The relationship between mathematics and physics is a topic that has always fascinated me, and today I noticed two interesting blog postings related to the topic. The first was Ben Webster’s posting inspired by a recent XKCD comic. The discussion in the comment section is well worth reading, especially the contributions from Terry Tao.
Over at Backreaction, Sabine Hossenfelder discusses an interview with Max Tegmark from the latest issue of Discover magazine entitled Is the Universe Actually Made of Math?. Much of the discussion is about Tegmark’s comments on how he dealt with the potential danger to his career caused by his unconventional publications on the “Mathematical Universe Hypothesis”. This says that
Our external physical reality is a mathematical structure
I’ve always had an extreme case of mixed feelings about this, thinking that Tegmark manages to bring together the extremely deep and the extremely dumb. He embeds this as “Level IV”, the highest level, of the multiverse, and multiverse mania is one reason he has gotten attention for this and not had it dismissed as crackpotism. The idea he is pursuing is that any mathematical structure can be thought of as a “universe”, and we just happen to be in some random one of these. This seems to me to be pretty much content-free, and the attempts to fit it into more conventionally popular multiverse studies don’t help.
At the same time, this does get at an incredibly deep problem, that of the relationship between mathematical structures and physical reality. Some of the central mathematical structures that mathematicians have discovered have turned out to be identical to those found by physicists pursuing models of fundamental physics. This has happened in several very striking ways over the years. Thinking of the universe as a mathematical structure has turned out to be extremely fruitful, both for mathematics and for physics.
What is important though is that not all mathematical structures are equally important, central, or interesting, and this is the crucial point that Tegmark seems to me to be missing. Once you learn enough mathematics, you find certain recurring themes and deep structures throughout the subject. What fascinates me is that these often also turn out to be central in theoretical physics. Tegmark just accepts every mathematical structure as equally important, creating a huge undifferentiated multiverse where we occupy some random anthropically acceptable point. But the evidence is that the mathematical structure we inhabit is a very special one, sharing features of the very special structures that mathematicians have found to be at the core of modern mathematics. Why this is remains a great mystery, one well worth pursuing from both the mathematician’s and physicist’s points of view.
An acquaintance of mine has a simple but deep criticism of Tegmark’s ideas. He says that all mathematical structures are built and based on sets. This is especially the case for all structures looked at by Tegmark. But on the other hand, it is not clear at all whether nature is a set. In fact, there are many reasons that point to the opposite conclusion.
He says that at high energy, it is not clear that a “set” is the correct description for nature, it rather looks as if it is not. If nature is not a set, Tegmark’s ideas lose their base.
I’m amazed that Max Tegmark seems to entertain the notion that mathematics is constituent of the physical world, somehow waiting to be discovered “out there”. Is it not obvious that mathematics is an invented language that we chattering African apes have evolved to describe the mysterious world we find ourselves in? No wonder its structure has evolved to closely resemble that of the physical world.
“Is it not obvious that mathematics is an invented language that we chattering African apes have evolved to describe the mysterious world”. It is not obvious, to say the least. Whole books have been written about this, and I recommend this one : “Matière à pensée”, a dialogue between Alain Connes and the neurologist Jean-Pierre Changeux. I can also point you to Omnès book “Converginf realities”.
I think that when you ask yourself what “real” means, what criterion permits to classify something as real, what comes to mind is logical consistency. Everything else seems to be ill-defined intuitions. I talked about this subject in more length here : http://math-et-physique.over-blog.com/article-1591626.html (in french). I’ll try to put a trackback but for some reason it never works…
All applied mathematics for real world physics is only approximate:
1. Newtonian physics only has exact analytical solutions for two-body interactions, whereas there are many bodies present in the universe. Poincare chaos arises for orbits of more than two bodies, where each affects (alters) the orbit of the other as it moves. There is also a quantum chaos from the random exchange of field quanta that causes the electromagnetic interaction between electrons and protons, which on small scales is random (on big scales the large number of field quanta interaction statistics smooth out to give the deterministic classical Coulomb law). This prevents deterministic calculation of electron orbits inside the atom.
