[FOM] repairing a bridge between mainstream mathematics and f.o.m.
pratt at cs.stanford.edu
Sun Jul 19 07:53:30 EDT 2009
Timothy Y. Chow wrote:
> I've reread your message several times, but I confess that I still don't
> understand what you think the disconnect between f.o.m. and mainstream
> mathematics is.
Tim, I hope by this that you mean you don't see Dunion's disconnect, as
opposed to any disconnect.
As near as I can tell, the concept of "mainstream mathematics" exists
only in the minds of those who take an active interest in "fringe
mathematics," either as a participant therein or as a spectator staking
out some position on it such as disapproval or wonderment.
Bill Thurston made the point a while back that communicating his
geometrical insights to analysts required a nontrivial shift in
perspective from when he was addressing topologists. Clearly there is
some sort of disconnect in practice, if not in principle, between
analysis and topology. Yet neither would be comfortable accusing the
other of doing "fringe mathematics" for fear that the combinatorialists
and number theorists and algebraic geometers might round on them both
and characterize them as, if not outright fringe, at least passe.
Mathematics is driven by its fashions no less than the runways of Paris
and New York.
The questions that pass for foundational these days seem somewhere
between old-fashioned and baroque. Who in the foundations business is
addressing the urgent question, why do the IMO, Putnam, and other
competitions so neglect the proven power tools of mathematics such as
abstract algebra? This seems far more relevant than worries over the
consequences of the obvious fact that aleph-1 cannot be split evenly in
half, which I would take as a good argument for not reading too much
into the relevance of uncountable ordinals to analysis.
Concluding from that reasoning that aleph-2 is a better ordinal for
analysis is like jumping out of the 300 degree frying pan into the 600
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