Mathematical platonism is the view that there are mathematical objects, not reducible to any physical object, that are independent of us and knowable by us. (To be platonistic, strictly speaking, they must not be merely irreducible to physical objects but more fundamental than them, and the knowability must be strictly by intellectual apprehension than by any other way. In practice, the term 'platonism' is often used here much more loosely just to indicate actually existing and knowable abstract objects. But the elements of the more purely platonistic versions are worth remembering.) The reliability problem is an argument that mathematical platonism is unable to give an acceptable account of how we have the right kind of epistemic access to these mathematical objects. It is occasionally called the Benacerraf-Field problem, because the standard locus for it is Hartry Field's modification of a dilemma proposed by Paul Benacerraf. The SEP article on mathematical platonism summarizes it as the following:
Premise 1. Mathematicians are reliable, in the sense that for almost every mathematical sentence S, if mathematicians accept S, then S is true.
Premise 2. For belief in mathematics to be justified, it must at least in principle be possible to explain the reliability described in Premise 1.
Premise 3. If mathematical platonism is true, then this reliability cannot be explained even in principle.
Obviously, the real question here is why anyone should accept these premises. (1), of course, is not true if taken very strictly; mathematicians are wrong all the time, it is easy to be wrong in mathematics, and almost all major advance in mathematics consists of mathematicians cooperatively working very hard to prove themselves and each other wrong. There is a huge amount of error-elimination in mathematics, which can only be if there is quite a bit of error. But we probably should not take it very strictly; the idea instead is that when mathematicians using mathematical methods converge on S, then S is true, in which case we have a much more plausible kind of reliability. In any case, (1) is not particularly controversial. Likewise, (2), or at least some version of it, is generally accepted. (3) is more tricky.
The most natural way of defending (3) is to say that if we know something, we must have some kind of causal relationship with it. I see a bottle of water -- how? There is a physical object, the bottle, located at a certain distance from me, and light is bouncing off of it to my eyes, which activates receptors in my retina, which stimulates my optic nerves, which stimulate my brain. But how does this work with the number 2? The number 2, itself, doesn't seem to be located anywhere in my vicinity, nor does light (nor sound, nor chemical reaction, nor direct physical contact) bring me any information of number 2 as such. Furthermore, abstract objects are often characterized as being causally inert. And so, if we need some kind of causal link to know these alleged mathematical objects, there seems to be no way we could know them at all, in which case the reliability of mathematics seems unexplained.
Formulated this way, there is no particular reason why any mathematical platonist needs to be worried. It is true that people commonly assume that abstract objects are causally inert, and it is even true that there have been mathematical platonists who have accepted such a position, but there is nothing in mathematical platonism itself that requires such an assumption. And indeed, if we look at how mathematics functions in explanation, it is not difficult to find cases in which we seem to be saying that some result arises due to the requirements of mathematics. It is difficult to make sense of much of physics without taking mathematical truths to have real-world results. Mathematics is not just used as a sort of precise bookkeeping, a super-accounting; we appeal to it to explain why bodies, waves, and the like work the way they do. There is no obvious reason why we should not regard this as counting as causation -- 'A necessitates real effect B' seems like a good candidate for A causing B. But even if one wanted to confine 'cause' to physical causation involving conserved quantities and the like, almost nobody has an account of this kind of physical causation that does not presuppose some kind of mathematical dependence. Thus the mathematical platonist can simply take the relevant sort of dependence to be more fundamental than, and presupposed by, what his critics call causation. You can have a regularity account of causation that does not take this to be true -- but regularity theorists are not the people to be pressing unexplained reliability as an objection against anyone. Everyone else takes the kind of causation described by physics to be analyzable at least in part in terms of mathematical relations that we know, not vice versa.
