the application of geometrical descriptions. I used enumeration as an example, but it is not the main focus. So the issue of a "unit determination" never arises. Perhaps you do not find the ideas radical enough—one can try to be more radical—but the notion that space-time has an intrinsic geometrical structure is coherent and consistent. Given the right geometrical concepts, one can also see how temporal structure can generate such a geometry.
The problem with "transcendental" arguments, at least as Kant deployed them, is that they were supposed to explain how we can have various sorts of a priori knowledge. But as it turns out, we just don't have that knowledge. So the transcendental approach does not fit with what we now know.
Regards,
Tim Maudlin
Lorraine Ford wrote on Mar. 25, 2015 @ 13:32 GMT
Tim,
You say that "If there are physical items so constituted as to be solid objects, held together by strong internal forces and resistant to fracture and to amalgamation, then they will be effectively countable"; and that "The relevance of the theory of integers for physics is unproblematic so long as the way that physical items are being counted is conceptually sharp."
But counting implies making distinctions (1) and counting is necessarily a multi-step procedure, - so who or what is performing the counting procedure?
Lorraine Ford
1. "What is a Number?" by Louis H. Kauffman, http://homepages.math.uic.edu/~kauffman/NUM.html
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Author Tim Maudlin replied on Mar. 25, 2015 @ 16:45 GMT
Dear Lorriane,
I would not say that a sharply defined enumeration requires any agent or anyone performing anything at all. What is required is the existence of a well-defined map from a physical situation to the integers, and the question is what physical characteristics the situation must have for the map to be well-defined. If I put a bunch of jelly beans in a jar and say there is a definite, exact number of jelly beans in the jar, that is true whether or not anyone ever counts them or goes through any multi-step procedure. It is because there already exists a definite number of jelly beans in the jar that if we want to find out how many there are, it does not matter who does the counting as long as they count correctly, or how they do it. The number they come up with will be the number that is already there before they start counting. If the jelly beans start to melt and merge then (again quite apart from anyone doing anything) the conditions required for a definite number may no longer obtain.
Regards,
Tim
Lorraine Ford replied on Mar. 25, 2015 @ 21:26 GMT
Hi Tim,
Re "there already exists a definite number of jelly beans in the jar":
Where or how does this number exist? Seemingly this number also has a category: "jelly beans in the jar". How is this 2-part (i.e. number and category) informational entity that exists interconnected to physical reality?
Cheers,
Lorraine
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Author Tim Maudlin replied on Mar. 26, 2015 @ 00:47 GMT
HI Lorraine,
I'm not sure how to take the question "how and when does this number exist?". There is a particular physical situation—jelly beans in a jar—and it is clear where and how that physical situation came to exist. It came into existence when the jelly beans were put in the jar by a certain physical act. The point is that the physical situation so created is, because of its physical character, one in which a particular number can be used to describe the situation. The number itself, the thing being used to as part of a representation of the physical state, is not the sort of the thing that exists anywhere or is created. It is, if you will, a Platonic entity. Wigner does not doubt the existence of such Platonic mathematical entities, he just wonders why some of them would be of any use in describing physical situations. The "interconnection" here is the connection between a representation and a represented object. The representation is mathematical and the object physical. It is not a physical connection between the number and the physical situation, if that is what you have in mind, but a representational connection.
Framed this way, the only remaining questions is why certain physical situations or entities have a structure that is usefully represented using particular mathematical entities. That is the question I am trying to clarify, particularly for geometrical structure.
Regards,
Tim
Thomas Howard Ray replied on Mar. 26, 2015 @ 01:06 GMT
If you'll pardon the intrusion ...
Tim, your jelly beans in a jar made me think of Mandelbrot's question about the length of the coastline of England.
Measured in uniform units of jelly beans, there would still be no definite answer, no pat number -- and that's even without assuming melting and merging. So I think you are quite right that the number exists independent of the counting units -- because it's scale dependent. No agent required, therefore -- the number is determined by scale of observation, not the observer's choice of measurement units.
Best,
Tom
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Author Tim Maudlin replied on Mar. 26, 2015 @ 01:20 GMT
Dear Tom,
I see your point about Mandelbrot, but here too we should be careful. If the coastline really were a fractal—which would be a physical feature of it—then there would be nothing that counts as "the length" of it. But if it is not fractal, and becomes smooth at fine enough scale or discrete at fine enough scale, then we could define the "true length" as the limit as scale gets finer and finer. In the discrete case, this bottoms out and in the non-fractal case there is a well-defined limit. So the nature of the dependence on scale is itself something that depends on the physical situation. The fractal would give one extreme sort of case.
Cheers,
Tim
Thomas Howard Ray replied on Mar. 26, 2015 @ 02:00 GMT
Nice, Tim, very nice. That could be a whole other very interesting discussion -- thanks!
Best,
Tom
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Laurence Hitterdale wrote on Mar. 25, 2015 @ 18:17 GMT
Dear Dr. Maudlin,
Perhaps you would be willing to say more about the relationship between the geometry of linear structures and the question whether time is asymmetric. I am concerned about the position, presented by Huw Price and others, that there is no intrinsic difference between the directions of past to future and of future to past. It seems to me that a geometry based on the concept of the line might be more helpful on this issue than a geometry based on the concept of the open set. In the first place, a linear-structure geometry could clarify the discussion. We could formulate one very basic issue as the question whether the mathematical structures more accurately corresponding to physical time are directed lines or undirected lines. Furthermore, as you point out, the theory of linear structures would be appropriate for either answer. So, just as with the question of continuity or discreteness, the mathematical language would not add its own weight to the scales when we are trying to investigate a problem about physical facts. As you said in response to an earlier comment, this neutrality is an advantage.
Best wishes,
Laurence Hitterdale
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Author Tim Maudlin replied on Mar. 25, 2015 @ 18:40 GMT
Dear Laurence,
As I see it, the situation is this. There are three possible positions on the direction of time: 1) there is no directionality to time at all—the direction from this event to the past is physically just like the direction from this event to the future—; 2) there is a directionality, but it is not fundamental but rather should be analyzed in terms of something else (e.g....
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Dear Laurence,
As I see it, the situation is this. There are three possible positions on the direction of time: 1) there is no directionality to time at all—the direction from this event to the past is physically just like the direction from this event to the future—; 2) there is a directionality, but it is not fundamental but rather should be analyzed in terms of something else (e.g. an entropy gradient). Note that on a view like this, using the entropy gradient to define the direction it comes out analytic that entropy never decreases, since the direction to higher entropy is the forward direction. The "direction of time" can flip around on this view if entropy behaves the right way. This is not what we normally say: we say that the entropy of (say) a gas in a box can go down, and even does go slightly done, even if on average it goes up. 3) The direction of time is a real, physical difference between the time directions, and does not get reduced to or analyzed in terms of anything else. Huw holds 1), I hold 3), and lots of people hold 2) offering different sorts of analyses.
If you hold 3), then the best one can ask for from mathematical physics is a clear mathematical representation of this directionality. Since we are talking about an intrinsic directionality in space-time geometry, one would need a geometrical language in which directionality can be naturally represented. In standard topology, this is not true. If I ask you to "put a direction" on an open set of points, it is not at all clear what I am asking you to do. But lines, in contrast, are exactly characterized by having only two directions on them. Indicating that these directions are physically different is just a matter of associating one of the two possible directions with the line. This is what can be done using Directed Linear Structures: if the lines have directions, the geometry becomes intrinsically directional. There is no standard topology analog for this at all.
Now we push further: if space-time has an intrinsically directed geometry, what is the source of the directionality, the directional asymmetry? In Relativity, there is a very natural answer: the directionality is produced by the asymmetric nature of time. Some pairs of events (but not all) can be characterized by an asymmetric temporal earlier/later distinction. And in Relativity (but not classical space-time) that distinction alone is enough to recover almost all of the space-time geometry: everything up to the conformal structure. The whole light-cone structure gets built in, and indeed a complete Directed Linear Structure can be defined. So the picture is that the fundamental asymmetry of time creates a fundamentally directed space-time geometry. And this can be done for both continuous and discrete structures.
In sum, if you think time is intrinsically asymmetric (unlike Price), standard topology provides no way to easily represent that feature of the geometry and the Theory of Linear Structures does. That is not itself an argument for the directionality. But it is a response to someone who says: "I don't see any time asymmetry in the math!". That, of course, depends on what mathematical language you are using to represent the physical situation. Maybe it is hard to see the directionality in the math because you are using math that does not have a simple way to represent directionality.
I hope this is some help,
Cheers,
Tim
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Laurence Hitterdale replied on Apr. 23, 2015 @ 01:10 GMT
Dear Dr. Maudlin,
Thank you. Your response is helpful. I tend to favor either the second or third possible position, probably inclining more towards the second of the three. I realize that the availability of a particular mathematical structure is not of itself an argument that an aspect of physical reality is one way rather than another. Still, it is important, I think, that we are not compelled to use a mathematical language which, if time is inherently asymmetrical, leaves that asymmetry out of the mathematical representation. It is good to know that there is an alternative language which would capture this important feature of the physical reality.
Best wishes,
Laurence Hitterdale
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Eckard Blumschein wrote on Mar. 27, 2015 @ 17:54 GMT
Dear Tim Maudlin,
When David Joyce commented on a previous essay of mine "it contains some interesting points", I was not sure whether at least he understood my observation that Dedekind replaced Euclid's 1-D notion of number as a measure (or as you are calling it a line?) by the 0-D point at the end of the distance from zero. I understand that it might be no opportune to question the fundamental of point set theory and point set topology. Did you deal with this perhaps historically decisive change?
Since I read Fraenkel 1923, I am sure to understand Cantor's logical flaw. My strongest additional argument is the indisputable fact that alephs in excess of 1 didn't prove useful.
Concerning my distinction between Relativity and relativity, see the essay by Phipps. My opinion that there is no preferred point in space but the natural zero of elapsed as well as future time might be too bewildering to those like you.
Because English is not my mother tongue, I had sometimes difficulties to clearly understand what you meant, e.g. on p. 5 with "sifting humor". On the same page, it would be helpful to find out where footnote 2 refers to and what conjugate points are meant.
Just an aside concerning Wigner: Von Békésy got a Nobel prize for his claim of a a passive traveling wave in cochlea, the mathematics of which was provided by Lighthill and was indeed unreasonably effective in the sense it was just fitted to measured data. Already Thomas Gold had argued that a passive traveling wave cannot work at all. Later on the cochlear amplifier was found.
Respectfully,
Eckard Blumschein
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Author Tim Maudlin replied on Mar. 27, 2015 @ 18:30 GMT
Dear Eckart Blumenschein,
I can reposed quickly to the questions about my essay. The term "sifting humor" was used by David Hume, and it means continuing to analyze some concepts even further. Readers familiar with Hume would pick that up, but probably if you had not read the passage in Hume it would sound odd even to a native speaker.
The conjugate points I have in mind can occur in models of General Relativity (but not Special Relativity) where distinct light-like geodesics intersect more than once. Call two such intersection points A and B. In such a case, each path is light-like even though A and B are the endpoints of two different lines (in my sense). So the criterion for a light-like geodesic that works well in Special Relativity fails in General Relativity. But this can be fixed, because each light-like geodesic can be subdivided into overlapping parts, each of which satisfies the simple definition. So the General Relativistic case can be covered by a simple amendment to the definition.
Regards,
Tim Maudlin
Eckard Blumschein replied on Mar. 28, 2015 @ 10:15 GMT
Dear Tim Maudlin,
Thank you for your quick response. Lee Smolin lost my admiration because up to now he did not even respond to my simple request whether he actually meant "off", what was not understandable to me, or simply "of". I wonder why nobody else admitted not having understood your term "sifting humor".
We all make mistakes. Misspellings of my name don't matter; here is no risk of confusion.
I am looking forward learning from your criticism of my admittedly uncommon arguments.
Regards,
Eckard Blumschein
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Eckard Blumschein replied on Mar. 31, 2015 @ 16:17 GMT
Dear Tim and dear Spencer,
Are mathematicians in position to at least partially correct mistakes? I was surprised reading in Wikipedia: "once known as the topology of point sets, this usage is now obsolete".
Of course, while it is possible to attribute a direction to a measure e.g. to a piece of a line, a point doesn't have a direction. Moreover, a set of continuous 1-D pieces can constitute any continuous line while Dedekind just begged to believe that a "dense" amount of 0-D points may constitute a continuous line.
I agree with Spencer on that the symmetry between past and future cannot be avoided just by means of the otherwise necessary return to Euclid's notion of number. I tried to explain in a discussion with Cristinel Stoica that the direction got inevitably lost due to abstraction from reality to model.
Serious criticism of my arguments is welcome.
Regards,
Eckard
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Author Tim Maudlin replied on Mar. 31, 2015 @ 16:47 GMT
Dear Eckard,
It is not clear what it means to apply a direction to an arbitrary measure—think of a volume measure, for example—an in any case we are here talking about geometrical structure that can be defined at a sub-metrical level, without any appeal to measures. Of course, I am not suggesting attributing a direction to a point but only to a line, and every line contains at least 2 points.
Dedekind was right, of course: the "real line" is just the set of real numbers, and the standard (Lebesgue) measure over the reals gives every individual number measure zero. And the set of reals forms a continuum by any reasonable definition. Since the reals can be put in 1-1 correspondence with the points in a Euclidean line, one can see how a continuum it constituted from 0-dimensional points. I'm not sure what problem you find with Dedekind here.
Cheers,
Tim
Eckard Blumschein replied on Mar. 31, 2015 @ 22:41 GMT
Dear Tim,
If I recall correctly, Euclid spoke of the Unity (1). I asked myself what he meant and perhaps it was my idea to translate it into measure because two units of distance, area, or volume are two according measures. A positive measure like length is naturally directed from smaller to larger. Maybe, I was inspired by one or several of the books and papers on history of mathematics I...
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Dear Tim,
If I recall correctly, Euclid spoke of the Unity (1). I asked myself what he meant and perhaps it was my idea to translate it into measure because two units of distance, area, or volume are two according measures. A positive measure like length is naturally directed from smaller to larger. Maybe, I was inspired by one or several of the books and papers on history of mathematics I read mostly in German. I mainly recall O. Becker and W. Gericke but also B. Bolzano, G. Cantor, R. Dedekind, H. Ebbinghaus, A. Fraenkel, F. Hausdorff, J. König, C. Lanczos, D. Laugwitz, Sh. Lavine, W. Mückenheim, D. Spalt, and H. Weyl. I forgot some names, in particular a Spanish sounding one and a Catholic mathematician. Dirichlet, Weierstrass, Heine, and others were more or less involved in the replacement of Euclid's geometric notion of number by the elder and more primitive pebble-like points.
