This a belated American Pi Day post. I say ‘American Pi’ day not for an opportunity to make Don Mclean puns but because March 14 only looks pi-like if you do the month/date thing. Elsewhere it was 14/3, which could be 1.43 day or less-good-square-root-of-2 day (Feb 14 being better but Dec 14 being best). For the day/month people July 22 makes a better Pi day as it gives the classic rational approximation of 22/7.
I was asked where I stood on the issue of Pi being invented or discovered. I’m firmly in the discovervented camp.
The ‘discover’ camp tends towards Platonism – the idea that mathematics is not just real but really real. Indeed mathematics in Platonism is more real than reality, which is just a crappy shadow of truth, goodness and all things geometrical. Discovered feels right but if you follow the reasoning you end up having to swallow a very big metaphysical pill.
The ‘invented’ camp tends towards Formalism – the idea that mathematics is the logical outcomes of arbitrarily chosen rules. I’m much more inclined towards formalism but it can feel a bit arbitrary. After all we could make up all sorts of self-consistent logical schemes and prove empty theorems about them but we don’t. The ones we study in mathematics not only tend to be relevant ones but have had an uncanny knack of BECOMING relevant.
Aristotle pitched his tent just a little way away from Plato. From him mathematics was something embedded in the universe but maybe not as transcendental as Plato took it to be.
From Kant and modern notions of evolution and modern psychology and neuroscience we get an alternate notion of embedding – that mathematics is something kind of built into us and the way we make sense of the universe.
I’ll play pick and mix with all that. I do tend towards FICTIONALISM as a model of mathematical truth, which marries nicely with formalism. That is a mathematical truth like 2+3=5 is true in a similar sense as ‘Sherlock Holmes lived at 221b Baker Street’. Mathematics is a kind of fiction, a work of imagination but as any writer knows your imagination is constrained by your experience and by your contact with reality and by what makes sense and by what is self-consistent.
Mathematics is like that as well. We invent it to describe the world around us but in doing so we create a space which we explore within rules and then discover things. If we invent the abstract notion of a ‘circle’ based on our real world experiences of circles and a concept of ‘diameter’ and ‘circumference’ then we discover in our inventions a little pi, like an alchemic homunculus. Which is all well a good but then we find the same little fellow starting out at us in our other inventions and we begin to suspect Plato was right all along.
But if mathematics is fiction, what kind of fiction is it? Well in modern times we associate it with science but in the past it has played nicely with mystical and religious disciplines as well. Mathematics is like some strange kind of genre, a genre that can encompass imaginary realities, alternative sciences but also magic and fantastical worlds all within a broader notion of the weird and the speculative. Mathematics is the ultimate SF/F.