Logic and Mathematicians

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3 total messages Started by Richard Zach Wed, 11 Nov 1992 20:46

#346

Author: Richard Zach
Date: Wed, 11 Nov 1992 20:46

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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

Re: Logic and Mathematicians

#376

Author: Jack Campin
Date: Fri, 13 Nov 1992 17:20

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zach@csdec1.tuwien.ac.at wrote:
> 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.

I think this gets the practice of both ordinary mathematics and logic wrong.
In both cases many of the major achievements are kinds of classification of
informal ideas: 19th-century analysis only got results by seeing that the
general idea of a function split into many, depending on continuity,
uniform continuity, degree of differentiability or integrability, and so on;
what Cantor did by breaking the pre-existing idea of "infinity" up into
cardinal and ordinal hierarchies is not all that methodologically different.
(And Cantor's work *did* start from a conventional mathematical problem, the
convergence behaviour of Fourier series).

Move on a bit: later work in logic tries to pin down the idea of "proof",
originally as vague as "function" was in 1800, and ends up with a vast
range of classifications by proof-theoretic complexity, degree of
constructivity, ontological assumptions about the infinite, strength of
choice principles employed, and so on.  Meanwhile in "normal" mathematics
you have a whole industry going on inventing new kinds of "geometric"
entity and classifying them: Banach and Hilbert spaces, topological spaces,
lattices,...  yes, these *do* come up in the context of solving existing
problems like stability of dynamical systems, but so does the definability
hierarchy of set theory arise out of conventional mathematics.  And the
notion of "algorithmic problem" was no more invented by the logicians of
the 30s than the idea of "curve" was by 19th-century differential
geometers; both are thousands of years old.  What recursion theory did was
to find *structure* inside that notion, the degree hierarchy.

Any takers for renaming mathematical logic as "hierarchy theory"?  It would
be a lot less misleading in some ways.  And a "Frege-free" history of the
subject might make it clearer what most logicians have actually been doing
with their time over the last few decades.

-- 
--  Jack Campin    room G092, Computing Science Department, Glasgow University,
17 Lilybank Gardens, Glasgow G12 8RZ, Scotland   TEL: 041 339 8855 x6854 (work)
INTERNET: jack@dcs.glasgow.ac.uk or via nsfnet-relay.ac.uk    FAX: 041 330 4913
BANG!net: via mcsun and uknet     BITNET: via UKACRL    UUCP: jack@glasgow.uucp

Re: Logic and Mathematicians

#492

Author: Randall Holmes
Date: Sun, 15 Nov 1992 22:03

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In article <1992Nov14.141234.26280@email.tuwien.ac.at> zach@csdec1.tuwien.ac.at writes:
>In article <1992Nov12.204959.17389@husc3.harvard.edu>, zeleny@husc10.harvard.edu (Michael Zeleny) writes:
>|> I offer Post's problem as a counterexample.  Also, it doesn't seem
>|> that there could be a solution for "P =? NP" that would render the
>|> problem silly in retrospect.
>
>Oh yes: Prove P=NP is independent of ZFC. It actuall HAS
>been shown independent in some weak arithmetics.
>
>-- 
>Richard Zach                         Technische Universitaet Wien
>[zach@csdec1.tuwien.ac.at]     Abteilung fuer Formale Logik 185.2

Wouldn't that actually answer the question?


-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu

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