2. General relativity’s stress-energy tensor uses an artificially smoothed distribution of mass and energy instead of representing the real particulate (discontinuous, i.e. atoms and quanta) distribution of matter and energy, to create an equally false smooth source for the Riemann curvature. It just ignores the QFT idea that gravity field quanta (gravitons) are exchanged in discrete interactions, not continuous acceleration (smooth curvature).
3. Even if you just consider simple addition, counting two electrons, you haven’t an exact mathematical model with 1+1=2 for two times the same thing, because the electrons are all slightly different in their motions and by the uncertainty principle in principle you can’t ever find their exact positions and momentums. So they will have slightly different velocities and therefore slightly different masses. So you’re not adding up exactly the same real thing. To make the point clearer, if you add up apples (or if you count sheep), you are adding up things which are approximately similar, but not exactly the same. Two similar looking items will differ at the atomic scale. So addition is only ever exactly true when dealing with tokens like money, an invention due to mathematics.
It is impossible even in principle to get exactly true input data in the real world from making measurements. Also, it’s impossible to make exact predictions, because all applied physics calculations for the real world involve making approximations. So the universe isn’t intrinsically mathematical. You can’t get completely exact input data, and – even if you did know exact initial conditions – the mathematics used to model real (complex) phenomena is an approximation only.
In order for the universe to be intrinsically mathematical, it would be necessary in principle for there to exist some way of exactly representing the real world using mathematics, instead of relying on approximations and statistical wave equations. Mathematics is in principle at best just an approximation to the universe, so the universe can’t – even in principle – be intrinsically mathematical in nature.
‘In order for the universe to be intrinsically mathematical, it would be necessary in principle for there to exist some way of exactly representing the real world using mathematics’.
Isn’t it what physicists are after ?
Anon, you seem to confuse two kinds of approximations : 1) the fact that our best theories are still approximations 2) the approximation in data. I don’t see any relation between point 2 and the issue of the true nature of physical objects. As for 1), what is your argument for saying that there is no such thing as an exact mathematical description ? We do not have such thing at hand right now, but the simplest way to explain the fact that there exist excellent mathematical models of reality is that these models are approximations of a complete mathematical description.
Finally, I think that one should distinguish between approximation and abstraction. Mathematical concepts are abstracted from ‘familiar objects’ (‘familiar objects’ might include other mathematical concepts), in a way I think is akin to chemical purification. Then these concepts are assembled to form a “synthetic product”. These synthesis can produce mathematical theories which (perhaps crudely) model parts of the “real world”. But both the “real thing” and the mathematical models are of the same nature : there are made of “atoms”, which in my metaphor are elementary mathematical concepts.
I speak about this “chemical metaphor” in the link I gave above.
Apparently, his philosophy is to equate the physical world with mathematics (yes, equate, not a sort of mapping between the two, nor a kind of approximation or representation or abstraction or whatever). If I understand Tegmark’s point correctly, in general terms he seem to argue that this direct equality solves the philosophical problem of whether there is an ultimate reality. He says there is one and it is pure mathematics.
I may have misunderstood it all since I have only read a simplified paper that he published some time ago. In any case, I didn’t find any of his arguments brilliant or convincing. Looks like a very bad philosophy to me.
The fact that we can describe physical phenomena through mathematical *reasoning* (no matter limitations concerning approximation of data, etc) is something much deeper to me and equating both is no solution (again, to me). It’s like turning a difficult question into a trivial one as the best way to actually avoid it.
Why am I being harsh?
My formative years in astrophysics were the 90’s, where I have seen cosmology turn into a real, quantitative science, the beginning of the “precision era” of cosmology. What I see today, however, concerns me deeply. I see well-known cosmologists, Tegmark and Sean Carroll being two examples that come easy to my mind, whose work I came to appreciate for many years, to start drifting into a speculative world, which has nothing to do with cosmology or science.
