Alan Baker, Does the Existence of Mathematical Objects Make a Difference?
Consider the following question, directly quoted from Baker’s paper
Surely it is obvious that mathematical objects—if acausal and non-spatio-temporal—make no difference to the arrangement of the concrete world?
What position would you say is supported by someone who answers ‘no’ to that question? Baker thinks it is the view that mathematical objects make no difference to the world. Or at least that’s what I think he thinks. None of this matters to the overall theme of the paper, which is better construed as looking at ‘yes’ and ‘no’ answers to its titular question. Baker thinks none of the arguments for either answer to this question are any good.
The first argument for ‘no’ is a direct appeal to what would happen if mathematical objects went away. This is fairly obviously question-begging. It isn’t obviously coherent either. If mathematical objects are non-spatio-temporal, how could they just go away at a particular time.
The second argument is Mark Balauger’s argument that mathematical objects make no causal difference to the world so they make no difference to the world. But there’s little argument for why causal difference is the only kind of difference that can make a difference.
Cheyne and Pigden argue for the ‘yes’ answer by using ‘mixed mathematical facts’ like There are three cigarette butts in the ashtray. But this doesn’t really show what we cared about, which is whether pure mathematical facts, like 2+3=5, could make a difference to the world.
Baker argues that the right answer to the question may depend on whether mathematical objects are indispensible for science. If they are dispensible, then by Lewis’s theory of world-similarity, the nearest world in which there are no mathematical objects will probably be just like this world in its ‘concrete’ aspects. In that case, it is reasonable to say mathematical objects don’t make a difference. But if they are dispensible, then there may be no fact of the matter about which is the ‘closest’ possible world in which there are no numbers, hence there is no fact of the matter about what things would be like if there were no numbers, hence there is no fact of the matter about whether the world would be different without mathematical objects. So the principle that mathematical objects make no difference to the world cannot be relied upon without a prior argument for dispensibility, which is bad news for those nominalists who want to so use it without proving dispensibility first.
I don’t know the background literature here that well, and I was a little sceptical about using Lewis’s theory of counterfactuals so far from the cases for which it was designed, but this seemed like a pretty good paper all in all.