A Few Notes on Platonic Perception

I want to note straight out that I, like Jim Brown, am deeply sympathetic to Platonism in mathematics. I want to be able to say that we can have a priori knowledge of mathematical truths, and it strikes me as tempting to say that we attain this knowledge through a process that is somehow analogous to perception. How would we come to know mathematical truths that cannot be given in experience, except by somehow ‘glimpsing’ them with another faculty? Brown thinks this is precisely what is going on, even if no account of what this faculty of mathematical perception is or how it works can be given. Now, on Brown’s view we can sometimes ‘glimpse’ the laws of nature in much the same way that we ‘glimpse’ that 2 and 2 are 4. And indeed it is not readily apparent why we should find Platonism a plausible view in the mathematical case but not in other cases of (apparent) a priori knowledge.

Now, one might object that we just cannot understand what it is to see abstract entities such as laws, numbers, sets etc. And Brown is more or less willing to concede that we really don’t understand how this sort of platonic perception works. Thus the analogy between platonic and ordinary perception breaks down. However, Brown doubts that this can really stand as an objection to his view. He points out that we really don’t have an adequate understanding of how everyday perception works either. So critics who demand a full account of how we make epistemic contact with abstract entities are simply demanding too much from the Platonist. Moreover, the mere fact that we don’t have such an understanding is not by itself evidence against the existence of this kind of perception (doubtless a lot of things exist of which we don’t have an adequate understanding).

John Norton thinks this is an unsatisfactory response to the worries of the empirically minded. Brown’s account allows that platonic thought experiments (even very good ones) can sometimes lead us into error. Such thought experiments afford apparent glimpses of the true laws of nature, but in some cases this is an illusory appearance. What appears to be an instance of a priori knowledge of natural laws can turn out to be no knowledge at all. Brown suggests that we need not be troubled by the fallibility of platonic perception any more than we are troubled by the fallibility of ordinary perception. The argument from illusion is hardly taken seriously in the latter case, so why should it be in the former.

Norton thinks this overlooks a serious disanology between the two cases, which he summarizes as below:

(3A) Ordinary perception is well understood insofar as it governed by an array of familiar regularities. They make the success or failure of ordinary perception controllable and intelligible in all but exceptional cases.

(3B) The mechanism of perception of Platonic laws is essentially completely mysterious. We have no understanding of why this perception succeeds when it succeeds and fails when it fails. ( "Are Thought Experiments Just What You Thought" 359-60)

I think it is difficult to deny 3A, without opening ourselves up to a somewhat unsavory skepticism about sense perception. We may not ultimately understand all the mechanisms at play in ordinary perception, but surely we know something about the conditions under which it is reasonable to trust our senses and those under which their deliverances are particularly liable to be deceptive. We have whole catalogues of perceptual illusions of which we are aware, and we can and do correct for them. This itself seems to indicate some understanding of the mechanisms in play, even if it is imperfect. The upshot is that sensory perception is familiar enough to us that we have a kind of system (not perfect but a system nonetheless) by which to gauge its reliability in particular cases. It may be true that in no particular instance can we be absolutely certain that our senses are not deceiving us. But if we’ve agreed not to take the argument from illusion seriously, this is no problem.

However, if 3B gives an accurate picture of platonic perception, this does strike me as a serious problem. Norton’s argument view can say quite simply that thought experiments fail when the argument of which they are but a picturesque presentation is either unsound or invalid. We can choose between thought experiments with different conclusions by weighing the comparative strength of the arguments they can be reformulated as. This is all very neat and convenient. But if platonic thought experiments are not arguments, but perceptions of abstract entities, and this perception is fallible, how are we to choose between them when their ‘findings’ conflict? It seems to me that if the Platonist view is to be persuasive, we need to be able to say that the working of platonic perception is not just mysterious, but that it is governed by something like the sort of regularities governing ordinary sense perception. Further, we need to be able to say something about these regularities, so as to ascertain the conditions under which a platonic thought experiment is most likely to be a genuine source of a priori knowledge. I am not ready to concede that the Platonist cannot meet these demands, but doing so would seem to require a more robust account of platonic perception than what Brown gives us.