the power and beauty of mathematics

February 2, 2013 9:14 AM

An eternity of infinities (via)

"The comparison of infinities is simple to understand and is a fantastic device for introducing children to the wonders of mathematics. It drives home the essential weirdness of the mathematical universe and raises penetrating questions not only about the nature of this universe but about the nature of the human mind that can comprehend it. One of the biggest questions concerns the nature of reality itself. Physics has also revealed counter-intuitive truths about the universe like the curvature of space-time, the duality of waves and particles and the spooky phenomenon of entanglement, but these truths undoubtedly have a real existence as observed through exhaustive experimentation. But what do the bizarre truths revealed by mathematics actually mean? Unlike the truths of physics they can't exactly be touched and seen. Can some of these such as the perceived differences between two kinds of infinities simply be a function of human perception, or do these truths point to an objective reality 'out there'? If they are only a function of human perception, what is it exactly in the structure of the brain that makes such wondrous creations possible? In the twenty-first century when neuroscience promises to reveal more of the brain than was ever possible, the investigation of mathematical understanding could prove to be profoundly significant."

posted by kliuless (23 comments total)

Wow. "Hottentot" is a pretty fucking loaded word, and it was awfully disconcerting to seen it thrown about willy-nilly there.
posted by mr_roboto at 9:49 AM on February 2 [2 favorites]

The question as to whether or not mathematical objects actually exist -- in some sense of the word "exist" -- or are essentially human constructs is an old and serious question in the philosophy of mathematics (although to tip my hand, I'm definitely in the latter camp; I see mathematics as a fundamentally human creative process). From the point of view of physics, I'd say that different sizes of infinity do not actually "exist" in a concrete sense of the word: from quantum mechanics we know that the universe is discrete and hence is finite. However, our physical models of the universe need not be finite and indeed math works much better when we don't restrict ourselves to the finite. So on the one hand we have reality (which is finite) and on the other hand we have mathematics and mathematical models (which use the infinite in a fundamental way), which are highly useful approximations to reality. See also Wigner's famous essay on the subject.

(As an amusing sidebar, it's kind of mind-blowing that Cantor's diagonal argument is still a popular target of attack among mathematical cranks, despite the fact that it's a cornerstone argument in modern mathematics. You can easily find all kinds of amazing/depressing "disproofs" by poking around on Google.)
posted by Frobenius Twist at 10:12 AM on February 2 [1 favorite]

I'll just throw this out there: the cardinality of all the physical laws had better be less than the cardinality of all the events in the universe, or else it's all just arbitrary (if you can match up each event with a "physical law" then it's all just miracles).
posted by jepler at 10:14 AM on February 2 [5 favorites]

we know that the universe is discrete and hence is finite

I don't understand the implication. We know, for example, that the natural numbers are discrete, and are infinite.
posted by Flunkie at 10:36 AM on February 2 [2 favorites]

Ah, very good point. Although the universe is discrete, current knowledge of the universe indicates that it may be infinite if it ends up being non-compact -- I misspoke. What IS true is that regardless of the curvature of space, the cardinality of the universe is no bigger than countable infinity.
posted by Frobenius Twist at 10:51 AM on February 2

Well, if the universe is countably infinite, then the power set of the universe is uncountable!
posted by Elementary Penguin at 11:08 AM on February 2 [1 favorite]

The “cardinality of the universe” seems like a fundamentally unknowable and ill-defined concept.

I think it should be unsurprising that the cardinality of any possible effective description of the universe is finite, since we have to write it down with a finite number of symbols in a finite time.

