If you look up an encyclopedia entry on the philosophy of mathematics you will usually find yourself presented with a list of competing approaches dating back to Plato. For someone of my temperament this is unsatisfactory. Well, it's fine having competing views, but which one (if any of them) is true?
With areas like ethics or art, the fact that there is no consensus may be explained by the very real possibility that these areas are largely subjective. But mathematics? Surely there is something that mathematics is, some broad understanding at any rate that we can agree on?
In the philosophy of mathematics there is a basic division between realists (or platonists) who believe that mathematical objects exist (but not in time and space); and anti-realists who don't believe this (seeing mathematics simply as a human activity, for instance).
Most mathematicians are thought to embrace some form of realism (mathematical truths are 'out there' to be discovered); but not all do. The distinguished mathematician Timothy Gowers is an anti-realist (aligning himself with Wittgenstein in this matter).
This basic division is just the beginning, however, as there is (as in just about any area in which philosophers are involved) a proliferation of arguments and counter-arguments resulting in an ever-proliferating list of divisions and subdivisions and so of positions to attack or defend.
Amongst which is the gloriously named neo-Meinongianism. (Could anything so called actually be true?)
A part of me says: 'Steer clear of all this, my dear fellow. Life is too short. And it's not about what it seems to be about. In part it's merely a perverse, self-perpetuating amusement for philosophers, in part an attempt by serious, religiously-inclined thinkers to defend a metaphysico-religious (or should that be religio-metaphysical?) view of the world.'
Unfair, no doubt. What of the serious, non-religious and/or anti-platonistic participants? Like Timothy Gowers.
In fact Gowers has himself asked the question of whether mathematics needs a philosophy, and I find his thoughts on the matter very persuasive.
I will probably not be spending a lot of time researching this area, but I would like to follow up on Gowers's views.*
And also, I intend to have a closer look at neo-Meinongianism. Why not?
* Gowers is a very political (and influential) figure within the mathematical and broader intellectual community and seems to have some fairly radical views about the ownership and distribution of ideas.