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Trick or Truth Essay Contest (2015)
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Mathematics is Physics by Matthew Saul Leifer
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Author Matthew Saul Leifer wrote on Feb. 26, 2015 @ 21:10 GMT
Essay AbstractIn this essay, I argue that mathematics is a natural science---just like physics, chemistry, or biology---and that this can explain the alleged "unreasonable" effectiveness of mathematics in the physical sciences. The main challenge for this view is to explain how mathematical theories can become increasingly abstract and develop their own internal structure, whilst still maintaining an appropriate empirical tether that can explain their later use in physics. In order to address this, I offer a theory of mathematical theory-building based on the idea that human knowledge has the structure of a scale-free network and that abstract mathematical theories arise from a repeated process of replacing strong analogies with new hubs in this network. This allows mathematics to be seen as the study of regularities, within regularities, within ..., within regularities of the natural world. Since mathematical theories are derived from the natural world, albeit at a much higher level of abstraction than most other scientific theories, it should come as no surprise that they so often show up in physics.
Author BioMatt Leifer is a visiting researcher at Perimeter Institute for Theoretical Physics. His research interests include quantum foundations, quantum information, and particularly the intersection of the two. He is hoping to break the world record for the number of FQXi essay contests won by a single individual.
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Roger Schlafly wrote on Feb. 26, 2015 @ 23:02 GMT
Your 2-page description of mathematics leaves out its most distinguishing feature. Mathematics has been axiomatized, and all mathematical knowledge is gained by proving theorems from axioms. That is what makes those abstractions work, and what gives math its autonomy. It is also what causes me to deviate from your naturalism, and I would say that
my essay is more directly opposite Tegmark's view than yours.
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Author Matthew Saul Leifer replied on Feb. 27, 2015 @ 01:48 GMT
In Sec. 2.3, I said, "As the formalists suggest, mathematical theories are just abstract formal systems, but not all formal systems are mathematics." Because mathematical theories are formal systems I accept that they are defined by axioms and that theorems are proved from those axioms. If that is what you are referring to then I agree with you.
On the other hand, if you mean that all of mathematics can be axiomatized starting from set theory then I don't agree that this correctly captures the nature of mathematics. This sort of foundation is rather retrofitted to mathematics at a later date. The fact that there are several competing foundations for mathematics indicates that there is nothing unique about the foundations. I don't think there is anything wrong with foundational work of this kind. We can view it as creating a massive hub in the structure of mathematical knowledge that a lot of other theories can hang off. It is rather that I think the informal ideas and theories come first and they are slotted into the formal structure later.
Roger Schlafly replied on Feb. 27, 2015 @ 19:46 GMT
Axioms are not just for formalists; the Platonists use them also. And you call it "retrofitted", but the axiomatic method goes back 2000 years.
There are competing axiom systems for set theory. Probably the biggest difference is the axiom of choice, that you mention. However that does not undermine the axiomatization of mathematics. It only means that some mathematicians want to make dependence on that axiom explicit.
So when you say that "Mathematics is Physics", you are not talking about Mathematics has it has been understood for 2000 years. You are only talking about some narrow empirical subset of mathematics.
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Author Matthew Saul Leifer replied on Feb. 27, 2015 @ 21:01 GMT
Yes, but why do mathematicians decide that some sets of axioms are more important than others? That is what I am really trying to get at.
Roger Schlafly replied on Feb. 27, 2015 @ 22:58 GMT
The major alternatives are
Zermelo–Fraenkel set theory,
Von Neumann–Bernays–Gödel set theory, and
Tarski–Grothendieck set theory. Those Wikipedia pages have some explanation of the advantages. But the vast majority of math papers are valid in any of the systems. If Mathematics were Physics, then maybe there would be some empirical difference, but there is not.
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Edwin Eugene Klingman wrote on Feb. 27, 2015 @ 03:48 GMT
Dear Matt Leifer,
Your essay describes your '
theory of mathematical theory building' and contrasts it with (o.d.l.) Tegmark's MUH. As I agree completely with you, let me focus on a few specifics. First, you say "…
our fundamental laws of physics are formulated in terms of some of the most advanced branches of mathematics…" I would probably have said that our fundamental...
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Dear Matt Leifer,
Your essay describes your '
theory of mathematical theory building' and contrasts it with (o.d.l.) Tegmark's MUH. As I agree completely with you, let me focus on a few specifics. First, you say "…
our fundamental laws of physics are formulated in terms of some of the most advanced branches of mathematics…" I would probably have said that our fundamental laws of physics
can be formulated in this way. The necessity to do so is not obvious to me.
Second, you say "
unless the platonist can give us an account of where the abstract realm actually is in physical reality, and how our brains interact with it…" My understanding is generally that the ideal realm they claim is outside of physical reality, but then, as you say, how can our brains interact with it?
I also agree with you that "
the search for a theory of everything is not fruitless; I just do not expect it to ever terminate." You conclude that "
theories are just convenient representations of the regularities… [iterated] of the physical world."
My essay begins with an approach to discovering these regularities and to deriving formal representations. This approach to a "theory of theory" is discussed in my first pages and in my endnotes. I believe it to be fully compatible with your ideas.
The meat of my essay however concerns Bell's mathematical formulation of a simple model of physics and his assumptions that are not clearly stated. In fact, after specifying quite clearly that the "hidden variables" can take almost any form, he introduces his basic assumption with one word, the conjunction 'and', followed by his equation (1).
As I agree with you on both the nature of math and on an approach to theorizing about theory building [as I understand you], and as I quote you in my endnotes to define the current state of quantum mechanics, I hope you will find the time to read my essay and to provide me with feedback on it.
My very best regards,
Edwin Eugene Klingman
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Author Matthew Saul Leifer replied on Feb. 27, 2015 @ 17:28 GMT
Let me first say that I do not generally respond to requests for me to read someone's essay made on this page, via email, or via any other method. I hope this does not offend you, or anyone else, but I get too many such requests and there are too many essays that I want to read anyway for it to be feasible for me to comply with all requests. Everyone who makes such a request claims that their...
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Let me first say that I do not generally respond to requests for me to read someone's essay made on this page, via email, or via any other method. I hope this does not offend you, or anyone else, but I get too many such requests and there are too many essays that I want to read anyway for it to be feasible for me to comply with all requests. Everyone who makes such a request claims that their essay is related to mine, so that's not really a good way of picking what to read either. Instead, over the next couple of months I will look over all the abstracts and decide what to read for myself, and I will of course give you my feedback if your essay is one of the ones I pick. That said, let me move on to addressing your comments.
Your first point is really just the under-determination of theory via experiment. There are, of course, all sorts of ways of writing down theories that are empirically equivalent to one another, in these sense of making the same predictions. These equivalent formulations will vary in the degree of sophistication of the mathematics they use. At the extreme, we could just write down a theory as a list of readings that our experimental equipment would display in every conceivable scenario. As another example, you could write down the predictions of general relativity without using differential geometry on a spacetime manifold, but instead introduce a series of forces within Newtonian spacetime that mocks up all the relativistic effects. Such a theory would require less mathematical sophistication, but I hope you would agree it would be far less elegant.
In practice, the under-determination problem does not come up that much. We may have a few equivalent formalisms for a physical theory that are all in common use, e.g. Hilbert space vs. path integrals for quantum theory. However, it is rare that the formalisms in common use differ radically in their mathematical sophistication. In my view, if our job is to structure knowledge efficiently in a scale-free network then that requires working at a certain level of abstraction, and hence of mathematical sophistication. If we used less mathematical sophistication then we would end up with a network with a lot of direct connections between seemingly disparate nodes, and if that happens I would argue that it is preferable to introduce a new node to explain the common structure of the connections, which inevitably makes the theory one level more abstract.
Regarding the platonic world of forms, I guess I could have been a bit clearer on this. Yes, the platonists think that the mathematical world is purely abstract and not actually part of our physical universe. However, naturalists think that all our thought processes must be explainable in terms of the activity of our brains, so if we have intuitions about the platonic world then it must be interacting with our brains in some way. If it interacts with our brains then it must be physically real, and hence cannot be purely abstract. This is just a reducto ad absurdium for platonism+naturalism, so one of them has to go, and I plump for retaining naturalism.
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Akinbo Ojo wrote on Feb. 27, 2015 @ 09:14 GMT
Dear Matt,
Congratulations on your thought provoking essay. As you would probably be entertaining other questions and have challenges on your time, I will have only one question for you:
Borrowing from different information sources, including cosmology,
can a Universe, either of the 'Physics is Mathematics' or 'Mathematics is Physics' variety perish?If the universe can perish, what is the possible implication for physics that mathematical/ physical objects are not eternally existing things but have a finite duration of existence?
Sorry, one question became two.
Regards,
Akinbo
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Author Matthew Saul Leifer replied on Feb. 27, 2015 @ 16:54 GMT
I don't think my view has any implications for whether or not the universe will perish.
To answer your second question, I think it is helpful to first answer a related question, namely, in a universe with different laws of physics than our own, would mathematics be different?
According to my view, mathematical theories are just abstract formal systems, but only those formal systems...
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I don't think my view has any implications for whether or not the universe will perish.
To answer your second question, I think it is helpful to first answer a related question, namely, in a universe with different laws of physics than our own, would mathematics be different?
According to my view, mathematical theories are just abstract formal systems, but only those formal systems that bear a suitable relationship with the physical world are counted as mathematics. Specifically, mathematical formal systems are the ones that would be developed by a society of finite beings via a process of abstraction and generalization from the physical world. This does not mean that mathematics requires society to exist. There may well be a fact of the matter about the sorts of mathematical theories a society would generate were it to be present in a given physical universe, so the mathematics of a universe may well be a property of its physics rather than of a society.
Given that mathematical theories are formal systems, it is of course possible to contemplate any formal system within any universe, so in that sense mathematics is not dependent on physics. However, there is the more important question of whether beings in a universe with different physics would ascribe the same roles to the same formal systems that we do. For example, it is conceivable to me that there could exist a universe in which modular arithmetic plays the same role that normal arithmetic does in our universe, i.e. addition of finite collections of things, even sheep, is always cyclic and gets reset to zero after you add a certain number of objects. It seems quite crazy, but nevertheless not logically inconceivable. Beings in that universe would regard modular arithmetic as the most basic theory of arithmetic and would derive our usual theory of arithmetic only as an abstract exercise in pure mathematics or perhaps for some specialized applications. In that sense, which is I think the more important sense, the mathematics of a universe is determined by its physics.
Given this, what happens if there is no universe to speak of? I think it is definitely the case that, in the second sense, there would then be no mathematics to speak of either. There would be no way of singling out certain formal systems as the interesting ones to study. I am not sure whether it even makes sense to contemplate the status of formal systems if there is no universe, so perhaps there is no mathematics in the first sense either, but I am not so certain about that.
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Christophe Tournayre wrote on Feb. 27, 2015 @ 16:57 GMT
Dear Marc,
It was a real pleasure to read your essay, thank you.
In your essay, you state that physical theories will become increasingly abstract and mathematical. Who will be able to discover them if it gets even more complex? Do you believe we are close to hitting a knowledge wall?
Regards,
Christophe
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Author Matthew Saul Leifer replied on Feb. 27, 2015 @ 17:36 GMT
Abstract does not necessarily mean complex. In fact, I would argue that we introduce more abstraction precisely to reduce complexity, i.e. it reduces the number of direct connections between seemingly disparate parts of the knowledge network.
I don't think we are close to hitting a knowledge wall. One can work quite effectively on just a small portion of the knowledge network, i.e. just a few nodes clustered around a hub. The abstraction prevents an individual from having to know everything about everything in order to make any progress.
However, I would argue that in order to make progress we need to be good at identifying connections between nodes in the network, especially if those nodes are far away from one another. Given that any individual only knows a small part of the network, this means we require interdisciplinary collaboration to find the strong analogies that will be later abstracted into more powerful theories. We may well be hitting a knowledge wall for what a single individual can do on their own, but I believe we are just at the beginning of developing new modes of collaboration that can get around this.
Christophe Tournayre wrote on Feb. 27, 2015 @ 17:01 GMT
Dear Matt, sorry for misspelling your name.
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Michael Rios wrote on Feb. 27, 2015 @ 18:20 GMT
Greetings Matt
I enjoyed your essay. A network view of evolving mathematical knowledge does indeed exist. Even now, we are connecting more nodes, and physics seems to be giving a "garden of insights", fueling this process.
What is interesting about our current era, is how advanced physics has become. In an attempt formulate a theory of quantum gravity, it is clear Riemannian...
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Greetings Matt
I enjoyed your essay. A network view of evolving mathematical knowledge does indeed exist. Even now, we are connecting more nodes, and physics seems to be giving a "garden of insights", fueling this process.
What is interesting about our current era, is how advanced physics has become. In an attempt formulate a theory of quantum gravity, it is clear Riemannian geometry must give way to a new geometrical structure. What this new structure is, will become one of the ultimate examples of the unity of mathematics and physics. For at present it seems current mathematics, in all its complexity, is not sufficient to describe the microscopic structure of spacetime.
