Zermelo Self-Evident
Prof. Shapiro's talk at the Philosophy Club this Wednesday was, as he himself said, mostly about philosophy of mathematics, "but with thirty seconds of political philosophy at the end". The topic was "We hold these truths to be self-evident: But what do we mean by that?" The first line is probably familiar to my American readers (if any), but for the rest - it is a line from the US Constitution. Thus, political philosophy.
The chief aim of his paper, however, was to elucidate the notion of self-evidence in the foundational project of mathematics. In particular, he presented a study of the notion in Frege and Zermelo. The latter of these is without doubt worth brief mentioning. On Shapiro's account, if a mathematical proposition p (Zermelo's is discussing the Axiom of choice) is unconsciously applied frequently and widely by professional mathematicians, then it is Zermelo self-evident. The notion that is attributed to Zermelo, was supposed to protect the Axiom of Choice from its many critics, who claimed that it was not selv-evident enough (in a non-Zermelo sense) to be an axiom.
The talk provoked a great deal of questions about the epistemic status of Zermelo self-evidence, whether or not such an empirical finding gives evidential warrant. Of course, examples like the theorem of the excluded middle seems to indicate that it is, at least, controversial to grant any epistemic significance to the notion. Furthermore, some of the commentators (among them Prof. Nolan) seemed unimpressed with the interpretational work on Zermelo (there were some translational issues).
At any rate, it was a stimulating paper, and turned out to be the talk with the largest audience so far this year. Prof. Shapiro's paper is not yet availabe online, but as he has promised to make it available, I will hopefully soon be able to give a link to it.
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2 comments:
It was definitely a good talk. But - of course - I'm still worried about what role apriority plays here. Actually, the worry goes for both Frege and Zermelo. In Frege's case: Are the self-evidently true axioms and self-evidentially truth-preserving rules of inference a priori? And in that case, how can Frege 'define' something to be a priori if it is deducible from a general law - which does not have to be logical?
In Zermelo's case: Is something a priori just because it's implicit in practice. What if we find out that what we've been implicitly using in practice is wrong? Can p be a priori and false at the same time (as opposed to its being a priori that p is false)?
Puzzled...
"We hold these truths to be self-evident" is from the US Declaration of Independence, not the Constitution.
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