FOM: constructive mathematics

Matthew Frank mfrank at
Sun May 28 23:15:37 EDT 2000

Apropos of Stevenson and Ketland:

I do think that Brouwer was subjectivist, but he may have been the last
(real) subjectivist intuitionist.  (I agree that Heyting's Int was a
subjectivist, but that was a fictional character.  The "Sign" of Heyting's
dialogue was supposed to be a significist; while I once found an
explanation of that philosophy, I can't remember it.)

Also, Sazonov asks:

> By the way, if another philosophy(?) called formalism really
> exists (which allegedly asserts that mathematics deals only with
> formalisms without any meaning), who are full-fledged formalists,
> personally? Or such a formalism is a philosophy without any real
> ``philosophical formalists''? 

I am a formalist, in that I believe:  1) it is essential to modern
mathematics that it can be formalized, and 2) for much of modern
mathematics the only meaningful notion of truth (if it is a notion of
truth at all) is provability in some formal system.

I am also a methodological formalist in that I am skeptical of discussions
of mathematical knowledge, mathematical objects, mathematical truth; I
generally prefer those discussions (if and) when they can be recast in
terms of formal systems.  I would not say that the formal systems are
without any meaning, but I would rather frame the discussion in terms of
the intuitions that we can associate to those formal systems.

I presume one could identify various others as formalists on the basis of
their publications, though I have not tried to do so.  So while I
generally recommend against discussing "constructivists" (since I am not
sure if there are any, and in any case they are very few), I do not have
those reservations about the term "formalists".


More information about the FOM mailing list