[FOM] Platonism and Formalism (a reply to Podnieks)

[Torkel Franzen]

Todd Wilson says:

  >To use formal systems
  >in these ways, it is not necessary to commit to the belief that every
  >mathematical statement concerning these systems, for example their
  >consistency, is decided or even meaningful -- or even that all of the
  >infinite number of formulas and proofs that the system engenders
  >exists as a complete totality -- except in the context of other formal
  >systems.

  In the philosophical literature, as far as I know only Wittgenstein
has consistently taken this attitude:

     But here I must make an important point: a contradiction is only
     a contradiction when it arises. People have the idea that there might
     at the outset be a contradiction hidden away in the axioms which
     no-none has seen, like tuberculosis...one day the hidden contradiction
     might break out, and then the catastrophe would be upon us. What I am
     saying is: to ask whether the derivations might not eventually lead to
     a contradiction makes no sense at all as long as I'm given no method
     for discovering it. ...There can be no such question as whether we
     will ever come upon a contradiction by going on in accordance with the
     rules. I believe that's the crucial point, on which everything depends
     in the question of consistency.

But of course Wittgenstein didn't actually work in logic or
mathematics or computing. At this level, "belief" is not really an
issue. We must examine how people actually work, how they argue, how
they apply mathematics, what kind of considerations seem to guide
their thinking in practice. For example, from a consistently
non-Platonistic point of view it makes no sense whatever to worry
about the possibility of there being an efficient algorithm, as yet
undiscovered, for factoring large numbers, or to think that the
likelihood of a computer program ever producing an output
conventionally described as "a computation showing that k is an odd
perfect number" is somehow related to the existence of an odd perfect
number, in some sense that is not defined in terms of actual
computations. A consistently non-Platonistic point of view with regard
to mathematics is much like a consistently skeptical view of human
knowledge: it has a certain appeal to the intellect, but it has no
apparent relation to how people, including professed non-Platonists or
skeptics, actually go about their business.

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