Vladimir Kazakov began his talk at ICTP-SAIFR this week with a variant of Tolstoy’s famous opening to the novel Anna Karenina: “Happy families are all alike; every unhappy family is unhappy in its own way.” Kazakov flipped the order of the quote, stating that while “Un-solvable models are each un-solvable in their own way, solvable models are all alike.”
In talking about solvable and un-solvable models, Kazakov was referring to a concept called integrability, the idea that in certain quantum field theories it’s possible to avoid the messy approximations of perturbation theory and instead jump straight to the answer. Kazakov was observing that these integrable systems seem to have a deep kinship: the same basic methods appear to work to understand all of them.
I’d like to generalize Kazakov’s point, and talk about a broader trend in physics.
Much has been made over the years of the “unreasonable effectiveness of mathematics in the natural sciences”, most notably in physicist Eugene Wigner’s famous essay, The Unreasonable Effectiveness of Mathematics in the Natural Sciences. There’s a feeling among some people that mathematics is much better at explaining physical phenomena than one would expect, that the world appears to be “made of math” and that it didn’t have to be.
On the surface, this is a reasonable claim. Certain mathematical ideas, group theory for example, seem to pop up again and again in physics, sometimes in wildly different contexts. The history of fundamental physics has tended to see steady progress over the years, from clunkier mathematical concepts to more and more elegant ones.
Some physicists tend to be dismissive of this. Lee Smolin in particular seems to be under the impression that mathematics is just particularly good at providing useful approximations. This perspective links to his definition of mathematics as “the study of systems of evoked relationships inspired by observations of nature,” a definition to which Peter Woit vehemently objects. Woit argues what I think any mathematician would when presented by a statement like Smolin’s: that mathematics is much more than just a useful tool for approximating observations, and that contrary to physicists’ vanity most of mathematics goes on without any explicit interest in observing the natural world.
While it’s generally rude for physicists to propose definitions for mathematics, I’m going to do so anyway. I think the following definition is one mathematicians would be more comfortable with, though it may be overly broad: Mathematics is the study of simple rules with complex consequences.
We live in a complex world. The breadth of the periodic table, the vast diversity of life, the tangled webs of galaxies across the sky, these are things that display both vast variety and a sense of order. They are, in a rather direct way, the complex consequences of rules that are at heart very very simple.
Part of the wonder of modern mathematics is how interconnected it has become. Many sub-fields, once distinct, have discovered over the years that they are really studying different aspects of the same phenomena. That’s why when you see a proof of a three-hundred-year-old mathematical conjecture, it uses terms that seem to have nothing to do with the original problem. It’s why Woit, in an essay on this topic, quotes Edward Frenkel’s description of a particular recent program as a blueprint for a “Grand Unified Theory of Mathematics”. Increasingly, complex patterns are being shown to be not only consequences of simple rules, but consequences of the same simple rules.
Mathematics itself is “unreasonably effective”. That’s why, when faced with a complex world, we shouldn’t be surprised when the same simple rules pop up again and again to explain it. That’s what explaining something is: breaking down something complex into the simple rules that give rise to it. And as mathematics progresses, it becomes more and more clear that a few closely related types of simple rules lie behind any complex phenomena. While each unexplained fact about the universe may seem unexplained in its own way, as things are explained bit by bit they show just how alike they really are.
4 gravitons 
Whoa! Extremely interesting on many levels! I haven’t yet read the Wigner essay (but will!), and Woit has an essay of his own about it to read. I do wonder about the role math plays. On some level, math seems an abstraction of the “real” world — there are no zero-dimensional points, nor one-dimensional lines, nor two-dimensional planes. But there does seem three-dimensional extension.
There is no perfect circle, and hence nowhere in physical reality for pi with all its transcendental strangeness. I’m still trying to wrap my head around the idea that, supposedly, every possible finite string exists somewhere in pi — and in all “normal” sequences. Every GIF, every PNG, every work of Shakespeare (in all possible translations and encodings). Math seems to lead us into some very strange territory.
And then there’s Gödel… what does that imply about math’s reality?
