[FOM] The irrelevance of Friedman's polemics and results

John Steel steel at Math.Berkeley.EDU
Thu Feb 2 16:39:54 EST 2006

Arnon Avron wrote:
why not choosing a good point
which is *not* arbitrary (like predicative mathematics) and devote
most of our efforts  to it? or at least wait until "core mathematicians"
come themselves across problems in which they need stronger methods,

As I understand it, predicative mathematics is not such a "good point", as 
the existence of least upper bounds for arbitrary sets of reals is not 
predicatively justified. Not many would want to revamp the way we teach 
undergraduate Analysis in a way that paid attention to predicativity. 
Certainly not our revered "core mathematicians".  So "core mathematicians" 
(whoever those people are-- the editors of Annals of Math?), who have 
learned the naive lub principle, are forever in danger of using it in a 
necessary way.

Is this a problem, an accident waiting to happen, so to speak?
Should we fight to reform our curriculuum, so as to encourage
only "safe" predicative mathematics?

Pure mathematicians get out ahead of applications. They find natural
structure, and later the less pure use it. They of course look for
applications, so as to see the structure they have found is important.
Mostly, these applications are to other parts of pure math. Much of what
Avron writes seems to be a criticism of this procedure in itself.

Finally, wasn't Lebesgue a "core mathematician" of his day, and
didn't he care about the Lebesgue measurability of projective sets?
One needs large cardinals to get much anywhere on this problem.
Perversely, it is only because we understand the question is
tied up with large cardinals that today's "core mathematicians"
can dismiss it. If we didn't know about large cardinals, every
theoretically inclined Analyst would take a shot at this wonderful
open problem.

John Steel

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