If I may interject for a moment, for the benefit of less math-savvy people out there: There are many kinds of numbers used in mathematics. When you were learning math, you first learned about the counting numbers, then extended that to the integers (which include negative numbers), then extended that again to the rational numbers (fractions), and then to the real numbers (which include irrational numbers like pi). In high school, you might have seen numbers extended to the complex numbers (which include imaginary numbers). Complex numbers are enough to deal with almost all real-world situations, so most folks never learn anything beyond that, but there are actually many, many other possible ways of extending the concept of “number”, some of which don’t even have any practical applications and which mathematicians just made up for fun.
Nonstandard integers and hyperreal numbers and so on, that folks have been talking about for the past few posts, are examples of that: Extensions of the number system that mathematicians made up just for fun. And while, sometimes, something mathematicians make up just for fun turns out to have some practical purpose somewhere, so far as I know that’s never happened yet for hyperreals and nonstandards.
Except that hyperreals weren’t made up just for fun, they were created to justify Newton’s original approach to calculus using infinitesimals. They provide another foundation for calculus—nonstandard analysis—that, though it hasn’t caught on completely at the college level, does have a following. There are calculus textbooks available using infinitesimals as their foundation in place of limits, now that we know it can be done rigorously.
Mathematicians have been struggling to deal with the implications of infinity, infinitesimal and limits probably as far back as the formulation of the area of a circle as half it’s circumference times its radius; the area being the limit of a hypothetically infinite number of infinitely narrow triangles. To which someone probably immediately objected by saying “But wouldn’t that be infinitesimally less than the full circumference?”.
The quest for rigor in mathematics has led to some very strange places.
So far as I know, that result originated with Archimedes, and he did handle it rigorously, using a sort of proto-limit. Basically, he used inscribed polygons to argue that any amount greater than 1/2 rC was greater than the area, and then he used circumscribed polygons to argue that any amount less than 1/2 rC was less than the area.
Poor Hardy liked to think that his mathematics was pure as the driven snow, with no conceivable applications - he contributed to physics, biology and to computer science considerably
I think it is understandable, especially during wartime, to worry about one’s work being adapted to make H-bombs (or any military applications), or by the GCSQ, even if one does not work for them or accept any funding from them.
Note that Feynman did not have a problem working on the Manhattan Project, while Hardy took pains to emphasise (via wishful thinking) that “real” mathematics at least does no harm and has no effects on war.
Feynman had a lot of regrets afterwards - in his mind he had signed up to stop Hitler and in retrospect wished he had stopped his efforts after V-E Day
Granted, though that’s a little different than being appalled that one’s work has any practical application. Wasn’t a case where he worked on some nuclear physics problem for its standalone interest and then got worked up when it was used for a bomb. He was working on the bomb the whole time.