FOM: definition of Platonism

[Randall Holmes]

I am M. Randall Holmes, a professional mathematician specializing in
the consistent subsystems of Quine's set theory "New Foundations" and
related systems of combinatory logic, and in computer-aided theorem
proving.  More information can be found in my signature.

I must question Simpson's characterization of Platonism (actually,
since I'm aware of the history, I prefer to say "mathematical
realism") as "the view that mathematical truth exists independently
of, and reveals itself to, passive consciousness".  Platonism is a
position about mathematical _objects_; there are possible positions (I
will describe one below) which privilege mathematical truth without
providing any mathematical objects!  Further, I don't think that
realism requires that mathematical or any truth about the
independently existing reality will "reveal itself" to a passive
consciousness; if we don't actively think about universals, we will
not become aware of them (the same is true of many if not all
particulars).

I outline a position which privileges mathematical truth without
providing mathematical objects.  To simplify matters, let's consider
the natural numbers only.  Let P be the conjunction of all the Peano
axioms (or your favorite axiomatization of arithmetic).  Let E be
Euclid's theorem that there are infinitely many primes.  The position
I have in mind is one that says that the real meaning of the theorem
expressed by E is "if there are objects (and relations/predicates)
which satisfy P, then there are objects which satisfy E", or "E is a
logical consequence of P".  The difficulty of this position is that
the meaning of "if...then..."  here cannot be material implication
(even with quantifiers): if there are actually only 10^100 objects,
the statement "Each system of objects and predicates which satisfies
axioms P satisfies ~E (the negation of Euclid's theorem)" is vacuously
true.  Another way to interpret "E is a logical consequence of P" is
as a kind of syntactical assertion ("there is a proof p that E is a
consequence of P").  But an appeal to syntax is not any better than an
appeal to actual natural numbers; expressions in your favorite
language are exactly as problematic as natural numbers (there are
infinitely many of either).  If we are not prepared to assume the
existence of at least a countable infinity of objects, then the nice
relationship between logical consequence and either models or proofs
(the latter considered as syntactical items) breaks down.

I'm not going to try to figure my way out of the finitist predicament
here; if we suppose that there are at least a countable infinity of
objects, the meaning of the implication in "E is a logical consequence
of P" has nice (and equivalent) interpretations taking both semantic
and syntactical approaches.  Notice that in this position it makes
sense to talk about mathematical truths, but mathematical truths are
not about any special objects.  E, on this view, does not say that
there are infinitely many prime numbers; it does not assert the
existence of any specific objects.  It is an assertion that if you
provide me with notions of "natural number", "zero", "successor",
"addition" and "multiplication" satisfying the Peano axioms, there are
infinitely many "prime numbers" as naturally defined in terms of these
notions.  (The axioms P might be thought of as the interface of an
abstract data type).

I don't necessarily maintain this position; I'm just pointing out that
it appears possible to defend and it privileges mathematical truths as
objective truths about the world while not providing any mathematical
objects.  It is not obvious that it is a Platonist position; the
infinite number of objects that it requires as a hypothesis could be
actual physical objects.  This position might be adapted to work
without the assumption of an actual infinity of objects, at the price
of figuring out how to make sense of counterfactual assertions of the
form "if P were true then E would be true" in a world where there
might indeed be only 10^100 objects.

Of course nothing I am saying here is original (or unassailable; I can
see lots of objections).

And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the       | Boise State U. (disavows all) 
slow-witted and the deliberately obtuse might | holmes at math.idbsu.edu
not glimpse the wonders therein. | http://math.idbsu.edu/~holmes