29 May 2016 6:47 PM (musing | mathematics)
What am I talking about?
In mathematics, one makes statements with great certainty about objects of whose nature one is ignorant. I say “There are infinitely many prime numbers.” and I know it to be true. I don't know how a number “exists”.
“There are infinitely many prime numbers.” is a sentence about the structure of the natural numbers (individual numbers are meaningless without a structure). It states that however large a natural number you pick, there are greater natural numbers that are prime. In classical logic, if there happen not to be such things as natural numbers, the statement is false.
For most of Western history, the majority of people who seriously investigated the properties of natural numbers thought the natural numbers were a real thing that existed. This might be because Western matheamtics was in large part founded by a mystic named Pythagoras who believed all the world was Number. The idea that abstractions Really Exist somewhere became known as Platonism since Plato originated and popularized the idea that you have Beauty, Goodness, The Ideal Circle, the Natural Numbers… out there somewhere more Really Real than the everyday world which might be thought of as a mingling of their shadows. It's also known as Realism, because it holds that mathematical abstractions are Real and Exist independently of anything else.
Standards of Beauty change over time; there are competing claims of Goodness, but everyone (Intuitionists and Constructivists accept fewer theorems, but they don't come to conclusions that contradict classical mathematics) has the same math. There's never been an instance of nature acting against mathematics; on the contrary, people invent newer, ever more abstract mathematical ideas and someone finds a way to apply them to the natural world. Thus, mathematical Realism stays while Beauty and Goodness are thrown into the seas of cultural contingency.
Realism invites questions. Exactly how do these things exist? Transcendently? Outside the universe? Beyond space and time? That's how it's usually taken. It avoids having to explain how something completely immaterial could exist in space and how something eternal could exist in time. It also matches the intuition that mathematical statements would be true whether anything existed or not.
The most famous and compelling argument for Realism, the Indispensability Argument of W. V. O. Quine, states that since our best physical theories rely on mathematical abstractions, we ought be as willing to accept the existence of those abstractions as we are electrons. This is compelling, but the physical theories also explain how electrons interact with each other and other charged particles. The second derivative operator does not interact with a moving rocket in the same way that the gravity of a planet does. Furthermore, there are mathematical abstractions that no physical theory depends on. A Realist position in which all statements about the Fischer-Griess Monster Group are true only if some physical theory is found that depends on them is not very Realist. This is the main reason why Quine and others referred to his stance as Empiricist.
There are more reasons to ask which abstractions are Real. Leopold Kronecker famously said "God made the natural numbers; all else is the work of man." Almost every Realist would say that the natural numbers exist, they feel too primitive and natural not to. While the natural numbers' naturalness is almost certainly a property of humanity rather than a property of the natural numbers, let us say they exist.
Let's throw in the integers and rational numbers, those are fairly uncontroversial. What about the real numbers? Do they live up to their name? The Finitists reject uncountable sets; some go further and, while accepting the existence of every natural number, reject the infinite set of natural numbers. If they are correct and real numbers do not exist, statements about them may be well-formed and provable but false. We might allow other things to exist: lambda calculi and set theories. Which set theories? ZF? ZFC? Something else? There are constructive set theories. There are set theories that have only a countably infinite number of finite sets. This is my biggest problem with Realism. Unless you accept ontological maximalism (everything that can exist does exist, where 'can exist' is usually some notion of 'follows from some consistent set of axioms'), you don't know whether anything you say is true. The things you're talking about might not exist and there's no way to find out, unless you accept Quine's formulation and its contingency of mathematical truth.
Realism is kind of weird. Let's make some conservative assumptions: the natural numbers, lambda calculi, and some set theories, logics, and other countable things exist. No uncountable anything. The Natural Numbers are Real, truly, in themselves. They also exist, Really, as multiple constructions in set theory, as multiple constructions in logic, and as multiple constructions in lambda calculi. They have to. If our Realist abstractions are to have any meaning, the Number Five has to Really exist in the Natural Numbers constructed from the Set Theory that Really exists just as much (if not moreso!) as it exists when we pile up groups of five rocks and see how many equal groups of rocks we can divide them into.
