First Year Seminar. The Intellectual History of Information: 1680-2001
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Date: Fri, 9 Nov 2001 00:08:31 -0500 (EST)

Class 19

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

To wit:

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 here.

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.

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