I was also fascinated by Smolin’s idea that time is real. I’ve been pondering lately (after really getting into Special Relativity for a series of posts I’m planning for March) about how SR conflates time with space, yet to really make it a dimension you multiply “t” by “c * sqrt(-1)” — the large constant doesn’t bother me, but the need for “i” is interesting — imaginary time?
I’m struck by how we can move about in space, but not time. And we’re all seemingly dragged along the time axis at the same speed, and there’s no going back. And the current moment seems (at least to us) to be a razor edge between a fixed past and an unfixed future (minus the idea of reality being fully deterministic or the idea of a “block universe”). If time is an emergent property of thermodynamics, what does that imply for SR?
And certainly number me among those very askance at the idea of the multiverse.
OTOH, I’ve always wondered about the meta-laws necessary for Smolin’s cosmological selection idea. I’ve been wondering if he still held that idea — apparently so. (I do have high regard for his idea of the necessity of philosophy in science.)
Lotta good stuff to chew on here! Thanks!!
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You shouldn’t get too hung up on euclidean geometry. 😉
You’re used to math as a good approximation: planes that aren’t really planes, points that aren’t really points, circles that aren’t really circles. This is an important purpose of math, but it’s not the only role math can take. To use JollyJoker’s example, 2+2=4. Take two apples, and add two apples, and you don’t get approximately four apples, you get exactly four apples. Similarly, there are six flavors of quarks: not approximately six, but really actually six. That six, that mathematical object, is something that exists in the real world.
(Godel’s kind of a sidetrack too, from my understanding. Godel proved that there is no one set of axioms that you can use to prove every truth about the natural numbers. Even if you extend this to math in general, it’s a restriction on what can be proven, not what can be observed.)
Time is legitimately different from space, I don’t think anyone would deny that. The question is, what properties are shared? “Time is imaginary space” lets time have very different properties in some ways, but still share many aspects with space. In particular, it means that both can be thought of in terms of geometry (imaginary directions aren’t very weird for geometers).
By the way, if “time is imaginary” is weird for you, a better way to think about it might be if you think about trig functions and exponentials. The only difference between them is that imaginary number i, but one is cyclic, and the other (for a negative exponent) fades away. You can think of that as meaning that things like to cycle in time, but are localized in space.
We’re only going the same speed in time if you measure speed as a ratio to time, which is kind of cheating. 😉 Even in something like special relativity, we’re moving through time at different rates.
My understanding is that if time is emergent, then you’d still need a reason why it “emerges” along the “imaginary” SR direction. I’d expect that most accounts of emergent time would include that in some way, but I’m not familiar enough with them to say for certain.
Funny story about the cosmological selection idea…when I first heard about it, I was a kid, and I didn’t know Smolin wasn’t a string theorist (or that the idea was Smolin’s). So I thought that cosmological selection was supposed to be part of string theory in some way. From a “pop science” perspective, this isn’t actually so ridiculous…at the center of a black hole, everything is around the string scale, so it’s conceivable that if a universe emerged from a black hole the compact dimensions would be rearranged, resulting in different laws of physics. Amusingly enough, this automatically solves the whole “meta-laws” problem, because string theory is quite clear on which laws would change and which would govern the whole process. Unfortunately, I suspect that the idea doesn’t actually fit in with a string theory understanding of black holes, though I don’t know enough about that part of string theory to be certain.
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Thanks for the detailed reply! I do get it and agree.
The thing about 2+2=4 is Kronecker’s famous statement: “God made the integers, all else is the work of man.” There is a reason “2” and “4” are called “natural” numbers. Even “a/b” is rational, but as we know, most numbers are irrational.
The thing about Gödel is he proved that even something as basic as Peano arithmetic cannot be complete and consistent. That seems to indicate something at least slightly non-real about mathematics. Euclid aside (and I take your point), there’s pi and other perfect abstractions throughout math.
Don’t get me wrong, I love math, and am fascinated by the underlying theory. I’m not trying to knock it down! And I’m not a constructionalist. I’m perfectly comfortable with square roots of negative numbers and infinity!
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I tend to take Godel’s statement in entirely the opposite way, really. If there are mathematical facts that are true but can’t be proven from first principles, that seems to mean that math must be “real” in some sense, independent of our construction of it.