No wonder people people say mathematical Realism sounds like weird religious mysticism! Start thinking that way and you'll fall into the Tree of Life with the Natural Numbers at the crown, flowing through lesser abstractions into the world. Except the Tree of Life has a top and a direction of flow. With Gödel numbering we can spin it around and put logics at the top. Their theorems and rules of inference would stand supreme, reflected in the natural numbers and flowing down into the world. We could throw out the tree of life entirely and have a try at Indra's net, with abstractions reflected in other abstractions, each complete in itself and constructible in others, shining upon the world. Georg Cantor, Master of Infinity, believed in the Absolute Infinite, the Infinite that contained all other infinities. Too infinite to be a number, each of its properties reflected in the things it comprised. He also thought the it was God. Cantor was a mathematical Realist and ontological maximalist. He believed that everything consistent (lacking internal contradictions) that followed from some axiomatic system was Real.
Squishy Organic Stuff?
Traditional mathematical Realism is dualist. There's matter, and there's math. Dualism has all sorts of philosophical problems, like how your two substances interact. Also, nobody takes it seriously. It's socially condemned. So, people come up with alternatives. One of the most recent, championed by George Lakoff (like most things favored by George Lakoff, it isn't very good), is the Embodied Mind theory of mathematics. This school of thought tries to explain mathematics as a behavior born of evolution and instinct. To the extent that this is true, it is trivial. Professor Lakoff tries to get rid of the idea of general reasoning over logical abstractions and reduce all of mathematics to a few basic metaphors related to interacting with the physical world.
It fails, for one, because it assumes that children learning their multiplication tables think and reason about the natural numbers in the same way and with the same internal abstractions as number theorists proving a theorem. Professor Lakoff's theory is written in terms of representation rather than relation. This is the biggest problem. As you can see above, there are many ways to construct one abstraction in terms of another, and Professor Lakoff's way of building mathematics from metaphors requires that each method of construction lead to a different mental object. This fails utterly at capturing how mathematicians actually think. It also violates the most fundamental attribute of mathematics: that its subject is structural and relational. Lakoff's account of the predictive power of mathematics shows where his entire notion of the Embodied Mind (even apart from mathematics) goes wrong.
He explains that, since humans evolved to survive in the physical world, they should expect that their minds and metaphors would be very well suited to modelling the physical world. This sounds like a very reasonable, logical answer. It's false. You, as a human, are very, very, very bad at probability. Astoundingly bad. You have crude heuristics for running away from things that might be snakes in the grass, but they're awful at making accurate predictions. This should be enough to kill off Lakoff's explanation. Humans can develop probability theory and build abstract mental machinery to make up for their more ‘embodied’ aspect's failure. Mathematics also works remarkably well at grasping quantum electrodynamics, which has nothing to do with the ancestral environment. The biggest flaw in the current crop of Embodied Mind theories is that they assume that (to borrow Daniel Kahneman's term) our minds comprise System 1 and nothing else. Embodied Mind theories may one day be quite valuable in education or predicting systematic errors, but their authors will need to do better than writing the word ‘metaphor’ repeatedly sprinkled with an occasional PHRASE IN CAPITAL LETTERS.
One of my favorite answers to the question of what mathematics is about is ‘Nothing!’. Hartry Field declared that mathematical objects do not exist and all statements about them are false. He called it fictionalism: the belief that mathematics is a useful fiction. I adore this theory, not because I believe it, but because of the work Professor Field did to support it.
In Science without Numbers, he recreated Newtonian mechanics and gravitation without numbers. Instead of numbers he used regions of space-time and notions of congruence and betweenness. It's a triumph and one of the most awesome things I've ever read. It makes me awfully happy, but I'm not convinced. For one thing, Field ends up with a very abstract, rigorous, and structured system. It doesn't look like a demathematicized science to me, it looks like a beautiful system of calculus invented by aliens. It is, too. It maps very well onto Calculus. Field attempted a proof that mathematics does not conflict with any purely physical theory. He thought the de-mathematicization and lack of conflict together could explain the unreasonable effectiveness of of mathematics. It doesn't work for me. To me, Field's demathematicization is math (also assuming the reality of space-time regions independent of anything else plus all the heavy logical machinery he used racks up a lot of metaphysical debt), while showing that mathematics is not inconsistent with known physical theories seems insufficient to explain why mathematics and the world should have anything to do with each other.
