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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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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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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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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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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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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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 R⁴ around that point. The directions of the directed lines in R⁴ at that point can be any subset of the sphere from R⁴ 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 R⁴ 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 R⁴ 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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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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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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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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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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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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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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"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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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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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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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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