Because I had a discussion with a Platonist last week, I've been thinking about it even more than I already was--it being something of a major theme in Dot, Dot, Dot: Infinity Plus God Equals Folly (which I'm hoping to publish pretty soon!). While listening to music earlier today, I hit upon a thought that added a lot of clarity for me regarding something Ian Stewart mentions in his Letters to a Young Mathematician: "the working philosophy of most mathematicians is a mostly unexamined Platonist-Formalist hybrid." Why is Platonism in there? Why does mathematical Platonism feel so reasonable?
Now, I want to clarify a matter before getting too involved with today's insight. That is that the discussion I had with the Platonist was essentially that mathematics is in nature, and not that mathematical objects simply exist in a Platonic realm of ideals. These two things, strictly speaking, aren't the same. Mathematical Platonists, for the most part, I think, are of the second sort more often than of the first, although things get a bit blurry there. Hopefully I'll be able to shed some light on this. It's also worth pointing out here that working mathematicians don't have an awful lot of need to bother with the philosophical foundations of mathematics--hence Stewart's use of the word "unexamined." We just don't need to bother to do mathematics, at least not very often.
My insight occurred while listening to a song that featured a very prominent bass (guitar) line, something I recognized as being a popular musical style from the late 1990s and early 2000s. Noting that it was popular then but not decades before got me thinking about musical styles in general, and then all of a sudden it popped into my head that there was absolutely nothing to have stopped, say, the Beatles from pulling off the exact bass line I was hearing from the Red Hot Chili Peppers. The bass guitar existed at the time, and the musical arrangement itself could literally have occurred at any point in human history. I wondered--did the Chili Peppers "discover" that style, or did they invent it? What precluded the Beatles or Beethoven--or one of their contemporaries--from discovering the appealing rhythm there?
The answer to that riddle doesn't matter much (but is pretty likely to come from the fact that musical styles and the features therein evolve over time), but it led me to an old analogy I used to think about with doing mathematics, revealing that at one point in time, I was indeed a mathematician with an unexamined Platonist-Formalist hybrid position. That analogy, a bit appropriately, is of a cave.
I used to think that doing mathematics (or science, maybe, but somehow differently) was a lot like exploring a cave. We might venture down this tunnel or that, and sometimes we'd find wide-open spaces filled with wonders, and sometimes we wouldn't. Some tunnels simply dead-end. Other tunnels constrict maddeningly and involve a lot of struggle for very little reward and, often, ultimately failure. The metaphor, though, kind of holds the idea that the whole cave exists, and we as mathematicians are exploring it, discovering vistas of mathematical interest as we go. That's mathematical Platonism.
And it's correct, except that it isn't possible to actually make it into full-blown Platonism, the kind where we say "oh, so this mathematics exists in nature and that's why mathematics works to describe nature." That brand of Platonism does not follow from my metaphorical cave mathematical Platonism. And as of today, I think I understand what's going on here.
Mathematics is a process of determining truth values for statements that follow from basic formalized axioms, when viewed from the formalist position (and from the intuitionist position, which hybridized with formalism and, I guess, some brand of Platonism is somewhat closer to the one from which I think). In other words, mathematics can be seen as the field of coming to understand certain classes of axiomatic systems (which I'll explain in the next paragraph). I'm going to go ahead here and generalize to axiomatic systems in general, in fact, since it applies to them all.
Axiomatic systems are systems of propositions, each of which is assigned a value known as a truth value. The mechanism by which we assign truth values to propositions is known as a "logic," and the underlying logical structure includes a definition of the number of truth values in that logical framework. In the West, we're mostly familiar with Boolean (binary, two-valued) logic: propositions are either true or false. Taoism employs a three- (or four-) valued logical system in which there are some statements to which "true" and "false" simply don't apply, like "the belly is yin," since yin and yang are inherently defined in relationship to one another. Other logics have more truth values, and any number, including infinitely many, are valid logical constructions. We call certain kinds of infinite-truth-value logics "fuzzy logics," and they let us work with statements like "that pile of rocks is now a heap." (To complicate matters, the work of Kurt Gödel shows us that all axiomatic systems, if coherent (without paradoxes) contain statements whose truth values cannot be determined.)
It's key to note here that axiomatic systems are built upon statements called axioms that are taken to be "self-evident." Generally, we accept the fact that axioms are baldly asserted, and since this is true of all axiomatic systems, it's not a strike against any of them simply to point that out. Axioms have to be judged against how "self-evident" they really are, how useful they are, how little they assume, etc.
An axiomatic system, then, is a system of propositions together with truth values that are determined by the underlying logic, which works a bit like a truth-value function, with respect to the underlying axioms. In this sense, truth values aren't universal phenomena--they're only valid against the axioms of the underlying axiomatic system. This is a big, but tangential, point: "truth" doesn't mean anything except against the underlying axioms. In other words, something is only as "true" about the world as the underlying axioms and logical framework are accurate assumptions.
Here, then, we can make sense of the cave metaphor I used earlier. Once we know which logic we're using, and once we've decided upon our fundamental set of axioms, the "truth" of every proposition that can be examined from within that axiomatic system is already determined--although usually we know very, very few of the truth values. Indeed, we know very few of the propositions worth examining! Within the abstract context of that axiomatic system (the imaginary cave), the whole thing exists once we choose the system (the underlying axioms and the logic we're employing on it).
It's exactly like the game of Candyland, in fact. In Candyland, once the cards are shuffled at the beginning of the game, the entire game is determined. There's no need to play at all, and nothing except the unfolding of the cards really happens. As one plays the game, it feels as if a game is being played (especially to children), but it isn't. The whole thing is determined from the initial shuffle of the cards. That whole "game" already exists, is done, before it started, but the unfolding of the game gives the illusion that it's being played. This is exactly the case with axiomatic systems.
So, mathematical Platonism, in a sense, makes some sense. Once we've chosen the axioms and logic, the whole thing is done. It's up to us to explore that system and discover the timeless truths contained within it, if we want to know them, and so it feels very much like those truths exist and we're discovering them. But this is because it's easy to lose sight of the facts that we made the whole system when we chose the axioms and underlying logic, and that those axioms and logic do not actually exist in reality. They are abstract statements made and held in the minds of thinking beings that create, in a sense, abstract realms that can be "explored" in the mental sense. That said, in the other sense, mathematical Platonism makes essentially no sense at all.