Lawvere and Leibniz

Lawvere and Leibniz

[David Roberts]

Hi all,

here is an philosophy article perhaps of interest to some on the list

http://www.newappsblog.com/2011/11/the-truth-about-infinitesimals.html

discussing the contrast between the 'old' solution (Weierstrass et al) to
non-rigorous calculus a la Leibniz and the new solutions, nonstandard analysis
and SDG, but mostly the latter. Not all ideas are credited, so only Lawvere
is mentioned, and not Grothendieck, A. Kock, or others who contributed to
the development of the philosophy of 'infinitesimals' in geometry.

I think it is a nice advertisement for category theory, even mentioning the
phrase 'The world or category of smooth spaces and smooth maps...'.


David


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Re: Lawvere and Leibniz

[Michael Barr]

Very interesting.  Curiously, nowhere in the post is its author
identified.  Some of the comments refer to a Dennis, but that is no one I
can identify.

Russell's view strikes me as especially inane.  Had Weirstrass not
produced his concept of limit, would Russell have simply discarded
calculus as nonsense?  An awful lot of problems were solved correctly,
even before Weierstrass--would Russell have discarded them.

Another point that bothered me was the tossing aside, without explanation,
of dx^2 because it was infinitesimal vis-a-vis dx.  Better to do what
Robinson did, divide by dx and then take the ordinary part as the answer.
Perhaps he could have introduced a different "equality" symbol.

Still, it was a nice mention of categories.

Michael


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