FOM: Formalists (reply to V. Sazonov)

[V. Sazonov]

Many thanks to Allen Hazen for his review on formalists.


>    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

"game formalism" and "deductivism" (or "...we don't know
what we are talking about..." )? I guess there is no
difference. Just a way to show one side of the medal.
The other side (related, say, with the questions "why
such and such  formalisms are considered?", "could they
be related one with another and with the real world?",
"how it is possible that some of formalisms are so useful,
say, in physics, engineering?") was probably either
implicit or not clearly understood/articulated. I think
that only one side of the medal cannot serve as the
whole philosophy.

Can anybody comment whether the other side was anyway
present in the ideas or writings of formalists? Is
it possible, that the other side is not discussed
just because the author consider it evident. (Cf. e.g.
comments to the next paragraph.)  But the opponents
blame him for ignoring (say, Platonistic) meaning.


>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.)

I think that a good teacher also explains what is the
goal and benefits of complex numbers and of corresponding
concrete rules adding/multiplying. It is exactly this the
required JUSTIFICATION. ("Discover RULES for such and such
new number system for which .......holds") The problem is
that this may be difficult to explain for beginners and
the teacher TEMPORARY suggests them to learn the rules
and postpone some question. (By the way, how were we
learning at the beginning school multiplication of 5*7,
etc., addition and multiplication of arbitrary decimal
numbers?) I consider this example with complex number as
a very good evidence that mathematics is essentially a
formal science which simultaneously has a very deep
meaning. (Philosophical Platonism, except some helping
intuition which any normal mathematician has and which
only looks like Platonism, is absolutely unnecessary!)

By the way, is imaginary number i (i^2 = -1) a "real" one
in the sense of Platonist or realist philosophy?


> 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.

Who "they"??? Cf. also the above notes on complex numbers.


> Frege's
> response was to try to say what number-words MEAN (=what numbers are),

And what they ARE? Was not this just reducing of
one game to another one, more general, of course
also having some intuition/reason as any good
mathematical formalism? Was any formalist *really*
against the intuition? ("Against Platonism" does not
mean "against intuition".)

It seems these questions are like to which religion
to believe or to be an atheist. It is strange that
in science we should discuss this at all.


> 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.

Thanks, I will try to find.


Vladimir Sazonov