Once upon a time there was a man named Meinong. He rejected the idea that you couldn't make true statements about nonexistent things. After all, unicorns have one horn. Nemesis is a twin star to the sun that caused the extinction of the dinosaurs. I can speak about the Natural Numbers whether they exist or not. He made existence a property something could have like redness or tallness. Some things happen to exist and some things happen not to exist. Some things are impossible (those either lacking properties that grant them mass and weight and extent in space and time, or those having contradictory properties).
The theory of nonexistent objects requires that all nonexistent objects…nonexist— Square circles, prime numbers with fifty divisors. This is what the phrase ‘metaphysically extravagant’ was made for. (No, really, it was!) Meinong's theory was that for every set of properties, there is an object. Some objects had the property of existence. I like ontological maximalism as much as the next guy, but, like Georg Cantor, Master of Infinity, I'm only interested in objects that are consistent. Bertrand Russell destroyed Meinong's theory, causing it to explode into a mess of paradoxes. There have been attempts to rehabilitate it, but they lack the appeal of the original.
Just playing around?
Mathematical Formalism is the belief that mathematically true statements are statements about the evolution and manipulation of formal systems. One variant, Term Formalism is concerned with syntactic manipulations of large (possibly infinite) vocabularies of primitive terms. It was best elaborated by by Haskell Curry. He defined mathematical statements as true if it would be possible to derive the associated relations of primitive terms from other true relations of primitive terms. This is elegant, but is weighed down by so much metaphysical debt in the form of reified logical machinery and primitive terms that it falls back into Realism. It's more interesting as a primitive base from which other things can be constructed than as a metaphysics of mathematics.
The other variant of Formalism, Game Formalism, defines mathematics as the manipulations of strings in accord with rules. A statement is viewed as true when an appropriate string manipulation yields it. This is the most popular escape from Realism. In retrospect, this is surprising. It doesn't explain why mathematics should describe the world so well. It doesn't bear any relationship to how mathematicians think. Mathematicians do not take an arbitrary string and apply arbitrary allowable manipulations to it. They think about sets and functions and shapes. Automata theorists think about string manipulations, but they think about them being done by abstract machines working under complexity bounds. Furthermore, statements are neither true nor false until someone has performed the appropriate string manipulation, and some theorems, like whether very large numbers are prime (large enough that to answer will take exponentially longer than the lifetime of the universe), will forever be neither true nor false.
I think the popularity of Game Formalism comes from people not thinking about it very much. They like the connection between proof and truth and don't grasp that the ‘proof’ in Formalism and the ‘proof’ in their heads have little in common. It lets them not be Realists with a minimum of effort. I also suspect that the intuitions of most Game Formalists tend toward what is actually Modal Structuralism but that they have never heard of Modal Structuralism. I might be biased.
I used to be a Game Formalist.
An Idea Objects to the Company I Make Him Keep
One night, as I was sleeping, a figment of my imagination came to me. He was a Realist and he was not very happy with me. For, you see, I think that the generalized continuum hypothesis is likely true. Kurt Gödel, Lord of Logic, proved that it could be proved neither true nor false within the generally accepted axioms of set theory. He believed it was false. Georg Cantor, Master of Infinity, hypothesized the hypothesis. He accepted Gödel's proof and thought his hypothesis was true. They were both Realists; they are allowed to believe unprovable things about mathematical abstractions.
The figment explained to me rather fiercely that I had no business claiming to be a Formalist, since I certainly didn't believe it. If I really believed it, I wouldn't have opinions on proved-undecidable hypotheses. That's the thing that anyone but a Formalist can do! By my stated beliefs it was not merely unknowable, it must be and must forever remain neither true nor false and I was a cad and a bounder who had just adopted what seemed like an easy way out of a mentally challenging question and I should be ashamed! (It was friendlier than you're probably imagining.)
Having been informed of my error, I spent some time reading and thinking about a way to believe in the truth of mathematical statements that would not get me yelled at by the other things I think about.
I settled on Modal Structuralism, the belief that a statement about some mathematical object is a statement about how any entity possessing the structural attributes defining that object must behave in any possible world in which it exists, while committing to the idea that at least one possible world has something possessing those structural attributes. So, if I make a statement about the real numbers, I am saying that in any possible world where something has the properties of the real numbers, that thing must behave the way I say it does, and that such a world is possible. It might be this world if space really is continuous and every straight line has the structure of the real line. If space is pixellated then it's some other possible world.