Now let me be clear on this. I have a deep passion for philosophy and a great appreciation for the great classical philosophers, and actually I firmly believe that science must go hand in hand with philosophy. So I have nothing *against* a cosmologist or anyone attempting a serious work in philosophy or any other kind of speculative adventure, up to the point that he or she makes that absolutely clear. One should not confuse one thing with the other. Philosophy and science can go hand in hand, but one must be clear of what he/she is talking about, and make no confusion between the two disciplines.
Concerning “the potential danger to his career caused by his unconventional publications”, I take a very pragmatic approach. If a PhD student of mine in cosmology (let us say) started to drift into philosophy to the point of interfering with his/her research, no matter how fascinating his/her ideas were, I would have a conversation with him/her in order to make it clear that he/she would have to make a choice between these fields in that point of their formative years. After his/her PhD, he/she could choose whatever he/she wants to do, it doesn’t concern me. I see no point in making so much fuss about this.
I’ve always liked a remark that physicist Sir James Jeans made back around 1935: “The more we study it, the more it all looks like a dream.”
What one does not put into the equation will not finally be given by the mathematics. James Franck
IOW if you begin with math and try to go backwards you end up with nothing.
You make a very crucial observation that deserves serious attention namely that physics appears to select mathematical structures to faithfully represent it that end up being recurrently isomorphic among themselves! This indeed is what makes Tegmark’s call for `mathematical democracy of all structures’ so totally sterile and silly! The notion of a “universe made of math” is not new: it is an old creed called pythagorianism, sure to be periodically revived by people who lack (better) ideas. Tegmark’s profligate version is also not original: it was advocated by the argentinan philosopher Mark Balaguer in his book
“Platonism and Anti-Platonism in Mathematics” as Full-Bloodied Platonism, though I believe Plato to be a lot less sanguine in his advocacy! You are right that this notion caters to the whole Landscape mentality that anything goes and it is basically a call to abandon all criteria of adequacy of mathematical speculation to empirical evidence, a notably self serving strategy for people who don’t care to have their “bright” ideas falsified.
The issue of whether (logical) self-consistency distinguishes mathematical structures that apply to physics from those that don’t has been raised by Hawking, Barrow and others but it seems clear that, if this may (or may not) be a necessary qualification, it surely cannot be sufficient — unless perhaps one revives Hillary Putnam’s proposal that logic may be an empirical science! There is indeed a very interesting set of questions surrounding this math-phys interface but the pythagorians seem very much more keen in obfuscating than addressing it…
I think its just that what mathematicians work on is inspired by the real world and in the real world you get analogous physical structures at different levels of physical reality.
‘It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of spacetime is going to do? So I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed, and the laws will turn out to be simple, like the chequer board with all its apparent complexities.’ – R. P. Feynman, The Character of Physical Law, 1965.
Feyman is not referring to the use of the path integral for summing and weighting all possible interactions in spacetime even for relatively simple interactions, e.g. two electrons. There are an infinite series of increasingly complex Feynman diagrams for even the simplest interaction. The perturbative expansion for the path integral is an infinite series of terms of increasing complexity. The math is summing an infinite number of possibilities for how the field quanta will be exchanged.
This calculus error reminds one of the exponential formula for radioactive decay, a continuous curve which ignores the fact that radioactive decays are discrete events, so that there should really be a step-wise quantized decay line (with quantum drops as individual atoms decay, and horizontal lines in between the decays). The math is a nice approximation when the decay rate is very large, but nature doesn’t strictly obey the equation at any time, and after the final atom decays the exponential law is obviously completely false and misleading.
Heh, if someon wanted to seek an independent research, I would recomend looking for a job in the federal governament, specialy in something burocratic. I don’t know how is it outside Brazil, but generaly here you get a enough free time to study, have some money to subscribe to a decent ISP and buy some books. Maybe sometimes one could travel and attend conferences, during vacations.
As a physicist, Tegmark’s ideas seem completely useless. I wonder if philosophers would find anything interesting. I don’t enjoy sounding harsh, but I suspect not. Any philosophers out there?
