Mathematics is Physics by Matthew Saul Leifer

Matthew Leifer

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.

Matthew Leifer

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.

Matthew Leifer

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.

Thomas Ray

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

Matthew Leifer

"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 Ray

"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?

Ed Unverricht

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

Tapio Salminen

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

Tommaso Bolognesi

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

Vesuvius Now

If math is a natural science then the correct logic to reason with should be quantum logic.

Sylvain Poirier

"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."

Philip Gibbs

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.

Sophia Magnusdottir

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

Matthew Leifer

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.

Joe Fisher

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

Ian Durham

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

Matthew Leifer

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.

Michel Planat

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

Matthew Leifer

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.

James Hoover

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

[deleted]

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.

Matthew Leifer

Something screwed up with my login credentials, but the previous post was me.

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