Here's a fun one: Carlo Rovelli - the renowned theoretical physicist whose name seems to be everywhere these days - published a small seven page paper in 2015 arguing against mathematical Platonism. It can be found here, and I'd encourage everyone to read it, for discussion's sake. It doesn't require a great deal of mathematical knowledge. The argument is interesting in itself, but I think it's even cooler coming from a physicist, given that physicists are usually those who are most enamoured with the idea of mathematical Platonism. Anyway, here's how the argument goes -

Rovelli begins with a simple definition of Mathematical Platonism, which "is the view that mathematical reality exists by itself, independently from our own intellectual activities." Now, he asks that we imagine a world M, which contains every possible mathematical object that could ever exist, even in principle. Not only does M include every mathematical object we have currently discovered (integers, Lie Groups, game theory, etc) it also includes every mathematical object we could possibly discover. M is the Platonic world of math. The problem, though, is that this world is essentially full of junk. The vast majority of it is simply useless, and of no interest to anyone whatsoever.

As he goes on to say, mathematics, as it stands, only includes an infinitesimal subset of M, and even if we could go on to discover M, we probably wouldn't, because most of it would be totally uninteresting. This, though, opens up a new question - what is 'interesting?' Well, interest simply is in the eye of the beholder: we develop some parts of M and not others because those parts help us do stuff, alot of the time. Rovelli asks: "Why has mathematics developed at first, and for such a long time, along two parallel lines: geometry and arithmetic?" And answers: "because these two branches of mathematics are of value for creatures like us, who instinctively count friends, enemies and sheep, and who need to measure, approximately, a nearly flat earth in a nearly flat region of physical space.... From the immense vastness of M, the dull platonic space of all possible structures, we have carved out ... a couple of shapes that speak to us".

Moreover, "there is no reason to assume that the mathematics that has developed later escapes this contingency". One way that Rovelli fleshes this out is by looking at examples of just how contingent our current mathematics is. He gives a few, but I'll focus on just one, the idea of natural numbers (1, 2, 3...). Rovelli basically asks: why does anyone think natural numbers are natural at all? We certainly find it useful to count solidly individuated items, but he notes that what what actually counts as 'an object' is a very slippery affair: "How many clouds are there in the sky? How many mountains in the Alps? How many coves along the coast of England?".

In order to make the point stick, Rovelli asks us to consider a species of intelligent beings evolved on Jupiter. Because Jupiter is fluid, most natural structures found of Jupiter would be continuous complex structures - flows, vortexes, currents, and so on. What kind of math might be developed in this environment? R: "The math needed by this fluid intelligence would presumably include some sort of geometry, real numbers, field theory, differential equations..., all this could develop using only geometry, without ever considering this funny operation which is enumerating individual things one by one. The notion of "one thing", or "one object", the notions themselves of unit and identity, are useful for us living in an environment where there happen to be stones, gazelles, trees, and friends that can be counted. The fluid intelligence diffused over the Jupiter-like planet, could have developed mathematics without ever thinking about natural numbers. These would not be of interest for her."

From all this Rovelli concludes: "Far from being stable and universal, our mathematics is a
fluttering buttery, which follows the fancies of inconstant creatures. Its theorems are solid, of course; but selecting what represents an interesting theorem is a highly subjective matter.... The idea that the mathematics that we find valuable forms a Platonic world fully independent from us is like the idea of an Entity that created the heavens and the earth, and happens to very much resemble my grandfather."

Discuss!