Mathematicians discovered a new [whole] number. It’s between six and seven and is called “bleen”.
When we’re talking about π, what are we talking about? What kind of object is it?
A number, your text book will tell you. Where is it? On the real line. Where is the real line and how long has it been around? It’s just there, and has been forever …
Your textbook may not answer these questions but philosophers have tried. Needless to say there is no consensus. They’ve come up with several answers, and in my humble opinion, all have something to offer, but all miss the mark.
I have an answer which you’ll be happy to hear I am going to share with you. Briefly, π and its friends live in our collective imagination. To be corny, in our hearts and minds.
Why is it important to understand the nature of mathematics? Isn’t it just useless philosophical hair-splitting that distracts us from the hard work of discovering new math? (Or inventing new math. Hmmm).
In fact it’s of great practical importance. For a start, how do we discover/invent new math if we don’t know what math is? How do we evaluate what we come up with if we don’t know what math is? And the big question is, why is math so darn useful, “unreasonably effective” as the saying goes.
Hands down the most popular philosophy is Platonism. According to Platonism, numbers, sets, functions etc live in the Platonic universe (PU) of mathematics (lately, set theory) independently of time and space. The job of mathematics is to discover facts about the PU, and logical rules that apply to it.
When you’re actually doing math, it’s hard not to think Platonicly. So it’s a good heuristic. But that doesn’t make it true.
What does it mean to exist independent of time and space? Nothing, as far as I can see. I was once at a UVIC philosophy seminar where this came up. I asked if, in the time of the dinosaurs, bounded linear operators already existed. Yes, I was told. I have no idea what this meant.
How do we find new math? By exploring the PU … (how?) How do we evaluate new math?… by verifying that it’s true in the PU (how?). Why is it so useful? – anybody’s guess.
The second most popular philosophy of mathematics is formalism. Formalism reduces mathematics to symbol manipulation. According to formalism, mathematics is working with formalized facts and rules. You have a set of axioms and some rules of inference, and you draw logical consequences – you prove theorems. So π is just a symbol that appears in the facts and rules and in the consequences derived.
Formalism was famously described by Hilbert when he said that “mathematics is a game played according to certain simple rules with meaningless marks on paper.”
How do we find new math? – symbol manipulation. How do we know it’s … true? By carefully following the rules of the game. Why is it math so useful? – a mystery.
Formalism certainly describes a lot of what mathematicians actually do. But not everything. Gödel’s incompleteness killed formalism as a philosophy by showing that there are always candidate theorems that can neither be proved nor refuted, and that such ‘undecided’ assertions arise no matter how many axioms you add. OK, maybe the process is never complete. But where do new axioms come from?
Intuitionism and Fictionalism
The other theories are minor players. According to intuitionism, mathematical objects are products of our mind, like characters in a novel. I agree, as far as it goes [see below]. But it comes with a lot of constructivist baggage that results in severe constraints on the forms of reasoning (it allows assertions that are neither true nor false) and a very reduced mathematical universe (e.g. no discontinuous functions). It’s not clear to me that it answers any of the three questions.
There is another approach that, like intuitionism, nearly gets it right. This is fictionalism. Fictionalism holds that the mathematical universe is a collective fiction, like Star Trek or Game of Thrones. That the Cantor set or the Borel Hierarchy is like the USS Enterprise or the Iron Throne. In fact Star Trek fans talk of the Star Trek universe and GOT fans of Westeros and they talk like these places really exist.
So far so good. And fictionalism dispenses with the constructivist baggage. Even better. But there’s a gotcha: according to fictionalism, all statements of mathematics are false. Because they refer to entities that don’t exist. Thus the statement “James Kirk is captain of the Enterprise” is false because Kirk and the Enterprise are both made up. New math? Anything you want. Good math? It’s all false. Why is it useful? Mystery.
What I’m proposing is a minor patch to fictionalism – allow statements about imaginary entities to be true as well as false. So the statement the statement “Spock is captain of the Enterprise” is still false but “James Kirk is captain of the Enterprise” is now true.
Strictly speaking fictionalism has a point: we’re using “true” and “false” in a nonstandard sense; but the solution is not to discard the distinction, but to make it clear that we’re talking relative to a fiction. Maybe call them “true.ST” or “false.GOT”.
What I’m saying is that the mathematical universe (MU) is humanity’s most ambitious fictional creation. There are tens of thousands of Trekkies and (?) Gamesters who know their respective universes inside out and have passionate discussions about Kirk being the best captain ever or Jon Snow being the rightful heir to the Iron Throne. But there are millions – hundreds of millions – who know the MU well and can answer (as true.MU or false.MU) questions about it.
I’m calling this revised fictionalism “neofictionalism” (catchy). It says that Math is like Star Trek but on a vastly greater scale, both in time to create it, in terms of extent (the MU is huge). And, most important of all, in terms of the precision with which it is described.
More on this next week.