File feyerabend/feyerabend.0602, message 1

Date: Sat, 11 Feb 2006 16:35:55 -0500
Subject: [PKF] Online mini-course in Goedel's Incompleteness Theorems

Dear All,

I am thinking of offering online mini-courses in selected areas of
mathematics, and of starting with a course whose topic would be
Goedel's two Incompleteness Theorems, a subject in which I
have particular expertise.   Since I have never taught  more than
one person at a time over the Internet,  I am going to offer the first
course for free -  which will help me test and debug many of the
uncertain aspects.

The Incompleteness Theorems, which were first published in the 1930s,
address  the possibility of constructing an adequate formal axiom
system for arithmetic  - i.e. a system of propositions that would
serve as an adequate formal definition of the set of positive
integers.  The first Incompleteness Theorem shows that however one
chooses such an axiom system, it will always be possible to construct
a proposition which is a true statement about the positive integers,
but whose truth cannot be proved from the given axiom system.   The
second Incompleteness Theorem shows that if a given axiom system for
arithmetic is consistent, its consistency cannot be proved within the
system itself.

The issues with which these theorems are involved, and the questions
they raise, are of great relevance to the philosophy of science and
to many of the issues with which Feyerabend was involved.   They
concern the nature of axiomatic systems;  the difference between logical
coherence, logical consistency,  scientific consistency, and the intrinsic
logic of reality; the difference between operational adequacy of
scientific theories and their adequacy to Nature; the relationship
between thought and Nature; the capability of Science to be more than
a model, or more than an axiomatic - to be an actual description of
reality; the nature of Mathematics and of Science.

The mini-course will not require any background in mathematics or
logic.   It will be taught by me live, using a conference-call
service in the US and a software whiteboard (which is like a remote
chat program, but permits the participants to  not just type text,
but also draw graphics).  It will consist of 10 "live" weekly
sessions, each between 1 and 2 hours long.  There will also be
asynchronous means of exchanging ideas such as a private Web forum
and/or an email list.  I envision the course to have 5 to 10
participants.   The only cost to the participants would be the
long-distance telephone charges, which can be as low as 4
cents/minute from most anywhere (if your long-distance service is
more than that, I can recommend a cheap long-distance service that
operates on a "local number plus access code" basis, and therefore
does not require you to change long-distance providers).

I would like to begin this course around March 15th.  Currently, my
rough "syllabus" is:  4 sessions to develop the methodological and
philosophical background, 4 sessions to discuss the 1st theorem, 1
session to discuss the 2nd theorem, and 1 session for summary and
general ramifications.   My aim to present a clear, comprehensible
and philosophically uncluttered idea of the theorems and their role
in mathematics.

If you are interested in participating in this course, please let me know
by writing to me at .   Tell me a bit about
yourself and how to get in touch with you by phone - and tell me, in
the most  excruciating detail you can muster, what days of the week,
and what times, would be acceptable to you - so that I can pick a day
and time that provides the greatest good the greatest number.

With best regards,


Malgosia Askanas
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