First Year Seminar. The Intellectual History of
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Date: Fri, 9 Nov 2001 00:08:31 -0500 (EST)
There are three things here:
1) Summary of our discussion of the essay quiz (exam?)
1) A summary of our initial discussion of Godel (and
Wittgenstein) on Wednesday.
2) Your assignment for next Monday on Hilbert
1) In class on Wednesday, November 7, we read aloud some
examples of essays from the exam and discussed them.
Remember that everyone is supposed to go to the Sweetland
writing center by early next week and take samples of their
writing to discuss with the writing advisors there.
2) We discussed the first section of [Davis] on Godel, from Chapter 6.
This concerns Godel's Ph.D. thesis which he did under Hahn. He worked on
one of the two open problems raised by Hilbert and his student, Ackermann,
in their little text book on Frege's first order logic. This is the
problem of showing that FOL is "complete". Demonstrating completeness
means showing that FOL encompassed all forms of valid logical deduction.
In particular, this means showing that if one makes valid (true)
propositions in FOL, that any valid conclusion reached from these
propositions can be shown to also be valid in FOL. Hilbert and other
mathematicians wanted a logical system that encompasses all possible types
of valid logical inference. Recall, in this regard, how Aristotelean
logic's syllogisms were shown by Boole and Frege to be very restricted and
unsatisfactory. Now, Hilbert wanted to have a proof that FOL was the real
thing, literally everything one could or would ever need.
Godel said he had to reject much of the program of the
Vienna Circle (i.e., of the logical positivists), and the
way they approached problems, in order to complete his Ph.D.
dissertation in which he proved Frege's first order logic is
complete . He concluded that the logical positivist
objections to non-finitary methods for solving problems
concerning the foundations of mathematics (such as
Russell-Whitehead's PM, and Hilbert's metamathematics) ARE
reasonable, but these restrictions are not relevant to
treating problems of logic itself. Proving FOL is complete
is not a problem in the foundations of mathematics, and so
there is no reason to restrict oneself to finitary methods
So, everyone else had handicapped themselves by sticking to
Brouwer and Weyle's very restrictive finitary
(intuitionistic-based) methods. According to [Davis], even
Hilbert, their main opponent, had tacitly given in because,
when he agreed to stick to Brouwer's finitary methods in
examining his own (Hilbert's) metamathematics. Hilbert had
apparently just assumed (unconsciously?) that these
strictures apply to logic itself as well. Only Godel (at
24?) had the clarity to see these are distinct questions.
After this, the actual proof of the completeness of FOL was
almost trivial, and even seemed very "old fashioned."
We also discussed a bit what the attitudes of the
empiricists and positivists were towards "explanations" in
mathematics and science. They were quite hostile to this,
referring to discussions of mathematical "truth" and
"explanations" in a derogatory manner as being
"metaphysics" (i.e., as if it is like practicing a
"religion" - just a "belief", something which cannot be
proven objectively and hence doesn't belong in science or
math). [Of course, there were and are other non-positivist
scientists who were either religious or completely
un-religious and thought it was fine to talk about
scientific and mathematical "truth".] We discussed how
Russell-Whitehead's success in PM, Frege's successes in
logic, and the works of Ludwig Wittgenstein were great
inspirations to the Vienna Circle.
Wittgenstein, the philosopher, in his Tractatus Logico-
Philosophicus, had been very preoccupied with language. In a
nutshell, he believed that thought is revealed in language,
and, so, if one understood what language was about, they
could understand thought (which is of course very important
to philosophers). The point here, for our purposes, is that
this led to a great deal of interest in language, and
Wittgenstein was especially preoccupied with the problem of
how could one analyze and talk about ordinary language
using language itself. Note how this preoccupation is
similar to the problems which with the metamathematicians
and logicians were concerned. In fact, Wittgenstein made
contributions to logic, and, in particular invented truth
tables (Boole had not done this as part of his propositional
logic, contrary to what we were led to believe from our
earlier readings). Wittgenstein was originally a student of
Russell's, but very quickly was treated by Russell as a
colleague and Russell respected him immensely for his
brilliance and ideas.
We started to take apart some examples of propositions and
conclusions so as to illustrate what was at stake in showing
FOL is valid. We'll continue this on Monday, 12nov, and then
on to undecidable propositions -- Godel's real opus!
3) Your assignment for Monday (emailed to me before class)
is to write a one page "encyclopedia" entry on Hilbert.
Single spaced. It should not be shorter, if anything, make
it a little longer, up to about 1 1/4 pages or so. Try to
cover all important facets of his life and especially his
work and program. Include a bibliography. This should be a
valuable resource for study later on.
Go back, up one level