My essay reminds of the contradiction between something every part of which has parts of non-zero measure (the continuum alias aleph_1) and something that has no parts (a rational, i.e., zero-measure element of aleph_0). Dedekind claimed having filled the gaps by creating irrational numbers. Actually he didn't create a single new irrational number. Dedekind's downward definition by a "cut" proved of no use in contrast to the feasible upward approach by Meray and by Cantor who merely ignored that it is impossible to single out an element from an infinite amount of them. Was it warranted to generalize known limits? Nobody doubts that the limit of 0.999999... is one. However, equivalence is not the same as identity, and the limit pi has no exact numerical correlate. A measure cannot be rational and irrational at a time. Finite and infinite exclude each other.
By the way, I see my reasoning confirmed in Wikipedia:
"Any closed interval [a, b] of real numbers and the open interval (a, b) have the same measure b-a".
I see the academic distinction between open and closed not justified because single points in IR don't matter at all. Their location in IR is not even completely addressable. Isn't this an obstacle for the bijection you are referring to? I see rational numbers as truncated real ones.
Cantor's transfinite cardinals remind me of his failed attempt to convince cardinal Franzelin of his infinitum creatum. So far, nobody even tried to object when I mentioned that only aleph_0 and aleph_1 proved useful. Cantor's naive point set theory seems to be just a historical burden. If Dedekind did also offer mistakes - and meanwhile I am sure he did - this is much less obvious.
Cheers,
Eckard
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Spencer Scoular wrote on Mar. 29, 2015 @ 03:58 GMT
Dear Tim:
This is a fantastic essay. And it is very well written.
As we all know, mathematics has been very effective in physics. Its weaknesses to date in modelling physical reality have been twofold:
1. Using open set theory, it does not model that time has an order (whether we interpret this as an order in forward time or an order in backward time);
2. It does not model which directed order (forward order or backward order) corresponds to the observed Arrow of Time.
Your Theory of Linear Structures addresses point 1 - and is therefore important. However, I do not believe it can address the second point. In particular, the initial end point of a line could represent a past instant of time and the final end point of the line could represent a future instant of time - but equally the initial end point of a line could represent a future instant of time and the final end point of the line could represent a past instant of time. The mathematics cannot differentiate between these two interpretations. So, the Theory of Linear Structures, while it can encode an order, will not I believe be able to encode which directed order of a line corresponds to the observed Arrow of Time. Instead, we would have to impose the direction of time from outside the mathematics on the solution - as we do now for open set theory.
What this would mean is that the Theory of Linear Structures - although important - will produce time-symmetric theories, as open set theory does now. We will unfortunately not be able to use it, for example, to have a mathematical theory of evolution. Nevertheless, your work is very good.
If you are interested, in my essay I explain more generally why mathematics cannot, in principle, model the Arrow of Time.
Thank you again for some great research and a clear essay.
Kind regards
Spencer Scoular
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Author Tim Maudlin replied on Mar. 29, 2015 @ 05:10 GMT
Dear Spencer,
Thanks for the careful reading. I don't think we disagree on this much: there is a real, physical distinction between the two time directions and it would be useful to have a way to describe geometries that include such a fundamental asymmetry. That is the part my language does. Then we ask a further question: can we give a more profound account of the physical nature of the asymmetry? This could turn out two ways: it could be a fundamental physical structure, and so not admit of further analysis, or there may be a deeper analysis possible. I am open to both possibilities, and it sounds like your work addresses the second.
My hope is that this mathematical structure can be of use for many different projects. Perhaps yours is one.
Cheers,
Tim
Michel Planat wrote on Mar. 30, 2015 @ 09:32 GMT
Dear Tim Maudlin,
I know from your work that you have a strong acquaintance withh Bell's work (B). I arrived at Bell/CHSH inequality from my investigation of Kochen-Specker theorem for multiple qubits mainly through Mermin' treatise (my ref. [19]). At some stage, I observed that the commutation diagram for a set of four observables involved in the violation of the inequality is just a...
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Dear Tim Maudlin,
I know from your work that you have a strong acquaintance withh Bell's work (B). I arrived at Bell/CHSH inequality from my investigation of Kochen-Specker theorem for multiple qubits mainly through Mermin' treatise (my ref. [19]). At some stage, I observed that the commutation diagram for a set of four observables involved in the violation of the inequality is just a square/quadrangle.
Hence my attempt to deepen the subject. My work on KS in dimensions 4, 8 and 16 (two, three and four qubits) is in 1204.4275 (quant-ph) [Eur. Phys. J. Plus 127,86 (2012)] where I also mention a paper of P.K. Aravind on BKS.
My 2013 FQXi essay [also 1310.4267 (quant-ph)] provides the details you ask for. The inequality in p. 4 of my present essay is that of Peres's book [(6.30, p. 164 of Quantum Theory: Concepts and Methods, Kluwer, 1995]. Replacing the dichotomic variables s_i by the appropriate (i.e. commuting like a square) two-qubit operators (or n-qubit operators) that have dichotomic eigenvalues +/-1) as in Peres, p. 174, the norm of C equals 2v2. With two-qubits, there are 90 distinct squares/violations, some involve entangled pairs of operators, others no (as in my example of Fig. 2a). B or KS is not a matter of entanglement but of contexts (compatible observables) as already recognized by many authors. Here I don't refer to an interpretation of QM but to a strict application of its domain of action.
Of course on can go ahead and try to discover a realm for squares and other finite geometries relevant for BKS as I started to do in the 2010 FQXi essay and my subsequent work. I have found that the application of Grothendieck's dessins d'enfants is very promising in this respect. I have been quite surprised that stabilizing a particular square from the two-generator index 4 free group is an instance of the smallest moonshine group (p. 5) whose structure amounts to that of the Baby Monster group.
I hope that it clarifies a bit what I wrote. I am currently working at your own ambitious essay and I intend to give you some comments in the coming days.
All the best,
Michel
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Author Tim Maudlin replied on Mar. 30, 2015 @ 11:23 GMT
Dear Michel,
Maybe it is because you came at this from KS that this seems unfamiliar. If one thinks of the sort of experimental arrangement that Bell had in mind, with observation on the two qubits being made very far apart, then the commutation structure you mention is obvious: any observable on one side must commute with any observable on the other, or else qm would violate no-signalling. And on the same side, the two possible observables cannot commute, or else you do not violate the inequality. But it is not the case that for any set of observables with the commutation structure you show that one can get the maximal qm violation of 2tr(2). so the norm you mention does not follow from the commutation structure you have written down. (Think or what happens as the two possible spin directions on each side approach each other: in the limit, the maximal value becomes 2, not 2rt(2).)
Regards,
Tim
Michel Planat replied on Mar. 30, 2015 @ 14:29 GMT
Dear Tim,
I agree with your point (i): commutation, in my diagram IX commute with XI and ZI but not with IZ, and vice versa.
I don't agree with point (ii) for qubits. I have checked that for all multiple qubit operators (starting with two qubits) one arrives at the maximal violation 2v2. It is the reason why a finite geometry like the Mermin's square (the 3 by 3 grid) for two qubits has nine proofs of Bell's theorem in it.
If one makes use of the dessins d'enfants the extension field involved is Q(v2).
You may have in mind another experimental scheme than the one I am using where the maximum violation does not apply stricto sensu.
For other type of violations of classical inequalities, there is the paper by Alexis Grunbaum and an optical experiment that I mention in his blog.
Michel
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Author Tim Maudlin replied on Mar. 30, 2015 @ 14:34 GMT
Dear Michel,
Consider the following experimental set-up. On one side, there is a choice between measuring spin in the z-direction and in a direction 5° away from the z-direction, and similarly on the other side. Since the two possible measurement on each side do not commute, and each on one side commute with both on the other, this satisfies your commutation square. But no quantum state gives the maximal value of 2rt(2). If you think one does, maybe you can specify what you think it is.
Cheers,
Tim
Michel Planat replied on Mar. 30, 2015 @ 15:01 GMT
Dear Tim,
The two commuting operators on an edge share their states and thus remove the degeneracy occuring in the 4 by 4 observable/matrix. Only these states are involved in the calculation. This is implicit in the norm. If we are talking about a two-qubit experiment I see no other way (and similarly for more qubits).
Michel
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Jonathan J. Dickau replied on Mar. 30, 2015 @ 20:43 GMT
Meaning no disrespect..
There has been a lot of heated discussion on various pages of the FQXi forum, regarding Bell's experiments and variations, in the context of Quantum correlations, including their measure and interpretation. Of course; this really stems from concerns raised by EPR, which Bell was hoping to decisively resolve. Unfortunately; there is some ambiguity or inconsistency in the paper by which Bell first articulated this, and successive interpretations have somewhat obscured that.
There was a comment by Michael Goodband on the thread for one of his previous essays - which I only partially recall - that makes this inconsistency clear, or the ambiguity obvious. One of the key variables is first introduced in Bell's paper as a tensor and thereafter used as a scalar, I think, which restricts the applicability of his conclusions. I see a sensitive dependence on precise definitions and interpretations, surrounding this question, with divergent outcomes for different choices of what principle is most fundamental.
It appears that you have a settled view of this subject Tim. But for some of the contest participants, at least, there are open issues surrounding Bell's theorem and experiments, how well they address the questions raised by EPR, how well these efforts characterize nature, or what is observed, and so on. The biggest question still remains why we see what we do. Perhaps linear structure theory can help us sort this out. We'll see.
All the Best,
Jonathan
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Michel Planat wrote on Mar. 30, 2015 @ 14:35 GMT
Dear Tim,
More on your comment: "as the two possible spin directions on each side approach each other: in the limit, the maximal value becomes 2": this is classical argument that seems irrelevant in the quantum (not spatial) scheme, either the spins are apart or the same.
Best,
Michel
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Michel Planat wrote on Mar. 30, 2015 @ 16:49 GMT
Dear Tim,
You write a very attractive essay about a speculated structure of the "physical space-time" that you call a Theory of (Directed) Linear Structures (DLS). " I will argue that standard geometry has been built on the wrong conceptual foundation to apply optimally to space-time". Your essay is based on your recent book at Oxford University Press that unfortunately I could not access...
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Dear Tim,
You write a very attractive essay about a speculated structure of the "physical space-time" that you call a Theory of (Directed) Linear Structures (DLS). " I will argue that standard geometry has been built on the wrong conceptual foundation to apply optimally to space-time". Your essay is based on your recent book at Oxford University Press that unfortunately I could not access yet. But I could find a 2010 paper of you: The Geometry of Space-Time (Tim Maudlin and Cian Dorr). If I am not wrong, your idea is that the DLS have to be structured in terms of open lines with an order of their points reflecting the succession of space-time events.
A DLS may also be discrete and you write in the aforementioned paper "For example, a space consisting of five points admits of 6,942 distinct topologies and 1,048,576 distinct Directed Linear Structures. These Directed Linear Structures generate 6,942 different topologies: every topology can be recovered from some underlying Directed Linear Structure (DLS), and most can be recovered from many different underlying Directed Linear Structures.", that is the relation between a topology and a DLS is not one to one.
I have a few remarks that may be are clarified in your book or in the next one to appear.
* The discrete LS can be seen as point/line incidence geometry and I am curious to see what kind of non-trivial geometries with a few poinst you recover. Is a DLS reminiscent of a Schreier coset graph?
* As you, I am interested in incidence geometries particularly those that arise from the coset space attached to dessins d'enfants (they are topological objects). I don't see these geometries useful for a space-time physical space time but as an observable space. Not surprisingly (as in QM) they have to do with finite projective spaces. For these geometries having three lines the Veldkamp space (the space of geometric hyperplanes) is isomorphic to a projective space. However, I do not see any reason why it has to with your approach.
* Can you classifial your geometries in terms of invariants like their genus, or the number of voids or an automorphism group? Could they be finitely represented (in terms of groups)?
* Are they Cantor sets in the continuous case?
Thanks in advance for your feedback.
All the best,
Michel
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Author Tim Maudlin replied on Mar. 30, 2015 @ 19:32 GMT
Dear Michel,
In the finite point case, the theory essentially reduces to directed graph theory. I don't see any deep connection to group theory and representations of cosets...the nearer analogs are differential geometry and (a bit) even non-commutative geometry (only the latter with strong restrictions). If you take as a target the space-times that are solutions to the General Relativistic Field equations, then there is not so much reason to focus on automorphisms.
The continuous case will include all standard Riemannian and semi-Riemannian structures. I can't see any connection to Cantor sets. I think that trying to connect this approach to group theory is not the right way to go.
Regards,
Tim
Jonathan J. Dickau wrote on Mar. 30, 2015 @ 19:25 GMT
Hello Tim,
I found a lot to like, in your essay. You articulated well, the problems I've encountered with point set topology, and its limitations for a realistic description of physical form. I have adopted a constructivist and emergentist view toward geometry, in my own research, which reproduces some of the features of your linear structure theory program. And intriguingly; my work linking Cosmology to the Mandelbrot Set necessitated a departure from the standard program, and conclusions similar to yours.
The Mandelbrot Mapping Conjecture states that the periphery of the Mandelbrot Set encodes the dynamism for the full range of physical processes, from the most to the least energetic. So if it is rotated from the conventional view such that (-2,0i) is on the bottom; it can be viewed as a thermometer. But in cosmological terms; the cusp at (.25,0i) is the moment of the universe's inception. What is clearly described, even when the argument is extended into the quaternion and octonion domains, is that initially spacetime was purely timelike and broken symmetry forced spacetime to evolve relativistically.
Philosophically speaking; we know that for structures to exist in time, they must have a time-like projection or duration. Accordingly; for spacetime itself to be an enduring feature, it must also exist in time and have a time-like aspect - hence it must possess linear structure. This is something I have attempted to articulate in several papers, but you have summed things up rather elegantly. I will have to take some time to re-read this paper, and fully digest it, before I make a determination. I have some issues to address, I think. But on first look it appears your program has much to recommend it.