Nor does the more specific complaint that there is no causal path from mathematical objects to us seem to do anything more than beg the question by assuming that the causal path must in some sense be sensory. A mathematical platonist like Gödel who holds that we have a special form of mathematical perception will obvious reject this. Gödel held, based on his own experience, that it is a psychological fact that we can perceive mathematical truths (although, of course, this is very different from saying that it is always easy to do so even with training, and is also different from saying that mathematics simply reduces to such a perception). We can in fact distinguish between the experience of simply accepting the mathematics as a procedural black box and being able to 'see' why it has to work the way it does in a given case. And if you hold that we have reason to accept that some such mathematical perception exists, why should one not regard that as an effect of what one perceives? If you have good reason to think that there is this mathematical perception, you already and immediately have reason to think you've discovered an effect of what you perceive. One can even begin to sketch out some of the features that the in-principle explanation would have (e.g., the ability to identify the negations of some of the things we see as contradictory and thus impossible), and by the nature of the argument it does not matter whether we can currently, or even ever in practice, fill in all of the blanks.
I think a bigger problem, though, is that it misconceives what it is to explain something's being reliable. Part of the motivation for thinking of (3) causally is in thinking that the most plausible account of the reliability of physics is causal, along the lines I noted for the water-bottle. But it's not so clear. What about this causal story actually explains the reliability of anything? It explains part of what it is to perceive a bottle of water, but nothing about this on its own explains anyone's ability to draw reliable conclusions about it. For one thing, all of the actual drawing conclusions here is obviously being left out, or at least treated as a black box. What the causal story is really doing is telling us what it means to say that it is objective, an object of perception. This helps to explain why it's not a bizarre accident that we are thinking about something that also happens really to exist in the world, as if you imagined that there was a building shaped like a tree, and, lo, it turned out by sheer happenstance that there was a real building exactly like it. Establishing that your conclusions are non-accidental is a big and important thing. But that your claims are non-accidentally representative does not imply that they are reliably true in the sense said by (1).
The confusion seems to arise in that non-accidental-ness does seem to be one reasonable requirement for an acceptable explanation of something's being a reliable in getting true claims. But it is not the only one, and what explains non-accidental representativeness is not what explains reliability, which is a kind of consistency. (You can have, for instance, something that is reliably false.) Reliability requires primarily an internal and structural explanation. In other words, it is the structure of a method that gives it its reliability, it is whether you are thinking logically that gives it its reliability, it is the features of a form of inquiry that give it its reliability; whether this reliability is a reliability useful for drawing true conclusions in particular will require bringing in non-accidental-ness. (If we want to say why something is reliably wrong, we want to understand why its structural reliability is not accidentally getting the wrong result, if we want to say why something is reliably useful, we want to understand why its structure and features make it not accidental that it is always of use).
Now, obviously this just moves back the problem, but notice what happens if we rephrase the premise to emphasize non-accidental-ness rather than the distinct issue of reliability:
(1) Mathematicians consistently reach claims that do in fact describe the way things are.
(2) For belief in mathematics to be justified, it must at least in principle be possible to explain why this is not a mere accident or happenstance.
(3) Mathematical platonism cannot explain why this is not a mere accident or happenstance.
But the explanation for why so many mathematical claims are not merely accidentally true is that we have good reason to think that it is impossible for them not to be true, that 'this claim (or set of claims) is false' often ends up being a contradiction. If I am getting A because ~A is impossible, then my conclusion's being true is obviously not a mere happenstance or accident.
Now, to be sure, one could make a fuss about the question of how we get necessities and impossibilities, but if we can reason about them at all, no matter how, then the challenge is answered: if we prove that it is a contradiction for X to be false, then it cannot be a mere accident that X is true, and it cannot possibly surprise anyone that we turn out to be right. And this is the sort of thing that mathematicians actually do. There is no reason to hold that mathematical platonists cannot give this explanation, and thus, whatever problems there may be with mathematical platonism, (3) is wrong.
Various Links of Interest
* Why Roman concrete is more durable than modern concrete in seawater.
* Darwin summarizes the Church's more-than-millenium-long struggle against the practice of dueling.
* The historical background to The Band's "The Night They Drove Old Dixie Down".
* A high school paper asked Secretary Mattis for an interview, and he gave it.
Isaac Asimov, Foundation and Earth
Mary Astell, The Christian Religion as Professed by a Daughter of the Church of England
John of St. Thomas, The Gifts of the Holy Spirit
Kenneth L. Pearce, Language and the Structure of Berkeley's World
Stephen R. Lawhead, Dream Thief