[FOM] Platonism, Realism, Formalism
JoeShipman at aol.com
Tue Oct 28 23:39:43 EST 2003
I agree strongly with Hazen's complaint about changing meanings of words. Perhaps the following will clarify matters and allow discussion to focus on issues of genuine philosophical content and avoid issues hinging on the use of words.
The modern usage of "Platonism" is "the view that mathematical objects exist **independently of the physical universe or of human minds**", the highlighted phrase henceforth to be abbreviated "IPUHM". This comes in several flavors. The strongest place no restrictions except consistency on the type of objects for which existence IPUHM is asserted, but is only coherent insofar as the objects one discusses are mutually consistent: for infinite cardinals, this is observed to be the case, but not for objects of other types, such as "a countably additive real-valued measure defined on all subsets of the continuum" (henceforth: "RVM") and "a bijection between the real numbers and aleph-one" (henceforth: "CH"). Weaker forms of Platonism assert the existence IPUHM of particular mathematical objects, such as the set of subsets of the real numbers or V_omega+omega.
The modern usage of "Realism" is "the view that mathematical propositions have objective answers (truth values), independently of whether human minds can discover them" (IWHMD). This also comes in several flavors, depending on the language and the type of propositions allowed. The strongest commonly seen uses the language of ZFC and places no restrictions upon propositions beyond their being expressible in the language of set theory; though coherent, it is too strong to be of much interest to ordinary mathematicians. This is because any proposition about the objects of ordinary mathematics which is follows from the existence (NOT just the consistency) of a large cardinal is also consistent with there not being any large cardinals; so it is of no consequence whether "a measurable cardinal exists" has an objective answer IWHMD. What we mostly care about is whether statements about the objects of ordinary mathematics (sets contained in a finitely iterated powerset of omega) have objective answers IWHMD. Weaker forms of Realism assert the objectivity of the answers to particular mathematical questions, such as RVM.
When framed this way, it is clear that "Platonism" entails "Realism", and similarly for the weaker versions of each. Mathematicians care about the answers to propositions, and "existence of objects" is just a means to that end. What I'm interested in is how Realism can fail to entail Platonism: how can we make sense of "the truth value" of a sentence like "there exists a measurable cardinal" (henceforth: "MC"), which we know is independent of any assumption necessary for "ordinary mathematics", without being Platonists?
The modern usage of "formalism" is "the view that the primary reality of mathematics is formal systems, and questions about existence of objects or truth of propositions make no sense except in the context of a specific formal system, when they reduce to questions about provability of propositions, an arithmetical or algorithmic notion". Again, there are several flavors, the strongest commonly seen denying "objectivity" to any propositions that have not been proven and which no finite amount of numerical evidence can settle. (This version of formalism would refuse to assert that the twin primes hypothesis has a determinate truth value; still stronger versions, which would deny an objective truth value to pi_1 or sigma_1 statements like "there are no odd perfect numbers", are not commonly seen.)
Having defined these terms, I will now abandon them in order to make a sharper point: we can usefully discuss the standard mathematico-philosophies most usefully in a very concrete context.
The propositions I wish to discuss are the following:1) No projective plane of order 12 exists (NPP12)2) The Riemann Hypothesis (RH)3) The twin prime conjecture (TPC)4) The Invariant Subspace conjecture (IVS)5) RVM6) CH7) MC8) GCH
For each of these propositions, the questions arise whether it is a) meaningful b) discoverable c) objective d) true. I will attempt a classification in my next posting, which will illustrate some useful points, but first I'd like to see what others have to say. (Technical point: RH is straightforwardly equivalent to a pi_0^1 statement; there is a Turing machine which halts iff RH is false. The same is true for the Poincare Conjecture, by the way.)
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