There a different approaches to quantum gravity, such as loop quantum gravity, Connes' noncommutative geometry, and string/M-theory. And although there has been a backlash in public opinion towards string/M-theory (our most promising theory of unified physics), one must remember it has its origins in hadronic physics, not quantum gravity. Moreover, pure bosonic string theory is only consistent in 26-dimensions, a fact which is very mysterious from a mathematical viewpoint.
If string/M-theory, in its complete nonperturbative form, proves to be the unique theory of quantum gravity, not only for our universe, but for all other universe in a multiverse system, it is a testament to the usefulness of exceptional mathematics, instead of general mathematical structures. General mathematical structures, hold no preference for a given n-dimensional space, with fixed value of n. Special values of n occur in the study of sporadic groups, exceptional Lie groups, and other "odd ball" mathematical constructions.
What seems to be occuring, at present time, is the unifying of these exceptional mathematical structures, using insights from string/M-theory, loop quantum gravity and noncommutative geometry. For example, it has been argued by Horowitz and Susskind that it is difficult to relate bosonic string theory in 26-dimensions to M-theory in 11-dimensions because there should be a 27-dimensional
bosonic M-theory that reduces to bosonic string theory, just as M-theory reduces to the 10-dimensional superstring theories by compactification. In other words, at strong coupling, the extra dimension "opens up" and there should be a stable ground state as the full extra dimension is restored.
Smolin (a father of loop quantum gravity), years ago, introduced a Chern-Simons type matrix model in 27-dimensions as a candidate for bosonic M-theory, that effectively unites loop quantum gravity with M-theory. The matrix model makes use of an algebra of observables, that was discovered as an exceptional case by Jordan and Von Neumann in 1934. The algebra makes of 8-dimensional variables called the octonions (found in 1843), which themselves are a maximal case of the so-called (finite dimensional) division algebras. The real, complex and quaternion algebras are the only three algebras of this type, existing in 1, 2 and 4-dimensions.
There is mounting evidence that bosonic M-theory may indeed be formulated as a matrix model in 27-dimensions. If this is the case, the dimension with value n=27, is a result of the maximality of construction of an algebra of observables over the octonions. Such an algebra, permits only 3x3 matrices, at most, which together close into a hermitian matrix algebra that leads to a sensible quantum mechanics. This hermitian matrix algebra, the exceptional Jordan algebra, is the self-adjoint part of a unique Jordan C*-algebra. This C*-algebra allows a noncommutative geometrical interpretation.
So here, a historical series of accidental mathematical and physics discoveries, could lead to a theory of unified physics, that unites all promising approaches to quantum gravity. Moreover, it seems likely sporadic groups, exceptional Lie groups, L-functions, Shimura varieties and noncommutative geometry can all be connected in the process. This is quite an embarrassment of riches considering the haphazard process by which it stemmed. This is not to say, ultimate reality is necessarily mathematical. For as Hawking once asked, “What is it that breathes fire into the equations and makes a universe for them to describe?” What is the structure of the ancient Pythagorean's apeiron? Is there a pre-mathematical description of reality? Is Nature, at its core, acausal? Hopefully, in the coming years, we can begin to address such questions in a sensible manner.
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Member Tim Maudlin wrote on Feb. 27, 2015 @ 18:45 GMT
Dear Matt,
I'm not sure I understand the sense in which mathematics is supposed to be "about the physical world" as you understand it. In one sense, the truth value of any claim about the physical world depends on how the physical world is, that is, it is physically contingent. Had the physical world been different, the truth value of the claim would be different. Now take a claim about the integers, such as Goldbach's conjecture. Do you mean to say that the truth or falsity of Goldbach's conjecture depends on the physical world: if the physical world is one way then it is true and if it is another way it is false? What feature of the physical world could the truth or falsity of the conjecture possibly depend on? Do you think the conjecture could fail to have a truth value at all? The formalists tried to reduce mathematical truth to theoremhood, but Gödel proved that won't work. We know that if Goldbach's conjecture is false then (in some sense) it is provably false by direct calculation (although the calculation might take more steps than there are elementary particles). But if it is true, it may not be provable from any acceptable axioms. It is very hard to make any sense of what we all believe about a case like this (the conjecture is either true or false) without, in some sense, being Platonistic. I can see you don't like Platonism, but I can't see how you deal with questions like these about mathematical truth. Could you provide an example of a purely mathematical claim whose truth depends on the physical world, and point to the feature of the physical world it depends on?
Cheers,
Tim
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Author Matthew Saul Leifer replied on Feb. 27, 2015 @ 21:27 GMT
"Could you provide an example of a purely mathematical claim whose truth depends on the physical world, and point to the feature of the physical world it depends on?"
No, because I don't think there is one. Mathematical theories are still formal systems in my view, so truth or falsehood is supposed to be decided by the usual methods of proof.
Titling my essay "Mathematics is...
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"Could you provide an example of a purely mathematical claim whose truth depends on the physical world, and point to the feature of the physical world it depends on?"
No, because I don't think there is one. Mathematical theories are still formal systems in my view, so truth or falsehood is supposed to be decided by the usual methods of proof.
Titling my essay "Mathematics is Physics" is mainly intended as a contrast with Tegmark's views, but you shouldn't take it too literally. What I mean is rather that which formal systems we decide to call mathematics, out of all the myriad of arbitrary axiom systems we might choose to lay down, is a matter of physics.
I take it you would agree that there is a fact of the matter in our universe about how collections of discrete objects like rocks and sheep behave when we combine them? This regularity is a matter of physics, and I claim that this is a large part of the reason why the theory of number and arithmetic plays the role that it does in mathematics. It does not explain how that theory should handle things like infinities, but I argued in the essay that this eventually gets determined by the relationship to other areas of mathematics that have been developed, such as geometry and calculus. Because each of these have, at their root, some physical origin, the whole structure is highly constrained by the physics of our universe.
Regarding Goldbach's conjecture, its truth does not depend on the physical world so long as we have already decided which axiom system to adopt, but I would argue that physics has played a role in our decision to adopt that system.
To understand in what sense I think mathematical truth is contingent on the physical world, it might help to look at my answer to Akinbo Ojo above. It is quite conceivable to imagine a world in which modular arithmetic plays the same role that the usual arithmetic does in our world. The theorems of the usual arithmetic are no less true in that universe than ours, but beings inhabiting that universe would regard the our theory of arithmetic as esoteric and they would cast modular arithmetic in that role instead. In that sense, mathematics is contingent on physics and, in this sense, whether 1+1=2 can be contingent on physics.
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Member Tim Maudlin replied on Feb. 27, 2015 @ 22:09 GMT
OK, but really this is a Platonistic answer. Of course, which mathematical systems can be usefully used to describe the world depends on how the world is! That is not in dispute. But your claim that settling on a axiom system settles Goldbach's conjecture (and does so independently of the world) is Platonistic. The notion of a "theorem" (i.e. something that follows, by application of rules, from a set of axions) is also Platonistic: you don't think that what the theorems are depends on the world. And this does not address the status of claims that are not theorems and whose negations are not theorems.
Regular arithmetic and modular arithmetic are different mathematical structures. Which is useful for describing things is of course contingent on those things, but the purely mathematical structure of the structures does not. What naturalistic fact determines whether or not Goldbach's conjecture follows from the Peano axioms, and what naturalistic fact determines its truth if neither it nor it's negation follow?
I recognize that this is just philosophy of math—and not really central to your essay, which can be read as about which mathematical systems we discover (or invent) and decide to deploy in physics and why. But a thoroughly non-Platonistic account of mathematics is both very hard to formulate (since one thinks of the axiom/theorem relation in a completely Platonistic way) and inconsistent with normal beliefs about mathematics (since truth cannot be reduced to theoremhood in any case).
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Author Matthew Saul Leifer replied on Feb. 28, 2015 @ 04:15 GMT
You are being too modest to call it "just" philosophy of math. It's important to get the background right and I am relatively naive in this area, so I appreciate your attempts to pin me down.
I am disturbed by your suggestion that I am a platonist. To me, platonism suggests the existence of an abstract world, independent of the physical world, to which we somehow have access through...
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You are being too modest to call it "just" philosophy of math. It's important to get the background right and I am relatively naive in this area, so I appreciate your attempts to pin me down.
I am disturbed by your suggestion that I am a platonist. To me, platonism suggests the existence of an abstract world, independent of the physical world, to which we somehow have access through mathematical intuition. I definitely want to deny that. I would be happy to be called a mathematical realist, in that I think there is a fact of the matter about what should be called a mathematical truth, but I want to cash that out in terms of the physical world rather than some abstract mathematical world. In any case, I think it is helpful to distinguish platonism from the broader concept of mathematical realism.
As I said towards the beginning of the essay, I have a somewhat pragmatic notion of truth, at least in the context of science and math, so I am prepared to accept something as "true" if it makes the system of knowledge hang together more efficiently than it would do otherwise. This notion is partly empirical, in the sense that empirical facts form part of our knowledge network and so our axiom systems have to be chosen to make those facts fit efficiently into the network, but it is also partly pragmatic, i.e. there are several choices for how to build theories that connect up our knowledge, but the "true" one is the most useful, by which I mean the most efficient. Because of this, even though the notion of a theorem is not itself naturalistic, the set of things we should be willing to call the true theorems of our world is.
Regarding claims that are not theorems, if they are true claims then they are theorems of a meta-theory. I would argue that the axioms chosen for the meta-theory are not arbitrary, but decided by the same sorts of considerations as the axioms of the lower level theories. I realize that there is an infinite regress here, but where to stop bothering about deciding truth claims in higher level theories is again, I would say, a pragmatic matter. For example, I want to say that there is a fact of the matter about whether we should accept the axiom of choice, because without it we can't do conventional analysis and that would be a disaster for many areas of mathematics and physics. For things that occur at a much higher level of abstraction there are two possibilities: either they do have a truth value that is not yet determined because we don't know how they impact out other theories yet, or they don't because they will never be hooked up to the rest of knowledge in any significant way. For any given statement, we don't know which category it is in, and I am prepared to say that there may be some statements that do not have well-defined truth values, but we can't know which ones those are.
By the way, I have been implicitly assuming throughout this that there is a unique "most efficient" encoding of our knowledge in a scale-free network, which is what we are trying to generate with our theorizing. I don't think it would matter too much if it was not quite unique, but could be modified without changing the overall structure too much. However, it is possible that there are two or more very different ways of generating an efficient knowledge graph that incorporates all of our empirical knowledge. If so, then truth claims would be relative to that. However, since under-determination rarely occurs as a practical problem, I doubt that this is the case in our world.
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Member Tim Maudlin replied on Feb. 28, 2015 @ 21:55 GMT
I won't get into deeper waters here, but just a note: I think you are being too scared of the idea of "mathematical intuition". An intuitive judgment is just one that you accept without further argument, so all of our reasoning ultimately relies on accepting intuitive judgments. This includes logical inference: Given "A" and "If A then B", you are happy to accept "B" by logical intuition. If you think you have to back up accepting the conclusion by further argument, you get into the situation in "Achilles and the Tortoise". So accepting non-empirical mathematical intuition is no more (or less) scary than accepting non-empircal logical intuition, which you are committed to. One might describe this as saying you believe in non-empirical logical facts that your mind can intuit. I see no problem with that at all. But if that's right, why a problem about non-empirical mathematical facts your mind can intuit as well? Of course, to do this well you have to be thinking in terms of sharp mathematical concepts, just as you have to understand the logical concepts. It is just that drawing a line between math and logic, with one obviously OK and the other somehow problematic, seems unmotivated.
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Vesuvius Now replied on Apr. 3, 2015 @ 22:50 GMT
"I would be happy to be called a mathematical realist, in that I think there is a fact of the matter about what should be called a mathematical truth, but I want to cash that out in terms of the physical world rather than some abstract mathematical world."
So the question is what is the physical mechanism by which "2 + 3 = 5" is true?
I suppose one could give a tautological definition that it is all the physical instances for which that equation is a description.
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David Lyle Peterson wrote on Feb. 27, 2015 @ 22:12 GMT
Dear Dr. Leifer,
I very much appreciated your essay on how abstract mathematics still possesses an empirical tether to the natural world (and I like that word, ``tether’’). Your figure showing mesh connections going to ``star’’ really helped in understanding your point. We don’t want our networks to have too many links. The abstractions of mathematics are easier for us than having way too many links. [``we introduce more abstraction precisely to reduce complexity’’ (your note of 2/27)]. And then we continue to do that for abstractions of abstractions.
Just a few little comments: The dividing line between natural versus ``spiritual’’ foundations for the Forms is slightly obscured by the possibility that the basic fundamental physical Forms are the quantum fields of the Vacuum – which are of course ``natural’’ but in a highly unusual non-classical and somewhat intangible way. This also makes Tegmark’s view a little more plausible in that these basic fields are highly mathematical and ``ethereal.’’ If the ancient platonic view was brought more up-to-date and modified for current relevance, then the math and physics Forms become more ``natural,’’ and the ``where’’ becomes various but numerous intelligences in the universe viewing nature produced from the ``Vacuum’’ (everything in quotes because our old words don’t quite fit).
Thanks again, Dave.
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Author Matthew Saul Leifer replied on Feb. 28, 2015 @ 03:31 GMT
Thanks for your comments. Regarding quantum fields, of course there is widespread disagreement about quantum theory and what can be said to exist in the quantum world. I like to say that the biggest problem we have with quantum theory is the problem of quantum jumps, i.e. quantum physicists are always jumping to conclusions.