All the Best,
Jonathan
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Author Tim Maudlin replied on Mar. 30, 2015 @ 19:38 GMT
Dear Jonathan,
Thanks for the comments, and I hope you find something useful in the program. I would be a bit surprised if it makes contact with what you are doing given that you start from fractals. In fact, my approach tends to make fractals more peripheral than the standard topological approaches, because by the standard topological definition of "continuous" fractal functions are continuous and by my definition they are not. I have a little hope that this might help for a path-integral formulation of the quantum theory, because in the standard approach the measure over path space tends to be dominated by fractals, with makes it something of a mess. But if this is any use for your approach, I would be very pleased,
Regards.
Tim
Jonathan J. Dickau replied on Mar. 30, 2015 @ 21:13 GMT
Bruce Lee once said..
Retain what is useful. I like that advice.
Regards,
Jonathan
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Jonathan J. Dickau replied on Mar. 30, 2015 @ 21:25 GMT
Mountains may be hard to define precisely..
But observers on adjoining peaks can clearly distinguish their positions from each other, and have a distance to travel to be in the same place. I find that pondering questions like 'what is the distance when traveling along the shore in Britain?' make life interesting for me.
All the Best,
Jonathan
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Michel Planat wrote on Mar. 31, 2015 @ 06:56 GMT
Dear Tim,
The number of topologies (transitive digraphs) on n labeled elements is Sloane sequence 1000798 that is 1, 1, 4, 29, 355, 6942, 209527, ... for n = 1, 2, 3, 4, 5, 6, 7. The number of DLS for n =5 is 2^20 = 1,048,576.
Do you always have the number of DLS equals to 2^p (for some p) and what is the sequence? I suspect a relation to a finite projective space PG(2,p-1), e.g. for n = 5 the number of DLS is 1+|PG(2,19)|. This is reminiscent of a Veldkamp space (set of hyperplanes of a finite geometry with 3 points on a line).
Regards,
Michel
ps/ Today an interesting paper about topologies on a finite set
http://xxx.lanl.gov/pdf/1503.08359.pdf
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Michel Planat wrote on Mar. 31, 2015 @ 08:27 GMT
Dear Tim,
I found your slides of a 2013 talk. The number of DLS is 1,4,64,4096,1048576 (Sloane sequence 1053763). It is the number of simple digraphs (without self-loops) on n labeled nodes. It also corresponds to the number of nilpotent n x n matrices over GF(2).
May be this property of nilpotency makes sense for a space time as it does for quantum mechanics (Rowlands). Your approach opens many perspectives.
Michel
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William T. Parsons wrote on Apr. 1, 2015 @ 20:36 GMT
Hi Tim--
Your essay is superb: rigorous yet readable. In particular, I thought that it was quite thought-provoking, which is the sure sign of any excellent essay.
Question: How does your theory of Theory of Linear Structures deal with closed time-like curves (CTCs) in General Relativity? To focus the question, consider the Gödel Cosmology, which you addressed, for example, on pp. 216-217 of Quantum Non-Locality & Relativity (3rd Ed.). As I understand it, Gödel formulated his cosmology to put a stake through the heart of time, specifically, the notion that time consists of well-ordered linear events. This seems to directly conflict with your theory. Of course, there have been many responses to Gödel’s Universe. Many a physicist has simply noted that the Universe does not appear to rotate. Others have said that the cosmology is so insanely vicious that it just can’t be right. What’s your view?
Two additional, non-substantive points:
First, don’t forget page numbers!
Second, on a very personal note, I would like to thank you for taking the time to respond to everyone’s questions. You are one of the most prominent contestants. (I know that because I own several of your books, including both the second and third editions of QNL&R.) We all know that you are a very busy guy. Nonetheless, you have taken the time to patiently and judiciously respond to all manner of posts. That is incredibly impressive.
Very best regards,
Bill.
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Author Tim Maudlin replied on Apr. 1, 2015 @ 21:08 GMT
Dear Bill,
Thanks for the kinds words!
The situation with respect to CTCs is interesting, and it goes like this:
If all you want to do is model a space-time with CTCs, you can do it. In fact, there is clean definition of a "simple loop" in this theory: a simple loop is a set of points that is not itself a line, but removing any point from the set yields a line. (Recall that...
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Dear Bill,
Thanks for the kinds words!
The situation with respect to CTCs is interesting, and it goes like this:
If all you want to do is model a space-time with CTCs, you can do it. In fact, there is clean definition of a "simple loop" in this theory: a simple loop is a set of points that is not itself a line, but removing any point from the set yields a line. (Recall that that lines are open lines, and the loop fails because it is closed.) When modeling a space-time, the natural thing is to use directed lines to represent the direction of time, and in any temporally orientable space-time (which includes Gödel's) that will be possible. So the language has the resources to describe CTCs.
But if the space-time includes CTCs, you lose a particularly lovely feature: that the whole conformal structure (and the whole Directed Linear Structure) is determined by nothing but time order among events. The problem, of course, is that time-like related events in the CTC have no definite time order: one can't say which happened before which, and not because they are space-like separated. So this particularly beautiful connection between the pure time order (the partial ordering of events by earlier/later) and the space-time geometry does not hold in Gödel space-time. It does hold in globally hyperbolic space-times, i.e. space-times that admit of a Cauchy surface.
My own view about this is that the only serious grounds we could have to believe in the physical possibility of CTCs is the existence of some actual, observed phenomenon that seems to require them. That is, if we don't see any direct evidence of CTCs, I see no reason not to assume that they just are not physically possible. We already do this with non-temporally orientable space-times: as a purely mathematical question, one can specify solutions to the GR field equations that are not temporally orientable, but no one concludes that we have to take them seriously as real physical possibilities. Why not? Because they contradict the nature of time itself. So I would be open to empirical evidence that CTCs exist, but absent that (or some very powerful theoretical argument) I am inclined not to take them seriously as physically possible. It is of particular note that (unlike, say, black holes) no one has ever suggested a means to make a CTC. The only models with CTCs have them put in "by hand".
I hope this is useful.
Cheers,
Tim
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Author Tim Maudlin replied on Apr. 1, 2015 @ 21:40 GMT
Dear Bill,
In case it wasn't clear: the point about simple loops is that in this approach the existence of a CTC is just the existence of a simple loop in the geometry. We get rid of all space-like lines and space-like geometry, and leave only lines that are everywhere time-like or null. So we can model such simple loops, but admitting them spoils a nice program for deriving the whole geometry from temporal structure.
Cheers,
Tim
Harry Hamlin Ricker III wrote on Apr. 1, 2015 @ 21:49 GMT
Dear Sir, When I read the first sentence of the abstract I am certain that your thesis is a tautology. Mathematics is a human invention, and as such is fallible. Your thesis doesn't seem to take account of that problem. In my view, any attempt to expound the thesis of the essay is bound to be problematic, because the proposed subject implies that mathematics is physics, or put differently that the universe is mathematical. Since all of the ideas involved are human inventions, they are likely to be completely wrong in conception. So far I have found nothing to convince me that the inventions that we humans have created do represent actual truth and so are not just fallible delusions of the human imagination.
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Pentcho Valev wrote on Apr. 2, 2015 @ 19:52 GMT
"Our understanding of the structure of time has been revolutionized by the Theory of Relativity. Intriguingly, the change from a classical to a Relativistic account of temporal structure is of exactly the right sort to promote time into the sole creator of physical geometry."
How did the theory of relativity "revolutionize" the understanding of time? By replacing the true tenet of Newton's...
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"Our understanding of the structure of time has been revolutionized by the Theory of Relativity. Intriguingly, the change from a classical to a Relativistic account of temporal structure is of exactly the right sort to promote time into the sole creator of physical geometry."
How did the theory of relativity "revolutionize" the understanding of time? By replacing the true tenet of Newton's emission theory of light, "the speed of light depends on the speed of the emitter", with the false tenet of the ether theory, "the speed of light is independent of the speed of the emitter":
"Relativity and Its Roots", Banesh Hoffmann, p.92: "There are various remarks to be made about this second principle. For instance, if it is so obvious, how could it turn out to be part of a revolution - especially when the first principle is also a natural one? Moreover, if light consists of particles, as Einstein had suggested in his paper submitted just thirteen weeks before this one, the second principle seems absurd: A stone thrown from a speeding train can do far more damage than one thrown from a train at rest; the speed of the particle is not independent of the motion of the object emitting it. And if we take light to consist of particles and assume that these particles obey Newton's laws, they will conform to Newtonian relativity and thus automatically account for the null result of the Michelson-Morley experiment without recourse to contracting lengths, local time, or Lorentz transformations. Yet, as we have seen, Einstein resisted the temptation to account for the null result in terms of particles of light and simple, familiar Newtonian ideas, and introduced as his second postulate something that was more or less obvious when thought of in terms of waves in an ether. If it was so obvious, though, why did he need to state it as a principle? Because, having taken from the idea of light waves in the ether the one aspect that he needed, he declared early in his paper, to quote his own words, that "the introduction of a 'luminiferous ether' will prove to be superfluous."
Pentcho Valev
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Author Tim Maudlin replied on Apr. 2, 2015 @ 21:32 GMT
Dear Pentcho Valev.
We know that the speed of light is not affected by the speed of the emitter by the observation (for example) of binary stars. If the speed of light emitted by the receding star were even slightly different from that of the approaching star, given the period of time that the light is en route, the apparent motions of the stars as seen from earth would be quite different from what we see. The motions would not appear to us to be regular. So the independence of the trajectory of a light ray in a vacuum from the motion of the emitter is empirically established.
Regards,
Tim Maudlin
Pentcho Valev replied on Apr. 3, 2015 @ 00:32 GMT
"The de Sitter effect was described by de Sitter in 1913 and used to support the special theory of relativity against a competing 1908 emission theory by Walter Ritz that postulated a variable speed of light. De Sitter showed that Ritz's theory predicted that the orbits of binary stars would appear more eccentric than consistent with experiment and with the laws of mechanics. (...) De Sitter's argument was criticized because of possible extinction effects. That is, during their flight to Earth, the light rays should have been absorbed and re-emitted by interstellar matter nearly at rest relative to Earth, so that the speed of light should become constant with respect to Earth. However, Kenneth Brecher published the results of a similar double-survey in 1977, and reached a similar conclusion - that any apparent irregularities in double-star orbits were too small to support the emission theory. Contrary to De Sitter, he observed the x-ray spectrum, thereby eliminating possible influences of the extinction effect."
Here is Brecher's paper:
K. Brecher, "Is the Speed of Light Independent of the Velocity of the Source?"Brecher (originally de Sitter) expects a system with unknown parameters to produce "peculiar effects". The system does not produce them. Conclusion: Ritz's emission theory (more precisely, the assumption that the speed of light depends on the speed of the emitter) is unequivocally refuted, Einstein's theory is gloriously confirmed.
Needless to say, refutations and confirmations of this kind can only be valid in Einstein's world. Note that they cannot be criticized - the fact that the parameters of the double star system are unknown does not allow critics to show why exactly the "peculiar effects" are absent.
Pentcho Valev
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Author Tim Maudlin replied on Apr. 3, 2015 @ 00:55 GMT
Actually, the Relativistic treatment of binaries has been quite strongly tested, apart from just general considerations about how gravitating bodes orbit, including precise predictions for changes in orbital period due to gravitational waves. And of course, the gravitational temporal effects are now confirmed using high-accuracy clocks even just in the Earth's gravitational field.
I am open to emendations to the Relativistic picture—indeed, I think quantum non-locality suggests it—but the basic Relativistic account of temporal structure has been severely tested in many distinct ways, and seems to be close to correct.No alternative does so well.
Eckard Blumschein replied on Apr. 3, 2015 @ 10:30 GMT
Dear Tim,
Should we just be open to emendations or may we possibly reveal truly foundational alternatives? Pentcho Valev has been persistently offering arguments for the emission theory without asking how to otherwise explain the weakness of Relativity. In my essay I support Leibniz' relativity but not Relativity. What about binary stars, I don't doubt that they disprove Newton's emission theory. However, does this mean they confirm Relativity (capitalized like God)? Why not trying the idea by Leibniz that space is just mutual distances? In principle, the disproved aether theory arose from Newton's idea as space as a body.
When I was trained as an EE, I was told that Maxwell's equations are definitely correct because they proved useful for decades. The strongest and meanwhile the lonely valid argument in favor of SR was its equivalence with these equations.
That's why several papers by Phipps are certainly a challenge to all believers in SR.
I am fully aware of swimming against the mainstream when questioning length contraction and naive point set theory. However, will science advance just by voting?
Best regards,
Eckard
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Author Tim Maudlin replied on Apr. 3, 2015 @ 14:25 GMT
Dear Ekhard,
I don't understand your attitude here. No one is saying that theories can't be questioned and alternatives considered. But if an existing theory make extremely good, verified predictions, then the alternative ought to make those predictions. There are observations that are directly relevant to the speed of light, and observations of clock behavior are directly relevant to temporal structure. Those should be guides to development of any theory.
Why not try Leibniz's idea? There have been lot's of attempts in that direction. Mach never actually built a proper theory. Julien Barbour has been working on ideas like that for years, and he gets lots of attention. But the reason Newton rejected it were the sorts of effects described by the bucket experiment, which are straightforward empirical facts. So the first thing any new theory should do is explain those facts. Leibniz himself never did.
Science does not advance by voting, but by developing clear theories and testing them against data.
By the way, I was advised by an editor to capitalize Relativity since it is a reference to a particular theory that goes by the name the Theory of Relativity. It is to avoid confusion with other theories, such as Leibniz's.
Regards,
Tim
Pentcho Valev replied on Apr. 3, 2015 @ 20:53 GMT
"the basic Relativistic account of temporal structure has been severely tested in many distinct ways, and seems to be close to correct"
That's what it was devised for - disfigured space and time form an efficient "protecive belt" around the false "hard core" of Einstein's relativity:
"Lakatos distinguished between two parts of a scientific theory: its "hard core" which contains its...
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"the basic Relativistic account of temporal structure has been severely tested in many distinct ways, and seems to be close to correct"
That's what it was devised for - disfigured space and time form an efficient "protecive belt" around the false "hard core" of Einstein's relativity:
"Lakatos distinguished between two parts of a scientific theory: its "hard core" which contains its basic assumptions (or axioms, when set out formally and explicitly), and its "protective belt", a surrounding defensive set of "ad hoc" (produced for the occasion) hypotheses. (...) In Lakatos' model, we have to explicitly take into account the "ad hoc hypotheses" which serve as the protective belt. The protective belt serves to deflect "refuting" propositions from the core assumptions..."
