An Investigation into the Nature of Mathematical Objects

My thesis primarily concerns metaphysics, the philosophy of mathematics and the philosophy of causation. A common assumption is that mathematical objects, if they were to exist, would be non-spatiotemporal acausal abstracta. But this allows for some classic problems. The Benacerraf challenge concerns how it is that we could know about mathematical objects, given their acausal nature. The “makes no difference” objection says that, given the acausal nature of mathematical objects, the world would be the same as it is whether or not they existed, so we need not believe in them. Indispensability arguments may respond by saying that we have to believe in mathematical objects, but they do not respond to the core of these issues. I aim to solve these problems by suggesting that mathematical objects are a kind of ‘in-between’ object that is neither abstract nor concrete, I call such objects “exotic objects”. I present a view that mathematical objects might be exotic by being non-spatiotemporal but causal. Mathematical objects are causal in virtue of constraining the physical world, i.e. the reason that 23 strawberries are indivisible between 3 people is because 23 is indivisible by 3. I argue that this constraint relation can be viewed as a kind of causation because it exhibits the relevant kind of counterfactual dependence, e.g. “Had 23 been divisible by 3 then…”. But such statements are counterpossible rather than merely counterfactual so I also offer an account of why we should treat counterpossibles as non-trivial. I argue that the theory I propose maintains the advantages of platonist theories of mathematics whilst avoiding the classic problems mentioned above, but also avoiding committing to nominalism.