Branko L Zivlak wrote on Feb. 27, 2015 @ 22:38 GMT
Dear M. S. Leifer,
It is very interesting your figure 2. It is very applicable to my article.
I would like to you to fill your hopes
Best Regards,
Branko Zivlak
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Pentcho Valev wrote on Feb. 28, 2015 @ 09:34 GMT
"Since mathematical theories are derived from the natural world..."
No. Just like physical theories, they are derived from initial assumptions (axioms, postulates) that could be arbitrary and false. In Big Brother's world, a new arithmetic theory has been derived from Big Brother's postulate "2+2=5":
"In the end the Party would announce that two and two made five, and you would have to believe it. It was inevitable that they should make that claim sooner or later: the logic of their position demanded it. Not merely the validity of experience, but the very existence of external reality, was tacitly denied by their philosophy. The heresy of heresies was common sense. And what was terrifying was not that they would kill you for thinking otherwise, but that they might be right. For, after all, how do we know that two and two make four? Or that the force of gravity works? Or that the past is unchangeable? If both the past and the external world exist only in the mind, and if the mind itself is controllable what then?"
There are paradoxes in the new arithmetic theory. Here is one of them (it can be juxtaposed with the twin paradox in Einstein's theory of relativity):
3(2 + 2) = 3x5 = 15
3(2 + 2) = 3x2 + 3x2 = 6 + 6 = 12
Pentcho Valev
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Author Matthew Saul Leifer replied on Feb. 28, 2015 @ 23:52 GMT
If you don't think physical theories are derived from the natural world, then I don't think we have much to talk about. Of course, other factors go into the construction of our physical theories, but if they are not designed to account for the empirical facts then what is the point of them?
Pentcho Valev replied on Mar. 1, 2015 @ 09:27 GMT
I think physical theories are derived from ASSUMPTIONS about the natural world that could be false. For instance, recently the assumption that light always travels at the same speed in a vacuum has been refuted:
"A team of Scottish scientists has made light rays travel slower than the speed of light.""Spatially structured photons that travel in free space slower than the speed of light"Pentcho Valev
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John C Hodge wrote on Feb. 28, 2015 @ 14:06 GMT
Your essay view of math follows mine nearly exactly.
Mine is a little stronger on the in the natural world because it is apart of the natural world like gravity. I also add the idea of fractals (self similarity) rather than hubs. Thus math developed out of our human scale and applies to other scales because the universe is fractal.
There are some problems with the abstractions that deviate with the observation of math such as irrational numbers, division, and infinity. These things are not observed in our scale and are part of the human introduced postulates that are false. Indeed, the current study of math allows the introduction of postulates and the reasoning from those postulates. It is called ``pure’’ but it is really only unjustified abstraction. This is not necessarily physics. For example, the introduction of non-Euclidean geometry is unreal - its use in cosmology is problematical because the universe has been measured to be flat (Euclidean).
My view allows the idea of using a mathematical structure that is observed such as by statistics or group theory to be considered real. For example, the periodic table was developed first by noting common characteristics of elements. A few holes were filled (predicted) by where the hole was in the classification scheme. Later, the causal underlying structure of atoms explained the periodic table. Indeed, the position of an element indicated something about the atomic structure. The same type of classification is true for the particle group models. Holes in the group model have been used to predict particles that were found. Can this be used to imply an underlying structure of particles? My model says yes.
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Thomas Howard Ray wrote on Feb. 28, 2015 @ 15:06 GMT
Matt,
Though I couldn't disagree with you more, I really enjoyed your essay.
We will have much to debate -- my view agrees with Max's, and my own upcoming essay deals with the issues of Godel and Goldbach that Tim raised.
Two things for the time being:
1. "There is no 'adding zeroes and ones to the end of binary strings' research group in any mathematics department. " Sure there is. Chaitin's number is maximally unknowable, and its algorithm cannot predict the next binary digit of the value. What's more, the value is dependent on the language in which the algorithm is written.
2. Hierarchical knowledge? What if knowledge is laterally distributed on multiple scales in the hub-connected complex network? No hierarchy -- which was Bar-Yam's solution to the problem of bounded rationality (Herbert Simon).
Tom
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Author Matthew Saul Leifer replied on Feb. 28, 2015 @ 23:48 GMT
1. I was referring to the specific (very boring) formal system that I had introduced earlier. There is no mathematical research group studying that. I initially thought to include a footnote that of course there are people studying the general structure of formal systems, which would cover the kind of thing you are talking about here.
2. I am not sure I understand all of the technical terms you are using here, but indeed I do expect that knowledge is distributed on multiple scales. I am not familiar with the literature on bounded rationality, but it sounds like something I should look into.
Thomas Howard Ray replied on Mar. 8, 2015 @ 16:03 GMT
Hi Matt,
I delayed replying until my essay entry was posted.
You should be able to find all you want to know about multi-scale variety, on the home page of the New England Complex Systems Institute.
Bounded rationality was formally developed by Herbert Simon, as a heuristic approach to decision problems. Because Bayes' theorem is also motivated by decision problems -- I...
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Hi Matt,
I delayed replying until my
essay entry was posted.
You should be able to find all you want to know about multi-scale variety, on the home page of the
New England Complex Systems Institute.Bounded rationality was formally developed by
Herbert Simon, as a heuristic approach to decision problems. Because Bayes' theorem is also motivated by decision problems -- I think it is quite natural that you would apply Bayesian solutions to problems of physics, and include your philosophy of mathematics in the solution. However:
In my own journey into knowledge of complex systems, it hit me right between the eyes, years ago, when I read Bar-Yam's seemingly innocuous statement -- "Ashby's law of requisite variety is a theorem in complex systems science." As a mathematician, it did not sit well with me, because 'theorem' has a specialized meaning. And because I knew that Bar-Yam was trained as a physicist I thought it brash and unprovable, even to the extent that I believed I could find a contradiction in the 'theorem.' To my surprise, I found that it really is a theorem, that opens up a whole field of *physical* solutions to problems of information and knowledge. Simon's convention, like yours, is the hierarchical framework. Combining Ashby's law with his own theory of multi-scale variety, though, Bar-Yam provided a compelling argument for laterally distributed information on every scale, This definition of local boundary, and lack of global boundary, convinced me that multi-scale variety is physics independent of any philosophy of physics. That sharp demarcation of philosophy from physics, is, in my opinion, what makes possible a rational correspondence of mathematical model to physical result.
Anyway, my abstract begins with the words, "Mathematics is not physics." I hope we have a meaningful dialogue over the extreme contrast in our views.
Best,
Tom
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Lawrence B Crowell wrote on Feb. 28, 2015 @ 20:24 GMT
Dear Matthew Leifer,
At certain times I take a stance similar to this. You might by way of comparison look at
Peterson’s paper for a different perspective. In
my paper I am primarily concerned with what I call mathematics that has “meat” or “body,” by which I mean things that are computed in some rather explicit way and that have reference to physical properties. I look informally at decidability issues, by treating this in a somewhat physical way, and make arguments with respect to the complexity of numbers.
There is what might be called the “soul” of mathematics, which is all of that Platonist stuff. I am not committed to saying this exists or does not exist. This is in some way connected to mind or consciousness, but connected in way that I don’t understand and I don’t think anyone else does either. Whether one want to argue for the existence of this “soul” is a matter of choice or almost what might be called faith. I don’t think there ever will be some decidable criterion whereby we can say Platonia exists or not. I will put on the hat of Platonism at times and at other times not wear it. In my essay I largely keep it off.
“On a dark night in a city that knows how to keep its secrets; on the tenth floor of the atlas building one man searches for answers to life’s persistent questions, Guy Noir private eye.” Garrison Keillor “Prairie Home Companion. That about states where the deep question about the relationship between mathematics and physics lies.
LC
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Sylvain Poirier wrote on Mar. 5, 2015 @ 14:20 GMT
Hello. This is the second essay I read after that of Lee Smolin, that tries to give a naturalistic philosophy of mathematics, as opposed to a Platonistic one. As I commented there, I have yet to see a coherent formulation of naturalism. The Stanford encyclopedia article you refer to on this point, admits it directly : "The term ‘naturalism’ has no very precise meaning in contemporary...
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Hello. This is the second essay I read after that of Lee Smolin, that tries to give a naturalistic philosophy of mathematics, as opposed to a Platonistic one. As I commented there, I have yet to see a coherent formulation of naturalism. The Stanford encyclopedia article you refer to on this point, admits it directly : "The term ‘naturalism’ has no very precise meaning in contemporary philosophy." It is remarkable to see that still after quite a long time that a majority of physicists and philosophers who care about metaphysics are trying to defend and develop naturalism, no clear formulation of its actual meaning could even reach a status of notability without being also loaded with big troubles (such as those of
Bohmian mechanics); while I would explain the lack of well-known coherent formulation of idealism by the lack of serious tries by competent physicists and philosophers, a gap I care to fill by
my essay.
The main difference I see between yours and Lee's exposition, is that Lee entered in specific details, especially about where he draws the line of existence between things. His claims are totally incoherent, both logically and with respect to existing knowledge in math and physics as I explained in my many comments there, however he has at least this merit, of taking that risk of making specific claims and thus expose himself to refutation. It seems you found a pretty good way to minimize the amount of criticism that you will get, by minimizing the quantity of claims contained in your essay. Indeed, most of the ideas there are already given in the abstract. You managed to approach the standard size of 9 pages not by adding up many ideas but rather by diluting the few you had. Anyway, by lack of effective matter to discuss from your essay, and also because I don't like to repeat myself, I invite you to read my comments to Lee's article (I'll still have more to write there), for the general remarks I made about this common topic between yours and his essay, of naturalism vs. Platonism, and for the specific analysis of his tries to effectively specify the claims of naturalism (I'm curious how would your options differ from his).
Now for your few specific claims :
You wrote: "Naturalism is the position that everything arises from natural properties and causes, i.e. supernatural or spiritual explanations are excluded. In particular, natural science is our best guide to what exists, so natural science should guide our theorizing about the nature of mathematical objects."
It is true that sciences of "natural" things (physics, biology) were extremely successful and these aspects of the world are a privileged field for scientific inquiry. However, like so many naturalist authors, you fail to distinguish between the scientific method and materialistic metaphysical prejudices, by using the ambiguous expression of "natural science" that may actually mean either the scientific method or the body of discovered knowledge in physics and biology, but where the adjective "natural" it contains can be irrationally played on in philosophical discussions to make it look as if the scientific method required to adopt such metaphysical prejudices. It doesn't.
"A naturalistic theory has no place for a dualistic mind that is independent of the structure of our brains. Therefore, if we have intuitive access to an abstract realm, our physical brains must interact with it in some way. Our best scientific theories contain no such interaction."
Our best scientific theories do not contain any explanation of anything psychological : what it may mean for a mind to understand something in general. Thus, there is no wonder why they do not explain either how we (I mean, some people) can understand mathematics in particular.
"The only external reality that our brains interact with is physical reality, via our sensory experience. Therefore, unless the platonist can give us an account of where the abstract mathematical realm actually is in physical reality, and how our brains interact with it, platonism falls afoul of naturalism."
The content of this argument looks like an absolute argument against platonism, to which I replied in comment to Lee's article. Strangely, for no reason I could see, the last phrase "platonism falls afoul of naturalism" looks as if it was only an argument relative to naturalism (thus void outside it). But then in this case, it implicitly seems to assume that a remark that "Naturalism contradicts X" should be taken as an argument against X, thus taking for granted that naturalism should be true. However I still beg for a rational argument for naturalism, in the name of which it would make sense to consider a view as being made less plausible just by its conflict with naturalism.
"The various attempts to reduce all of mathematics to logic or arithmetic reflect a desire [to] view mathematical knowledge as hanging hierarchically from a common foundation. However, the fact that mathematics now has multiple competing foundations, in terms of logic, set theory or category theory, indicates that something is wrong with this view." ; and in your comments: "What I mean is rather that which formal systems we decide to call mathematics, out of all the myriad of arbitrary axiom systems we might choose to lay down, is a matter of physics."
Well no. There is, for example, a clear absolute sense of which rules can form a valid and complete system of proofs for first-order logic, and satisfy the
completeness theorem. It is clearly independent of physics. Now for the choice of an axiom system to best serve as the foundation of mathematics, I do not see such a competition between possibilities as you assume. Instead, as I explained in
my work on the foundations of maths which I developed showing its intrinsic necessities independently of physics, I rather see a coherent whole of complementary parts of the foundations, with a dynamic articulation between them. Where there are alternative possibilities in competition for the same purpose (such as axiomatizations of set theory) I see a situation where in some aspects there is a preferred kind of formalization, and in other aspects such as the continuum hypothesis there is a remaining real diversity of acceptable systems corresponding to a real diversity of acceptable realities, that can often be understood as due to the real fact of ambiguity of the powerset of infinite sets.
"Firstly, in network language, the concept of a “theory of everything” corresponds to a network with one enormous hub, from which all other human knowledge hangs via links that mean “can be derived from”. This represents a hierarchical view of knowledge, which seems unlikely to be true if the structure of human knowledge is generated by a social process. It is not impossible for a scale-free network to have a hierarchical structure like a branching tree, but it seems unlikely that the process of knowledge growth would lead uniquely to such a structure. It seems more likely that we will always have several competing large hubs..."
