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.

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

    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.

    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.

    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.

    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

      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.

      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.

      "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.

      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).

      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.

        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

        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.

        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.

        "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

          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.

          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

            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