From: rz@mips.complang.tuwien.ac.at (Richard Zach)
Subject: Logic and Mathematicians
Reply-To: zach@csdec1.tuwien.ac.at
Organization: Technische Universit"at Wien
Date: Wed, 11 Nov 1992 20:46:55 GMT

I just had an interesting discussion about the
fear mathemicians have of logic (and logicians):

Many, if not most, "great" logicians did not
"solve problems", as one usually does in mathematics,
in that someone states a problem, and a the you go
ahead and solve it (positively or negatively);
furthermore, their solutions are more
instructive in themselves than the original problems
where and they not only develop new concepts
to solve an existing problem (this is commonplace
in mathematics), but rather invent a new concept which
they think is interesting IN ITSELF, prove something about it,
and incidentially some open problem follows as a
corollary. In fact, more often than not, they do
not solve problems in the usual sense, but they
say something about the problem itself, eg, that
it was mis-posed, mis-conceived, or is meaningless.

Examples: G"odel's Incompleteness Theorems:
He INVENTED the concepts of undecidability of
a sentence in a theory and of incompleteness.
The interesting sentence (from the logical, and
also from G"odels standpoint) is the FIRST
incompleteness theorem (Why else did G"odel
entitle his paper "On undeciadable propositions"?
If he were interested more in consistency,
he would have titled it "The Unprovability of
Consistency" or something). From the mathematician's
viewpoint, the proof is trivial: A simple
diagonalization argument (in Kreisel's words).
Yet, Hilbert (and others) were very much taken by
surprise. In a sense, incompleteness says:
"The concept of consistency is not important"

The consistency of CH: is a corollary of
the consistency of V = L. And L is G"odel's
invention. Also it says (together with Cohen's result),
that the question of whether CH holds or not  
is meaningless.

The undecidability of predicate logic from
Turing's work: The really important part of his
paper were the "Computable Numbers", not 
the "Application to the Entscheidungsproblem".
And he MADE UP Turing machines etc. Again,
easy theorems, difficult concepts requiring
deep insight.

Gentzen's sequent calculus: his invention. Totally
disconnected from anything that had been done before.
Required deep insight into the nature of logic
to come up with. The "problems", eg, Herbrand's
Theorem, were easy corollaries of the Hauptsatz,
which, in itself, was easy to prove. But how do
you come up with the Hauptsatz in the first place?

I'm not sure myself what I'm trying to get at,
nor what I'm trying to say, so please take what I say
cum grano salis. Some questions:

(1) Is this really a difference in the way logic works
to how mathematics works? (I realize that most of what's
done in logic is not of the kind described above,
but more "mathematical", in this sense)

(2) If yes, is that because logic is a relatively
young field? Were the mathematical couterparts
of these results the early results, say the
invention/discovery of: variables/algebra/the infinite/etc?

(3) Are there more recent examples of such fundamental
findings which cast a problem in a different
light, maybe made it seem a silly problem to pose
in the first place?

-- 
Richard Zach                         Technische Universitaet Wien
[zach@csdec1.tuwien.ac.at]     Abteilung fuer Formale Logik 185.2
Newsgroups: sci.logic