To be a little superficial, the Platonist position in philosophy of mathematics is that the number 17, and all of its friends and relations down to distant cousins, exist quite independently of us. Mathematical truth is discovered in something closely analogous to the way rare jungle fauna are discovered by exploring zoologists. The truths of mathematics are not invented, stipulated, or constructed, not directly, not indirectly, not even very indirectly.
An important mathematical conjecture, called the continuum hypothesis, appears as an interesting test case for Platonism. The continuum hypothesis can be expressed as an (apparently) perfectly straightforward mathematical question. Is there a set intermediate in size (cardinality) between the set of all the integers and the set of all the real numbers (including, e.g. π)? That the reals are more numerous than the integers was proved by Cantor in 1874. The proof is a human achievement of the first magnitude. You will not overvalue it if you mention it in the same breath with Beethoven’s Ninth or Picasso’s Guernica. I strongly recommend to you Cantor’s 1894, “diagonalization” version of the proof. It can be found easily on the web and can be followed by the average high school graduate, so long as that graduate is willing to concentrate a little more intensely than he or she did in Ms. Watson’s algebra class. (http://planetmath.org/cantorsdiagonalargument is fine, although perhaps a little condensed if you left off math years ago. Taking things slower is https://www.youtube.com/watch?v=qGYDQWm49wU.)
The interesting thing about the straightforward question about the existence of our intermediate sized set is that it has been shown to be consistent with (Gӧdel) and independent of (Cohen) standard axiomatizations of set theory. You can add either the continuum hypothesis or its denial to your favorite set theory and go about your set theoretic business just fine. It looks as if it might be just a matter of taste, and de gustibus non est disputandum.
For a long time my reaction to the continuum hypothesis was, “Damn it, either there is a set of that intermediate size or there isn’t.” I took this to be a Platonist intuition. I was convinced that there was a fact of the matter about the continuum hypothesis whether or not human beings would ever know it to be true or false.
My confidence dropped, at least below the level of swearing about it, when I realized that there might be something to a conjecture of Skolem’s that predated even Gӧdel’s consistency result. What Skolem said was that our concept of set might not yet be so well developed as to answer the continuum hypothesis question. That is, the problem might be not with the things to be counted, but with the gathering them together to be counted.
A little chastened by this thought, I nonetheless want to respond along the following lines, “Well, the question is this: is there any way of putting together a bunch of things so that the bunch has cardinality greater than the bunch of integers but yet less than the bunch of all the reals. I don’t care whether the bunch is a set, a herd, or a gaggle. If there is any way of producing any kind of such collection, aren’t we going to want to say that the continuum hypothesis is flatly false? And wouldn’t we have excellent reason for reconstructing our concept (and axiomatiztion) of “set” to correspond to this way of putting together a bunch?
I still think, albeit with reservations, that there is a straightforward, if perhaps never to be discovered, answer to the intermediate size question. If that is right, it supports Platonism. If there is no answer, that is, if there is no fact of the matter as to whether there is something intermediate in size between the integers and the reals that would at least tend towards anti-Platonism. Platonism, however, might still sneak in the back door, and, if not, which of the many alternatives to Platonism would be most benefited would still be to be worked out.
The one possibility for there being no fact of the matter that I can get my mind around is again Skolemesque. Even if we look past any deep ambiguity in our concept of “set,” there might yet be a similar ambiguity in our concept of cardinal size. Perhaps “one to one correspondence” is in the overlap area of two different cardinality measures. When we get beyond familiar cases, where one to one correspondence works well, and try to apply the concept to exotic cases, we will find that there are two different natural extensions, on one of which there is an integer/real intermediate cardinality and on the other not.
The Platonist, passing through the back door, could say in this eventuality that what we will have discovered is that there are two different mathematical objects corresponding to the two different ways of disambiguating cardinality. When non-Euclidean geometries were discovered that did not doom Platonism; it showed only that a favorite example of Platonists, going back to Plato himself, of the accessibility of geometry to intuition needed to be expanded to the accessibility of different geometries to intuition.
Yet this also suggests that there might be something here that could as easily be called “invention” as "discovery." Won’t the exotic objects that show us either the ambiguity of our concept of sets or our concept of cardinality, or both, likely be introduced by a mathematician deploying some startlingly creative new construction? Could it be that all of us who feel Platonism to be congenial are seduced by concentrating on versions of non-Platonism that are too crude? Structures that are human creations may still have properties that are far beyond the ken of their creators. In such cases both the language of invention and the language of discovery will be perfectly appropriate, one in one context, one in another.
For those of us for whom the ontology and epistemology of High Church Platonism are deeply suspect, there is a motivation to explore whether our Platonistic intuitions in such cases as the continuum hypothesis might turn out to be consistent in the end with understanding mathematical objects to be structures invented and explored by humans in our attempt to understand the natural world.
Still, I cannot free myself from the suspicion that the intermediate size just does not exist, full stop.