But for example, I’m not aware of any serious, predictive physical theories that don’t make use of continuous space and time variables. So until someone comes up with a discrete model that explains the emergence of space and time, it seems like there are uncountable infinities lurking everywhere in our models, which if we’re lucky we can tame into a finite description of some kind.
posted by mubba at 1:30 PM on February 2 [2 favorites]

Slightly sick, so I'm not able to be sure - but isn't the cardinality of the universe related to its total entropy at heat death?
posted by IAmBroom at 4:32 PM on February 2

A friend of mine has the continuum hypothesis (2^ℵ₀ = ℵ₁) tattooed on his back. One of the uninitiated saw us when we were swimming once and asked him what it was. After explaining, he was questioned. "What if they find out it's false, man?" he was asked.

His retort? "Well, I just go back to the tattoo parlor and get them to put a slash through the equal sign."
posted by King Bee at 5:23 PM on February 2 [4 favorites]

"What if they find out it's false, man?" he was asked.

His retort? "Well, I just go back to the tattoo parlor and get them to put a slash through the equal sign."

But it's independent of ZFC, so the dude is basically just screwed.
posted by mr_roboto at 5:37 PM on February 2 [5 favorites]

But it's independent of ZFC, so the dude is basically just screwed.

To be fair, he could get away with putting a question mark over the equal sign.
posted by ILuvMath at 7:38 PM on February 2 [2 favorites]

I think it should be unsurprising that the cardinality of any possible effective description of the universe is finite

How do you define the cardinality of a description of the universe?
posted by empath at 8:17 PM on February 2

> The question as to whether or not mathematical objects actually exist -- in some sense of the word "exist" -- or are essentially human constructs is an old and serious question in the philosophy of mathematics (although to tip my hand, I'm definitely in the latter camp; I see mathematics as a fundamentally human creative process).

I don't think you're correctly representing the consensus - I think that the overwhelming majority of people who've seriously studied the subject believe that mathematical objects are discovered, not created by humans.

It does depend on whether you think science, as opposed to technology, is discovered or created. I think pretty well everyone thinks that science is not created by scientists - but I have talked to some people who (more or less) think that "doing science" is a spell that creates science, and that if a person of great conviction had had different ideas in the past, then science would be different.

I don't believe it - and I don't even think this idea is science, since it isn't disprovable - but I can't actually refute it. (This btw is also the key point underlying the novella "Waldo" by Robert Heinlein, which also gives its name to the remote manipulator, first mentioned in that story.)

So I'm only speaking to you if you believe science is discovered and not invented.

But why do you think science is discovered, but, say, fashion is not? Or, if I started some new activity, how would you determine if what I were doing was science or not?

In both cases, you would point to the scientific method - you start with objective observations that can be reproduced by any other interested individual; you formulate hypotheses to explain these observations; you devise experiments to test these hypotheses; you accept or reject them based on the results of the experiments; if you can explain a lot of observations, and all the experiments test out, you have a theory.

You can see this coming, but the process of doing mathematics is exactly the same. No one just sets out to prove a theorem - instead, they exercise mathematical objects and observe their behavior, and after a while form some hypothesis like "there are infinitely many prime numbers" or "every event number is the sum of two primes".

Now that you suspect this hypothesis is true, you need to verify it. In mathematics, proofs correspond to the experiments of science - and like science, people work on proofs all the time that are true but fail to solve the problem they set out to - most proofs "fail" just like most experiments "fail".

In fact, I'll bet you're not going to be able to pose any consistent definition of science that excludes mathematics. Sure, you can try to, say, include "The physical world" or some such in your definition - but mathematics makes all sorts of amazingly testable predictions about the physical world, and humans learn these from a very early age ("If John has three apples...") or to quote someone smart, "If you think mathematics doesn't constrain the real world, try making a sixth Platonic solid.")

You're also going to have trouble if you try to bar "things that don't exist". We all agree in the existence of doorknobs and fish, but is "a quark" (an item that can never be observed and has a highly indeterminate position) any less real than "the number 2"? What about "evolution"? Can I buy an evolution in the store? What about evolution makes it exist more than the number 2?

Many people accept that argument but don't find it satisfying. But there are other reasons to strongly suspect that mathematics does exist.

