[FOM] Goedel, computers in mathematics, etc.
Jeremy Avigad
avigad at cmu.edu
Wed May 31 11:00:18 EDT 2006
Friends and colleagues,
I have posted online a few things that may be of interest to some in the
FOM community. They are on my web page:
http://www.andrew.cmu.edu/~avigad
If I may be forgiven the shameless self-promotion, I'd like to describe
them here.
First, under "Surveys," there is the text of a lecture, "Goedel and the
metamathematical tradition," that I recently presented at the ASL 2006
spring meeting in Montreal. The lecture describes the metamathematical
tradition emerging from Hilbert's program, and characterizes Goedel's
work as fitting squarely in this tradition. Nonetheless, one often finds
Goedel expressing dissatisfacation with the associated methodological
orientation. I explore this tension, and try to explain what lies behind it.
Second, under "Surveys," there is an article called "Computers in
mathematical inquiry." This will ultimately appear in a collection, *The
Philosophy of Mathematical Practice*, that is being edited by Paolo
Mancosu. The collection will also contain essays by Ken Manders, Marcus
Giaquinto, Paolo Mancosu and Johannes Hafner, Mic Detlefsen, Jamie
Tappenden, Colin McLarty, and Alasdair Urquhart.
Each contributor was assigned a topic and asked to write an introductory
article, followed by a more focused research piece. "Computers in
inquiry" is my introductory article. I survey the broader
epistemological issues that arise in discussions of the use of computers
in mathematics, and note that they tend to fall into two categories:
first, there are assessments of the ability of computers to provide
"evidence" for mathematical assertions, and second, there are
assessments of the ability of computers to deliver appropriate
"understanding." The goal of the article is, for the most part, simply
to lay out and clarify the issues. I do, however, take a somewhat
critical stand against various attempts to make sense of the notion of
mathematical evidence, or assessments of "likelihood" or "plausibility"
in mathematics. My second contribution to the collection will address
the notion of mathematical understanding, along the lines laid out in
another article I have written, "Mathematical method and proof" (also on
my web page, under "Research").
Third, I would like to note that a book called *Architecture of Modern
Mathematics: Essays in History and Philosophy*, edited by Jose Ferreiros
and Jeremy Gray and published by Oxford, is now in print. According to
Amazon and Oxford USA, it is "not yet released," but Oxford UK has it
listed:
http://www.oup.com/uk/catalogue/?ci=9780198567936
My copy arrived a couple of weeks ago, and it is a very nice collection.
"Modern mathematics" refers to the style of mathematics that emerged in
the nineteenth century, encompassing algebraic, axiomatic,
set-theoretic, and nonconstructive methods that are common today. The
book's essays provide a multifaceted perspective on these important
developments, and a better understanding of features that are now
central to mathematical thought. My own contribution, on the development
of Dedekind's theory of ideals, is posted on my web page, under "Research."
Finally, a few months ago I announced a review of two books on the
history of logic and foundations in the early twentieth century. At the
request of the editor of the Mathematical Intelligencer, this has been
split into two separate reviews, both of which can be found under "Reviews."
Best wishes,
Jeremy
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