FOM: FOM/Time Magazine

JC Beall beall at
Tue Feb 27 18:32:27 EST 2001

on 2/27/01 12:39 PM, Harvey Friedman at friedman at wrote:

> So what of the future of real f.o.m. in the 21st century University? I see
> only two viable alternatives.
> 1. As a joint enterprise between Mathematics and Philosophy, facilitated by
> University Administrations.
> 2. As the leading component of a new Academic structure - Foundational
> Studies.
> ...[SNIP]
> We must continue to work hard for these two alternatives. But above all,
> there is one thing that we must not do. We cannot think for the moment that
> genuine foundations of mathematics - of the glorious kind that had such a
> special impact on 20th century intellectual life - is being adequately
> continued by current developments in mathematical logic, philosophy of
> mathematics, or computer science.

I think that Professor Friedman's proposed task is important; the given
alternatives should indeed be pushed.  Moreover, his assessment of the
current state of "real f.o.m." is probably accurate.  What I wish to note,
however, is that there may well be drama (and hard work) in the future of
f.o.m., and especially on the very issues that had such a special impact on
20th century intellectual life, namely the very "limitative results" of
Goedel and Tarski (and related work in computability).  What I have in mind
is the increasingly discussed idea that some contradictions --- some
sentences of the form A&~A --- may be true.  While such true contradictions
are likely to be rare, the chief candidates for such entities involve
self-reference, diagonalization, or the like --- liar-like sentences, and
even the pivotal Goedel/Tarski sentences utilized in proving the relevant
theorems.  In the most natural way to model such true contradictions,
Goedel's theorem comes out true (but also false).  [E.g., Priest's 1979 LP,
suitably extended to full first-order, yields such a result.]

So what?  The only point is this:  Foundations of Mathematics is precisely
the place wherein the "very idea" of true contradictions ought to be
explored.  If there are true contradictions (forced upon us by our best
theories of truth, paradox, or the like), then this would certainly seem to
be one of the most significant discoveries since Goedel's "limitative"

All the best,


JC Beall
beall at

Pluralism Conference:

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