If things went as you describe here, we would not have got the amazing success of mathematics in physics we had, with an amazingly universal agreement on what is the right fundamental theory of gravitation on the one hand, of microscopic physics on the other hand, how amazingly well, each on their side, they indirectly explain so many things on almost all physical processes that could be tested. So I do not see your philosophy coherent with the state of science that could be observed.
"...and that some aspects of human experience, such as consciousness and why we experience a unique present moment of time, will be forever outside the scope of physics." What a nicely non-naturalistic claim ! ;-)
"I have argued that viewing mathematics as a natural science is the only reasonable way of understanding why mathematics plays such a central role in physics." Only under the assumption of naturalism. But, if, as I hold, naturalism is irrational, and the only reasonable way of understanding science is an idealistic one (more precisely a mind/mathematics dualism), then your conclusion fails.
Now for the general ideas : your account of mathematics and its success is much too vague, and fails to relate to the effective contents of how successful is mathematics for physics. You present an abstraction of explanation for an abstraction of a problem. But the difference between mathematical abstractions with their success in science (as abstractions of real problems), and the abstraction of your approach to the problem of the success of mathematics in physics, is that the success of mathematical abstractions depends on their care and success to keep effective logical articulations (as you say, "tether") with the real problems they are abstractions of, but your abstraction of approach to the issue of usefulness of mathematics, is a non-mathematical abstraction that fails at keeping a tether to the reality of this usefulness. If the success of mathematics in physics could be explained as simply as what you describe, there would have been no amazement at this success in the first place.
Now, of course, it would make little sense to only express this objection in the abstract: to justify it, I need to enter the specifics about how your approach fails to account for the specific success of mathematics. So here are a few specific details of how you fail.
First, you explain mathematics as "the study of regularities, within regularities, within ..., within regularities of the natural world". In this case, we have a hierarchy of different levels of abstractions, where some concepts are more abstract than others, as they are not directly natural objects but abstract generalities about wide ranges of natural objects, or generalities about generalities about natural objects. When abstract generalities are developed as abstract generalities, they are not themselves the natural objects. That structure of knowledge you describe can only be expected to be useful as a path of reasoning through which we might discover new natural objects by first going up to the abstraction and then down again from the abstraction to concepts of new objects that would be particular objects inside classes described by these abstractions: the new fundamental objects we discover should display this same dependence to mathematical abstract generalities as the objects of our experience do.
But this is not what we observe. Instead, what we observe is that what we discover as fundamental objects of physical reality are some very abstract mathematical objects themselves. And what I mean here as "abstract mathematical objects", does not look like any generality like a general description of regularities among a range of many particular objects in a naturalistic sense. You may make this confusion because some of the most famous mathematical concepts, such as category theory, precisely look like generalities describing a wide range of particular cases (it is a generality of generalities). But this is not the case for all highly abstract mathematical concepts, and this is not how it goes for the success of mathematical concepts used in physics. In short, "high abstraction" and "generality" are not synonyms.
And to explain how the concepts of "abstraction" and "generality" differ, I need to take a specific example. I would like you to consider the case of the Dirac equation. This is a particular case of equation of a particular object (electrons or other spin 1/2 massive particles), but nevertheless a very abstract one.
This equation describes the field of electronic presence as taking values in the space of bispinors of space-time. To say roughly, we can define this space as the sum of 2 spinor spaces (with conjugate types), where a spinor space is a 2-dimensional complex space E such that the space of hermitian forms on E is identified with the tangent space to this point of space-time. Namely, tangent vectors (x,y,z,t) to this point are identified with Hermitian forms on E with matrix
(t+x y−iz)
(y+iz t−x)
since the determinant of this matrix coincides with the relativistic invariant (t
2−x
2−y
2−z
2).
This is very abstract, but not any kind of "generality of things" like what category theory does by describing regularity classes of particular systems that may go down to objects that look "natural" in a naturalistic sense. Now what is amazing with the success of mathematics is that this spinor space E was found to be "what electrons (and other fermions) are actually made of". Yet its connection to space-time, as described above, is... quite abstract. Spinors are not "made of" space-time connections, since any spinor (element of E) would correspond to a light-like direction of space-time but any physical description by such a direction would fail to fix the phase of this spinor. Concretely, a big problem with a spinor is that its phase is reversed when you apply to it a rotation with angle 2pi.
So, unless you provide a naturalistic explanation of how an object can be reversed when applying a rotation with angle 2pi, and how such an amazing thing as the Dirac equation can be relevant to physics, I must consider that naturalism fails to account for the mathematical aspects of the physical reality as we observed.
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Author Matthew Saul Leifer replied on Mar. 8, 2015 @ 18:04 GMT
There is a lot of food for thought in your comments, and I don't have time to answer all of them in one go. This is just the first of several replies and I will address the rest of the issues you have raised in due course.
For now though, I just want to comment on "naturalism". I think it is unfair to criticize the term on the grounds that it does not have a clear and unique meaning. If...
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There is a lot of food for thought in your comments, and I don't have time to answer all of them in one go. This is just the first of several replies and I will address the rest of the issues you have raised in due course.
For now though, I just want to comment on "naturalism". I think it is unfair to criticize the term on the grounds that it does not have a clear and unique meaning. If you look at any similar philosophical term, such as even your favoured "idealism", you will find that they typically refer to a broad church of views that have a main theme in common. The term itself serves as code for this set of views and it may be used when the distinctions between the sub-varieties are not too important for the issues under discussion. Add to this the fact that FQXi essays are supposed to be pitched at a general audience, and I don't think I have been too vague in using this term.
As far as my stance is concerned, I mean two things by the term "naturalism". The first is that the results of scientific enquiry are not to be ignored when they are relevant to a philosophical enquiry. This is a fairly innocent claim of methodological naturalism, that I think is fairly mainstream in western philosophy. So, for example, an enquiry about consciousness should take into account the results of modern neuroscience. For mathematics the main implication of this brand of naturalism is that, since modern theories of physics use advanced mathematics in an indispensable way, we need to find a theory of mathematics that explains why this is so rather than leaving it as an unexplained miracle.
However, I do also adopt a stronger version of naturalism, which you may want to brand "materialism", but would more properly be called "physicalism". Here, the idea is that our best guide to what fundamentally exists in the world is physics, so I want a metaphysics that does not posit entities beyond those of physics, or at least one that does so only minimally. Note that, just as with empirical science, the implications of this view are revisable. If a scientist one day discovers reliable evidence for ESP, or is able to reliably detect "mind particles" or some such entity, then the idea that we have to explain everything in terms of what we nowadays call physical entities would have been shown to be incorrect. The point, however, is that this is a matter for science to decide, and we should not go around positing such entities purely for the purposes of our metaphysics.
As for psychology and the like, physicalism is perfectly consistent with the concept of emergence, of which the emergence of thermodynamics from statistical mechanics is a prime example. The emergence of consciousness, human psychology, sociology, etc. is supposed to be explained in a similar way, but since they are much more complex than thermodynamics, we cannot boil this down to a few simple equations and relationships. Nevertheless, we have no good evidence that they require positing any new fundamental entities beyond those of physics.
You are right that I do not enter into a defense of naturalism or physicalism in my essay. The starting point is to assume these things as premises and to see what theories of mathematics they are compatible with. That is why "platonism falls afoul of naturalism" is appropriate in the context of this essay. I do think naturalism and physicalism are fairly mainstream positions in philosophy of science, so I don't think it is inappropriate to argue from them. However, I agree with you that, in a broader context, these notions require a defense. Getting into the general debate would take us into deep waters that are probably not too relevant for the philosophy of mathematics specifically. However, it is important for naturalists to come up with a viable theory of mathematics. If we cannot do this, as you seem to think we cannot, then that means that we would have to abandon naturalism, at least for mathematics, so this is an important issue we should debate further.
Finally, regarding the comparison of my essay to Lee Smolin's, you say I have been vague but I don't think so. Lee thinks that, for physics, we can get away with number, geometry, and maybe a few other things. For him, these things are straightforwardly "real" and the rest of mathematics just a formal game. Lee's views are actually quite close to those of Quine on this issue. Both of them think that it is possible to ring-fence some areas of mathematics as the "physical" ones, and not be too bothered with the rest. I do not agree with this position as I think that more advanced mathematics is truly indispensable for modern physics and it leaves the applicability of such mathematics to physics a total mystery.
In contrast, for me, there is no distinction to be made between "physical mathematics" and the rest of mathematics. ALL of mathematics is derived from the natural world, and ALL of it is real in exactly the same sense. I hope this is a clear enough statement of my main thesis and how it differs from Lee's. There are, of course, subtleties. I am at least partially a pragmatist about scientific truth, so when I call something "real", what I really mean is that it is useful, indispensable, etc. to the entities who practice science. Therefore, to defend the idea that mathematics is real I have to explain how it is constructed and why it is useful, rather than trying to locate it explicitly in the physical world as a nominalist would. This applies to all of mathematics, including the basic concepts like numbers as well as more advanced branches of mathematics.
I'll get into some of your more specific criticisms later, but I hope I have at least clarified my main position.
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Thomas Howard Ray replied on Mar. 9, 2015 @ 13:27 GMT
Matt, you write, " ... I want a metaphysics that does not posit entities beyond those of physics, or at least one that does so only minimally."
I think this could not be a clearer statement of an anti-rationalist viewpoint. Those entities that are metaphysically real -- such as the moon when no one is looking at it -- is not just minimally real. It is real or it is not.
" ... when I call something 'real', what I really mean is that it is useful, indispensable, etc. to the entities who practice science."
What is useful or indispensable about the moon when no one is looking at it, to a scientist or to anyone else?
Tom
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Author Matthew Saul Leifer replied on Mar. 9, 2015 @ 15:31 GMT
"I think this could not be a clearer statement of an anti-rationalist viewpoint. Those entities that are metaphysically real -- such as the moon when no one is looking at it -- is not just minimally real. It is real or it is not."
"Minimally real" is not a terminology I have introduced or at all relevant to what I am saying. The atoms and molecules that make up the moon are real. There is a particular arrangement of those atoms and molecules that we call "the moon". That is also real.
"What is useful or indispensable about the moon when no one is looking at it, to a scientist or to anyone else?"
The pragmatist criterion of "usefulness" is very often misunderstood. It is intended in a very broad sense. Theories of the solar system that say that the moon is real are more coherent and tell a more consistent story than those that do not. A scientist who goes around thinking that the moon is real will have a far easier time reasoning about what goes on in the solar system than one who does not. It passes the pragmatic test of "usefulness".
In fact, I would argue that realist theories are pragmatically preferred in general, as they provide a better explanatory framework than anti-realist theories.
Thomas Howard Ray replied on Mar. 9, 2015 @ 17:44 GMT
"There is a particular arrangement of those atoms and molecules that we call 'the moon'. That is also real."
Physically or metaphysically? If physically real, how does one demonstrate it without disturbing the arrangement?
"The pragmatist criterion of 'usefulness' is very often misunderstood. It is intended in a very broad sense."
Then it could mean anything, understood only in the private context of the understander.
"Theories of the solar system that say that the moon is real are more coherent and tell a more consistent story than those that do not."
Really? What theory of the solar system says the moon is physically real? I hope you're not thinking of general relativity, where spacetime is physically real and guides the motion of the planets. The planets themselves are metaphysically real objects of the field dynamics, not independently physically real. Or perhaps you are thinking of quantum field theory -- are the particles real, or the field? No field theory, in fact, is dependent on real objects.
"A scientist who goes around thinking that the moon is real will have a far easier time reasoning about what goes on in the solar system than one who does not. It passes the pragmatic test of 'usefulness'."
It would, if science were a pragmatic enterprise, rather than a rationalist enterprise.
"In fact, I would argue that realist theories are pragmatically preferred in general, as they provide a better explanatory framework than anti-realist theories."
Zeilinger is anti-realist. Do you think he is handicapped by his philosophy? Do you think that he thinks his explanatory framework is inferior to that of a realist?
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Sylvain Poirier replied on Mar. 20, 2015 @ 08:42 GMT
"
on "naturalism". I think it is unfair to criticize the term on the grounds that it does not have a clear and unique meaning. If you look at any similar philosophical term, such as even your favoured "idealism", you will find that they typically refer to a broad church of views that have a main theme in common."
I was quoting what you gave as reference for "naturalism". If you have...
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"
on "naturalism". I think it is unfair to criticize the term on the grounds that it does not have a clear and unique meaning. If you look at any similar philosophical term, such as even your favoured "idealism", you will find that they typically refer to a broad church of views that have a main theme in common."
I was quoting what you gave as reference for "naturalism". If you have your favorite specific version of it, please give your reference. As for my idealism, I cared to precisely define it in my essay, to make it specific without reference to tradition.
"
As far as my stance is concerned, I mean two things by the term "naturalism". The first is that the results of scientific enquiry are not to be ignored when they are relevant to a philosophical enquiry. (...) However, I do also adopt a stronger version of naturalism, which you may want to brand "materialism", but would more properly be called "physicalism"."