Compare what Tegmark says with pages 42-49 of the July Scientific American, where the question of fundamental constituents of spacetime is answered differently. This site has the complete article free, for online reading:
They get deSitter space to self-assemble from no prior geometry. Instead of being made of mathematics, their universe is more like a flock of birds.
Is there a connection between Tegmark’s “The universe is math” contention and Seth Lloyd’s “The universe is computation” contention? I find the computation analogy or mapping more compelling, but I am curious.
The statement that the Universe is made out of Math first emerged in Plato’s philosophy. It is a proposition that is untestable and, as such, it does not belong to science.
Don’t forget Bee’s hilarious posting on Backreaction when Tegmark’s book came out.
I think it bears on this topic to mention that the physiology of the brain itself causes an internal creation of complex geometric representations. When deprived of external stimuli the brain can become aware of its own geometric background signal:
It’s furthur interesting to note that experimental subjects who have these experiences often recall the recognition of these patterns as
extremely important or all encompassing realizations of the universe.
This is not to say that our view of the universe is nothing more than a hallucination, but without doubt human brain function is very geometric at fundamental levels. To this day, one of the classic images we assign to cosmology is that of Kepler’s Platonic solid model of the solar system (Mysterium Cosmographicum).
Centuries later String theory seems much more sophisticated, but it remains inseparable from its imbedded geometric allusions.
A big question could be to what extent do we have the freedom or ability to transcend the very limitations of our own mental machinery.
Instead of “the universe is math,” wouldn’t it be more accurate to say that the universe is built on evolvingly complex patterns? We then translate the patterns into the “language” of math.
He says all mathematical structures are created equal, but I’m pretty sure that 2>1.
I thought all this stuff was settled long ago — especially by Russell. Anyway, if you want to say something objective about real world relations (or any relations) or describe what’s happening you have to use math, since that what math does. And no, it is not a separate language — everything in math can be expressed in English or Russian, etc.
“Paper in white the floor of the room, and rule it off in one-foot squares. Down on one’s hands and knees, write in the first square a set of equations conceived as able to govern the physics of the universe. Think more overnight. Next day put a better set of equations into square two. Invite one’s most respected colleagues to contribute to other squares. At the end of these labors, one has worked oneself out into the doorway. Stand up, look back on all those equations, some perhaps more hopeful than others, raise one’s finger commandingly, and give the order ‘Fly!’ Not one of those equations will put on wings, take off, or fly. Yet the universe ‘flies’.”
John A. Wheeler
Consider your conscious experience as a trajectory through the n-dimensional space of possible conscious experiences.
It clearly has a certain coherency; the world does not radically change on a Planck-time-by-Planck-time basis.
However, if all possible mathematical structures are equally “true”, then for each Planck time, there are an infinite number of discontinuous universes that you could suddenly find yourself in that aren’t this one, but are otherwise perfectly acceptable. That is, mathematically, despite living in the world you think you’ve been living in all your life, it is perfectly acceptable mathematically that in the instant after you read this, you’ll find yourself on a planet Vulcan, as in, the Star Trek planet. The universe discontinuously shifts around you, but math has no problem with discontinuous shifts.
Therefore, if all possible mathematical structures were equally real, the probability of use experiencing such a clean, ordered universe are infinitesimal. Perhaps we are simply that infinitesimal bit of the mathematical structure that actually, factually experiences such a simple and orderly universe, but it is far more likely that Tegmark is simply wrong, and there really is some sort of mysterious dividing line between “real” and “unreal”.
Has no one thought to look at this using the discipline of cognitive science? I think it could offer some perspective that is lacking within the frameworks of physics and mathematics.
Mathematics is just a practice of studying relationships between patterns. We come across these patterns in individual experiences, and and in human thought.
It is no surprise that patterns in human experience are related to the structure of the universe, or that the mind is structured (both through nature and through nurture) to pay particular attention to patterns that bear upon physical existence. But mathematics is a map, rather than a territory.
The spam software is having trouble distinguishing between the short, not very informative comments and quotes people are posting here and the ones coming from spambots, and I have to admit I can’t completely blame it. Well thought-out comments that add something significant are encouraged, but please resist the temptation to add to the noise level here.