Imre Lakatos, Falsification and the Methodology of Scientific Research Programmes: "All scientific research programmes may be characterized by their 'hard core'. The negative heuristic of the programme forbids us to direct the modus tollens at this 'hard core'. Instead, we must use our ingenuity to articulate or even invent 'auxiliary hypotheses', which form a protective belt around this core, and we must redirect the modus tollens to these. It is this protective belt of auxiliary hypotheses which has to bear the brunt of tests and get adjusted and readjusted, or even completely replaced, to defend the thus-hardened core."
Banesh Hoffmann is quite clear: the Michelson-Morley experiment confirms the variable speed of light predicted by Newton's emission theory of light unless there is a protective belt ("contracting lengths, local time, or Lorentz transformations") that deflects the refuting experimental evidence from the false constant-speed-of-light postulate:
"Relativity and Its Roots", Banesh Hoffmann, p.92: "Moreover, if light consists of particles, as Einstein had suggested in his paper submitted just thirteen weeks before this one, the second principle seems absurd: A stone thrown from a speeding train can do far more damage than one thrown from a train at rest; the speed of the particle is not independent of the motion of the object emitting it. And if we take light to consist of particles and assume that these particles obey Newton's laws, they will conform to Newtonian relativity and thus automatically account for the null result of the Michelson-Morley experiment without recourse to contracting lengths, local time, or Lorentz transformations. Yet, as we have seen, Einstein resisted the temptation to account for the null result in terms of particles of light and simple, familiar Newtonian ideas, and introduced as his second postulate something that was more or less obvious when thought of in terms of waves in an ether."
Pentcho Valev
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Author Tim Maudlin replied on Apr. 4, 2015 @ 22:05 GMT
I am a big fan of Lakatos's account of methodology. But the hard core vs. protective belt distinction he makes is not relevant here. That is called into play when a theory seems to make bad predictions, in order to deflect the blame from the fundamental tenets to auxiliary assumptions. I was referring not to cases where the General Relativity seems not to make good predictions but to cases where it makes strikingly accurate predictions. The gravitational effects on atomic clocks, for example, that have only become testable with great advances in technology. The gravitational effects have been confirmed. It is this success, which the theory could not have been designed for (since the experiments were not done until decades after the theory was formulated) that give confidence that the theory is on the right track.
Eckard Blumschein replied on Apr. 4, 2015 @ 22:49 GMT
Dear Tim,
Concerning Newton's bucket argument I quote from my essay:
"While Leibniz argued in favor of understanding space as merely distances between locations, i.e., as RELATIONS, Clarke on behalf of Newton kept space and time for being ABSOLUTE, being substances. Leibniz and Newton merely agreed on that acceleration is an absolute quality. Let’s illustrate Newton’s mistake with the metaphor of an unlimited to both sides box [14]. Only if there is a preferred point of reference, it is possible to attribute a position to it. In space, such point is usually missing. Newton believed having demonstrated with his bucket experiment that space is ABSOLUTE. His background was in theology, alchemy, and the old fluentist view of moving indivisibles. Leibniz criticized Newton’s ABSOLUTE space as too restricting (to God). When he replaced fluxions by the derivative dx/dt, he made calculus more attractive by pragmatically calculating with fictitious infinitesimal quantities. Neither Newton nor Leibniz realized that the rotation of the bucket defined a point of reference. For the same reasons Michelson’s 1881/87 null result was not understood but kept for at odds with the Sagnac effect [15]." Endquote
I wonder if I am the only lonely one who considers the speed of light in vacuum not related to emitter, medium, or observer/receiver but to the distance between the relative locations of the emitter at the moment of emission and the receiver at the moment of arrival divided by the time of flight.
Regards,
Eckard
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Pentcho Valev replied on Apr. 4, 2015 @ 23:03 GMT
"The gravitational effects on atomic clocks, for example, that have only become testable with great advances in technology. The gravitational effects have been confirmed. It is this success, which the theory could not have been designed for (since the experiments were not done until decades after the theory was formulated) that give confidence that the theory is on the right track."
Hm... The Pound-Rebka experiment can be called classical:
David Morin: "The equivalence principle has a striking consequence concerning the behavior of clocks in a gravitational field. It implies that higher clocks run faster than lower clocks. If you put a watch on top of a tower, and then stand on the ground, you will see the watch on the tower tick faster than an identical watch on your wrist. When you take the watch down and compare it to the one on your wrist, it will show more time elapsed. (...) This GR time-dilation effect was first measured at Harvard by Pound and Rebka in 1960. They sent gamma rays up a 20m tower and measured the redshift (that is, the decrease in frequency) at the top."
But:
Albert Einstein Institute: "One of the three classical tests for general relativity is the gravitational redshift of light or other forms of electromagnetic radiation. However, in contrast to the other two tests - the gravitational deflection of light and the relativistic perihelion shift -, you do not need general relativity to derive the correct prediction for the gravitational redshift. A combination of Newtonian gravity, a particle theory of light, and the weak equivalence principle (gravitating mass equals inertial mass) suffices. (...) The gravitational redshift was first measured on earth in 1960-65 by Pound, Rebka, and Snider at Harvard University..."
Anything embarrassing?
Pentcho Valev
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Author Tim Maudlin replied on Apr. 4, 2015 @ 23:20 GMT
No, what is embarrassing there at all? An experiment done in 1960 cannot have a result that the Field Equation was designed to give, whether one calls an experiment "classical" or not. The more detailed predictions of GR (which go beyond the Equivalence Principle) have been repeatedly and exactly verified. These all go beyond the three initial tests. I can't imagine what you find embarrassing in these quotes. All of the gravitational effect you cite were not tested for almost half a century after the formulation of the theory, which is "decades". The detailed observation of binaries as well.
I honestly can't follow what you think these citations show, or how they are supposed to cast doubt on GR. They just don't.
The gravitational time dilation effect, the effect on clocks, is not the same as the redshift. Its obvious that moving a clock in a gravitational field in itself has nothing to do with redshifting light: the redshift refers to the frequency of a single light ray that climbs out of a gravitational potential. Lifting a clock and then bringing it back down is a completely different physical situation. A redshift can be calculated using principles of energy balance and the relation of light frequency to energy. The clock experiment cannot.
Pentcho Valev replied on Apr. 4, 2015 @ 23:44 GMT
"The gravitational time dilation effect, the effect on clocks, is not the same as the redshift."
Yes but, in my view, gravitational time dilation can only be measured by measuring the redshift, as in the Pound-Rebka experiment. If I am right, juxtaposing the two quotations in my previous posting should cause embarrassement and even frustration in any Einsteinian. If I am wrong, please explain how else the gravitational time dilation is measured.
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Author Tim Maudlin replied on Apr. 4, 2015 @ 23:49 GMT
Put two high-precision atomic clocks on the floor together Synchronize. Lift one up on a table. Wait a while. Return to the floor and compare synchronization. This has been done. The clocks go out of syntonization, and the amount out is a function of how long the one is up on the table. No redshift or light involved. Experiments at this precision have only been possible recently. Flying atomic clocks around the world was done some decades ago.
Pentcho Valev replied on Apr. 5, 2015 @ 00:09 GMT
"Put two high-precision atomic clocks on the floor together Synchronize. Lift one up on a table. Wait a while. Return to the floor and compare synchronization. This has been done."
Reference (available online please)?
Pentcho Valev
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Author Tim Maudlin replied on Apr. 5, 2015 @ 01:16 GMT
http://www.nist.gov/public_affairs/releases/aluminum-atomic-
clock_092310.cfm
Pentcho Valev replied on Apr. 5, 2015 @ 06:55 GMT
This is just an improved version of the Pound-Rebka experiment. They measured the frequency difference, that is, the redshift, and concluded that the tick rates are different:
Optical Clocks and Relativity, C. W. Chou, D. B. Hume, T. Rosenband, D. J. Wineland, Science 24 September 2010, Vol. 329, pp. 1630-1633: "Differences in gravitational potential can be detected by comparing the tick...
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This is just an improved version of the Pound-Rebka experiment. They measured the frequency difference, that is, the redshift, and concluded that the tick rates are different:
Optical Clocks and Relativity, C. W. Chou, D. B. Hume, T. Rosenband, D. J. Wineland, Science 24 September 2010, Vol. 329, pp. 1630-1633: "Differences in gravitational potential can be detected by comparing the tick rate of two clocks. For small height changes on the surface of Earth, a clock that is higher by a distance Δh runs faster by Df/fo=gΔh/c^2."
But the frequency difference they measured did not indicate difference in tick rates! Just as in the Pound-Rebka experiment, this difference was caused by the acceleration of photons in a gravitational field, as predicted by Newton's emission theory of light:
University of Illinois at Urbana-Champaign: "Consider a falling object. ITS SPEED INCREASES AS IT IS FALLING. Hence, if we were to associate a frequency with that object the frequency should increase accordingly as it falls to earth. Because of the equivalence between gravitational and inertial mass, WE SHOULD OBSERVE THE SAME EFFECT FOR LIGHT. So lets shine a light beam from the top of a very tall building. If we can measure the frequency shift as the light beam descends the building, we should be able to discern how gravity affects a falling light beam. This was done by Pound and Rebka in 1960. They shone a light from the top of the Jefferson tower at Harvard and measured the frequency shift. The frequency shift was tiny but in agreement with the theoretical prediction. Consider a light beam that is travelling away from a gravitational field. Its frequency should shift to lower values. This is known as the gravitational red shift of light."
Albert Einstein Institute: "One of the three classical tests for general relativity is the gravitational redshift of light or other forms of electromagnetic radiation. However, in contrast to the other two tests - the gravitational deflection of light and the relativistic perihelion shift -, you do not need general relativity to derive the correct prediction for the gravitational redshift. A combination of Newtonian gravity, a particle theory of light, and the weak equivalence principle (gravitating mass equals inertial mass) suffices. (...) The gravitational redshift was first measured on earth in 1960-65 by Pound, Rebka, and Snider at Harvard University..."
Pentcho Valev
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Author Tim Maudlin replied on Apr. 5, 2015 @ 13:36 GMT
No it isn't. The Pound-Rebka experiment is a measurement of redshift for a photon that starts closer to the center of the earth and moves up. That is not what is happening here. Once the clocks are in place, nothing is moving up: each clock just sits at its location. The word "frequency" is used in both cases, but the physical phenomenon is different. Pound-Rebka can be predicted by energy balance, E = mc^2 and E = h(nu) with a classical argument: as the photon climbs the gravitational potential, it must lose energy, which gives a frequency shift. But as the clocks are just sitting there, no similar argument can be made. Not every frequency difference is a redshift. By that definition, the changing frequency of a grandfather clock is a redshift. Try to make this prediction by energy balance similar to the argument I just gave: it can't be done.
You can compare the tick rates just be looking at the clock outputs. Redshift doesn't come into it.
Pentcho Valev replied on Apr. 5, 2015 @ 16:08 GMT
"You can compare the tick rates just be looking at the clock outputs. Redshift doesn't come into it."
?!? Unfortunately Chou et al do not explain how precisely the measurement is done, but suggest that their experiment is an improved version of previous experiments, including the Pound-Rebka one:
"For example, if two identical clocks are separated vertically by 1 km near the surface of Earth, the higher clock emits about three more second-ticks than the lower one in a million years. These effects of relativistic time dilation have been verified in several important experiments (2–6)..."
The text that follows only states that this 1 km has been reduced to the record 0.33 m, but otherwise the experiment is analogous to previous ones. So essentially the frequency difference is measured, and this is exactly the redshift.
Pentcho Valev
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Author Tim Maudlin replied on Apr. 5, 2015 @ 16:18 GMT
You need to stop and reflect at this point. Two clock ticking, one above the other in a gravitational field, is not the same experimental situation as sending a light ray up from on to the other. The fact that the clocks get progressively more out of synchronization (comparing by bringing them back together) has no analog in the redshift experiments. The 1 km experiment they mention is also not a redshift experiment like Pound-Rebka. The reason that the classic gravitational redshift observation (spectral lines) is not a strong test of GR is because the prediction can be made (as I mention above) via energy considerations. Try that for this experiment. You can't do it.
Pentcho Valev replied on Apr. 5, 2015 @ 17:34 GMT
"comparing by bringing them back together"
I don't know how one can come to this conclusion - it is almost obviously absurd. The movement involved in bringing the clocks back together would spoil the experiment. I have been interested in this problem for quite a long time but so far have not seen even a hint that the clocks are compared by bringing them back together.
Pentcho Valev
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Author Tim Maudlin replied on Apr. 5, 2015 @ 19:43 GMT
Then you have to reflect on these experiments more. The differential in elapsed time between the clocks is a function of how long they are apart. Any small effect due to separating them and bringing them back together will be swamped by that factor. Think of this experiment: Start with three clocks together. Lift 2 up in the gravitational field. After a while, bring one down and compare. The later bring the second down. Whatever effect the lifting and returning have will be the same for the two clocks, but there will be a larger gap in synchronization for the clock left up longer, and the amount will scale with the time they are apart. The movement does not spoil the experiment at all.
Pentcho Valev replied on Apr. 5, 2015 @ 20:30 GMT
Your argument would be valid if the clocks were standard but these are not - just atomic oscillators that do not keep record of the time elapsed. So there is no point in bringing them together - there is nothing to compare. One can only measure the frequency difference (redshift) and ponder whether it is due to gravitational time dilation or variation of the speed of light with the gravitational potential:
Banesh Hoffmann: "In an accelerated sky laboratory, and therefore also in the corresponding earth laboratory, the frequence of arrival of light pulses is lower than the ticking rate of the upper clocks even though all the clocks go at the same rate. (...) As a result the experimenter at the ceiling of the sky laboratory will see with his own eyes that the floor clock is going at a slower rate than the ceiling clock - even though, as I have stressed, both are going at the same rate. (...) The gravitational red shift does not arise from changes in the intrinsic rates of clocks. It arises from what befalls light signals as they traverse space and time in the presence of gravitation."