As I explained in my general
review of ideological divisions in this contest, I subscribe to your first meaning of "naturalism", which I call "scientism", while I reject physicalism as directly refuted by science, so that I classify it in the opposite side, that of "obscurantism".
"
Here, the idea is that our best guide to what fundamentally exists in the world is physics, so I want a metaphysics that does not posit entities beyond those of physics, or at least one that does so only minimally"
Physics was remarkably successful, however I see a logical gap in your idea: did you mean "our best guide to
all what fundamentally exists in the world is physics" ? Physics was remarkably successful for many things, however it does not mean that everything we can understand can be traced to it. I consider that much of psychology can't.
As for a "metaphysics that does not posit entities beyond those of physics", what about the entity "measurement" that appears necessary for the formulation of quantum physics ? The only coherent way I see to dismiss it that really does not
care about such spiritualist things as the subjective appearance for observers is the Many-worlds interpretation, however I
saw on your blog that you reject this interpretation, for reasons which, precisely, come down to such an attachment to subjective appearances, which amounts to give a fundamental ontology to conscious experience.
"
I do think naturalism and physicalism are fairly mainstream positions in philosophy of science, so I don't think it is inappropriate to argue from them."
It can be indeed mainstream, however this does not mean that there is any rational ground for this mainstream view, other than collective irrational prejudice. As was pointed out by David Chalmers in his article
Consciousness and its Place in Nature, p. 31:
"
Many physicists reject [the mind makes collapse interpretation] precisely because it is dualistic, giving a fundamental role to consciousness. This rejection is not surprising, but it carries no force when we have independent reason to hold that consciousness may be fundamental. There is some irony in the fact that philosophers reject interactionism on largely physical grounds (it is incompatible with physical theory), while physicists reject an interactionist interpretation of quantum mechanics on largely philosophical grounds (it is dualistic). Taken conjointly, these reasons carry little force, especially in light of the arguments against materialism elsewhere in this paper."
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Alexei Grinbaum wrote on Mar. 6, 2015 @ 13:54 GMT
Dear Matt,
Good to hear you're a naturalist, not a pragmatist. Or maybe both? :)
Anyway, I take your argument to be more about mathematical physics than mathematics per se. But even if we restrict ourselves to mathematical physics (which is a branch of mathematics, of course), what about the problem of multiplicity? To follow your argument, the objective world of mathematics is just the physical world, but we know that the physical world admits competing mathematical descriptions (e.g., different formalisms of the same quantum theory: operators, path integrals, geometric approaches, etc.). Why would this multiplicity arise if the mathematical world and the physical world were one?
Cheers,
Alexei
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Author Matthew Saul Leifer replied on Mar. 8, 2015 @ 16:29 GMT
I am both a naturalist and a pragmatist, and I think that is important for answering your questions.
Our physical and mathematical theories are both highly constrained by the natural world, but they are not completely determined by them. There is also the constraint that our knowledge derives from a social process and must be represented in a form that is useful to that society (this is the pragmatism part). The fact that there may be several different such representations is therefore not much of a problem for me. I am not saying that the mathematics literally is the physical world.
I also think you misunderstand me if you think I am talking just about mathematical physics. It could be read that way for sure, but I really intend it as a theory of all of mathematics. I recognize that this is a harder thesis to defend, but I wish to defend it.
Vesuvius Now wrote on Mar. 7, 2015 @ 23:58 GMT
Would you say that another way to say 'regularities within regularities' is 'math theory X surrounds math theory Y'?
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Author Matthew Saul Leifer replied on Mar. 8, 2015 @ 16:11 GMT
That's not a bad way to put it I suppose, but the topology of the situation may be a bit more complicated than one theory being at a lower level than another.
Ed Unverricht wrote on Mar. 14, 2015 @ 22:00 GMT
Dear Matt Leifer,
Is it 1. "
our universe is nothing but a mathematical structure and that all possible mathematical structures exist in the same sense as our universe."
or 2. "
mathematics is a natural science—just like physics, chemistry, or biology—albeit ... fundamentally a theory about our physical universe and, as such, it should come as no surprise that our fundamental theories of the universe are formulated in terms of mathematics.Very interesting argument. I guess one idea that would support your side is that it is hard to imagine people developing things as simple as numbers and counting, if there was nothing to count or organize into classes and groups in the real world, ie. you need something to count to invent the concept of counting...
Enjoyed your essay, thanks.
Regards,
Ed Unverricht
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Tapio Salminen wrote on Mar. 16, 2015 @ 15:53 GMT
Hi Matt,
Enjoyed reading your essay, thanks for posting it! I find the network idea appealing and hope you're pursuing it further.
I must say though that the essays I've read so far already show that the phrase "the only" in your Conclusions is quite untenable.
"I have argued that viewing mathematics as a natural science is the only reasonable way of understanding why mathematics plays such a central role in physics."
All the best,
Tapio
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Tommaso Bolognesi wrote on Mar. 19, 2015 @ 16:26 GMT
Dear Saul,
very simple and elegant idea, very convincingly expressed. For me, the text has appealed to visual intuition even more than the pictures. Another strong plus is that your essay is one of the few that hits the central question of the Contest right on the head.
One observation. You stress in various ways (e.g. with your first image) that your approach is opposite to...
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Dear Saul,
very simple and elegant idea, very convincingly expressed. For me, the text has appealed to visual intuition even more than the pictures. Another strong plus is that your essay is one of the few that hits the central question of the Contest right on the head.
One observation. You stress in various ways (e.g. with your first image) that your approach is opposite to Tegmark’s. In my opinion, they can still coexist. If reality is ultimately a complex mathematical structure (let’s not worry about the multiverse aspects), what’s wrong with imagining homo-sapiens building the knowledge network for describing that external reality just as you indicated? Your meta-theory works independent of the origin or status of that reality.
Another point. I notice your prudence in envisaging the eventual formation of a final, mega-hub at the top. Indeed, there are phenomena and driving forces in nature, as found in our evolving biosphere, that seem to escape a precise mathematical formulation, and to resist the math-based game of regularity finding/aggregation/abstraction. Hence, the scenario implied by your meta-theory is one in which multiple separate disciplines - physics being one - will keep existing and developing, forever separated from one another by their degree of math-friendliness.
Is some stronger unification possible/desirable?
Perhaps it is, by looking not only at the
top of knowledge but also at the
bottom of reality. Let me explain. Your essay is extremely effective in covering the evolution of human knowledge (the observer side), and suggests a unification process going upwards, looking at
the top of knowledge, while the observed object - the universe - remains passive and static. But if we viewed it as dynamic, and managed to find the seed at the
bottom of reality, and that seed turned out to be a simple algorithm (as some crazy people dare suggesting), from which everything would emerge - fields, matter, but also biospheres - then we would have a more unified scenario: that seed would certainly deserve a very special place in the network of human knowledge. Perhaps the top, although it seems to me that it would not be a hub exactly as you conceive them...
Best regards
Tommaso
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Vesuvius Now wrote on Mar. 19, 2015 @ 19:14 GMT
If math is a natural science then the correct logic to reason with should be quantum logic.
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Philip Gibbs wrote on Mar. 20, 2015 @ 13:41 GMT
I see a lot in common between your scale-free network and my view of universality. It is all about the things that are in common between different topics that are the most interesting. These form the subject areas that mathematicians like to study.
It is curious that emergent, self-organised structures have this scale-free, self-similar, fractal form. You mention how this is related to category theory and that is how I see it too.
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Sophia Magnusdottir wrote on Mar. 21, 2015 @ 10:15 GMT
Hi Matt,
it is an interesting idea, but I don't think it's very well defined. I don't know for example what you mean with "knowledge" or a "theory" to begin with. Besides this, as you probably know, none of the real-world networks that you list are truly scale-free. They are just approximately scale-free over some orders of magnitude. I am not even sure that knowledge is fundamentally a discrete thing. We arguably use a discretization in reality (chunks of papers and websites and so on).
In a nutshell what you seem to be saying is that one can try to understand knowledge discovery with a mathematical model as well. I agree that one can do this, though we can debate whether the one you propose is correct. But that doesn't explain why many of the observations that we have lend themselves to mathematical description. Why do we find ourselves in a universe that does have so many regularities? (And regularities within regularities?) That really is the puzzling aspect of the "efficiency of mathematics in the natural science". I don't see that you address it at all.
I don't think that consciousness and the nature of now will remain outside physics for much longer, but then that's just my opinion. There may be aspects of our observations that will remain outside of our possibility to describe them with math though, I could agree on that.
I quite like your essay because you're a good example for the pragmatic physicist of my essay. Maybe you like to pick a philosophy from the categories in my essay? :)
-- Sophia
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Author Matthew Saul Leifer replied on Mar. 28, 2015 @ 20:32 GMT
If you read my writings on the foundations of quantum theory, you will see that I am not Pragmatic Physicist. I am somewhat of a pragmatist in the philosophical sense, but I define "usefulness" more broadly than you do in your essay. A concept or idea that has explanatory power is useful even if it is not, strictly speaking, needed in order to predict the observations. In other words, a...
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If you read my writings on the foundations of quantum theory, you will see that I am not Pragmatic Physicist. I am somewhat of a pragmatist in the philosophical sense, but I define "usefulness" more broadly than you do in your essay. A concept or idea that has explanatory power is useful even if it is not, strictly speaking, needed in order to predict the observations. In other words, a physicist who believes in the reality of theoretical entities like electrons, quarks, etc. is better placed to make progress than one who does not. I tend to think that realist explanations are more useful than anti-realist ones so, as a good pragmatist, I care deeply about what our theories tell us about the nature of reality. Pragmatists need not be logical positivists or operationalists and I think James, Dewey, et. al. would be rolling in their graves at the suggestion. Anyway, I think I am a "mathematical constructivist" in your categorization, but I am not quite sure.
I will happily admit that my theory is vague. I think FQXi essays are a good venue for considered speculation, so that is what I was aiming for. I just wanted to make it vaguely plausible that knowledge could be represented by a network and that there might be processes that would make it scale-free. There is a lot more work to be done to pin down exactly what the nodes and links in the network are supposed to represent. However, I do think that discreteness is justified because we are discussing human knowledge, and humans tend to understand things in discrete chunks. I do not think any of our theories are direct representations of reality, but rather representations viewed through the lens of a human social process, so it does not matter whether or not reality is truly carved up into discrete domains, whatever that might mean.
I agree that my theory does not explain why the regularities are there to be found in the first place. It is supposed to explain the use of advanced mathematics in physics by showing that the processes which generate mathematics and physics are more closely related than they appear at first sight. It seems to me that most answers to the question of why mathematics is useful for physics pose their own questions at a higher level, and my proposal is no different.
Let us admit that our universe has regularities, but it could have much more regularity (as the binary string 111111111111... does), or much less (as a random binary string does). So, I think that before answering the question of why there are regularities, we should try to pin down exactly what degree of regularity our universe has. One way of doing this would be to try to run a simulation of the knowledge gathering process in toy universes that are governed by rules with differing degrees of regularity. By observing which networks have the same properties as our own knowledge network, e.g. same power law exponents, this might give a clue as to how regular our universe actually is. This still does not answer the question of why there are regularities in the first place, but it is at least a problem we can address scientifically. In the same sense that theories of consciousness cannot ignore the results of neuroscience, this might provide useful data that could constrain the possible answers.
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Joe Fisher wrote on Mar. 21, 2015 @ 19:57 GMT
Dear Dr. Leifer,
I posted a comment at your site that was unnecessarily contemptuous and devoid of the civility all contributors are entitled to. I deeply regret having done so, and I do hope that you can forgive my slurring of your fully deserved reputation.
I suspect that I may be suffering a relapse of Asperger’s Disorder. While this might explain my distasteful action, it cannot in any way justify it.
Respectfully,
Joe Fisher
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Member Ian Durham wrote on Mar. 21, 2015 @ 21:06 GMT
Hey Matt,
I thoroughly enjoyed this essay and find I agree with most of your points. I am not entirely convinced that human knowledge is devoid of hierarchy, though. Take your comment about sociologists believing that knowledge is a social construct, for example. It would seem to me that the successes of modern science and the fact that many discoveries are independently and often unknowingly verified by different people in entirely different social settings, directly counteracts that argument. Perhaps even stronger evidence might be some of the basic mathematical and physical concepts that can be independently grasped and indeed "discovered" by other species.
Anyway, my point is that if there truly is an objective reality out there (which I personally believe there must be), then it would seem that there ought to be at least some, albeit rough, hierarchy to our knowledge of it. We can abstract away from that objective reality in any number of ways, but all have the commonality that they are looking for either regularities within regularities within regularities, or regularities on TOP of regularities on top of regularities. Either way, objective reality is the starting point and it would seem to me that certain fields are closer to that objective reality than others.
I have one other minor quibble, though it is not necessarily with you. I know that it is traditional to view logic, set theory, category theory, etc. as competing theories (on an essentially equal footing) for the foundations of mathematics. But it seems to me that there is an undercurrent of what I might call "intuitionist" logic (not quite "informal" logic, which is an actual field) common to all of them. I mean, think for example about the very process of "creating" category theory or set theory: start with some basic premises and a few axioms, and reason from there. That in and of itself is reason enough to think that there is some deeper, singular foundational "truth" to mathematics that underlies everything.