Unfortunately, the debate about whether or not Math forms the underlying fabric of the Universe leads nowhere. Statements cannot be proven true or false and the discussion is entirely scholastic. Despite what Tegmark advocates, Math is a tool , a rational collection of symbols helping to describe Nature, “a map rather than a territory” to paraphrase Joel.
I read the Discover article a month or so ago but if I remember my reaction, it was that the guy over-rates his insight. I didn’t think he was out-to-lunch; he was just too full of himself. I kind of find the reaction here to be a bit too much as well. This guy is esoteric but I think he acknowledges that fact which is better than some. One of the things that struck me was his comment about what turned him in his current direction. Something about the wave-function collapsing when a particle is observed & that he felt this was wrong in some sense. My thoughts were much the same in 3rd or 4th year undergraduate physics & I think many other physic students felt the same way. Intuitively the Schrödinger wave function interpretation feels wrong. It works but it seems to imply an underlying unreality that makes me uncomfortable. Interestingly my reaction was to treat it like a math tool that that helps solve physics problems but may or may not be a description of reality. So for me math was always a tool that I could make do what ever I wanted it to do while physics was something real. This guy seems to have gone in the opposite direction embracing the potential unreality. For him math is the reality & physics the abstraction. Strange but if he can get people to pay him to write like this – good luck to him.
Of course everything else seems to be ill-defined intuitions – it’s not consistent, so it’s not likely to look good under logical analysis. Which means you’re begging the question. The real point is whether or not it is necessary for “real” to be logically consistent, or if that is just a convenient assumption. Since logical consistency is something (in my view) dreamed up by humans, dependant on a number of assumptions, and not an external truth, I see no reason other than that of pragmatism for making such an assumption.
The whole “universe is maths” idea raises as many questions as it answers. For people, maths is a mental activity, where is the unversal mind that thinks the universe? Even if you discount the mental nature of maths, what medium exists to reify the maths? Just because I come up with a mathematical structure in my head, doesn’t mean I’ve created a universe of it that describes all the results of that structure beyond even what I’m able to trace. Or perhaps he wants to go back to platonism and the theory of forms.
We create models of our environment, our models are at most countably infinite (they can be expressed in language, or at least encoded in our brains), but there’s no reason to believe that the environment itself (rather than our model of it) does not require uncountably infinite symbols to accurately represent it.
It’s much more sensible to say that our maths is at best a model that approximates reality rather than that reality approximates our models.
Oh and for full accuracy, that “increasing purity” comic, should have had a philosopher to the right of the mathematician, and then a psychologist to the right of that, etcetera, ad infinitum.
Roger Penrose in The Road to Reality has a triangular diagram, that expresses the idea that some of our mental activity is mathematics; some of mathematics describes physical reality and finally, some of physical reality is our minds.
This diagram illustrates a mystery and it seems to me that Tegmark seeks to banish the mystery by collapsing Penrose’s triangle into a point. If the universe is mathematics and mathematics is the universe, then every thought of ours is also mathematics. Our minds are mathematics. Even thinking that Tegmark’s idea is absurd is mathematics. Agreeing with him is mathematics, too!
>Even if you discount the mental nature of maths, what medium exists to reify the maths?
What do you mean by “medium” ? In this debate, I noticed that one very often reintroduce in a way or another, under a name or another, the intuitive but rather wrong idea that what is real is made of matter, and matter is made of small hard spheres (made of what ?). I would be very interested if someone could come up with a definition of reality that excludes mathematical structures but not quantum fields, which do not possess any of the macroscopic qualities (such as hardness, localisation, etc…) with which we mentally endow “real things”. What is reality is certainly an old metaphysical question. I tend to think that these kind of questions cannot receive an answer as such because their terms are not well-defined. Logical consistency=reality is a definition of reality that has the advantage of not being polluted by our macroscopic prejudices.
>Just because I come up with a mathematical structure in my head, doesn’t mean I’ve created a universe
Certainly not. Mathematical structures do not live in spacetime, so they can’t be created. They just are. You can think about one, it does not mean that you have created it (as with everything else you can think about).