Pentcho Valev
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Author Tim Maudlin replied on Apr. 5, 2015 @ 22:43 GMT
Again, you are not understanding how these experiments work, or even more general aspects of gravity and GR. The GPS system need relativistic correction that have nothing to do with redshift and accumulate through time:
We note that this post-launch rate comparison is independent of frame or observer considerations. Since the ground tracks repeat day after day, the distance from satellite to ground remains essentially unchanged. Yet, any rate difference between satellite and ground clocks continues to build a larger and larger time reading difference as the days go by. Therefore, no confusion can arise due to the satellite clock being located some distance away from the ground clock when we compare their time readings. One only needs to wait long enough and the time difference due to a rate discrepancy will eventually exceed any imaginable error source or ambiguity in such comparisons.
see http://www.metaresearch.org/cosmology/gps-relativity.asp
The GPS system works through time stamps, not just measuring frequencies.
Pentcho Valev replied on Apr. 5, 2015 @ 23:50 GMT
The problem was: Did Chou et al bring the clocks together in order to compare them or not? If not, what conclusions can be drawn?
The relation GPS - relativity deserves a separate discussion.
Pentcho Valev
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Eckard Blumschein replied on Apr. 6, 2015 @ 21:20 GMT
Dear Tim,
You wrote: "... the bucket experiment, which are straightforward empirical facts. So the first thing any new theory should do is explain those facts. Leibniz himself never did."
Leibniz could not reply because he died in 1716, Newton in 1727. Is my quoted on Apr. 4 explanation wrong?
Regards,
Eckard
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Author Tim Maudlin replied on Apr. 6, 2015 @ 21:36 GMT
Dear Eckard,
The problem was not that Leibniz died, but that his doctrine, as expressed in the letters with Clarke, has no room for "absolute acceleration". If you consider Newton's globes example, from the end of the Scholium, the point is that in the rotating globes system there is no change in the relative position of any material object with respect to any other material object, hence by Leibniz's account of the nature of space, time and motion there can be no motion at all. Hence no acceleration.
Leibniz was, of course, aware of the bucket argument. The odd thing is that Clarke only brings it up late in the correspondence, but when he does Leibniz's response is both inadequate and inconsistent with what he has been asserting. If space and time are merely relational, and the relations among objects are unchanging, then there is no motion (that was Leibniz's whole point!). But the tension in the cord connecting the globes is indicative of motion, and in particular of acceleration. Leibniz had no account of this.
Regards,
Tim
Eckard Blumschein replied on Apr. 7, 2015 @ 07:04 GMT
Dear Tim
On Apr. 4 I wrote: "I wonder if I am the only lonely one who considers the speed of light in vacuum not related to emitter, medium, or observer/receiver but to the distance between the relative locations of the emitter at the moment of emission and the receiver at the moment of arrival divided by the time of flight."
You wrote: "If space and time are merely relational, and the relations among objects are unchanging, then there is no motion (that was Leibniz's whole point!)"
My definition of speed relates to a possibly changing relation between emitter and receiver but not to absolute positions, just to a difference. Is it wrong?
If I recall correctly, Newton and Leibniz agreed on absolute acceleration.
Regards,
Eckard
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Author Tim Maudlin replied on Apr. 7, 2015 @ 12:42 GMT
Dear Eckard,
Newton and Leibniz did not agree about absolute acceleration. I don't know what you have in mind there. Leibniz never considers distance between objects or events at different times. If he did, then he could define even the speed of a single particle, not with reference to anything else, i.e. an absolute speed, by reference to a distance between where it is now and where it was before. He strongly rejects this.
Your definition is basically Newton's. Leibniz rejected Newton's account completely.
It is essential, to understand the debate, to distinguish spatial relations (distances) between a pair of objects at the same time (Leibniz accepts these) from distances between different objects—or even the same object—at different times (Leibniz rejects these). You need the latter for your definition.
Regards,
Tim
Eckard Blumschein replied on Apr. 7, 2015 @ 18:02 GMT
Dear Tim,
I trusted in my reference [14] philosophy.hku.hk/.../files/Newton_vs_Leibniz.ppt who wrote on their p. 24 "Leibniz Admitted absolute acceleration Yet denied that it is related to absolute space". While I got the impression they slightly modified their presentation since I quoted it, they didn't change this statement.
What about my definition, I still wonder if it is correct and already written elsewhere. Of course, it could theoretically fit to Newton's conjectured body of absolute space. However, it may also apply if there is no such body and accordingly no preferred point of reference in space, even if Shtyrkow's measurement might be reasonable. If space extends in excess of its border of observability then we shouldn't refer it to it.
Someone who declares Relativity wrong blamed me for supporting Relativity. Actually, I consider my reasoning more basic. Einstein was perhaps not wrong when he shared the old argument that there is no naturally preferred point in space. "Give me a fix point, and I will level out the world". I know only one natural reference, the zero of elapsed and future time.
Regards,
Eckard
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adel sadeq wrote on Apr. 3, 2015 @ 00:00 GMT
Hi Tim,
My idea is also based on the concept of line, however the line in my system has a very simple interpretation, its the difference between two quantities that is all. The quantities have to be random otherwise you will not get our reality. Do you have some implementation of your system so I may compare it to mine.
I get real physics from my system.If you don't have the time just read the electron mass section and run the program (click "program link" at the end of the section) , it will execute in less than a minute.
EssayThanks and good luck.
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Author Tim Maudlin replied on Apr. 3, 2015 @ 00:25 GMT
Hi Adel,
Part of this project is to see how much of the geometrical structure of a space can be represented without there being any "quantities" at all, that is, it is done at a sub-metrical level. So if you are starting with quantities that have well-defined differences, your starting point is quite different from mine, and the approaches probably will not coincide. In fact, one of my goals was the opposite of Dedekind's. Dedekind wanted to get all reference to geometry out of his theory of numbers, and I want to eliminate all reference to numerical structure (including differences) from my account of geometry, or at least to have a very clear understanding of how any numerical structure gets in.
Cheers,
Tim
Alexey/Lev Burov wrote on Apr. 9, 2015 @ 15:16 GMT
Dear Tim, in the paragraph before last, you write:
"Wigner’s question is this: why is the language of mathematics so well suited to describe the physical world? A proper answer to this question must approach it from both directions: the direction of the mathematical language and the direction of the structure of the physical world being represented. In order for the language to fit the object in a useful way the two sides have to mesh."
I do not think that the way to answer Wigner's question lies on the plane expressed in this excerpt, which seems to reflect your approach. Mathematics suggests various rational structures. Physics tries them out for the role of laws of nature. Neither physics nor mathematics is suited to answer the 'why' question of Wigner: they both deal with 'how' problems only.
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Author Tim Maudlin replied on Apr. 9, 2015 @ 15:57 GMT
Dear Alexey,
"Why" is used in lots of ways. If I ask "Why did the bridge collapse?" it typically means exactly the same as "How did it come about that the bridge collapsed" or "What are the conditions that account for the bridge's collapse?" and an answer such as "Because the bolts rusted through" is fine. This is the sense of "why" I had in mind. I think it is also what Wigner had in mind: recall that he had no problems about certain mathematical structures being useful for physics. So your distinction between "why" and "how" questions does not seem to me to be appropriate to Wigner's concerns.
Regards,
Tim
En Passant replied on Apr. 9, 2015 @ 16:16 GMT
Well, Lo and Behold, as I was about to post my finished comment, I saw Tim’s answer. Since I already wrote it, I am posting it anyway…
Dear Alexey/Lev Burov,
Not that Tim needs any help in responding to this, but he is likely to take a different approach than the one I would like everyone to be aware of.
The issue I am addressing here is your claim that: “Neither physics...
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Well, Lo and Behold, as I was about to post my finished comment, I saw Tim’s answer. Since I already wrote it, I am posting it anyway…
Dear Alexey/Lev Burov,
Not that Tim needs any help in responding to this, but he is likely to take a different approach than the one I would like everyone to be aware of.
The issue I am addressing here is your claim that: “Neither physics nor mathematics is suited to answer the 'why' question of Wigner: they both deal with 'how' problems only.” The word “why” is kind of tricky. Some people use it in a sense comparable to how others use “Ultimate Truth.”
So let’s talk about “why.” For thinking and planning entities, “why” can refer to the objective they are trying to achieve (as in “why did you do that?”). For physical entities that don’t plan (at least we hope they don’t), “why” refers to causal factors that preceded the event about which you ask “why.” (This latter usage can also apply to planning entities.)
These two usages define (and limit) the format of your answer. Either the answer will talk about causes, or it will talk about intentions. Employing the word “why” outside of these two usages results in ill-posed questions (and, likely, ambiguous answers).
The word “why” in Wigner’s quote should be replaced (and perhaps he even intended it so) with the following: “What aspect of physics, or mathematics (or something else) makes the language of mathematics so well suited to describe the physical world?” The answer to that is unlikely to invoke causality, and (unless, ehm, God planned it that way) it certainly will not involve intent.
To get the answer, one will have to delve into the nature of each (math and physics) and figure it out from there. “How” may very well be the only consideration available in formulating the answer.
As you see, I am only clearing the landscape. The actual determination of the answer to Wigner’s question (even rephrased, as above) is for you to discuss. I think this exchange ought to be interesting, so please don’t drop the ball.
En
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Eckard Blumschein replied on Apr. 9, 2015 @ 17:03 GMT
In order to less speculatively answer Wigner's question I would like to remind of Otto de Guericke who argued for experiments instead of scholastics. Leifer's cut between useful and unwarranted speculative mathematics might become increasingly necessary.
In order to give an example I quote from Daniel Braun's essay:
"Set theory is part of mathematics, and applying set theory to itself is known to quickly run into trouble." Endquote.
Braun is nonetheless perhaps almost the only one who tried to ascribe physical meaning to aleph_2 and aleph_3.
Well, transfinite numbers evade experimental verification. Having critically read Fraenkel 1923 I localized Cantor's mistake and agree with Galileo Galilei on that infinity is a property that must not be used as a number: oo+1=oo.
Eckard
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Author Tim Maudlin replied on Apr. 9, 2015 @ 21:59 GMT
Dear Eckard,
Obviously there is no natural number or Peano number that is infinite, if that is what you mean. But also obviously, the set of natural numbers is infinite, because there is no greatest such number. If you object to calling the cardinals "numbers" then it looks like a purely semantic dispute. Cantor does demonstrate that for every cardinal there is a larger one (in the sense that the elements of a set with the larger cardinality cannot be put in 1-1 correspondence with the elements of the smaller, but a subset can). There is no error in Cantor's proof. One can make clear sense out of oo + 1 = oo: if you add new element to a set with an infinite cardinality, then new set has the same cardinality. That's a correct claim.
One can define some operations analogous to arithmetical operations on sets, and so define a fragment of an arithmetic for cardinals, although it has limitations. For example, "addition" is always well-defined but "subtraction" is not. But the proof of the infinite hierarchy of cardinals does not contain any mistakes.
Of course there can be nothing like a direct observational verification of the existence of a set of physical items with transfinite cardinality. But there is no obvious problem postulating them. By the same token, no experiment can prove that the total set of physical items is finite. So we ought to be prepared for either to be the case.
Regards,
Tim
Alexey/Lev Burov replied on Apr. 9, 2015 @ 22:12 GMT
Dear Tim, dear En,
I agree that sometimes the question 'why' is equivalent to a question 'how'. However, I do not think that this is the case for the Wigner's question.
Since Aristotle, three ways to answer 'why' are known: a reference to law, accident or purpose. Wigner expresses his wonder as to why the laws of nature are described mathematically at such huge range of parameters and with great precision. He is asking why the laws of nature have this specific feature—to be both accurate and so simple that we, humans, were able to discover them. Since the laws per se are under question, one more law cannot be the answer. Just logically you cannot answer this way. That's the difference between Wigner's question and your example with the bridge, Tim.
Thus, Wigner's question can be answered only by one or another of the two remaining Aristotelian options: accident or purpose. The first of them is actually known as the full-blown multiverse, most clearly presented by Max Tegmark as his 'mathematical democracy' principle. In
our essay we are refuting that option. Thus, we have only one choice among the three to answer Wigner's question.
Alexey.
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En Passant replied on Apr. 9, 2015 @ 22:57 GMT
Dear Alexey/Lev Burov,
Propriety indicates that I should not continue to be involved in this discussion on Tim’s page. I am only saying this so you would not think I am ignoring you.
If you wish to have a parallel discussion with me directly about this or another topic, I would be happy to accommodate that on my page.
I still have not read your essay, but it seems (now that we have “met”) that I ought to, and will try to make time for it.
En
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Author Tim Maudlin replied on Apr. 9, 2015 @ 23:01 GMT
Dear Alexey,
I would be cautious about assuming that Aristotle was the final word on anything: that was sort of the point of the Scientific Revolution. But in any case, Aristotle offered four basic answers to "why" questions: formal cause, material cause, final cause and efficient cause (Posterior Analytics). And in his view, mathematics is the study of form without matter (Metaphysics)....
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Dear Alexey,
I would be cautious about assuming that Aristotle was the final word on anything: that was sort of the point of the Scientific Revolution. But in any case, Aristotle offered four basic answers to "why" questions: formal cause, material cause, final cause and efficient cause (Posterior Analytics). And in his view, mathematics is the study of form without matter (Metaphysics). That is, he thought that matter can be structured in a way that allows distinct items to have the same structure, and mathematics studies those structures. The physical world is describable mathematically if it has the appropriate structure.
WIgner, as is clear, was not puzzled about why some mathematical concepts—specifically the geometrical ones—might describe the physical world since we develop those concepts via interaction with the physical world. His explicit question was why mathematical concepts not developed that way—mathematical concepts developed out of pure mathematical motivations, as it were—still often turn out to be useful for physics. This seems to be answered by the unity and interconnection of mathematical fields, which exists quite independently of any physical considerations. There can be no explanation of "why" these mathematical interconnections exists, because they are not contingent: they have to be as they are. (Again, Aristotle would approve: necessity is the ultimate explanation).
The logic of your argument seems to be this: explanations of one of the three kinds eventually end, at which point you have to switch to a different kind. Once "law" explanations run out, there must be either "accident" or "purpose" explanations of the laws. But where this ends up depends on the order one takes them up in. By this logic, the "purpose" explanations must end, and to explain the existence of the purposes one must invoke another category: say "law". And once the "law" explanations run out, all that is left is "accident". So ultimately, everything is accidental!
That can't be right, because by the exact same reasoning I can prove ultimately everything has a purpose or ultimately everything is determined by law. So the logic of your argument is self-undermining. There have to be ultimate explanatory factors that do not admit of further explanation at all. (Again, Aristotle would approve, see again Posterior Analytics).