Cheers,
Ian
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Author Matthew Saul Leifer replied on Mar. 27, 2015 @ 20:15 GMT
Let me reply to your second point first, as that is the easier of the two. I agree that there is a core of what we might call "informal logic" that is common to all foundations of mathematics. That core is what most mathematicians actually use in their daily work of proving theorems, and indeed it is what we all use when we try to make rational arguments. This informal logic is a massive hub in...
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Let me reply to your second point first, as that is the easier of the two. I agree that there is a core of what we might call "informal logic" that is common to all foundations of mathematics. That core is what most mathematicians actually use in their daily work of proving theorems, and indeed it is what we all use when we try to make rational arguments. This informal logic is a massive hub in our knowledge graph, compared to which the different formal foundations of mathematics are all parochial backwaters. This indicates to me that these formal theories are not the real foundations of mathematics, but rather specialized theories that attempt to make the informal foundations more precise. However, this just bolsters my argument though by suggesting that mathematics is not really about or reducible to such formal foundations. I am prepared to be much more free-wheeling about the nature of proof etc., which I think is decided more by the nature of physical reality and pragmatic considerations rather than some watertight rigorous foundation.
On your first point, I admit the existence of an objective reality, but I do not see this as a barrier to also believing that knowledge is partially a social construct. To avoid misunderstanding, I am not a social constructivist, but I do think that the structure of knowledge itself is reflective of the process that generates it. There are two aspects to this. Firstly, it is determined by the fact that knowledge is discovered by a social network of finite beings. This would, presumably, be the same for an alien society as for ours, so this, on its own, does not make knowledge culturally relative. It is possible that an alien society would inevitably be led to the same connections as we are. In this respect, knowledge is still objective, but we shouldn't view it as a direct reflection of reality, but rather as the best encoding of what a society of finite beings can learn about it. Secondly, I also think it is undeniable that at least some of the structure of the knowledge network is influenced by the specific history and beliefs of the agents who generate it. The relative importance of various concepts or the preferred mathematical formulation of a theory that has multiple equivalent formalisms may vary from society to society.
I think it is helpful to think of the role of reality and our observations of it in knowledge construction as analogous to constraints in a constrained dynamical system. Such constraints imply that otherwise prima facie valid solutions cannot actually be realized, but there is still a choice to be made between the solutions that do satisfy the constraints. Similarly, many network structures are ruled out because they do not satisfy the constraints that come from our experience of reality. This may even be enough to determine the broad outlines of what the network must look like, but nonetheless there are still several possible choices for what the details can look like, which are determined by the specific trajectory that our knowledge gathering has taken.
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Michel Planat wrote on Mar. 25, 2015 @ 16:42 GMT
Dear Matt,
Seeing the whole human knowledge as a scale-free network (like the WWW, Internet, cellular and ecological networks: your ref. [13]) seemed to me first counterintuitive but its scientific soundness got into me gradually. I realize how much the network of subfields I met in my career had an impact in my today research.
Putnam's solgan: 'meanings' just ain't in the head, that he develops in his twin earth thought experiment, also gave me something to think about and I now start to understand why you see mathematics as a natural science, not just as subfield of cognitive science.
Going back to the power law of a scale-free network, the words of Henri Poincaré in Science and hypothesis, came to my mind
"We are next led to ask if the idea of the mathematical continuum is not simply drawn from experiment. If that be so, the rough data of experiment, which are our sensations, could be measured. We might, indeed, be tempted to believe that this is so, for in recent times there has been an attempt to measure them, and a law has even been formulated, known as Fechner’s law, according to which sensation is proportional to the logarithm of the stimulus. But if we examine the experiments by which the endeavour has been made to establish this law, we shall be led to a diametrically opposite conclusion."
It may be that, as in Poincaré's quote, if you zoom into the netwoork, you get a different structure like the resonant bubbles in Hamitonian chaos (e.g. resonances between subfields of maths and physics, of chemistry and life sciences, philosophy and language and so on that would create voids in the network). But even if the network is not scale free it is quite interesting to see the human knowledge as a complex and entangled system.
Best,
Michel
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Author Matthew Saul Leifer replied on Mar. 28, 2015 @ 19:59 GMT
The hypothesis that knowledge forms a scale-free network should certainly be put to empirical test, as it could certainly have some other type of structure. However, I am led to the scale-free hypothesis not just from the structure of web links and citations, which can be viewed as very rough approximations to the knowledge network, but also because of several toy models for how knowledge might...
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The hypothesis that knowledge forms a scale-free network should certainly be put to empirical test, as it could certainly have some other type of structure. However, I am led to the scale-free hypothesis not just from the structure of web links and citations, which can be viewed as very rough approximations to the knowledge network, but also because of several toy models for how knowledge might grow. In my essay, I described theory building as a process of adding new hubs. I do not actually know what criteria this process has to satisfy in order to generate a scale-free network, so that would be an interesting problem to study. However, aside from theory building, there is also the more common activity of solving problems within existing fields. I hypothesize that this would add new nodes to the network with a preferential attachment mechanism because people are much more likely to use concepts that are common knowledge within their fields in order to solve a problem rather than something obscure. If this preferential attachment can be show to exist then that would support the scale-free hypothesis.
Of course, the processes going on within the knowledge network are in general more subtle than just adding new hubs and terminal nodes, and I expect the knowledge network to only be approximately scale-free. However, if it can be shown that this is a good first approximation to what is going on then that would support the scale-free hypothesis.
Finally, I note that several people have tried to understand the sense in which our theories are "simple" in terms of concepts like algorithmic complexity. I think that is a bit of a cartoon of how we represent knowledge, as the process of knowledge gathering is surely more organic than that, but nonetheless the idea that we are trying to achieve a representation that is somehow "efficient" is one that I support. There are several efficient error correction codes that are based on scale-free networks, so that indicates that they are useful for storing information robustly with minimal redundancy. I don't know if there are also efficient data compression algorithms based on scale-free networks, but if so then that may help explain how we can achieve theories with low algorithmic complexity via a social process of knowledge gathering.
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James Lee Hoover wrote on Mar. 26, 2015 @ 20:27 GMT
Matt,
I also believe that math "connects to the physical world via our direct empirical observations."
You do not expect the search for a theory of everything to ever end, but is not fruitless? In 1000 or 2000 years when we enter the realm of a type 2 civilization, do you think that perspective might change? Is "one new hub" necessary for a theory of everything?
My "connections" speaks of some of the same ideas but less eloquently.
Jim
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Anonymous replied on Mar. 28, 2015 @ 17:27 GMT
I think that the search for a theory of everything is a search for connections at ever deeper levels within the knowledge network. I don't expect it to end because the network itself is always evolving.
I don't think the scale of energy available to a civilization has anything to do with the fundamental theory of how knowledge grows. So long as we are talking about a society of finite beings, our knowledge will reflect that structure. The only thing I can imagine that might change things is if we evolve to a borg-like entity with a single consciousness, but even then we are still talking about a network of finite entities interacting with one another, so maybe this would just speed the process up without changing its overall structure.
I don't quite know what you mean by "one new hub", but a theory of everything would be a single hub to which everything is connected heirarchically, i.e. in technical terms the knowledge graph would be a tree. It is not impossible for a scale-free network to have this structure, but I just don't think this is the structure of our actual knowledge network.
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Author Matthew Saul Leifer replied on Mar. 28, 2015 @ 17:28 GMT
Something screwed up with my login credentials, but the previous post was me.
adel sadeq wrote on Apr. 2, 2015 @ 01:54 GMT
Hi Matt,
My system is a counterexample to your thesis. 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.
P.S. That was a nice one you pulled on Lubos, some people actually believed it.
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Joe Fisher wrote on Apr. 6, 2015 @ 14:59 GMT
Dear Saul,
I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.
All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.
Joe Fisher
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Bob Shour wrote on Apr. 7, 2015 @ 01:34 GMT
Dear Matthew Saul Leifer,
I read and enjoyed your essay. You presented your ideas well analytically and logically, I thought.
I did not think of my essay contribution as advancing the point of view of mathematics as naturalistic, but it seems to me that would be a fair characterization. In fact, your remark in your conclusion that "Mathematics is constructed out of the physical world" accords with the idea in my essay that the rate at which it is constructed can be measured and quantified (using C log (n) ).
On your point at page 2 that human knowledge has the structure of scale-free network, that may be so if we can say that energy distribution in the universe has the structure of a scale-free network. To utilize energy that is distributed like a scale free network recipient systems would have to model themselves the same way. At least one might argue that. In light of your remarks about networks, you might find some of the references in my essay of interest.
Thank you for having taken the time to write your interesting essay.
Best wishes,
Bob Shour
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Member Ken Wharton wrote on Apr. 7, 2015 @ 16:49 GMT
Hi Matt,
I really enjoyed your network perspective, and it's even changed my worldview about math somewhat. I think you're absolutely right that some new mathematical "nodes" are developed for exactly the reasons you describe here. (Strong analogies between existing nodes make it more natural to build a new node, leading to a more-efficient common structure, etc.)
But is this the...
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Hi Matt,
I really enjoyed your network perspective, and it's even changed my worldview about math somewhat. I think you're absolutely right that some new mathematical "nodes" are developed for exactly the reasons you describe here. (Strong analogies between existing nodes make it more natural to build a new node, leading to a more-efficient common structure, etc.)
But is this the *only* way that new mathematical "nodes" are developed? It seems to me that an equally-important process is simple exploration/extrapolation from existing nodes, tweaking the premises that lead to one bit of math, and seeing what new bits might result. Sure, this isn't going to necessarily lead to the efficient network structure that you describe, but it would supply a bunch of "raw material" which your efficiency-driven process could then organize.
One big distinction between the nodes generated by an "exploration process" (rather than your "efficiency process") is that there's generally no requirement that they hold together self-consistently. This is what I had in mind when I talked about different "caves" in my own essay, even though you're right that eventually your efficiency-driven process tends to merge the caves together into much larger networks... maybe even to the point where you can claim that there is one "pragmatically-true" type of math when it comes to things like the axiom of choice.
If you're with me so far, then I think your comment on my essay reflects that we're trying to answer two somewhat different interesting questions. You are addressing the mystery of why the most-efficient network groups of math tend to map to physics; I'm trying to address the mystery of why so many separate explorative-nodes tend to find a use in physics.
I think there are some aspects that make your task harder than mine, and some that make them easier. On the easier front (which I think you hinted at), mathematicians know about physics when they are looking for efficiency-nodes. When faced with mathematical choices about the details of differential geometry, say, I'd be a bit surprised if mathematicians refused to think about GR. But while GR could potentially inform part of a subsequent efficiency-process, it certainly didn't inform the original development of non-Euclidean geometry; that's basically the mystery that I was trying to address.
On the other hand (as you noted on my page), given a collection of unrelated but internally self-consistent axiomatic systems, it shouldn't be terribly surprising that *some* of them find a use in physics. (Indeed, that's a key part of my argument.) But if mathematicians truly think that some of these network-systems are more "correct" than others, then it is quite interesting that the deemed "correct" systems are more useful in physics than the equally-logically-valid but "incorrect" ones (as judged by the overall mathematics community). I guess the big question here is just how much influence higher physics has had in framing the judgment of mathematicians (as per my GR example), or if it really is all built up from counting apples, and very basic stuff like that. Do you see higher-level physics as impacting higher-level math, or is it all one-way-influence at that level?
Thanks again for a thought-provoking essay!
Ken
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Author Matthew Saul Leifer replied on Apr. 7, 2015 @ 20:28 GMT
I do not think that much "pure" exploring actually goes on in mathematics. If that's what mathematicians were really doing then why would they not just explore arbitrary axiom systems? Mathematicians typically identify two strands of mathematical work: theory-building and problem solving. In reality, these are not completely separate from each other. A theory may need to be built in the...
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I do not think that much "pure" exploring actually goes on in mathematics. If that's what mathematicians were really doing then why would they not just explore arbitrary axiom systems? Mathematicians typically identify two strands of mathematical work: theory-building and problem solving. In reality, these are not completely separate from each other. A theory may need to be built in the service of solving a problem and vice versa. As an example of the former, consider the P=NP problem. Someone notices that although they cannot prove a separation directly, they can if they introduce a modified model of computing with some funny class of oracles. After that, they start exploring this new model, partly because it is expected to eventually tell on the P=NP problem, but also perhaps because it has its own internal elegance and might have practical applications different from the problem for which it was originally invented to solve. I think the vast majority of exploration/extrapolation is of this type. It does not exist in a vacuum, but is closely tied to the existing structure of knowledge and the existing goals of mathematicians.
You are right that there are other processes going on in the knowledge network other than the replacement of analogies with new hubs. The latter would be a process of pure theory building, whereas in reality there is a mix of theory building and problem solving going on. I emphasized the theory-building process primarily because I think it is key to explaining why abstract mathematics shows up in physics. But for my explanation to work, I have to argue that the other processes that add new nodes, like problem solving, do not screw up the scale-free structure of the network or my argument.
