FOM: Re: comments on Steel's "large cardinals needed/apprecator:!:
Lee J. Stanley
ljs4 at CS1.CC.Lehigh.EDU
Fri Nov 21 23:08:21 EST 1997
I would like to offer what I hope would be recognized as a friendly amendment
to Steve's post. I would suggest inserting "currently" before
most occurrences of "applicable", in which case one of his
(and Feferman's) main theses, that all of (currently) applicable
mathematics can be done in W becomes a reasonably precise, though
ultimately sociological, statement, for which there is probably
a fair amount of evidence.
However, to go from there to the suggestion that it might be ok
for (currently) scientifically applicable mathematics and pure mathematics
to go their separate ways seems dangerously ahistorical, and
probably, ultimately more dangerous to the progress of science
than to the progress of pure mathematics.
Even if, after the fact, it is possible to scrape away and get
down to W to save most currently applicable mathematics, that
is NOT the route which led to notions like "group" just happening
to be lying around when physics grew to the point of being to use
them (need them?). And the WAY THINGS HAPPENED has some importance,
I think, and is related to the robustness issue I raised in my last
post. Would anyone seriously contend that the development of mathematics
within W would have been anything near as successful as the development
we have known in a more "freewheeling" setting (setting aside, for
the moment, the unlikelihood of ever formulating such systems except
as a result of analysis of practice in more robust and natural systems)?
This said, identification of systems such as W as sufficient to account
for fairly large, fairly coherent pieces of current mathematical practice,
which correlate well with what is currently applicable, has to be of
interest in the mapping out of the axiomatic terrain. This has always seemed
to me to be one of the main concerns of foundations.
What is objectionable is to identify these pieces with all the
mathematics that ever-will/is-ever-likely-to be applicable to science.
More information about the FOM