Mathematical structures do not live in spacetime, so they can’t be created. They just are. You can think about one, it does not mean that you have created it (as with everything else you can think about).
What determines which mathematical structures “just are” and which “just are not”? E.g., why haven’t we seen supersymmetry so far?
As it happens I’m much closer to a Berkleian idealist than a believer in small hard round spheres.
If you’re going to say things like “Maths just exists”, perhaps you should give a clear and unambiguous definition of what you actually mean when you say maths. It’s quite possible that this is just a difference in use of terminology.
I’m very confused where you get “Logical consistency=reality is a definition of reality that has the advantage of not being polluted by our macroscopic prejudices.” from. It seems to me that belief in consistency is one of our biggest macroscopic prejudices.
You seem a little confused there, Arun. As a mathematical construct, supersymmetry has as much claim to exist in a Platonic sense as any other mathematical construct. Whether this construct can be given an interpretation that brings it into successful confrontation with observation (both passive and active, ie, experimental) is a separate question. So far, the answer seems to be no.
Of course this just indicates the fundamental wrongheadedness of Tegmark’s formulation. All sorts of mathematical constructs exist (in an appropriate sense) in our heads, or in objective form (following Karl Popper) in books, journal articles, implementations of algorithms in computer hardware and software, etc. (For all we know they also exist widely among the cultures of sentient beings across our galaxy and beyond.) However, in the empirical sciences we’re concerned with inference from certain premises that leads to specific, empirically testable conclusions about the physical world. We generally do this in a conceptual framework that organizes and motivates the inferences, but the relevance of logic and mathematics comes first and foremost from their power in helping us draw those inferences. Admittedly, in physics, various areas of mathematics have also contributed increasingly to the development of the conceptual frameworks, but this is ultimately to serve inference; we hope they can help us draw far-reaching and unambiguous conclusions that we can test.
I would suggest that the relevance of mathematics as a source of patterns comes ultimately from the simple fact of some regularity and stability in the world. Patterns are useless in describing experience if their appearances are completely ephemeral. In fact, noticing patterns in time is arguably where science begins. The fact that we can reproduce experiences of certain sorts, or merely note their regular repetition, is the starting point for the application of inference, and ultimately mathematical inference, via mediating abstractions (starting with simple idea of number).
One could say that the central mystery in nature is the interplay between pattern and structure on the one hand, and chaos and disorder on the other. It seems to involve a delicate balance; nowhere is this more evident than in quantum mechanics. John Stachel has argued that Einstein’s objections to quantum theory were misconstrued. He didn’t object to indeterminism itself so much as the lack of any explanation for why just so much, and no more, should be admitted. In other words, if the laws of physics do not fully determine the future from the past, why should they exist at all? Why should there be any structure in the world?
[By the way, your earlier comment—mentioning Penrose’s book—reminds me of my first encounter (in my teens) with B. F. Skinner’s Beyond Freedom and Dignity. I was entranced for a while with the idea that one could give a behaviorist account of the objections to behaviorism. What a great way to make an idea invulnerable to criticism! Over the course of the next year and a half I started to grasp how pernicious and sterile that tactic was. I’ll leave it someone to say whether Skinner was actually indulging in it.]
Thanks, Chris W. You made pretty much the point I was leading up to.
For me the universe-as-math question raises 2 primary questions:
1) Questions of the ultimate structure of the universe aside, it seems a tautology that the ‘ultimate structure’ of the brain must be a subset of the universal one. This imposes limits on what we as a species can perceive or even theorize about.(Tom W. makes this point above, with a link to a very interesting article.)
2) Why then the “unreasonable effectiveness of mathematics” in the natural sciences? – exactly what we’re dealing with here.
The shared element that connects these two questions is Logic, and the connection occurs at a fundamental level, pretty mush as deep as it gets. For this reason, I believe there is genuine merit in Hilary Putnam’s view of Logic as an empirical science.
