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.

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 essay

Best regards

Janko Kokosar

    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!

    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

    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

      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

      2 months later

      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

        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.

        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.

        9 months later

        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