Mathematics is Physics by Matthew Saul Leifer

Matthew Leifer

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.

Thomas Ray

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

Matthew Leifer

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 Ray

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."

Ken Wharton

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.

Alexey/Lev Burov

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.

Matthew Leifer

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.

Alma Ionescu

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

Matthew Leifer

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

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!

Alma Ionescu

Sorry, I just realized I forgot to rate your essay so I fixed that now. Good luck!

Vesuvius Now

I've probably misunderstood something, but in virtue of which physical processes is the mathematical equation 2 3 = 5 true?

James Hoover

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

Janko Kokosar

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

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 essay

Best regards

Janko Kokosar

Mohammed Khalil

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

pjackson

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

Sylvia Wenmackers

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

Matthew Leifer

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.

Jeffrey Schmitz

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

Marc Séguin

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

Matthew Leifer

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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