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I’ll have to read Wigner’s essay. I feel like I’m missing the point completely whenever it’s referred to. Maths is what’s logically possible and the universe must obviously follow that. Would anyone expect the real world to break 2+2=4?
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I don’t think any of these people are expecting the universe to break the rules of mathematics. I haven’t read all of Wigner’s essay either, but from reading other people with this opinion I think there are two main points:
First, some of these people are surprised that the universe has any rules at all. That doesn’t really make sense to me (the universe is complex in a way that seems to entail some sort of rules, just based on our experience with complexity), but if you think that the universe could just be filled with grues then I guess this is an option.
Second, it may surprise some people that the universe is not only described by math, but by interesting math. I feel like this falls to the same objection (that the world is complex to a rather impressive degree), but it’s a bit fuzzy depending on what the person arguing it means by interesting.
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Well, it’s a bit surprising to me that the world is described by math that’s still undiscovered to a significant part. OTOH it seems obvious that the low energy limit of physics is simpler than the full thing and most too-complex versions would prevent thinking beings from arising.
Perhaps the fact that we can start from simple Newtonian physics and actually make progress as far as we have without hitting some complexity barriers where things become impossible to calculate without exponentially increasing resources is unlikely in some way.
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This is always a fun topic, and it’s interesting to see how diverse the opinions are among theoretical physicists about the nature of mathetmatics and the relation between mathematical theory and the physical universe.
Readers here might find this article interesting:
“Reasonably effective: I. Deconstructing a miracle”
Frank Wilczek
Click to access reasonably1_406.pdf
I found this article (which was posted in the hallway of the physics building by K. Likarev) to offer a somewhat alternative view on the matter. We should shorten Wigner’s statement to say that (analytic) mathematics is unreasonably effective for some things in natural sciences, and it happens that those things are the fundamental things. I love the example Wilczek gives – the properties of ultra pure, ultra cold semiconductors can be explained by precise elegant mathematical theory, therefore, they are studied heavily by theorists. Other things, such as high temperature semiconductors, or turbulent flow, are not so easily described (although, I suspect this is due to lack of the correct mathematical aparatus, not an intrinsic issue as Wilczek suggests). Particle physics turns out to be amenable to beautiful theory. Rereading Wilczek’s essay, he points to symmetry and locality as possible explanations for this. I find this “explanation” for the unreasonable effectiveness to be circular – are not symmetry and locality also properties of the mathematical theory itself? While Wilczek adds some insight, the mystery of the unreasonable effectiveness remains.
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Wilczek’s “helix” argument is important here, though: symmetry and locality are mathematical concepts, yes, but they are in some sense much simpler than the structures they give rise to. I think intuitively it feels a lot more reasonable that we live in a symmetric and local universe than that we live in a universe of fiber bundles…but on the other hand, this is partly just due to the fact that “fiber bundles” sounds weird and symmetry and locality don’t, even if from a mathematical perspective neither is really more baroque.
I feel like chaos and turbulence and the like are a distraction, or at least connected to a slightly different topic. In those cases we know the underlying laws, we just can’t compute their consequences. I think that Wigner’s point is more to do with fundamental laws being mathematically interesting, rather than mathematically interesting systems providing useful predictive models for the real world. That said, the ability of mathematics to provide predictive models for the real world is also something people might find unreasonable.
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I’ve been thinking about 2+2=4 and apples… It’s true that there are precisely four apples and precisely two apples being added to precisely two apples, but both two and four are still idealizations. No two apples are exactly alike, so how is it we conflate them into a class, called apples, such that we can count them at all?
Isn’t it because we have an abstract class, apples, to which members can belong or not belong? If I add two apples to two oranges, to give me four fruits, now I’m dealing with the abstract class fruits. If I have four non-similar objects in my pocket, to count them at all, I must reduce them to a very abstract class, objects.
So I’m not sure that even the integers get us away from the idea that math is an abstraction and idealization. OTOH, there seems a “so what” element to that. Aren’t most tools — even hammers, saws, and screws — based on idealized concepts?
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That’s definitely a relevant point. It’s a little odd from a physics perspective because fundamental particles generally are exactly the same, not merely different objects in the same class. In a way, that becomes another example of the real world being surprisingly mathematical in scope.