If one is obliged to explain why the laws are as they are, then equally one would be obliged to explain why any purposes are what they are. In fact, the only one of your three categories of answers that seem to resist this is accident: accidents happen for no reason at all, in a sense, by definition. So either you have to stop your supposed explanatory regress or rising yourself to the idea that everything is accidental.
Cheers,
Tim
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Alexey/Lev Burov replied on Apr. 10, 2015 @ 00:34 GMT
Dear Tim,
Aristotle’s philosophy aside, if you wish to add the fourth final cause to the three we’ve discussed, it would be very interesting to hear. Meanwhile we have only the three to work with.
We seem to be in agreement that another law can not serve to explain laws per se. Only chance and purpose remain. You claim that the purpose explanation contains the same problem as a law, namely, “one would be obliged to explain why any purposes are what they are.” However, that is incorrect. As we mention in our essay, the final cause, a terminus of explanation, has to be a totality: something whose specificity need no longer be brought under question. So, laws are always specific, but chance isn’t. Purpose, at first glance, is specific too, but purpose originates from consciousness, which is essentially self-referential. So to answer why consciousness is the way it is, we can say, “because it made itself so.”
Now that we that are again faced with a choice between chance or purpose, we can try and see if chaos indeed could be the answer. Based on the discoveries already made, in our essay, we prove that such a possibility is ruled out.
Lev
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Author Tim Maudlin replied on Apr. 10, 2015 @ 00:46 GMT
Dear Lev,
Self-causation ("because it made itself so") is evidently incoherent as an end to explanation: nothing can "make itself" because it has to already exist to do the making. The idea that consciousness is "self-referential" is obscure: in what sense is (say) a toothache, a paradigmatic conscious state, "self-referential"?
Of course no law can explain "laws per se" if that means explaining itself along with all the other laws. And no purpose can explain "purpose per se" it that means explaining itself along with all other purposes. You somehow want to carve out an exception to the possibility of self-explanation for consciousness, but there is no grounds for this.
And again Aristotle would agree! His answer to the regress was not self-explanation, which he regarded as logically impossible for the reason just given.
Regards,
Tim
Alexey/Lev Burov replied on Apr. 10, 2015 @ 03:14 GMT
Dear Tim,
Self-referentiality is not only a coherent, but a required attribute of an explanatory terminus. So not to repeat, I quote ourselves: "Wittgenstein criticizes a silent acceptance of a composite and special mathematical structure as the ultimate explanation of the world. Such explanation barred from further questioning and not subject to
reasonable ground of its own existence is an affirmation of unreasonableness of this ground.” The ultimate source has to be the reasonable ground of its own specificity. It is specificity that is in question, not existence, such as “why this specific law not another?”
Yet your example of a toothache misses the point. We are not presenting our own limited consciousness with toothaches and rusted bolts, but consciousness per se, the ultimate mind, which defines itself completely.
Aristotle’s self-explanation, as rejected by him, is of the form,
it is so because it is so. It is very much different from self-creation, as his terminus was “thought which has itself for its object.” But to avoid extraneous tangents, perhaps it’d be useful to mention that it isn’t Aristotle we aim to be in agreement with, but the truth.
Lev
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Eckard Blumschein replied on Apr. 10, 2015 @ 08:29 GMT
Dear Tim,
The “proof of the infinite hierarchy of cardinals” is flawed: Fraenkel 1923 revealed to me the cardinal mistake with cardinality: Cantor’s second diagonal argument is based on the wrong assumption that there is a set of all natural numbers which can be considered as fixed. Cantor himself may have understood the infinity better when he compared it with an abyss. An ideal abyss has no bottom. While according to the stolen from Archimedes axiom of infinity an infinite set is conceivable, the axiom of extensionality is only valid for finite sets. Galileo understood: The relations smaller than, equal to, and larger than are only valid for finite quantities.
Transfinite cardinality was not even supported by Cantor’s friend Dedekind. It has never proved useful. Cantor himself listed the following opponents of his theory of actually infinite and simultaneously distinguishable numbers; Cauchy, Cavalieri, Fischer, Fontenelle, Fullerton, Galilei, Gauss, Gerdil, Goudin, Guldin, Gutberlet, Hegel, v. Helmholtz, Herbarth, Kant, Kronecker, Leibniz, Moigno, Newton, Peresius, Pesch, Renouvier, Sanseverino, Sigwart, Tongiorgi, Toricelli, Wundt, and Zigliara.
We may blame Weierstrass for supporting Cantor against Kronecker and Poincaré for not consequently proving Cantor wrong. Nobel perhaps felt; Leffler-Mittag supported something wrong. A majority of mediocre mainly German mathematicians were keen to have a putatively rigorous justification for treating the continuum as if it was discrete: Cantor’s paradise.
Formally, aleph_0 ^aleph_0=aleph_0, not aleph_1. Continuity may nonetheless be denoted aleph_1. However, it is a different quality, not a different quantity. Cantor and Dedekind were wrong in that. The latter begged for believing his claim without evidence; Cantor impressed by arrogant mysticism.
Regards,
Eckard
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Author Tim Maudlin replied on Apr. 10, 2015 @ 11:29 GMT
Dear Eckard,
Of course there is a "set of all natural numbers that can be considered as fixed". What could it mean for the set not to be fixed: that the set changes, and admits new natural numbers? The axiom of extensionality just says that the identity conditions for sets is having all the same elements. If you want to deny it, you either think that there are two distinct sets event though an item is an element of one if and only if it is an element of the other or that one and the same set can have different elements. That's not what we mean by "set".
The long list of names is not a proof of anything. Cantor provides a proof. You have not indicated any flaw in it. And simply using insulting descriptions ("mediocre","arrogant mysticism") is not an argument.
No set can be put into 1-1 correspondence with its power set. That is provable. Given the definition of "same cardinality" it follows that there is an infinite hierarchy of "larger" sets, by the definition of "larger". Lists of names and insults do not change that. The definition of "same cardinality" and "larger cardinality" is given above, and it applies to infinite sets. And by the definition, some infinite sets are larger than others.
If you think that the set of natural numbers cannot be "considered as fixed" can you please indicate an item whose membership or non-membership in the set is somehow uncertain, i.e. an item such that you are not sure whether or not it is a natural number? If you can't, then it is not clear what you are even asserting.
Regards,
Tim
Eckard Blumschein replied on Apr. 10, 2015 @ 23:09 GMT
Dear Tim,
You are reiterating what students still have to learn, and there is almost no interest in a clarification because the idea of cardinality is obviously of no use, and generations of mathematicians failed to disprove Cantor’s proof. Among the listed and also the not listed opponents were excellent minds like Aristotle, Galileo, Leibniz, Newton, Cauchy, and Poincaré. Cantor was...
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Dear Tim,
You are reiterating what students still have to learn, and there is almost no interest in a clarification because the idea of cardinality is obviously of no use, and generations of mathematicians failed to disprove Cantor’s proof. Among the listed and also the not listed opponents were excellent minds like Aristotle, Galileo, Leibniz, Newton, Cauchy, and Poincaré. Cantor was certainly arrogant when he declared them wrong altogether. They understood that an infinite quantity or an infinite series is by definition of infinite as the opposite of finite an unfinished process, not something that has been completed via counting. If we speak of “all” natural numbers then this means excluding the possibility to add a larger number, not because already all natural or rational numbers are already occupied but because, as formulated by Archimedes and the axiom of infinity, the property of being infinite means being open to unlimited enlargement.
When Cantor and Fraenkel postulated a fixed series of “all” natural numbers, they appealed on thinking in terms of counting discrete elements and they implicitly denied that being infinite is not at all a quantity but a quality. Cantor fell back into primitive mistakes by Albert of Saxony (1316-1390) and Bernhard Bolzano (1781-1848) who attributed points to a space or a Menge (a set) to a line, respectively. Cantor managed to humiliate Kronecker because the latter also intended but failed to make the continuum rigorously algebraic. It was already and is still undisputed that the expression infinity must not always be algebraically treated. For instance, it is impossible to increase or decrease infinity by addition, subtraction and other operations. Likewise the evidence “2^aleph_0 is larger than aleph_0” by Cantor/Hessenberg is based on treating infinity like a number. Mediocre mathematicians were and are perhaps still unable to think beyond the mathematical formalisms they learned. Therefore they could not even disprove naïve set theory.
Already Galileo used bijection in order to show that there are not more natural numbers than squares of it because both series are endless which implies they are uncountable in the original sense. This is logically convincing to me.
Cantor confused the world with uncommon definitions. He defined the natural and rational numbers countable in the sense the latter can be put into 1 to 1 correspondence with the natural numbers. Obviously, his Mächtigkeit (cardinality) of countable (according to Cantor’s definition) infinity is nothing else than the property of being discrete and therefore numerically distinguishable. This is the logical opposite of the property of being continuous.
What about the axiom of extensionality, I see the same problem as with Dedekind’s smaller than, equal to, or larger than relation for the continuum of real numbers (elements of measure zero).
I didn’t intend insulting any proponent of Cantor’s naïve set theory when I mentioned its mysticism. The attribute naïve stems from those who tried to circumvent the logical inconsistencies of Cantor’s set theory and make it seemingly less mystic while even less concrete.
In order to get an impression how Cantor impressed the experts, one may read how lecturing he reacted to justified question. Emmy Nöther reported how Cantor theatrical answered the question how he imagined infinity as follows: He directed his view towards the sky, his eyes starred to infinity, after a while he performed a slow wide movement with his hand, and spoke with pathos: I see it an abyss. The more insane he got, the more he was considered a genie.
Don’t forget: Point Set Theory didn’t lead to anything of value although Bertrand Russell meant: “The solution of the difficulties which formerly surrounded the mathematical infinite is probably the greatest achievement of which our age has to boast.” No comment.
Regards,
Eckard
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Author Tim Maudlin replied on Apr. 10, 2015 @ 23:21 GMT
This is obviously not the place to continue discussion of your views. But leaving aside all the historical references, the general proof that the power set of a set cannot be put in 1-1 correspondence with members of the set—which is just a reductio—does not require "treating infinity like a number", whatever you might mean by that. The proof holds for all sets: finite and infinite, so the notion of infinity does not come into it. But given an infinite set (such as the set of natural numbers) it proves that the power set has a higher cardinality. Since both sets are infinite, it proves that some infinite sets have a higher cardinality (by the definition of what that means) than others,
Eckard Blumschein replied on Apr. 12, 2015 @ 07:55 GMT
Dear Tim,
My attitude towards mathematical evidence may be similar to that by Oliver Heaviside who is nonetheless no longer my idol after I read Phipps.
So many proofs of the existence of God cannot be wrong even if they may contradict to each other. Nonetheless I will try my best and deal with Cantor 1874 and 1891 in my own thread as soon as I have the required time for that.
My own reasoning will start with the old idea that a line is a continuum every part of which has parts. In so far you could be interested too.
Leaving your thread, I would like to again highly appreciate both your refreshing attitude toward topology and your readiness to helpfully deal with my arguments even if they contradict current tenets.
Best,
Eckard
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James Lee Hoover wrote on Apr. 10, 2015 @ 02:12 GMT
Tim,
The physical world will not change at our command but is made malleable by changing the mathematical language used to formulate physics.Is the Theory of Linear Structures only one example of how this can be done? Time w/o the application of the theory creates the geometry of space-time? Is space a reaction to mass over time then?
I found theoretical physicist, Ilya Prigogine's quote from 2004: "I believe that what we do today depends on our image of the future, rather than the future depending on what we do today. We build our equations by our actions. These equations, and the future they represent, are not written in nature. In other words, time becomes construction." Is this relevant to your Theory?
It seems germane to me but I doubt that I have a full understanding.
Jim
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Author Tim Maudlin replied on Apr. 10, 2015 @ 03:06 GMT
JIm,
I don't think that the mathematics we use to describe the physical world can change the world itself. The question is rather whether the mathematics we choose to use is well-suited to the structure of the world. Different mathematical structures fit with different physical structures, and we are seeking the right one. The Theory of Linear Structures provides a novel sort of language to describe geometry, and particularly the geometry of space-time.
Using this language, one can analyze the whole space-time structure (and so anything one would like to say about space) as determined by purely temporal structure. In this sense, "space" arises out of time, or more exactly out of the Linear Structure created by ordering events in time in a Relativistic way.
Prigogine's quote sounds like it concerns human action. Human action can, of course, determine many facts about the future: whether there is climate change of a certain kind, or war, etc. And how we act depends on our beliefs and desires about the future. But the basic laws of physics, and the fundamental physical constituents of the universe, are not within our power to change. The best we can hope for is to figure out what they are an describe them accurately.
Regards,
Tim
James Lee Hoover replied on Apr. 13, 2015 @ 22:22 GMT
Tim,
As time grows short, I revisit those I have read. I find that I did not rate yours, something I usually do to those that impress me, so I am rectifying that. Hope you get a chance to look at mine.
Jim
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Alma Ionescu wrote on Apr. 13, 2015 @ 15:53 GMT
Dear Tim,
I enjoyed a lot your witty and refreshing essay. I too have mentioned the proof of Fermat's Last, but your most amusing and insightful conclusion about a companion to Wigner's paper didn't even cross my mind.
I think your treatment of time as fundamental is exciting and aiming to fill a gap in the foundations of today's physics. I know you have a book on the topic and I am planning to read it, however I am reasonably sure that the book, as any other topic-focused reading, does not answer a curiosity of mine regarding your view. I would like to know where is your intuition about time stemming from. I mean, when was that moment that your intuition crystallized and what caused it to happen? If you did write about it, I'd be grateful to be pointed to the paper in which you're describing it.
Your work is very attention-worthy and I certainly hope it receives the recognition it deserves. Should you have enough time to take a look at my essay, your comments are very welcome.
Warm regards,
Alma
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Author Tim Maudlin replied on Apr. 13, 2015 @ 20:55 GMT
Dear Alma,
Thanks for the kind words. It took a long time to arrive at these ideas, which basically come from two different directions. One was teaching the Theory of Relativity year after year, learning to think of it in purely geometrical terms rather than in terms of coordinate systems (Lorentz transformations, etc.). Eventually, one gets intuitions about how the geometrical structure works. The second strand came, while teaching on of these classes, from the realization that the standard way to approach topology is very hard to get a clear conceptual grip on. (I have had many, many students say that this is their reaction when they first learn the theory). I realized when I was trying to teach the standard approach that it does not at all correspond to how I would think of things, and asked myself whether a more visualizable theory could be created. This is the outcome of years of work in that new direction.