Modal Structuralism has a lot going for it. When I think about sets or the real line or functions, I'm thinking about sets, the real line, or functions because the structure is what matters, not the construction. It addresses the predictive power of mathematics. Theorems about a mathematical object predict the behavior of some aspect of the world when that aspect of the world models the structure of that mathematical object. It requires one reinterpret every mathematical\n statement to be about structures modeling something in a possible world, but I don't mind that. More concerning: what the heck is a possible world?
Possible worlds evolved as a tool in logic to evaluate statements that involve the world being other than it is. The statement “If there were a present king of France he could be named Louis.” is true if, in at least one possible world that is pretty similar to ours but in which France has a king, that king is named Louis. Possible worlds aer usually defined as complete descriptions of a world with consistent propositions and histories. Modal Structuralism also has one of the problems of Realism: what do I admit as possible? We're back to the same arguments over whether to admit everything consistent, like Georg Cantor, Master of Infinity, or to reject anything infinite like Leopold Kronecker. I, personally, side with Cantor. (By now, you've probably guessed that I like Georg Cantor, Master of Infinity, a lot.)
Why is there all this stuff here?
I once read a book called Why Does the World Exist?. It was a wonderful whirlwind tour of all sorts of mad ideas trying to explain why there's anything at all. We start with the idea that Nothing is such a strong force for annihilation that it eventually annihilates bits of itself and creates something. Others argue that nothing is impossible. The one that influenced me most was the argument that the world exists because of a primordial need for goodness.
This gentleman claimed that what made any world at all exist was of much less interest than why this world in particular exists. Thus, he said, all worlds containing beauty and goodness were the ones that came into being. I don't buy this idea because holding beauty and goodness as objective values is absurd. It made me think, though. We know that the world exists, so having more worlds exist isn't an extravagant leap. Some physical theories already suggest multiple, non-interacting universes. (Each such theory would be its own ‘world’.) Positing something that sifts the possible worlds and actualizes some of them is much more extravagant than multiplying the number of worlds. Thus, the best answer, with respect to Occam's Razor, to “Why does this world exist and not some other?” or “Why does this subset of worlds exist and not some other subset?” is “What makes you think that? All possible worlds exist.” I later discovered this belief is called Modal Realism.
So, Real Things?
It took me longer than it should have to realize it, but believing both an ontologically maximalist form of Modal Structuralism and Modal Realism compelled me to believe in the Reality of all mathematical objects. This came as a shock, considering how much effort I'd put into avoiding mathematical Realism. Now, my confidence in Modal Realism is fairly low for a belief I claim to have. It's like I'm tidying up my mental house and want everything in its place, and Modal Realism seems the neatest and tidiest for now.
Without realizing it, I had run head-first into Max Tegmark's Ultimate Ensemble. Tegmark claims that not only do all mathematical objects exist, but nothing but mathematics exists. Having arrived by the scenic route, this seems more plausible than when I first heard of it. It was a natural step. I'd committed to all these possible worlds containing all these things that model mathematical objects derived from axiomatic systems, and there was only one way to make it simpler.
Electrons, photons, quarks, and gluons, Ws, Zs, taus, and muons, neutrinos, gravitons, and the Higgs all have no internal life. Each exists only in its interactions. We have internal lives, they're visible in our behavior. They're, to some degree, measurable through examinations of our brains. We ourselves are made of electrons, quarks, photons, and gluons with the odd W or Z popping into being and a neutrino zipping away. Everything that happens in the universe is a matter of structure and relation.
We need the structure. We don't need the, well, stuff. Things have no essence, only relation, and the solution to the Dualism inherent in mathematical Realism is to throw out everything but mathematics. The answer to what puts the fire in the equations is that all equations have fire pre-installed. Burning. Somewhere.
You win this round, figment.
 Prime numbers are natural numbers divisible only by one and themselves. Euclid, an ancient Greek explorer who wrote the definitive text on the geography of Flatland, proved that however many primes you have discovered, there must be at least one more.
It's simple and elegant and goes like this. Take all your primes and multiply them together. We'll call that The Product. Add one to The Product and you'll get The Sum. It might be the case that The Sum is prime. If it is, you're done, because it couldn't have been on the original list of primes.
If The Sum isn't prime, then it must have a Prime Factor. If that Prime Factor were on the list it would have to divide The Product, but to divide both The Product and The Sum, the Prime Factor would have to divide one. And it can't. So it isn't. Therefore, how many primes you may have, there are always more.