Whatever else the universe is, it is an entity that allows Logic, and so far, seems to demand that we completely constrain our thought processes to those which are logical if we wish to model it. This is the origin of Pythagorean mysticism, Plato’s idealism and our own personal intuitions regarding the relationship between abstract mathematics and physical science.
Math is built from logic – Russell/Whitehead showed us this, Godelian limitations notwithstanding. When we are marveling over the relationship between math and science, we are really remarking on the fact that nature has, to some significant extent, a logical substructure. This is a lesson we were strongly encouraged to learn, as a species, and taking these lessons to heart our most reliable forms of thought are logical ones.
I think Physics, as a discipline, is approaching the point where it will begin to ask questions like: “what is the nature of the fundamental structure of the universe such that it supports Logic as a mode of computation?”.
Or something like that.
It seems to me that in all this comment about logical consistancy of math and its relationship to the world individuals have left out the most important relationship of math to the world – and that is the conservation rules. While geometry has its important place in math, it deserves a position one rung lower than conservation laws when it relates to physics. More mathematically oriented physicists seem to continue to forget that.
In the universe geometry is related to the geometry of energy. Energy, as far as we know, is the only substance that always remains and is finite in the universe even as the geometry of its forms change. For some reason which I can’t fathom theoretical physicists, as a generalization, seem to be the most likely people to forget it and are the ones who must continually strive to remember it. The relationship of how those forms change geometrically while continuing to conserve energy is where the answers lie in physics. This is not true for math.
The relation between conservation laws and symmetries is an absolutely fundamental aspect of how physics and math are related. In quantum mechanics this can be identified precisely with the mathematics of the unitary representations of Lie groups. I’ve gone on in many places about how important I think this is, and it’s exactly the sort of thing that I think Tegmark misses (along with people who see this as an issue about logic, or the human brain, or any number of other very different questions).
Dr. Woit, can you please elaborate on why Tegmark is wandering in the darkness, despite the illumination of Amalie Emmy Noether, who (as thumbnailed in wikipedia) the German Jewish mathematician [March 23, 1882 – April 14, 1935] known for her seminal contributions to abstract algebra. Often described as the most important woman in the history of mathematics, she revolutionized the theories of rings, fields, and algebras. She is also known for her contributions to modern theoretical physics, especially for the first Noether’s theorem which explains the connection between symmetry and conservation laws [Ne’eman, Yuval. “The Impact of Emmy Noether’s Theorems on XX1st Century Physics”. The Heritage of Emmy Noether. Ed. M. Teicher. Israel Mathematical Conference Proceedings. Bar-Ilan University/American Mathematical Society/Oxford University Press, 1999. OCLC 223099225. ISBN 978-0198510451. pp. 83–101.]?
My bias is that the 2nd most important woman in the history of 20th century mathematics, Olga Taussky Todd [30 August 1906, Olomouc, then Austria-Hungary – 7 October 1995, Pasadena, California] Czech-American Jewish mathematician, gave me her personal and professional take on Emmy, whom she’d met. Similarly, I’ve heard great talks at symposia in the memory of Emmy Noether and Olga Taussky Todd.
Please, though I agree with you, I’d like your more detailed rationale.
I’m talking about symmetries in quantum mechanics, not classical mechanics. See my book for more details.
Suppose it were possible to formulate a theory that was independent of topology and the number of dimensions. Then suppose you could use a procedure analogous to gauge fixing to work out the physical predictions of the theory in 3+1 dimensions and arrive at something that described the world we see. Then what we see as the physical world and it’s contents, including ourselves, is merely a result of a gauge fixing. (You’d have to show that choosing any other gauge i.e. dimensionality and topology produced results that wouldn’t support observers.)
Anyhow the point is that the world you end up describing would necessarily reflect the mathematics used for the “gauge fixing”. Then physics and mathematics would be inextricably linked and Tegmark’s view of mathematics would arise naturally.
It’s not “gauge-fixing” if physical observables depend on it. All you have done is decided (illegitimately…) to call the mathematical structure of physics “gauge-fixing”, you haven’t addressed at all why it is one structure and not another.