The curious thing is that several mathematicians have remarked how natural this way of setting things up is, and can't believe it hasn't been tried before. But so far, no one has pointed out anything similar. It is such a simple idea, in the end, that I am puzzled as well. Maybe it will turn out that someone thought of the basic approach long ago.
Once you put together a completely geometrical understanding of Relativity with this new way to think about geometry, the role on time in Relativity just jumps out at you. That was not something I was aiming at. But this mathematical language fits Relativistic space-time geometry like a glove. It is hard not to think that there is something significant in that. And time takes center stage as the basic ordering principle.
Cheers,
Tim
Alma Ionescu replied on Apr. 19, 2015 @ 13:26 GMT
Dear Tim,
Thank you for your answer as it's a very interesting insight regarding how new theories come to be. I don't find it particularly odd that no one thought of your idea before, because most people don't feel the need to innovate as long as there exists a functional theory, all the more since the effort of creating a logically sound new approach is considerable. Also I'm sure your unique intuition has to do with this, because I think I understand that your approach allows for discreetness, not necessarily requiring infinitesimals.
Cheers,
Alma
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En Passant wrote on Apr. 13, 2015 @ 23:46 GMT
Tim,
I only learned about you existence from watching the “time” video where you indulged the doctrine of Julian Barbour.
I knew ahead of time (no pun intended) that your philosophy would be self-consistent (no surprise there).
So this is just my “plug” to aid in judging your essay. We need people capable of composing the whole picture in an internally logical way, and Tim is one of those people. (Don’t get any ideas that I agree with everything you say.). Not that you care.
En
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Author Tim Maudlin replied on Apr. 14, 2015 @ 01:23 GMT
Dear En,
I'm glad you liked the video...it was fun to do. And I never imagine anyone agrees with everything I say!
Cheers,
Tim
vincent douzal wrote on Apr. 16, 2015 @ 16:08 GMT
1. Dear Tim,
Your essay was for me the occasion to discover your theory of linear structures, and it was an enchantment. I think it won't be long till I buy your book.
Your text is particularly clear, too.
If I really like your style and your contributed linear structures, I have not really found an answer as to the effectiveness of mathematics, or, as you aptly...
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1. Dear Tim,
Your essay was for me the occasion to discover your theory of linear structures, and it was an enchantment. I think it won't be long till I buy your book.
Your text is particularly clear, too.
If I really like your style and your contributed linear structures, I have not really found an answer as to the effectiveness of mathematics, or, as you aptly rephrase it, how mathematics meet the world.
2. I refrained from commenting Wigner's article, because I see too many irrecoverable conceptual errors in it, for instance: ``elementary mathematics [...] entities which are directly suggested by the actual world''.
- The actual world suggests nothing.
I'd like to quote Ferán Adria, the famous Spanish chef; in answer to journalist saying that he was finding inspiration on the Costa Brava, he stood on the shore, staring at the sea and cliffs before a camera, and after a while said: ``No veo nada.'' There is nothing in clear in the landscape. Where is that inspiration?
- Wigner makes what I think is a huge mistake; implicitly, he suggests that there are things in the world that we can access in an absolute way. So, in the objects that I talk about, some are `for sure', and some are hypothetical. If you really try seriously to draw the border between the latter and the former, you come to understand that it does not exist, and that all perceptions are hypothetical. Only, some of them are usually, frequently confirmed, and we deem them solid ground. But, read Karen Blixen report of a earthquake in Out of Africa, and you will have a vivid case of a solid ground becoming moving and alive. Witness how people can be in a state of shock after a large earthquake; Haroun Tazieff used to say that their whole mental system is broken into pieces after experiencing the ``disappearance of space'', in the most concrete sense. Read Gregory Bateson's report of completely loosing ground after his experiments with sensorily puzzling effects at Adelbert Ames Jr. laboratory (in Steps to an ecology of mind). After that, read Richard Gregory's Perception as hypotheses (In The Oxford companion to the mind for instance). And read about Bach-y-Rita's experiment (guess where). You should convince yourself that the most fruitful representation is to
consider uniformly all perceptions as hypotheses.
By the same token, you cannot talk about the `actual world', if you mean by that things you have a privileged, certain, absolute access to. All that you know about the world is relative to your perceptions, and is built through interaction with it, in action-perception loops: you impulse transformations into the world, and record the effects, and build pragmatically a whole world of hypotheses. You act, to abstract.
So, when you note after Wigner about the concepts of elementary geometry that ``those very concepts were developed via interaction with the world'', there is no solid way to circumscribe only them, because all your concepts, you eventually develop via interaction with the world. (You certainly agree: ``claims about [...] must be considered just as conjectural and fallible as all other physical claims.'')
(By the way, defining exactly what `elementary geometry' should be is not trivial, and finally not univocal because it poses a problem of foundation, and what we think is the historical, or natural genetic approach is not obviously better. Shall we just say that rays of light are our concept of a straight line? Shall we say that sets and groups are at the root? Shall we say that topos is a more elementary foundation?)
3. In the words of a common metaphor, Wigner sins by confusing the map for the territory.
(Korzybski for instance has repeated much this dictum.)
All what we do in our theories is language, and the world is not language.
The trains of spikes in your neuron chains are not labelled `vision', 'touch', `warmth', etc. These concepts are built afterwards. They do not exist substantially in the world, ready to be channeled to your perception. If you excite the retina by an electrical impulse, the subject reports a flash of light —the usual hypothesis for what comes from the retina. The most striking demonstration is in Bach-y-Rita experiment (refer to my essay for a pointer).
These facts should make clear that all we operate on is within the `maps' world. All we build in our maps is hypothetical constructs, that we try to fit to the interactions we engage with the territory that we know is out there.
So to be accurate, we must always maintain that we talk indirectly about the territory, through a representation. Any statement made directly about the territory, the world is problematic because it cancels this relativity to the representations of the subject. A statement about the territory itself, being absolute, must be completely right. Therefore, it cannot be scientific. It fails to satisfy Popper's criterion. If a statement fits Popper's criterion, then it is hypothetical.
It has nearly always been that progress has occurred when the operation of representation was made explicit and stated clearly. For instance, a correct definition of what a measure is, in mathematics, could not be derived until after there was a clear separation between the measurement, and the scale used to represent measurement, that is, a usable theory for the real numbers. I took an even simpler example in my essay, with the confusion between numbers-of something, and numbers `pure', and how until the late 19th century (!) not being able to separate clearly numbers from what they could be used for —counting objects or representing quantities in this case— was impeding comprehension and progress in using them.
The conclusion is that to be able to address properly the question of the effectiveness of mathematics in representing the world, we have to include the stage of representation of the world in our framework. That is, we have to include the stage of perception and cognition in our representation, and that means including explicitly the cognitive subject.
4. I have the feeling that you have fallen in the same trap as Wigner, when you write:
``claims about the geometrical structure of physical space or space-time must be considered just as conjectural and fallible as all other physical claims.''
``A proper answer to this question must approach it from both directions: the direction of the mathematical language and the direction of the structure of the physical world being represented.''
In both cases, if I am not misinterpreting, by `physical space' and `physical world', you are not talking about the space or the world, as seen by physics, but about the world itself. You have no final word about the structure of the world. You have all latitude to propose hypothetical
physical structures that prove fruitful for various uses.
5. Your title is particularly accurate and promising, because the key issue is exactly the meeting of mathematics and the world. I completely agree with it, and it is precisely what I have just said above: the issue is on perception. Perception is fundamentally a meeting, at any considered scale: Subject and event, sensory organ and object, sensory neuron and stimulus, etc.
If you think you talk about perception, and see no meeting, then you are certainly talking about things that take place during perception, but not the fact of perception itself, properly regarded.
Regards
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Author Tim Maudlin replied on Apr. 16, 2015 @ 16:51 GMT
Dear Vincent,
Thanks for the extensive comments.
You seem to object to the idea of some immediate, certain access to the structure of the physical world via perception. Of course that is correct. I'm sure I have not written anything that suggests otherwise. Physical theories are always conjectural.
But it is also clear that some mathematical structures (say Euclidean geometry) were suggested fairly directly by interaction with the world and perception, while others (octonians) were arrived at by abstract considerations fairly far removed from perception. This observation is compatible with the fallibility of perception, perceptual error, etc.
Yes, by "physical space" or "physical space-time" I mean what we wave our arms and legs around in! Physical theories are proposals for the structure of that, and its contents. The aim of physical theory is to give an accurate account of physical space (or space-time) and its contents. What we can alter are the theories, not the object that the theories are designed to describe. And of course no one has "the final word" about whether a physical theory is an accurate description of the physical world. That is the point of being fallibilistic about all theories.
Cheers,
Tim
Pentcho Valev wrote on Apr. 17, 2015 @ 15:26 GMT
Tim Maudlin Apr. 4, 2015: "Put two high-precision atomic clocks on the floor together Synchronize. Lift one up on a table. Wait a while. Return to the floor and compare synchronization. This has been done. The clocks go out of syntonization, and the amount out is a function of how long the one is up on the table. No redshift or light involved. Experiments at this precision have only been possible recently."
You abandoned this discussion after realizing that no such experiment has ever been done (no lifting and then returning to the floor). Gravitational time dilation has always been measured by measuring the gravitational redshift but the redshift actually confirms the variable speed of light predicted by Newton's emission theory of light:
Albert Einstein Institute: "One of the three classical tests for general relativity is the gravitational redshift of light or other forms of electromagnetic radiation. However, in contrast to the other two tests - the gravitational deflection of light and the relativistic perihelion shift -, you do not need general relativity to derive the correct prediction for the gravitational redshift. A combination of Newtonian gravity, a particle theory of light, and the weak equivalence principle (gravitating mass equals inertial mass) suffices. (...) The gravitational redshift was first measured on earth in 1960-65 by Pound, Rebka, and Snider at Harvard University..."
Pentcho Valev
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Author Tim Maudlin replied on Apr. 17, 2015 @ 18:08 GMT
I stopped the discussion because you do not know the situation with respect to either the predictions or tests of General Relativity. The basic scheme of actually comparing clocks has been going on for over three decades. These are not gravitational redshifts. Everyone knows the gravitational redshift is a weak test, but the tests moving and comparing clock readings are just different. Your insistence that they are not just demonstrates your mack of understanding of the situation.
Pentcho Valev replied on Apr. 17, 2015 @ 19:13 GMT
"The basic scheme of actually comparing clocks has been going on for over three decades. These are not gravitational redshifts. Everyone knows the gravitational redshift is a weak test, but the tests moving and comparing clock readings are just different."
Such tests do not and cannot exist. As I have already said, these clocks are atomic oscillators that do not keep record of the time elapsed. So there is no point in lifting one of them and then bringing them together - there is nothing to compare. One can only measure the frequency difference (redshift) and ponder whether it is due to gravitational time dilation or variation of the speed of light with the gravitational potential.
Of course nobody is going to check which of us is telling the truth - you are right by definition. So... good luck!
Pentcho Valev
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Author Tim Maudlin replied on Apr. 17, 2015 @ 19:33 GMT
1971
http://en.wikipedia.org/wiki/Hafele–Keating_experiment
Such
tests have exited for 44 years.
That is a fact.
If you think they "do not an cannot exists" then you have been proven wrong.
Pentcho Valev replied on Apr. 17, 2015 @ 21:05 GMT
Initially you wrongly referred to this experiment:
http://www.nist.gov/public_affairs/releases/alumi
num-atomic-
clock_092310.cfm
as an example of lifting one of the clocks and then bringing the two clocks together in order to compare them. Then you took refuge in the GPS system, and now you want to discuss the Hafele-Keating experiment which was not meant for measuring gravitational time dilation. Let us stop here.
Pentcho Valev
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Member Sylvia Wenmackers wrote on Apr. 21, 2015 @ 20:49 GMT
Dear Tim Maudlin,
I will comment mainly on the first part of your essay. (The second part of the essay, in which you outline your new foundations for geometry, is interesting in it's own right, but it does not seem to address the original question as directly.)
One aspect that I liked is that you distinguish the surprise due to discovering a connection between seemingly distinct parts of mathematics from the wide applicability of mathematics to empirical sciences.
In the first part, you also think about how the world needs to be in order for the counting numbers to be relevant. You give mountains and cells as examples of concepts that are not (always) sharp enough (conceptually) for counting, and atoms as an example of a concept that is. In my view, however, the sharpness of our concepts is a matter of degree. The concept of an 'atom' is still a vague one, to some degree; hence the question of whether something is an atom or not -and whether to count something as an atom or not- may still be ambiguous in certain situations. For instance, one can imagine an electron and a proton: they will always 'feel' each other's electrical field. When, exactly, do they form a hydrogen atom? This question is also relevant in foundations of chemistry (with competing partition schemes in computational chemistry to identify 'atoms in molecules').
Viewed in this way, there are many degrees of freedom in applying mathematics to empirical findings - something which is rarely discussed in relation to the perceived effectiveness of mathematics.
Best wishes,
Sylvia Wenmackers - Essay
Children of the Cosmos
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Author Tim Maudlin replied on Apr. 22, 2015 @ 04:52 GMT
Dear Sylvia Wenmackers,
Thanks for the comment. I do completely agree with what you say, which is why also the term "atom" is not used in the formation of fundamental physical law. First it was replaced by reference to electrons, protons and neutrons and then by reference to electrons and quarks. Since atoms are bound states of these things, the concept is only as sharp as "bound state", which is somewhat vague. At this point, "electron" and "quark" appear to be fundamental, and admit of not further analysis. But we may be wrong about that.
Cheers,
Tim
Member Sara Imari Walker wrote on Apr. 22, 2015 @ 06:46 GMT
Dear Tim, I very much enjoyed your essay and hope you can clarify one point for me. It seems that your argument suggests that there may be equivalent frameworks for describing physical systems (such as topology or linear structures) but that only one corresponds to physical reality. This suggests that we could be using the wrong mathematics to describe reality purely because distant branches of mathematics that seem unrelated are actually deeply related. How would we ever know then that we had the "right theory"?