It has very often happened that multiple individuals at different times and places have rediscovered the same mathematical ideas. Would this really happen if mathematics were invented?

There is also the unreasonable effectiveness of mathematics in the physical sciences. That classic paper is also extremely readable and thoughtful, and points out that the same mathematical patterns appear in science over and over again, work surprisingly well, and were often discovered in science long after those patterns had been discovered by humans. (And if you claim that these isn't discovered in science but that that mathematical pattern was imposed on science by humans, I'd point and science and say, "It works!")
posted by lupus_yonderboy at 8:53 PM on February 2

Frobenius Twist: "from quantum mechanics we know that the universe is discrete and hence is [at most countably in]finite."

I'm unaware of any quantum theory that doesn't use uncountable infinities in a completely fundamental way. If we literally can't describe reality accurately without them, I'm not sure what sense it makes to say that they don't "exist".
posted by Proofs and Refutations at 10:33 PM on February 2

How do you define the cardinality of a description of the universe?

Well, I guess that was nonsense (only sets have cardinalities), but what I meant was that an effective description is a finite sequence of symbols in some finite alphabet, so the set of all possible descriptions of the universe is countable.
posted by mubba at 10:44 PM on February 2

Well, I guess that was nonsense (only sets have cardinalities), but what I meant was that an effective description is a finite sequence of symbols in some finite alphabet, so the set of all possible descriptions of the universe is countable.

There are plenty of uncountable sets of functions that could theoretically describe the universe, though -- any of them that include the set of reals, for example. Sure you could theoretically replace any real number with a variable theta(or whatever) in the final equation, but unless you define the real number (of which there are an uncountable number, with uncountable definitions), you can't really say that's a description.
posted by empath at 11:48 PM on February 2

> I'm unaware of any quantum theory that doesn't use uncountable infinities in a completely fundamental way.

Really?! Do tell!

It seems hard to believe - if you have uncountable infinities then you can encode an infinite amount of information in a single state. But the Uncertainty Principle explicitly limits the amount of information that exists...
posted by lupus_yonderboy at 12:37 AM on February 3

There are plenty of uncountable sets of functions that could theoretically describe the universe ...

To me, a description means something I can write down and send to you, a sequence of bits. When we need to communicate a specific real number, we write down an approximation to some number of digits, or an equation that lets you calculate them.

Part of the “unreasonable effectiveness” is exactly that this works. The series converges or whatever, so you can stop adding terms at some point, and the inexact result you have is good enough for all practical purposes, and can be communicated by one finite being to another.

But like you said, the real numbers are there in the formalism, clearly representing some property of the universe that looks a lot like uncountable continuity in space and time, so it seems severe to say the universe is countable or finite, even though maybe it can be effectively described as finite in a given region of spacetime, within the limits of measurement by a finite being embedded in the universe. (Would you ever need to know a position to within 1/1000 of a Planck length, even if you could?)
posted by mubba at 9:17 AM on February 3

Presumably what Proofs and Refutations Is referring to is the fact that a fundamental starting point for quantum mechanics is a vector space over an uncountable field (eg the real or complex numbers). From this one derived various discrete states for the universe (by taking eigenvectors for appropriate linear operators) but the point is that to get to the useful (discrete) model, the linear algebraic setup requires a nondiscrete starting point.

And it's for this reason that I personally don't think the mathematical model "exists." It's an idealized abstraction of reality that helps us to understand reality, but it isn't equal to reality. The map is not equal to the terrain.
posted by Frobenius Twist at 9:18 AM on February 3

> But like you said, the real numbers are there in the formalism, clearly representing some property of the universe that looks a lot like uncountable continuity in space and time,

But it doesn't actually look very much like "uncountable continuity". It isn't continuous or anything at all like continuous at lower but still perfectly observable level; but more, there's nothing there at all that's at all "uncountable". You could do all your arithmetic with rational numbers, arbitrary precision fractions - this space is countable and would work exactly as well as the real numbers do for quantum mechanics, where only a limited precision is needed or even exists.

