3 ms·

So, he likes what is known as the completeness property of the reals. Good. Glad he likes that! One remark is "Calculus is the elementary consequences of the

by graycat 13y ago

So, he likes what is known as the completeness property of the reals. Good. Glad he likes that! One remark is "Calculus is the elementary consequences of the completeness property of the reals.". And, yes, calculus is nice stuff. Newton did nice things with calculus but may not have understood the completeness property, i.e., just took it for granted that his derivatives and integrals did yield numbers. Some of what is true about the reals is bizarre, nearly beyond belief, e.g., as in Oxtoby, Measure and Category. The main reason for being careful with fine details going from the natural numbers to the integers, rationals, algebraics, and reals is a theme that started near 1900 of trying to clean up the logical basement of mathematics and, there, build everything on just sets so that, if believe in sets, then could believe in everything that could be based on sets, that is, hopefully all of mathematics. This goal seemed okay for about 10 minutes until B. Russell thought of the set A = {x|x is not an element of x} so that A is an element of A if and only it is not. Bummer. Seventy years later the gods of axiomatic set theory came out with a lot of work that made set theory look more solid. Okay, that's curious work, and on a good day with four quarters will cover a $1 cup of coffee.

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 ago
  My 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?

by graycat 13y ago

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 ago
My 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?