FOM: wider cultural significance, part 1

[Stephen G Simpson]

This is the first in a series of FOM postings about the wider cultural
significance of f.o.m.

1. POSTMODERNIST ABUSE OF G"ODEL'S INCOMPLETENESS THEOREM

Humanist academics of the postmodern stripe routinely cite G"odel's
incompleteness theorem in order to buttress their neo-Marxist points.
It appears that almost all of the big names in postmodernism have
indulged in this.  They are all charlatans.  We f.o.m. professionals
ought to find ways to combat this kind of abuse.

Mic Detlefsen 16 Feb 1999 11:36:32 cited two recent essays:

 > (1) 'Reflections on the legacy of Kurt Goedel: Mathematics,
 >     Skepticism, Postmodernism', John Kadvany, THE PHILOSOPHICAL FORUM,
 >     1989, 20: 161-181.
 > 
 > (2) 'Goedel's Theorem and Postmodern Theory', D. Thomas, PUBLICATIONS
 >     OF THE MODERN LANGUAGE ASSOCIATION, 1995, 110: 248-61.

Normally I skip anything with postmodernism in the title, but just for
the heck of it I looked at these two papers.

The Thomas paper (2) is incredibly bad.  I can't refute it, because
there is no way to refute gibberish.  One thing I can say is that the
exposition of G"odel's incompleteness theorem is chock full of
howlers.  The paper as a whole reads remarkably like Sokal's famous
`Social Text' paper, a deliberate parody of postmodernist
gobbledygook.  (To be fair to Mr Thomas, I must note that many papers
published by the MLA have a similar ridiculous quality.)  Poor Mr
Thomas.  He is apparently a typical humanities grad student flunky.
It sometimes sounds as if he is actually trying to say something, but
his lit-crit education has undermined his command of language, to the
point where he can't express himself properly.  If I had to guess his
meaning, I would make the following stab: `G"odel's theorem isn't
completely compatible with postmodernism, because it's based on some
old-fashioned assumptions.'  Or maybe, `Postmodernism is in a mess,
because it's a lot like G"odel's theorem, which is also in a mess.'
I'm really not sure.

2. G"ODEL'S THEOREM AND SKEPTICISM

The Kadvany paper (1) is actually pretty interesting.  I would like to
get Kadvany into a discussion here on FOM, but unfortunately he is not
(yet) a subscriber.

Kadvany is appropriately disgusted by postmodernism, but he doesn't
say much about it directly.  Instead, he discusses arguments from an
ancient school of philosophy called Pyrrhonian skepticism, drawing
parallels to G"odel's incompleteness theorems.  This is much more
fruitful than trying to deal with deliberately obscure, postmodernist
mumbo-jumbo.

Among the skeptical arguments mentioned by Kadvany is the `problem of
the criterion'.  This says that the idea of a criterion of truth leads
to infinite regress, because another criterion is needed in order to
validate the first criterion.  According to Kadvany, this resembles
progressions of theories obtained by adding consistency statements.
(He could have mentioned truth predicates.)  There is also the
`peritrope' or self-destructive nature of skeptical arguments,
highlighted by the fact that the consistency of T is assumed as a
hypothesis in both the first and the second incompleteness theorem.
Finally there is Hume's `schizophrenic skepticism', to the effect that
skepticism is true but nobody can live by it, reflected in the 20th
century mathematician's split personality: formalist on weekends,
realist on weekdays.

In an attempt to defend reason against skeptics and postmodernists,
Kadvany attacks G"odel's theorem.  In particular, he raises the vexed
issue of the need for a `canonical' or `natural' proof predicate.  In
my opinion, Kadvany is barking up the wrong tree here, because there
is no serious challenge to the naturalness of the standard proof
predicate, and if any doubt remains, the Hilbert-Bernays derivability
conditions take care of it.

While I don't agree with everything Kadvany says, I applaud his
recognition that there is a need to cope with the destructive or
skeptical overtones of G"odel's theorem.  In my opinion, the way to do
so is to (a) accept the G"odelian conclusion that there is no *purely
mathematical* secure foundation of mathematics, and (b) seek another,
better foundation by integrating mathematics with the rest of human
knowledge.

One approach to (b) would be to argue that PRA is consistent because
the physical world provides a model of it, and then to justify at
least a significant fragment of mathematics by reducing it to PRA, a
la Hilbert's program of finite reductionism.  I develop some aspects
of this approach in my paper `Partial Realizations of Hilbert's
Program' <http://www.math.psu.edu/simpson/papers/hilbert/> and in my
book `Subsystems of Second Order Arithmetic'
<http://www.math.psu.edu/simpson/sosoa/>.

-- Steve