> Presumably what Proofs and Refutations Is referring to is the fact that a fundamental starting point for quantum mechanics is a vector space over an uncountable field (eg the real or complex numbers).

Again, if these were really points in a real or complex vector space, each point would contain infinite amounts of information.

> And it's for this reason that I personally don't think the mathematical model "exists." It's an idealized abstraction of reality that helps us to understand reality, but it isn't equal to reality. The map is not equal to the terrain.

"The map is not equal to the terrain" has nothing to do with whether maps actually exist or not. Indeed, were your argument correct it would prove also that maps don't exist.
posted by lupus_yonderboy at 9:47 AM on February 3 [1 favorite]

You certainly can't excise the infinities from quantum mechanics just by taking a few rational approximations. For a start the transcendental numbers e and pi are inextricably linked to the theory (we are after all talking about waves/rotations). You've got to be able to perform calculus, which pretty much requires the real numbers. And on top of that you've got Hilbert spaces of almost always uncountably infinite dimension. Then when you finally extract your prediction, quantum theory often provides you with a probability, and you are by no means guaranteed a rational probability.

That final point is particularly challenging, if your measured proportion keeps approaching something like 1/sqrt(2) with more accuracy the more experiments you perform and shows no sign of stopping, exactly what justifies excluding the reality of the irrational solution?
posted by Proofs and Refutations at 12:40 PM on February 3 [2 favorites]

if your measured proportion keeps approaching something like 1/sqrt(2) with more accuracy the more experiments you perform and shows no sign of stopping, exactly what justifies excluding the reality of the irrational solution?

But that's not actually what happens in a real experiment. In a real experiment, you measure the quantity you're after with greater precision until either you're overwhelmed by flaws in your apparatus ("systematic errors") or you discover a new effect that makes some small difference.

For example, suppose you want to confirm that the diagonals of a unit square really have length √2. (I can't think offhand of a quantum-mechanical experiment whose answer is 1/√2; usually the square roots appear in the wavefunction and go away when you compute probabilities. I'm sure somebody else will come up with one straightaway.) So you go to the hardware store and get some 2x4s, cut them so they are exactly the same length, build a square-ish frame, and square the frame until the two diagonals are exactly the same length. That length should be √2 times the length of the sides, right? This is when you'll discover that you haven't overlapped all four of the corners in the same way. Fix that and you'll realize that you shouldn't have built your frame vertically, because the vertical members are compressed under the load of the top piece and so the four lengths are no longer the same. Build it horizontally and you'll realize that all 2x4s have a little bit of a bow to them, so that the length you measure is a little bit different depending on whether your tape measure rides the convex part of the bow or not, and that the bow gets bigger when you put your frame under tension.

So now you design a new frame with titanium sides, and you get to worry about new instrumental effects. What is the mechanical tolerance associated with machine screws? What, exactly, is the "unit side" now that you have a more complicated joint system --- do you measure the corners from the center of a bolthole, or can you make some better monument to measure from? How much will your titanium sides expand or contract as they heat or cool --- do you need thermometry and thermal regulation? You start to fantasize about getting away from all these materials. Maybe your unit square could be a laser interferometer, and you can measure distances by sending fast pulses and doing a good job with timing resolution. You build it and you rediscover that a Mach-Zender interferometer is sensitive to the rotation of the Earth. Now your measurement of √2 is mixed up with the astronomical definition of a day.

There's another famous experiment you can do where you take a needle and toss it on a piece of lined notebook paper. Sometimes the needle crosses a one of the lines and sometimes it doesn't, and the ratio of the frequencies of these two outcomes (or something like that, I'm remembering dimly) turns out to be π. But it only turns out to be exactly π if the spacing between the lines is exactly the same as the length of the needle, and if the spacing of the lines is completely regular, and if there is no thickness to the lines on the paper and no parallax issues in deciding whether the needle is touching a line or not. Every time you toss the needle, in this experiment, and have so say, "hmmm, really touching the line there or not? pretty close", you are biasing your result.

