Maddy has written some of the most sensible philosophy of science (mostly mathematics) I have ever read. She combines talking about modern set theory (a viewpoint from what is called the Cabal, sort of UCLA, UCI, and UC Berkeley mathematicians, Maddy and some others), and discussions of the axioms of set theory and the large cardinals, with reference to the usual gang of philosophical (dead white men).
Her discussions of set theory are quite wonderful, and in more recent work she has been developing a philosophy of mathematics that is much more recognizable to mathematicians, less totally influenced by too simple examples and too abstract questions. (I'm sure she would not approve this past sentence.)
Floating around in the background are attempts to settle the Continuum Hypothesis, one way or the other. Is the set of all sets of natural numbers the size of the continuum, or not. Steel of Berkeley says yes, Woodin of Berkeley says it (the set of all sets) is much larger. I don't pretend to understand much of this. Tony Martin of UCLA is one of this group, and he has written very sensibly on the nature of mathematics.
The most curious feature of this endeavor is lies in the realm of what is called descriptive set theory. It is possible to develop a hierarchy-1 of sets (the large cardinals), and that hierarchy-1 corresponds to a hierarchy-2 of descriptions of sets that would seem to not have anything so directly paralleling hierarchy-1, but in fact they seem to follow each other precisely. What's also amazing to me is that these various large cardinals are useful for other parts of mathematics, and in Woodin's hands they lead to a claim about the Continuum Hypothesis that is not proved yet ("just" lacking a proof of the "Omega Conjecture").
No comments:
Post a Comment