I think it is plausible that problem solving adds new nodes to the network with a preferential attachment mechanism, so it is an example of the usual type of process that generates scale-free structure. To see this, one needs to again recognize that knowledge growth is a social process. If you are trying to prove a new theorem, you are much more likely to use techniques and ideas that come from network hubs rather than from more obscure parts of the network. This is because you are much more likely to know the contents of a hub, and also it is easier to communicate results to others if you can express them in a common language, and the hubs are the most commonly known parts of the network. (To see that this effect is real,just consider how much mathematicians complain when asked to verify a ~100 page proof by an unknown mathematician who claims to have solved a big open problem, but does so entirely in their own personal language and terminology.) Further, in collaborations between different researchers, the researchers are much more likely to have hub knowledge in common than anything else, so these methods will get used first. Thus, you are always likely to try ideas from hubs first, so new results are much more likely to get connected to hubs rather than more obscure parts of the network. This is nothing other than preferential attachment, which is the classic mechanism for generating a scale-free network.
So, I don't think that adding other types of mathematical exploration will change the structure of the network. In fact, it gives a better theoretical argument for scale-free structure than the process described in my essay. However, what these processes do change is that now not all the nodes at the edges of the network are directly connected to empirical reality. I still don't think this changes much. Theories are now built out of regularities in regularities etc. both in empirical reality and in the consequences of theories generalized from empirical reality. It is still no surprise that such theories should be useful for describing empirical reality.
Finally, I think that higher physics does have a big influence on at least some fields of mathematics. One just has to read Peter Woit's essay to see that. However, it is fairly unsurprising if some ideas in number theory that were originally motivated by quantum field theory later go on to have applications in quantum field theory. There is definitely a two way street. However, the more surprising thing is how often abstract mathematical ideas that at first seem to have nothing to do with physics later show up in physics. Gauss, Reimann, et. al. got the essential ideas of differential geometry right a long time before GR. Einstein simply had to allow for non positive-definite metrics to adapt the theory to spacetime. Since that time, people may well have used GR as inspiration for new ways of doing differential geometry. I am not an expert on that, so I don't know. Regardless, the main thing we need to explain is why differential geometry showed up in GR in the first place, and that is the type of thing I am trying to account for.
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Thomas Howard Ray replied on Apr. 8, 2015 @ 18:22 GMT
Matt,
You write "I do not think that much 'pure' exploring actually goes on in mathematics. If that's what mathematicians were really doing then why would they not just explore arbitrary axiom systems?"
Because axiomatics -- the logically deductive framework for mathematics -- is only a small part of the discipline. As Godel explained.
" ... the main thing we need to explain is why differential geometry showed up in GR in the first place, and that is the type of thing I am trying to account for."
I agree with you -- so did Einstein -- because general relativity is true only *up to diffeomorphism.* It could not be a final theory of gravity, then, and Einstein explicitly didn't intend it to be. What I am at a loss to understand, is why you invest so much interest in problem solving ("All life is problem solving," said Popper, a sentiment which I much appreciate), and yet don't attempt to differentiate the problem and solution:
"Mathematicians typically identify two strands of mathematical work: theory-building and problem solving. In reality, these are not completely separate from each other."
For course they are. Were they not, one could not differentiate theory from result in any rigorous way; no theorem could be a true, logically closed judgment. By such reasoning, how would one *ever* account for the role of differential geometry in Einstein's theory? -- the fact that space and time (Minkowski space) are not physically real except in a "union of the two," is the principle that motivates physical applications of differential geometry (the physics of continuous functions). Vesselin Petkov explains it as eloquently as anyone I've ever read.
Best,
Tom
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Author Matthew Saul Leifer replied on Apr. 8, 2015 @ 19:58 GMT
I am not saying that theories are not distinct from problems. Of course they are. All I am saying is that the two different activities that mathematicians consider themselves to be engaged in are not entirely separate. One may need to build a new theory in order to solve a problem and one has to solve problems in the course of theory-building.
Thomas Howard Ray replied on Apr. 8, 2015 @ 20:11 GMT
Matt, an example would be helpful here:
"One may need to build a new theory in order to solve a problem and one has to solve problems in the course of theory-building."
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Member Ken Wharton replied on Apr. 9, 2015 @ 21:51 GMT
Hi Matt,
Yes, your point is well taken; the "exploration process" I mentioned is almost always driven from problem-solving goals (at least in the cases I'm familiar with). And I agree that it therefore shouldn't change your scale-invariant structure. The biggest difference in the nodes that result from this process, as you note, is:
" However, what these processes do change is...
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Hi Matt,
Yes, your point is well taken; the "exploration process" I mentioned is almost always driven from problem-solving goals (at least in the cases I'm familiar with). And I agree that it therefore shouldn't change your scale-invariant structure. The biggest difference in the nodes that result from this process, as you note, is:
" However, what these processes do change is that now not all the nodes at the edges of the network are directly connected to empirical reality. I still don't think this changes much. Theories are now built out of regularities in regularities etc. both in empirical reality and in the consequences of theories generalized from empirical reality."
It's this "generalization" issue that worries me, and makes me wonder how tethered to reality one might expect such generalizations to be. Is there anything you'd suggest I might read on this front?
Related to this issue, the only real daylight between us might then be addressed by the following thought experiment. Suppose that some group of mathematicians decided to do "pure exploring", to no problem-solving purpose, and then came up with some new fields of mathematics that weren't directly connected to anything that had come before. (Although of course your efficiency process might later find deeper analogies with known mathematics.) Is it your contention that such theories would be much less likely to find use in future physics, as compared to theories generated by a problem-solving-style motivation?
I'd be surprised if this were true. So long as the "pure-exploration" theories were self-consistent and of comparable complexity, I'd think they would be just as likely to find use in physics as theories that had been inspired by some purposeful, problem-solving process. So for me, it doesn't particularly matter whether the "generalization" process loses the root-level link to empirical reality; that link can be re-established at a higher level, if physics finds a given theory useful for some real-world problem.
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Alexey/Lev Burov wrote on Apr. 10, 2015 @ 22:30 GMT
Dear Matthew,
You are suggesting an argumentation as to why it is natural to expect mathematics being suitable to physics. Essentially your point is that mathematics is a natural science, so a compatibility between the two natural sciences is also natural. I would not argue against this logic, but I do not see how it is an answer to Wigner's point of wonder.
Wigner's wonder about the relation of physics and mathematics is not just abut the fact that there are some mathematical forms describing laws of nature. He is fascinated by something more: that these forms are both elegant, while covering a wide range of parameters, and extremely precise. I do not see anything in your paper which relates to that amazing and highly important fact about the relation of physics and mathematics. This makes a difference between your paper and one by Tegmark.
Regards,
Alexey Burov.
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Author Matthew Saul Leifer replied on Apr. 13, 2015 @ 16:40 GMT
I was meaning to discuss elegance in my essay, but I removed that section for lack of space, and because I thought it best to focus on the main argument. It is hard to pin down exactly what one means by an aesthetic notion like "elegance" in the context of mathematics and physics. My own feeling is that it has to do with economy and compactness of representation. That is, if we find a compact...
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I was meaning to discuss elegance in my essay, but I removed that section for lack of space, and because I thought it best to focus on the main argument. It is hard to pin down exactly what one means by an aesthetic notion like "elegance" in the context of mathematics and physics. My own feeling is that it has to do with economy and compactness of representation. That is, if we find a compact representation of physical laws that we can write down on a t-shirt, such as Einstein's equations, Maxwell's equations, or the Dirac equation, then I think we would tend to call that "elegant". As you suggest, the fact that we have laws of nature that are economical, work over a wide range of parameters, and are extremely precise, is what needs explaining.
Many of my computer science/information theory colleagues are wont to describe the elegance of physical laws in terms of algorithmic complexity, i.e. an elegant law is essentially the shortest computer program capable of generating the empirical data. Now, this cannot be anything more than a cartoon of what is actually going on. Algorithmic complexity is uncomputable in general and physicists are not in the game of writing short computer programs, at least that is not what they think they are doing. However, if I can argue that the social process that generates human knowledge would tend towards generating such a compact representation then we have our answer.
Now, codes for data compression and error correction that asymptotically achieve Shannon capacity have been developed which have the structure of scale-free networks. Since data compression provides an upper bound on algorithmic complexity, which is exact in several cases like i.i.d. data, it is at least possible that the human knowledge network has the structure of an optimal compression of the empirical data, or that it is tending towards such a structure. (This is part of what I meant by describing the knowledge network as "efficient" in my essay.) For this I have to argue that the social process of knowledge growth would tend to generate such a structure. It is fairly easy to argue that a compact representation of at least the hubs of a network is something that people would spend a lot of effort trying to achieve, since compact representations encompass a wider range of phenomena in a small number of laws. They are also more reliably generalizable, because each law encodes a larger number of empirical regularities that have been observed to hold.
My argument then is that the laws of nature are not elegant because of any special property of physics, but rather that, in any universe, physicists would spend a lot of effort trying to compactify their description of whatever the physics is. For example, although Maxwell's equations look very compact in their modern Lorentz covariant form, in fact there are a lot of background mathematical definitions and concepts required to state them in this form. Without those, they look more complicated, as in their original integral form. This is a clear example of a case where the same physical laws went through a very conscious process of compactification. I would argue that something like this is playing a role every time we develop an "elegant" set of physical laws.
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Alma Ionescu wrote on Apr. 13, 2015 @ 15:35 GMT
Dear Matt,
Your well written and well argued essay looks like it's close to breaking that record mentioned atop of the page and with good reason. That math is derived from the observation of the physical universe is quite a compelling idea and your witty sense of humor can only benefit the exposition. It's a very good observation that you underscore at the end, about how your point of view is different from the Tegmarkverse, where the universe varieties can only be weakly interacting. I found the idea of regularities within regularities being quite striking as I considered it in my essay as well, only just naming it modularity instead. I also enjoyed the idea of knowledge as a scale free network. I am wondering what are the consequences of such a treatment and if a part of this network can undergo a change when the number of points in the region crosses a certain threshold - what I have in mind is creativity or the generation of new ideas that complete the pattern.
Thank you for a very good read and wish you best of luck in the contest! Should you have enough time and the curiosity to read my essay, I'd appreciate your comments.
Warm regards,
Alma
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Author Matthew Saul Leifer replied on Apr. 13, 2015 @ 16:07 GMT
I think it is difficult to see what the consequences of knowledge as a scale-free network are right now. This is because the idea is only a sketch at the moment. Work is needed to determine whether this really is a good representation of human knowledge and what the processes are by which it grows and changes. Nonetheless, if all this can be established rigorously then there is certainly scope for all sorts of statistical mechanics phenomena such as phase transitions to occur within the network. The idea that a "paradigm change" might be represented by a phase transition in the network is quite appealing.
Alma Ionescu replied on Apr. 19, 2015 @ 12:08 GMT
I was thinking either of a paradigm change (knowledge pertaining to society) or a model for how brains generate ideas. The former is maybe more difficult to asses as two individuals with the same number of connections may have a different general impact whereas the latter is perhaps something on the lines of the minimum number of information points needed to deduce a new piece of the puzzle, as related to the complexity dimension of the concept; as in how much information did you need so the idea of a knowledge network can pop into existence :) Anyway excellent idea!
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Alma Ionescu replied on Apr. 19, 2015 @ 12:10 GMT
Sorry, I just realized I forgot to rate your essay so I fixed that now. Good luck!
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Vesuvius Now wrote on Apr. 13, 2015 @ 23:49 GMT
I've probably misunderstood something, but in virtue of which physical processes is the mathematical equation 2 + 3 = 5 true?
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James Lee Hoover wrote on Apr. 14, 2015 @ 17:27 GMT
Matt,
As time grows short, so I am revisiting essays I’ve read (3/26) to assure I’ve rated them. I find that I did not rate yours, so I am rectifying that. I hope you get a chance to look at mine: http://fqxi.org/community/forum/topic/2345
Jim
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Janko Kokosar wrote on Apr. 18, 2015 @ 09:20 GMT
Dear Matthew Saul Leifer
Your essay has some positive aspects (+) and some where I disagree (-).
(+) Your thoughts are shown very concise, especially your figure 4.
(+) I never liked to be said that ''math is a thing of axioms, not a thing of intuition.'' For instance, the question, which number of hairs is a border between baldness and non-baldness is a thing of intuition....
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Dear Matthew Saul Leifer
Your essay has some positive aspects (+) and some where I disagree (-).
(+) Your thoughts are shown very concise, especially your figure 4.
(+) I never liked to be said that ''math is a thing of axioms, not a thing of intuition.'' For instance, the question, which number of hairs is a border between baldness and non-baldness is a thing of intuition.
(+) I like naturalism, because I read at Smolin that formulae exist in time, not in timeless environment.
(+) At least, you mentioned consciousness, because ignorance of it is not good.
(-) About consciousness, I have a similar standpoint as Poirer and still many people in this contest. Consciousness causes movements, thus it is part physics. If one philosophy of physics does not include consciousness, is not good.
(-) You said to Poirer that panpsychism disagrees with physicalism. This is not true. My model includes physicalism and reductionism. It does not need supernatural and spiritual in the first intoduction. Even, panpsychism is defended also by Koch and Tononi. But emergentism has not yet answered anything.
(?) In my essay I gave also speculation about Pythagora theorem, that it is a consequence of energy law, that Euclidean law is a consequence of physics. What is your opinion?
(?) I gave also an example where physics adjusts to math, this is dimensionless nature of Planck spacetime. What is your opinion?
