[FOM] BANNING impredicative mathematics

Nik Weaver nweaver at math.wustl.edu
Wed Feb 15 02:49:59 EST 2006

Harvey Friedman implied that I am out to "ban" all impredicative
mathematics.  I forcefully denied this and insisted that nothing
I have written suggests anything of the sort.

Incredibly, Friedman neither apologized for misrepresenting
me, nor quietly let the matter drop.  Instead, he posted a new
message containing a farfetched argument which purports to show
that I really am out to ban impredicative mathematics, despite
the fact that I've never said this and indeed strongly denied
it.  As if to shout me down, he has now begun writing BAN in
all capitals.

Here's his case:

> It is just a weaker form of banning. You seem to argue

("Seem to"?)

> that it should be banned if one says that one is doing mathematics
> with a clear philosophical basis. Many people want to have a clear
> philosophical basis, or think they have a clear philosophical basis,
> for the mathematics that they do. The practical effect for them of
> what you say is: to BAN their work.

and he deftly adds

> Do you want to BAN some of the work of that [unnamed] Fields
> Medalist [the star of an impenetrable earlier anecdote] and his
> colleagues?

Let me get this straight.  People want to think that their work
has a clear philosophical basis.  If I succeed in convincing them
that it doesn't, they will abandon that work, hence I have in
effect BANNED the work in question.

That seems like pretty nefarious activity on my part.  Perhaps
I should be BANNED from foundational discussions?

Wait a minute, I think I *am* being BANNED!  After all, I want
to think that my discussions of foundations will have some
positive effect.  But now Friedman informs me that the effect
is harmful.  The "practical effect" of what he says is to BAN
me from debate!

Enough farce.  What is my actual stand on impredicative mathematics?
I agree with Paul Lorenzen (Differential and Integral, p. 37):

"Logically, there can be no objection to the erection of axiomatic
theories (in this case axiomatic set theory).  Even if the axiom
system of the theory is not known to be consistent, the deducing
of theorems from the axioms is an unobjectionable preoccupation
... it is nobody's business here to permit or to forbid ... All
anyone can do is to decide for himself what he wants to do."

Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA
nweaver at math.wustl.edu

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