 As you can see from the above, this statement could be written more formally as "At least one natural number is prime and for any subset of the natural numbers all of whose members are prime there exists a natural number which is prime and not contained in that subset." The first half of the conjunction is important. If there were no prime numbers at all any statement we might make about all prime numbers or all sets of prime numbers would be true.
 I wonder if anyone has considered an Imminent Realism where mathematics pervades all of space and time in some immaterial sense. It's unclear what that would mean, but it's unclear what it means for them to be beyond space and time. Intuitively, Transcendent Realism feels a better match for the idea that all mathematical abstractions exist, aloof, outside reality. Imminent Realism would mesh better with the idea that only those demonstrated in physical law exist, something of a match for the Indispensability Argument.
 There are people called Ultrafinitists who reject the existence not only of the set natural numbers, but who reject the existence of very large natural numbers. They write very interesting philosophy papers but not much interesting mathematics.
 While I reject George Lakoff's ‘embodied mind’ analysis of mathematics as not actually being very good at describing mathematical reasoning, it is quite true that what one is used to, including embodiment and the environment, strongly influences ones notions of naturalness. I could imagine minds living in a gaseous or liquid world (maybe a particularly runny gel) whose most ‘natural’ number system is the real numbers. Their ancient civilizations might invent wonders of analysis without any idea what a prime number is. It might take millennia for anyone to discover that a counterintuitive, unnatural subset of the ‘natural’ numbers exists and is of interest. The basics of number theory could, for them, be post-doctoral level material.
 To be fair I reject most things written by George Lakoff.
 A long time ago there was a man named Georg Cantor, Master of infinity. He was the first to rigorously define infinity. His most famous insight was that some infinities are more infinite than others. The natural numbers are the least infinite and were called countably infinite. He proved that the real numbers were uncountably infinite by a very clever trick which I will let Vi Hart explain.
 Gödel numbering, invented as part of the machinery for Gödel's famous incompleteness theorem, allows one to turn theorems, or anything else that can be represented as a finite string of finitely many symbols, into a natural number. One assigns a number to every possible symbol. To encode a string, raise the number for the first symbol in the string to the power of the first prime, the number for the second symbol in the string to the power of the second prime, and so forth, then multiply them all together.
Once you have a Gödel numbering set up, properties about theorems in a logical system then become number theoretic properties and rules of inference become functions. Gödel numbering, by providing a mapping between various things and the natural numbers, also provide a convenient way to prove that various things are countably infinite. Alan Turing, Dreamer of Machines, used it to prove the countability of the computable numbers, for example.
 One of the few lightning-affine deities who isn't an embarrassment.
 Note that Cantor, Master of Infinity, believed the Absolute Infinite was an inconsistent idea. Something that, by his definition of mathematical freedom, was beyond mathematics. He also believed that his work on transfinite sets was communicated to him from Heaven and that he had been chosen to reveal it to the world. He was Catholic and did not, as some claimed, try to ‘reduce God to a number’, saying instead that transfinite numbers were ‘at the disposal of the Creator’ just like everything else.
 After whom the programming language Haskell is named (his wife once mentioned that he didn't like his first name, to the chagrin of Haskell's developers). He also gave his name to currying, turning a function taking multiple arguments into a function taking one argument and returning a function which takes the second argument, and so on aside from the last which returns the value. The logician Schönfinkel invented the concept before Curry did, so some people (but not very many, for obvious reasons) use the term ‘schönfinkelization’ instead.
 Wouldn't you be if you were a figment of someone's imagination?
 The generalized continuum hypothesis states that infinite cardinalities come in neat succession one after the other, that the first infinite cardinality is that of the natural numbers and the second infinite cardinality is that of the power set of the naturals and is also the cardinality of the reals, and that the third infinite cardinality is that of the power set of the reals and on and on with nothing in between. Its negation allows cardinalities between those of the natural numbers and the cardinality of their power set; specifically that the cardinality of the reals may be less than the cardinality of the power set of the natural numbers.
 Cynicism is, in the modern day, probably the single most common sign of moral and intellectual failure. Also, the Surgeon General would like to warn you that it greatly increases your risk of developing soul cancer.
 I once saw an interview with Peter Higgs in which he referred to it as ‘the particle that happens to bear my name’ with an annoyed air. He is even more unhappy about it being called ‘The God Particle’.