(?) One important question is simplicity of fundamental physics. Smolin does not believe in simplicity, but what about you?
(?) What do you think about positivism, like this of Roger Schlafly? I support it, but 100% positivism is not correct, by my opinion.
My opinion is that merging of fundamental physics causes that the number of axioms is reduced. Thus math is foundation of physics, but not on the same way as Platonic math.
Maluga wrote, that the essence of math is to describe physics more simply, because our brains have not capacity to think on all parameters. My addition is that the goal of math is to show that physics is simple. Here I also obtained answer, why Economy is not simple. Answer: Economy is a part of physics, thus simplicity is home in physics.
My essayBest regards
Janko Kokosar
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Janko Kokosar replied on Apr. 22, 2015 @ 15:33 GMT
Dear Matt Leifer
I should to add, that modeling with consciousness is very similar to physicalism. But otherwise physicalism is in contradiction with panspchism. This also does not mean that I do not defend scienfic approach to this question.
My essayBest regards
Janko Kokosar
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Mohammed M. Khalil wrote on Apr. 18, 2015 @ 20:09 GMT
Dear Prof. Matthew,
Wonderful essay! I enjoyed reading it, and we seem to agree in many points, as my essay reflects, especially that mathematics is a study of regularities in nature. I would be glad to take your opinion in my
essay.
Best regards,
Mohammed
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Peter Jackson wrote on Apr. 20, 2015 @ 12:43 GMT
Matt,
You argued hard for your interesting and original hypothesis but I was left with a few obvious questions.
Your 'main proof' (that we've been getting more mathematical) is wholly circumstantial, so based on other assumptions which were unsupported, i.e. that physics is not getting ever more mathematical purely due to being confounded by logical analysis. And that may perhaps be the WRONG direction for improving understanding. That may be right or wrong but seems equally possible.
You don't seem to consider the case of genius and advancement by those who never even learned maths, or perhaps anything beyond basic arithmetic, and haven't used it in their achievements. It seems most of the greatest achievers in history a fall into that category! Physics is after all only a small slice of humanity and it may be argued that slice has made less not more progress in recent times!
You seem to accept all maths as 'correct' per se and don't highlight or even seem to refer to cases where mathematics we employ does NOT model natures mechanisms and can mislead us (which I address in my own essay). Do you not think we should take better care of HOW we employ mathematics?
I look forward to your responses, but a well written and presented essay with an original hypothesis.
Best of luck hitting you record number of FQXi wins.
Peter
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Member Sylvia Wenmackers wrote on Apr. 21, 2015 @ 20:45 GMT
Dear Matt S. Leifer,
Is this is correct summary of your main thesis (in section 4)? : "First, humans studied many aspects the world, gathering knowledge. At some point, it made sense to start studying the structure of that knowledge. (And further iterations.) This is called mathematics."
Although I find this idea appealing (and I share your general preference for a naturalistic...
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Dear Matt S. Leifer,
Is this is correct summary of your main thesis (in section 4)? : "First, humans studied many aspects the world, gathering knowledge. At some point, it made sense to start studying the structure of that knowledge. (And further iterations.) This is called mathematics."
Although I find this idea appealing (and I share your general preference for a naturalistic approach), it is not obvious to me that this captures all (or even the majority) of mathematical theories. In mathematicians, we can take anything as a source of inspiration (including the world, our the structure of our knowledge thereof), but we are not restricted to studying it in that form: for instance, we may deliberately negate one of the properties in the structure that was the original inspiration, simply because we have a hunch that doing so may lead to interesting mathematics. Or do you see this differently?
On the other hand, the picture you describe does ring true with my subjective experience that learning supposedly 'difficult' theories never really turns out to be that difficult when you actually try: somehow, you keep dealing with combining a limited number of 'things' according to specific rules. To arrive at the relevant 'things', however, may require a long chain of explanation and abstraction. Your idea of replacing connections with a new hub seems to be related to the steps in that chain. So, I do think that this ability of 'going meta' is crucial to mathematics - not just to philosophy. ;-)
But as I indicated before, it is not clear to me that this is sufficient to consider mathematics as an empirical science. Or maybe it is possible, if you consider a large part of our knowledge to be suppositional knowledge?
I also like the conjecture of section 3 that human knowledge may be a scale-free network, which goes against the hierarchical view but still explains how it may appear to us like that. In this picture, I guess the old ideal of a universal human would translate to focussing on multiple hubs: still a great way to 'keep in touch' with the actual network structure of knowledge.
In addition, I think your essay is accessible for a general audience. And you actually thought of adding pictures. :-)
Great job. My vote is 9/10.
Best wishes,
Sylvia Wenmackers - Essay
Children of the Cosmos
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Author Matthew Saul Leifer replied on Jun. 10, 2015 @ 14:01 GMT
Apologies for not replying sooner.
I certainly did not intend my essay to provide a comprehensive analysis of all forms of mathematical enquiry. I was just trying to make it plausible that there is a process of mathematical abstraction that keeps enough of a tether to empirical reality to explain the later use of those theories in physics. I agree that there are other processes going on...
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Apologies for not replying sooner.
I certainly did not intend my essay to provide a comprehensive analysis of all forms of mathematical enquiry. I was just trying to make it plausible that there is a process of mathematical abstraction that keeps enough of a tether to empirical reality to explain the later use of those theories in physics. I agree that there are other processes going on in the mathematical knowledge network at the same time. Once such process is problem solving. Another is the kind of free play with axiom changes that you mention. However, I don't think this play is entirely free. It is curious that mathematicians can usually agree on which axioms are the ones worth changing. For example, why drop the parallel postulate in geometry rather than some other? I think these choices have to do both with physical intutions, i.e. which changes are likely to lead to the most applicable theories, and with the internal structure of theories, e.g. which axioms changes are likely to leave me with a consistent theory that shares enough structure with its parent to be interesting. Both of these are to do with the structure of the knowledge network, so are ultimately constrained by empirical reality. I admit that this rough idea needs to be considerably fleshed out.
I note that in your essay you point out that the vast majority of mathematics is actually irrelevant to physics. However, instead of doing a total page count, I would be inclined to weight the pages by the number of links that the corresponding piece of knowledge has to other nodes in the knowledge network, as Google does in the Page Rank algorithm. We can admit that much of the published corpus of mathematics is "failed mathematics" in some sense, just as the majority of pages of theoretical physics produced today are probably irrelevant to reality. Using Page Rank is tantamount to defining "important mathematics" as "applicable mathematics", so I may be accused of circularity, but ultimately I think we agree that applicability is implicitly at least part of the definition of what we mean by a successful mathematical theory.
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Jeffrey Michael Schmitz wrote on Apr. 22, 2015 @ 14:24 GMT
Dear Matt,
Thank you for the essay. Mathematics is not a science because there are no physical experiments that will change a proof. Mathematics is not a lesser thing because it is not a physical science, some would say it is more because it is more "pure". There are many examples of outdated or just wrong physics were the mathematical systems are still valid (equations in classical physics easily have particle going faster than light and not following the rules of quantum mechanics). Any gravitational interaction between more than three particles does not have a true solution in current mathematics. The relationship between math and physics is a wonderful and useful intersection, not the creation of either.
All the best,
Jeff
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Member Marc Séguin wrote on Jun. 9, 2015 @ 19:58 GMT
Dear Matt,
I had read your essay while the contest was underway but never got to comment on it. Better late than never!
In my essay, I side with Tegmark’s view that can summarized as “Physics is Math”, so I was naturally intrigued by your claim that the opposite is the case. After reading your essay, I think it all comes down to the different way we define mathematics. In your...
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Dear Matt,
I had read your essay while the contest was underway but never got to comment on it. Better late than never!
In my essay, I side with Tegmark’s view that can summarized as “Physics is Math”, so I was naturally intrigued by your claim that the opposite is the case. After reading your essay, I think it all comes down to the different way we define mathematics. In your essay, you write:
“ [F]or me, abstract mathematical objects can be called real insofar as they are useful for our scientific reasoning”
and
“[M]athematical theories are just abstract formal systems, but not all formal systems are mathematics. Instead, mathematical theories are those formal systems that maintain a tether to empirical reality through a process of abstraction and generalization from more empirically grounded theories, aimed at achieving a pragmatically useful representation of regularities that exist in nature.”
If I read you correctly, you essentially define mathematics so that,
(1) if it not ultimately derived from the generalization of physics, it is not math ;
(2) if it not even remotely useful for reasoning about physics, it is not math.
If mathematics is defined in this way, I fully agree with you that mathematics is (a subset of) physics.
On the other hand, if we define Mathematics in a wider sense that encompasses all abstract formal systems, including those that are too big, too complex or too irregular to be grasped and studied by human-level minds, I think it is quite possible that “All-of-Physics” (in the sense of all possible physical realities, observable or not), is “generated” by “All-of-Math”. With this larger definition, it is Physics that is (a subset of) Mathematics!
That said, I really like the way you explained how abstract mathematical theories can arise:
“The main idea is that when a sufficiently large number of strong analogies are discovered between existing nodes in the knowledge network, it makes sense to develop a formal theory of their common structure, and replace the direct connections with a new hub, which encodes the same knowledge more efficiently. […] In this way, mathematics can become increasingly abstract, and develop its own independent structure, whilst maintaining a tether to the empirical world.”
This essay contest cast a very wide net by asking about the relationship between math and physics, and it is fascinating to see the diversity of answers that were proposed. Thank you for contributing a very interesting essay, and all the best!
Marc
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Author Matthew Saul Leifer replied on Jun. 10, 2015 @ 13:43 GMT
I don't disagree too strongly with your characterization of my position, but I would modify it somewhat:
(1) It doesn't have to be physics per se unless one takes the position that all of natural science and all of human knowledge can ultimately be derived from physics. I don't take that position. It has to be derived from generalizations of observations of the natural world. Many of...
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I don't disagree too strongly with your characterization of my position, but I would modify it somewhat:
(1) It doesn't have to be physics per se unless one takes the position that all of natural science and all of human knowledge can ultimately be derived from physics. I don't take that position. It has to be derived from generalizations of observations of the natural world. Many of these, perhaps the majoirty, will be in the realm of physics just because we have defined physics to encompass the most fundamental and easily mathematizable aspects of the world, but not necessarily all. For example, if there are regularities in the way that societies behave that are not ultimately derivable from physics then these can form the basis of mathematics as well.
(2) Modulo the above, yes, but it is important to bear in mind that I intend a broad interpretation of the word "useful". If one theory is much more elegant and easy to think about than another, but their differences have no physical implications, then I would still call the first theory more useful than the second. For example, by this, I think I can justify that there may be a "correct" way of handling infinities in a mathematical theory, or at least ways that are more correct than others, even though this has no immediate physical consequences.
I admit that my view is logically compatible with yours. At least, I cannot rule out a very large mathematical multiverse containing all possibilities. However, my arguments are intended to at least undercut the reasons for thinking that this might be true. It seems to me that the main arguments for the maxiverse are simplicity (Occam's razor) and the ability to account for the role of mathematics in physics in a naturalistic way. I am not too impressed with Occam's razor arguments. It seems to me that, where it applies, its applicability can be derived from other more fundamental principles, and that there are areas where it doesn't apply, i.e. there are phenomena that are just complicated and messy. So, I'm not prepared to accept a hypothesis just because it is simple. There has to be more to it than that. For the naturalistic argument, I still think it is problematic to understand why our universe has the specific mathematical structure that it does, i.e. I don't think you are particularly successful in deflating the measure problem because it seems to me that there are still plenty of bizarre universes that are not incompatible with the continued existence of my consciousness. I also have trouble understanding why what *we* call mathematics should be what reality is fundamentally made of. I did emphasize mathematics as a social activity of finite beings after all.
My position is actually much closer to Sylvia Wenmackers' than to yours, although I don't agree with all her arguments for it. But basically we agree that reality is just what it is, we are in it, and our task is to figure out why the mathematics that we develop is at all relevant to it.
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Brian Balke wrote on Mar. 3, 2016 @ 21:41 GMT
Dear Matt:
I would find this meal more substantial if the distinguishing characteristics of mathematics as a formal system were mapped to what we know about physical reality. Playfully, thinking about realities in which mathematics would not apply might be revelatory. Let's consider a reality in which you had as many fingers as toes when you went to bed and woke up with half as many of the former. Or where a sheep arrived overnight (like flies on meat in the theory of spontaneous generation) - except that as you looked at it, it seemed to be more like a llama.
This guides me to a suggestion that the existence of an irreducible scale (quantization), fermion number conservation, and locality (charge cancellation and finite velocity in the propagation of effects) might all be critical to mapping mathematics to physics.
But I do like you idea that mathematics is an abstraction of physical reality. I see an explanation, in fact, of the observation that physics becomes more and more like mathematics every day. Physicists are reasoning about things that they cannot see. Their grasp of reality has become wholly Platonic. Therefore abstractions upon abstractions is all that they are left with. Rather like theologians arguing about angels dancing on the head of a pin, they have grasped the prop of the legitimacy of classical dynamics, extended it into the world of the unseen through quantum field theory, and then just gone off on a lark with group theory, discarding entirely the proposition that the complexity we observe might be explained by positing additional structure (as it always was in the past).
Brian
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