Every baby is born the same way (namely, as a baby).
And every mathematician is born the same way, too: as a baby as a Platonist.
It usually begins something like this: What exactly are triangles? In a cosmos where no plane is perfectly flat, no segment perfectly straight, and no corner perfectly sharp, are Euclidean triangles in any sense “real”?
Yes, says the Platonist. Tangible? No. Physical? No. But the Platonist has tasted enough of math to know, in her bones, that math is more than just human whim. Math must, in some independent sense, exist. The Platonist believes (in the grand tradition of Seinfeld) that mathematical objects are real, and they’re spectacular.
Other philosophies may come later. Structuralists evolve, like birds from dinosaurs. Formalists are trained, like soldiers in boot camp. Intuitionists emerge, like neo-reactionaries from economic recessions.
But every mathematician begins in the same sweet swaddle of Platonism.
Then, one terrible day, some nosy would-be-Socrates starts poking his grubby fingers inside your thoughts.
When you say these mathematical objects exist, what do you mean by “exist”?
Well, it’s not a physical existence. It’s a conceptual existence.
I see. Where exactly do concepts exist?
There is no “where.” They’re not made of matter. They don’t inhabit space.
So, these immaterial, nonphysical, nonspatial concepts of yours… how do they come to influence our physical reality?
I don’t know. There must be some kind of causal mechanism.
Via the pituitary gland, no doubt?
Hey! Stop making fun of me!
And by the way, how many Platonic forms are there? Is there a Platonic form for the set of all sets that don’t contain themselves, or only for non-contradictory objects? By the way, is the Platonic form of a Gödel statement true or false? And what about the real numbers — what formalization is the Platonic one? Is it Dedekind cuts, or are the True Real Numbers equivalence classes of Cauchy sequences? Either way, isn’t it weird that the Platonic Form of the Reals is a complicated set-theoretic construction over the Platonic Form of the Rationals? And have you considered…
Stop! Stop!
This phase lasts somewhere between 30 seconds and 1 lifetime.
Sooner or later there comes a rebirth of energy. You stop caring about the metaphysics, and recommit yourself to the thing that always mattered most.
The math.
Maybe you read the actual capital-P pragmatists: James, Peirce, Dewey. Or maybe you come by your pragmatism honestly: a few trips around the sun are enough to teach you that the highest form of Truth is whatever gets you through the day, whatever helps you make progress and find meaning on this mixed-up planet of rocks and love and AI slop.
So you give up on the question of whether math is real.
But the question remains: Whether or not math is real, how should I think about it?
So you ask an expert.
Not an expert philosopher. Your mentor hasn’t read any more Wittgenstein than you have. Rather, this is an expert solver of math problems, an expert thinker of mathematical thoughts, an expert in precisely the kind of life you want to live.
This expert is a pragmatist, just like you. This expert has walked your path. This expert knows what kind of outlook will help get you through a day of epsilons.
This expert tells you. And finally, wisdom is yours.
You know how to think about mathematical truth.
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