[FOM] repairing a bridge between mainstream mathematics and f.o.m.

Vaughan Pratt 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 
degree fire.

Vaughan Pratt

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