The quantum theory of electromagnetism is currently laid out like this. At the coarsest approximation, QED explains that we have electricity because charged particles are always exchanging virtual photons with each other: one-photon exchange gives the standard 1/r² Coulomb force. But that virtual photon has some small probability to fluctuate into a particle-antiparticle pair for a while en route. It's most likely to fluctuate into an electron and positron, but any other particles are also fair game. This is called "vacuum polarization" and means that the electric force is slightly stronger between particles that are very close together, and that how much stronger depends, slightly, on the mass of the heaviest charged particles that exist. The magnetic versions of these measurements have actually been used as a way to search indirectly, for new particles. They're called "g minus 2" experiments since, in the one-photon approximation, each particle has a magnetic property called "g" that is exactly 2.

Frobenius Twist said it nicely above: infinities in our models of reality don't necessarily imply that there are real infinities, or real transcendental numbers, with manifestations in the physical world.
posted by fantabulous timewaster at 9:46 AM on February 4

Does probability come from quantum physics? - "One answer to this problem has been to add a new ingredient to the theory: a set of numbers that tells us the probability that we are in each pocket universe. This information can be combined with the quantum theory, and you can get your math (and your calculation of the mass of a neutrino) back on track. Not so fast, say Albrecht and Phillips. While the probabilities assigned to each pocket universe may seem like just more of the usual thing, they are in fact a radical departure from everyday uses of probabilities because, unlike any other application of probability, these have already been shown to have no basis in the quantum theory."

Why quantum mechanics is an "embarrassment" to science - "Ninety years after the theory was first developed, there's still no consensus on what quantum physics actually means."

Weinberg on quantum foundations - "Considerable progress has been made in recent years toward the resolution of the problem, which I cannot go into here."

In Mysterious Pattern, Math and Nature Converge - "Universality is thought to arise when a system is very complex, consisting of many parts that strongly interact with each other to generate a spectrum. The pattern emerges in the spectrum of a random matrix, for example, because the matrix elements all enter into the calculation of that spectrum. But random matrices are merely 'toy systems' that are of interest because they can be rigorously studied, while also being rich enough to model real-world systems, Vu said. Universality is much more widespread. Wigner's hypothesis (named after Eugene Wigner, the physicist who discovered universality in atomic spectra) asserts that all complex, correlated systems exhibit universality, from a crystal lattice to the internet."

The Unreasonable Effectiveness of Mathematics in the Natural Sciences - "It is this 'what if we took this equation seriously?' factor that is, to my mind at least, the spookiest thing about the unreasonable effectiveness of mathematics in physics."

Leonard Susskind lectures on cosmology - "Leonard Susskind introduces the study of Cosmology and derives the classical physics formulas that describe our expanding universe."

Richard Feynman on Quantum Mechanics - "Photons: Corpuscles of Light."

What if technology makes scientific discoveries that we can't understand? - "A computer program known as Eureqa that was designed to find patterns and meaning in large datasets not only has recapitulated fundamental laws of physics but has also found explanatory equations that no one really understands. And certain mathematical theorems have been proven by computers, and no one person actually understands the complete proofs, though we know that they are correct."

Paradoxes of Randomness - "This leads to my first result on the limits of axiomatic reasoning, namely that most numbers are uninteresting or random, but we can never be sure, we can never prove it, in individual cases. And these ideas culminate in my discovery that some mathematical facts are true for no reason, they are true by accident, or at random. In other words, God not only plays dice in physics, but even in pure mathematics, in logic, in the world of pure reason. Sometimes mathematical truth is completely random and has no structure or pattern that we will ever be able to understand."
posted by kliuless at 4:56 PM on February 28