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.

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