FOM: Formalists (reply to V. Sazonov)

[Allen Hazen]

   The people Frege bashed as formalists were his German contemporaries, E.
Heine and J. Thomae.  Resnik discusses them and also such (more
sophisticated, one likes to think!) 20th century figures as Hilbert and
Haskell B. Curry.
   Heine seems to have been a "game formalist": making a mathematical
statement (in, e.g., the course of a proof) is simply making a move in a
rule-governed game, and no more "meaningful" than a move in chess.
   Thomae seems to have been closer to being a "deductivist" (what Hilary
Putnam, in his "Philosophy of Logic" [originally a pamphlet, treated as a
paper and reprinted in the SECOND edition of P's "Mathematics, Matter and
Method"] or "The Thesis that Mathematics is Logic" [originally in R.
Schoenman, ed., "Bertrand Russell: philosopher of the century"; repr. in
Putnams "MM&M"] calls "if-then-ism."  This recognizes that mathematical
statements have the FORM of meaningful sentences, and that logical
expressions in them are treated as having their ordinary meaning, but holds
that content-expressions (point, line, number, set...) need not have any
independent meaning; mathematics is just the activity of deducing
consequences from (arbitrarily chosen) "axioms".  This was summarized in B.
Russell's witticism, "Mathematics is the study in which we don't know what
we are talking about or whether what we say is true."
   Both of these ideas are, I think, still alive, and may be part of the
rhetoric mathematics TEACHERS use to emphasize the abstract nature of
modern mathematics.  ("Don't try to worry about what the square rood of a
negative number IS, and don't try to JUSTIFY the rules for
adding/multiplying complex numbers on the basis of some IDEA: the rules are
just RULES, chosen for reasons that needn't concern you.  Just learn them."
My best high-school math teacher, who on other occasions talked philosophy
and psychology with us, said pretty much this.)  The big philosophical
objection is that they  rule out the question  of what makes one game, or
one set of axioms, more worth playing (with) than another.  Frege's
response was to try to say what number-words MEAN (=what numbers are), so
he could go on to argue that mathematical axioms of interest were true as
descriptions of numbers.
    Everyone contributing to FOM has opinions about Hilbert.  Two of the
nicer expositions/philosophical discussions of his "program" are by Gentzen
and Herbrand, reprinted in their respective collected papers.
    Curry wrote a little book, "Outline of a formalist philosophy of
mathematics."  There is a nice discussion of his views by Bob Meyer (the
relevance logician) in "Curry's philosophy of formal systems," in
"Australasian Journal of Philosophy" v. 65 (1987), pp. 156-171.