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Author Tim Maudlin replied on Apr. 22, 2015 @ 12:17 GMT
Dear Sara,
I don't think we will ever we absolutely sure we have the right theory. All we can do is formulate different theories, using different mathematical resources, see what sorts of testable, observable behavior they predict, and try to gather data to test them. But there will always be different theories that agree with all the data we have. Choice among these different theories that are consistent with observations will either be made on grounds of simplicity, elegance, etc. or else not be made at all: we will just have to admit that we don't know for sure which, if any, if these theories is correct. But whatever we do, we have to acknowledge that we cannot be certain that any particular theory is right. That's just the situation we are in.
Cheers,
Tim
ABDELWAHED BANNOURI wrote on Apr. 22, 2015 @ 12:12 GMT
Dear Tim :
Your essay is very interesting,
In my opinion, mathematics that reflects reality, remains a priority for the current science.The Bi-iterative calculation could be the right one, because it has all the characteristics of a pure mathematics.
“NUMBERS”
first are integers
second are geometric shapes
third are physical entities, with...
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Dear Tim :
Your essay is very interesting,
In my opinion, mathematics that reflects reality, remains a priority for the current science.The Bi-iterative calculation could be the right one, because it has all the characteristics of a pure mathematics.
“NUMBERS”
first are integers
second are geometric shapes
third are physical entities, with infinite degrees of freedom
fourth maintain always the aspect ratio
fifth are elastic ... ..
Sixth are compact, and unseverable by two egual part.
We take an example:
1 + 5 + 7 + 12 =
= 1 + 5 + 7 + (6 + 8 + 10)
= 1 + 5 + 7 + (3 + 4 + 5) + 8 +10)
= 1 + 5 + 7 + (3 +4 +5) + (1 + 6 - 9) + 10
= 1 + 5 + 7 + (3 +4 +5) + (1 + 6 - 9) + 10
= 1 + 5 + 7 + (3 +4 +5) + (1 + (3 + 4 + 5) - 9) + 10
= 13
=2197
Of course, we speek about a cubic equazion.
Recursive calculation, the Fibonacci series and Fractal are wrongly linear, because they consider the space as a bi-dimensional sheet.
The system's Bi-iterative calculation is different, it consider the space, multi-directional and multi-dimensional. In fact, exist only set and sub-set. with (1) we mean (1 * 1 * 1).
Because we live in a complex and varied reality, only a struture able to transform can describe it.
The human brain is divided into two halves, the left hemisphere is masculine, active, called "rational". The right hemisphere is feminine, passive called "irrational".
The most important thing is that these two opposing positions should coexist. (x + 1) and (x - 1) are two limits , i meam the linear structure and the non-linear structure are inseparable.
A system that can not stop,follow the arrow of time. Or rather, it coping itself, reflecting itself. So, we have (x + 1), + 1, + 1, + 1 ....
"The universe was born like that, one bit after another, and continues to do so even now".
(See ANNEX)"The teorem"
sincerly yours
Bannouri
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attachments:
1_1_Theorem_1.jpg,
1_1_Theorem_345.jpg
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Peter Jackson wrote on Apr. 22, 2015 @ 16:05 GMT
Tim,
A very interesting essay. I was skeptical at first but your explanation gradually resolved most of my reservations. Then I found just as it was getting to the climax with 'discrete Relativistic space-times' the last chapter was missing!
If you've read any of my recent essays (all finalists) you'll see various aspects of such discrete 'space-time' systems or 'fields' explored. I...
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Tim,
A very interesting essay. I was skeptical at first but your explanation gradually resolved most of my reservations. Then I found just as it was getting to the climax with 'discrete Relativistic space-times' the last chapter was missing!
If you've read any of my recent essays (all finalists) you'll see various aspects of such discrete 'space-time' systems or 'fields' explored. I hope you may direct me where I may find your own last paragraph!
A few points;
1. I noticed your reply to Pentcho ref Hafele Keating above. If you'd read Hafele's NY lecture you wouldn't be so sure. He expressed his concerns that the data did NOT confirm theory as expected. It was only when the paper first didn't pass review that any such reference and data disappeared. The NY lecture text is also difficult to find (I can provide quotes if you wish). Nonetheless most still cite the work as the key excluder of many otherwise possibly valid theoretical approaches. Perhaps some gold plated 'confirmation bias'?
2. A question. Does your linear sequence model allow, say, expanding helical paths, and such paths which may form cascades on 'interaction requantizations' (via say atomic absorption and re-emission/'scattering', or perhaps 'decay' in particle terms) By 'cascades', consider a massive particle forming a shower through an atmosphere. The linear' structure would then have branches. Does your theory particularly exclude or support such models?
3. You say Wigners "..claim is indisputable', as commonly assumed. I'd like to challenge that. Lets say he had a friend 'd'Espagnolet' and they derived an 'inequality', later used by a chap called Bell, who cited a friend's odd red and green socks to prove its mathematical limits. All fine. But now let's give Dr Bertelmann a pair of reversible green socks with red linings. After their squash game they had to dress in the dark, so either foot could then have either colour! - giving a kind of 'stacked twin pair' probabilities, independent to each foot. Now we can have a perfectly physically feasible set of findings NOT constrained by Wigners 'amazingly effective' maths, or requiring 'quantum wierdness', or feet that can see in the dark and talk to each other! Some maths may then give the right bottom line but won't model the actual physical mechanism faithfully.
I propose the Wigner comment may then not prove indisputable, but that the situation he correctly identifies in the paragraph below applies - that there may be other truths. (A full description including complementarity referenced in my essay).
If we know that there are inconsistencies in current assumptions and theories is it right that we so consistently reject alternatives to doctrine? I just feel your own model may be on a track I recognize as being more widely consistent. Is "track" also consistent with your "linear", I've never considered 'lines' to exist any more than points!?
I do hope you'll also be able to read, consider and discuss my own essay, even after the score deadline.
Many thanks.
Peter
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Author Tim Maudlin replied on Apr. 22, 2015 @ 16:31 GMT
Dear Peter,
As I'm sure you know, a complete discussion of the presently available evidence in favor of GR would be quite extensive, and go far beyond the three "classic tests", including the rotational rate of binaries, all sorts of precise clock and timing experiments, as everyone always mentions corrections needed for GPS etc. GR should at least be recovered as a limit of a more...
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Dear Peter,
As I'm sure you know, a complete discussion of the presently available evidence in favor of GR would be quite extensive, and go far beyond the three "classic tests", including the rotational rate of binaries, all sorts of precise clock and timing experiments, as everyone always mentions corrections needed for GPS etc. GR should at least be recovered as a limit of a more fundamental theory that may imply corrections, but the discussion with Valev was not productive, and irrelevant to my paper in any case. The fact that he continues it despite being asked to stop (see below) is already indicative of a certain state of mind.
There is no problem at all about converging or diverging sets of trajectories in my account of space-time geometry. Such "branching paths" will be ubiquitous in the geometry.
Your "reversible socks" scenario is obviously a completely local physics, and no sets of observations of socks put on in the way you describe will violate any Bell inequality. So I can't really see what you are trying to get at with the example. If it is more direct, try to come up with a theory that predicts the GHZ statistics using whatever kinds of reversible socks you like, but where the socks are examined at space-like separation. No such theory will make the predictions of standard quantum theory. (It is easier with GHZ because we don't have to worry about long-term statistics.)
By the way, although the 3-polarizer experiment is, indeed, often presented as a "paradox", there is nothing quantum-mechancial about it, and the phenomenon can be received by purely classical models. So, unlike the 2-slit interference effects or, more strikingly, violations of Bell's inequality, it is not an experiment that displays any particularly quantum-mechanical behavior. Curiously, I even remember a TV show long ago showing how to get 3-polarizer behavior using a rope and some blocks of wood with rectangular channels in them that stand in for the polarizers. If you line up two of the blocks with the channels are right angles to one another and put the rope through, vibrations of the rope at one end all get damped out by the blocks. But if you add a third block in between, with the channel running at 45°, some vibration of the rope gets through all three blocks. Clearly, this is a purely classical system which shows the same sort of phenomenon.
Cheers,
Tim
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Pentcho Valev replied on Apr. 22, 2015 @ 17:00 GMT
"the discussion with Valev was not productive, and irrelevant to my paper in any case. The fact that he continues it despite being asked to stop (see below) is already indicative of a certain state of mind."
Thanks. Kind of you. I did stop the discussion but just found references (see below) disproving the following text of yours:
Tim Maudlin replied on Apr. 4, 2015 @ 23:49 GMT: "Put two high-precision atomic clocks on the floor together Synchronize. Lift one up on a table. Wait a while. Return to the floor and compare synchronization. This has been done. The clocks go out of syntonization, and the amount out is a function of how long the one is up on the table. No redshift or light involved. Experiments at this precision have only been possible recently."
Pentcho Valev
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Peter Jackson replied on Apr. 24, 2015 @ 18:26 GMT
Tim,
I understand your views, all conventional. You seem to take agreement with convention as always a priori falsification, so are happy to ignore anomalous findings and assume they'll disappear. I do understand as an educator that's the easiest position to maintain.
I must say my approach differs, seeking out and trying to remove or rationalise anomalies and paradoxes. It proved a...
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Tim,
I understand your views, all conventional. You seem to take agreement with convention as always a priori falsification, so are happy to ignore anomalous findings and assume they'll disappear. I do understand as an educator that's the easiest position to maintain.
I must say my approach differs, seeking out and trying to remove or rationalise anomalies and paradoxes. It proved a bigger task than I anticipated. i.e one of the toughies is the increased weight of spinning objects, trillions of times more than GR predicts! I long studied GPS and Galileo and wrote a paper on GPS showing most of the 'relativity proof' to be nonsense. GPS can just as easily be said to DISprove SR! Don't get me wrong; I've found absolutely NO evidence against Einstein's postulates, so disagree with Pentcho, however there's a stack suggesting a flawed original interpretation, indicating an adjusted interpretation far more in line with his 1952 paper. It's only theoretical inertia that's stopping it being countenanced not the evidence, which is ALL more consistent with the adjusted one.
The very same simple 'QG' mechanism hypothesized as underlying space-time is what reproduces the predictions of QM 'quasi' classically (as it unifies the descriptions it would do!) You're quite wrong about the 'sock switch con' not producing the correlations from 'entanglement' which we call 'quantum non-locality'. Again you seem to just dismiss the possibility a priori without looking! (Bell's 'sleepwalking') - but the derivation is exactly what Bell predicted! Sure there are 7 concepts to track when human brains alone have a limit of ~3, but I'm sure a man of your intellect can follow it. I'd be delighted if anyone falsified it but nobody who'se looked has done, yet. Most don't look. Ken Wharton identifies the same narrownness of vision!
Do have a go. The draft paper is web-archived here.
P Jackson, J Minkowski; https://www.academia.edu/9216615/Quasi-classical_Entanglemen
t_Superposition_and_Bell_Inequalities._v2. My previous (top 10) essays from 2011 also cover much of the ground of this 'discrete field' model.
The classical 3 polarizer effect for instance can equally be taken as a hint that QM MIGHT possibly have a classical 'type' solution as Bell predicted. It's just one of the 7 'jigsaw puzzle' pieces that you'll find fit to produce to coherent picture. Unlikely? Of course. So is the chance of the ground swallowing you up, but it happens!
Do report back on my blog, or direct to; pj.ukc.edu@physics.org
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Author Tim Maudlin replied on Apr. 24, 2015 @ 19:37 GMT
Peter,
The link you post is dead. In any case, I was making a couple of points, which you seem happy to dismiss with talk of convention, which I cannot even follow. One is that the so-called "3-polarizer paradox" is not a phenomenon at all difficult to replicate and completely understand even in a classical setting. In that sense, there is nothing paradoxical about it at all, and certainly nothing that could shed any light on quantum theory. The second is that no amount of putting on and taking off reversible socks, using any local physics, can replicate the predictions of GHZ and, in the long term, the statistics of spin experiments on entangled pairs in different experimental conditions. These are mathematical theorems, and they contain no mathematical errors. There is really no point in producing your theory...however many concepts it has... without first identifying an error in the theorem. My confidence in Bell's result arises from understanding the theorem. Other people in the contest who claim to "refute" Bell also demonstrate lack of comprehension of what he proved.
The GHZ case is simpler than the standard spin/polarizer cases because one does not have to worry about statistics. There are four possible experimental conditions and predictions about what will be observed under all four. And it is demonstrably impossible to reproduce those predictions using local physics. Since your socks—whether I put them on in the dark or not—are governed by local physics, they cannot reproduce the predictions.
There is actually nothing in your FQXi paper that explains how these socks are supposed to work, so nothing to analyze: just an assertion that everything is OK. If there really were a flaw in Bell's proof, the obvious thing to do is actually explain what is it, and if there is a local physical model violating the inequalities the thing to do is explain it. That is how a theory gain credibility.
Tim
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Pentcho Valev wrote on Apr. 22, 2015 @ 16:07 GMT
Tim Maudlin wrote on Apr. 17, 2015 @ 18:08 GMT: "I stopped the discussion because you do not know the situation with respect to either the predictions or tests of General Relativity. The basic scheme of actually comparing clocks has been going on for over three decades. These are not gravitational redshifts. Everyone knows the gravitational redshift is a weak test, but the tests moving and comparing clock readings are just different. Your insistence that they are not just demonstrates your mack of understanding of the situation."
I do understand the situation - you don't:
"A new paper co-authored by U.S. Energy Secretary Steven Chu measures the gravitational redshift, illustrated by the gravity-induced slowing of a clock and sometimes referred to as gravitational time dilation (though users of that term often conflate two separate phenomena), a measurement that jibes with Einstein and that is 10,000 times more precise than its predecessor."
"Einstein's relativity theory states a clock must tick faster at the top of a mountain than at its foot, due to the effects of gravity. “Our performance means that we can measure the gravitational shift when you raise the clock just two centimetres (0.78 inches) on the Earth's surface,” said study co-author Jun Ye."
Pentcho Valev
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Janko Kokosar wrote on Apr. 22, 2015 @ 20:01 GMT
Dear Tim
I never learn topology, so I do not understand everyting. First I need to learn some background. But, it is known, that special relativity (according to Newtonian physics) defines causality. The now thing is also unsimultaneity. How can you connect your explanation with unsimultaneity?
I think that every new explanation from new aspect can tell a lot, so it seems to me that you have a good essay.
In my essay I speculated, that Pythagora theorem is consequence of kinetic energy conservation in ortogonal direction. Do you have opinion about this?
I thing also that Planck's dimensionless nature of space and time tells a lot ... Maybe still your approach fails to complete explanation.
My essayBest regards
Janko Kokosar
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