That said, I also think there’s a difference between “math is an abstraction” and “math is an approximation”. On the one hand, a pancake isn’t really a plane. On the other, you really do have four objects in the category “apple”. The fact that you made up the category, that it is somewhat arbitrary and based on your own perspective, doesn’t change the fact that per the category you made up you have exactly four apples.
I’m not a mathematical Platonist, but I don’t think of mathematics as something unique to humans either. Mathematics isn’t just composed of abstractions, it tells you about abstractions, or catalogs universal features of abstractions, whoever is making them.
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I absolutely agree mathematics isn’t solely human. From a philosophical perspective — Platonist or not — math seems to be something we discover rather than invent. I’ve heard often that, should we ever meet aliens, mathematics would form the initial basis of communication because of it universality. Many “first contact” stories involve, for example, a string of a primes expressed in some base-one format (beeps or whatever).
It seems like, once you do create classes of things, you do create counting, and all the rest seems to follow. There’s a very obvious trail that leads from naturals to integers to rationals. Triangles lead you to square roots, which leads you to irrationals, and a close look at what pi is leads you to transcendental irrationals.
Surely any civilization that learns to count discovers prime numbers.
Good point about particles. It is interesting how, as you drill down into reality, things get more and more math-y. Some theories suggest it’s all math at the root.
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“It is interesting how, as you drill down into reality, things get more and more math-y. ”
My first reaction to that was “Shouldn’t it be obvious?”, but putting my finger on why is a bit hard.
The best explanation I came up with in 10-15 minutes of thinking is that larger scales deal with collections of particles and approximations of their behavior, therefore is a bit fuzzy and inexact. This explanation would require the assumption that the fundamental truth is exact mathematics and that our current theories are very close to that fundamental truth; math isn’t just an approximation to reality anymore.
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Yes, that’s the tricky part. After all, suppose it had turned out that Descartes and/or Liebniz were right, and there were no fundamental particles, just tides of infinitely divisible “stuff” arranged in different ways. In a world where it’s fluid dynamics all the way down, you never get out of approximation-land.
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Yet another thought that struck me. String theory has been hard to approach mathematically and might be an example of where the Unreasonable Effectiveness bumps into a complexity barrier and maths can’t in practice give us useful predictions. In particular, if our world is a KKLT construction and no way exists to narrow down 10^500 vacua, we have a very explicit example of something we can’t ever calculate.
If nothing else, this helps understand in what ways maths could fail to be effective for physics and if this is the general case, the previous effectiveness of mathematics really may be unreasonable.
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There’s two different ways mathematics could be unreasonably effective: either it’s unreasonably effective in specifying the basic laws, or it’s unreasonably effective in calculating their consequences.
The latter seems to be what you’re talking about here, and I think Wilczek’s article (linked by MoreIsDifferent) makes a pretty convincing case that that’s just selection bias. Mathematics has already had a really tough time with fluid dynamics, and protein folding has been shown to be NP-hard. If string theory ends up in the same sort of category, of situations where we know the rules but can’t do any of the calculations, I’m not sure it says anything one way or another about whether we should be surprised by those cases where mathematics does work well.
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“Mathematics itself is “unreasonably effective”. That’s why, when faced with a complex world, we shouldn’t be surprised when the same simple rules pop up again and again to explain it. That’s what explaining something is: breaking down something complex into the simple rules that give rise to it. And as mathematics progresses, it becomes more and more clear that a few closely related types of simple rules lie behind any complex phenomena.”
Just as good an explanation is that we only have a few simple rules available, and can only explain the complex phenomena that arise from those simple rules. Everything else we just give a probability distribution, I guess: if we can’t predict it we can still count how often it turns out in different ways.
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Eh, yes and no. You’ve got some things that really do appear to be irreducibly probabilistic, and there it’s not really a matter of ignorance. Luckily, probabilities are also complex consequences of simple rules.
The other side are things that we legitimately don’t understand yet and thus can only count, like the distribution of quarks inside a proton. These are a pretty clear example of the other side: phenomena we can’t yet explain that appear arbitrary and messy, but that with the right mathematical understanding will likely yield to the same sorts of tricks that other things do.
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