3 ms·
There was an article linked yesterday (sorry, don't have the link) which discussed the Reals in considerably more detail and sidetracked into a reformulation of
by Tloewald 13y ago
There was an article linked yesterday (sorry, don't have the link) which discussed the Reals in considerably more detail and sidetracked into a reformulation of the Reals that rigorously allows for Newton's infinitesimals without being "complete" in the "least upper bound" sense (and indeed, depending on the classes of functions you care about, doesn't even require an uncountable set).
- graycat 13y agoMy view is that infinitesimals were a mistake, a long, festering sore in math and some of its applications. Can think intuitively about infinitesimals if you want, but I see no need to mess up the now very clean situation on the foundations of the reals. Indeed, as a grad student, I had some results and went to see a world famous prof to show him what I had and ask him if my work seemed new to him. At one point he suggested an improvement, and, curiously, as a high end pure mathematician in analysis did his first cut, intuitive reasoning in terms of infinitesimals! Fine. But Dedekind cuts, Cauchy sequences, etc. make a rock solid foundation for the reals, and now can we just get on with, say, Fourier theory and the rest of analysis and their applications?