[FOM] Goedel, computers in mathematics, etc.

[Jeremy Avigad]

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