The more I think about it, the more I think Benacerraf-style epistemic challenges to mathematical platonism aren’t very strong as stated. The point is that it’s supposed to be unclear how we could come to have knowledge of mathematical entities if they are supposed to be acausal and non-spatio-temporal. But to really put the argument forth like this, it seems that you have to have a fully worked-out epistemological theory that requires causation or location or something else that mathematical objects are supposed to lack. Now I’m no expert in epistemology, but I think if you really had a compelling positive position like this about what knowledge requires, then that in itself would be quite an impressive achievement. Sure, causal theories have been proposed, but from what I understand, the arguments in favor of them are primarily negative and don’t obviously apply to objects that are supposed to be non-physical anyway. And more importantly for me, I don’t really understand causation anyway, and I’m not convinced that it really is a part of a mind-independent reality, so I’m hesitant to require it in any of my philosophical positions.
Burgess and Rosen, in their book A Subject with no Object, point out that in fact it seems plausible that we might be able to have non-causal knowledge even of physical things. For instance, physics might predict that in the last five minutes before the end of the universe, particles called “eschatons” will appear, as part of a unified theory of the cosmos. Perhaps in a more down-to-earth example, it seems that physicists were in a position to claim knowledge about top quarks even before one was ever produced in a position to trigger a causal chain ending with a physicist. This is because the theory as a whole predicted them, and the theory was supported by its variety of other predictions that were causally verified.
I take it that a supporter of a Quine-Putnam argument in favor of mathematical realism could say that our knowledge of mathematical objects is similar, in that they play a role in the scientific theories that predict the phenomena we can observe. (I think that she even might be able to say that they do causally interact with us – is it really clear that the sun interacts causally through Newton’s law of gravitation, but that the gravitational constant and the squaring function don’t?) However, I take it that Hartry Field’s version of physics without mathematical entities would block this sort of argument. If we can construct a nice alternate theory with the same predictive power as Newton’s that doesn’t involve mathematical objects, then the knowledge we have of our theories could at best limit us to the two cases – one with the objects and one without. This doesn’t seem to be knowledge of the objects, even if they do exist.
Burgess and Rosen ask why the nominalist has to both give epistemic arguments against platonism and reconstruct a nominalistic science – if the epistemic arguments were compelling then the reconstruction would be at most a later task for scientists, and if the reconstruction is better than the original then the epistemic argument is unnecessary. However, it seems to me that on the reading I give in this post, the epistemic argument is incomplete without the reconstruction (as long as one admits the possibility of a non-causal theory of knowledge due to holistic support for a theory), and the reconstruction doesn’t obviously give one grounds to prefer the nominalistic theory without the epistemic arguments. Once you’ve got the nominalistic reconstruction, it’s not clear whether the (somewhat more complicated) nominalistic theory is to be preferred over the (more ontologically committed) platonist one, until reflection on the epistemic argument (suitably bolstered by the nominalist reconstruction) shows that even if the objects do exist, we can’t have knowledge of them. Thus, the two arguments work together to point towards nominalism, even though neither alone is sufficient.
I think Burgess and Rosen are right if their point is that this epistemic argument doesn’t fully motivate nominalism, as some nominalists think it does. This is my point in the first paragraph of this post. But I think some strengthenings of the challenge may be able to. The fullest development that I have seen is due to Oystein Linnebo in his paper “Epistemological Challenges to Mathematical Platonism”. Rather than assuming some sort of causal theory of knowledge, the challenge is to ask the platonist to give some sort of story (explainable in platonist terms) of how our judgements about mathematical objects come to be reliable. Our judgements about physical objects are seen to be reliable because of the way patterns of reflected light and vibrating air impinge on certain sensory organs, which are appropriately connected to parts of our brain that form our judgements. It’s really not at all clear where the platonist can begin to give such a story (though Penelope Maddy at least begins in “Perception and Mathematical Intuition”), but such a story should eventually be forthcoming if the theory is correct. This is because although there may not be any sense in which the correlation between our mathematical judgements and mathematical reality (according to the platonist) is counterfactual-supporting, this correlation is still extensive enough to require explanation.
But at any rate, I think that Field doesn’t need an epistemic argument to motivate nominalism – he just needs it to support nominalism given a nominalist reconstruction of science.