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
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
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
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
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.
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.
Something screwed up with my login credentials, but the previous post was me.
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.
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.
Hi Matt,
My system is a counterexample to your thesis. If you don't have the time just read the electron mass section and run the program (click "program link" at the end of the section) , it will execute in less than a minute.
Thanks and good luck.
P.S. That was a nice one you pulled on Lubos, some people actually believed it.
"I would be happy to be called a mathematical realist, in that I think there is a fact of the matter about what should be called a mathematical truth, but I want to cash that out in terms of the physical world rather than some abstract mathematical world."
So the question is what is the physical mechanism by which "2 3 = 5" is true?
I suppose one could give a tautological definition that it is all the physical instances for which that equation is a description.
Dear Saul,
I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.
All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.
Joe Fisher
Dear Matthew Saul Leifer,
I read and enjoyed your essay. You presented your ideas well analytically and logically, I thought.
I did not think of my essay contribution as advancing the point of view of mathematics as naturalistic, but it seems to me that would be a fair characterization. In fact, your remark in your conclusion that "Mathematics is constructed out of the physical world" accords with the idea in my essay that the rate at which it is constructed can be measured and quantified (using C log (n) ).
On your point at page 2 that human knowledge has the structure of scale-free network, that may be so if we can say that energy distribution in the universe has the structure of a scale-free network. To utilize energy that is distributed like a scale free network recipient systems would have to model themselves the same way. At least one might argue that. In light of your remarks about networks, you might find some of the references in my essay of interest.
Thank you for having taken the time to write your interesting essay.
Best wishes,
Bob Shour
Hi Matt,
I really enjoyed your network perspective, and it's even changed my worldview about math somewhat. I think you're absolutely right that some new mathematical "nodes" are developed for exactly the reasons you describe here. (Strong analogies between existing nodes make it more natural to build a new node, leading to a more-efficient common structure, etc.)
But is this the *only* way that new mathematical "nodes" are developed? It seems to me that an equally-important process is simple exploration/extrapolation from existing nodes, tweaking the premises that lead to one bit of math, and seeing what new bits might result. Sure, this isn't going to necessarily lead to the efficient network structure that you describe, but it would supply a bunch of "raw material" which your efficiency-driven process could then organize.
One big distinction between the nodes generated by an "exploration process" (rather than your "efficiency process") is that there's generally no requirement that they hold together self-consistently. This is what I had in mind when I talked about different "caves" in my own essay, even though you're right that eventually your efficiency-driven process tends to merge the caves together into much larger networks... maybe even to the point where you can claim that there is one "pragmatically-true" type of math when it comes to things like the axiom of choice.
If you're with me so far, then I think your comment on my essay reflects that we're trying to answer two somewhat different interesting questions. You are addressing the mystery of why the most-efficient network groups of math tend to map to physics; I'm trying to address the mystery of why so many separate explorative-nodes tend to find a use in physics.
I think there are some aspects that make your task harder than mine, and some that make them easier. On the easier front (which I think you hinted at), mathematicians know about physics when they are looking for efficiency-nodes. When faced with mathematical choices about the details of differential geometry, say, I'd be a bit surprised if mathematicians refused to think about GR. But while GR could potentially inform part of a subsequent efficiency-process, it certainly didn't inform the original development of non-Euclidean geometry; that's basically the mystery that I was trying to address.
On the other hand (as you noted on my page), given a collection of unrelated but internally self-consistent axiomatic systems, it shouldn't be terribly surprising that *some* of them find a use in physics. (Indeed, that's a key part of my argument.) But if mathematicians truly think that some of these network-systems are more "correct" than others, then it is quite interesting that the deemed "correct" systems are more useful in physics than the equally-logically-valid but "incorrect" ones (as judged by the overall mathematics community). I guess the big question here is just how much influence higher physics has had in framing the judgment of mathematicians (as per my GR example), or if it really is all built up from counting apples, and very basic stuff like that. Do you see higher-level physics as impacting higher-level math, or is it all one-way-influence at that level?
Thanks again for a thought-provoking essay!
Ken
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.
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
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.
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."
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.
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.
