24
votes
8answers
2k views

Complex analysis is more “real” than real analysis

In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself ...
0
votes
3answers
62 views

Should the notion of continuity, usually ascribed to Cauchy, be ascribed to Leibniz?

In his text, Deleuze and the History of Mathematics, Simon Duffy writes: Leibniz also thought the following to be a requirement to continuity: "When the difference between two instances in a ...
4
votes
2answers
194 views

Why do imaginary numbers work (somewhat philosophical question)?

Asking as a layman, I've always puzzled over imaginary numbers and how they can be used to solve problems involving real numbers or quantities only (e.g. contour integration methods or Fourier ...
24
votes
4answers
1k views

Is mathematical history written by the victors?

The question is the title of a recent piece in the Notices of the American Mathematical Society, by twelve authors (of which I am one). The contention is that traditional history of mathematics is ...
4
votes
0answers
72 views

Constructivism implied or not

Let me take up some details in the answer of another question. Submitted by user hyg17: Heading: All real numbers can be expressed as a limit of rational numbers? The question was: Let $C$ be a set ...
4
votes
2answers
191 views

Gray's “Plato's Ghost” - a curious mistake

I am currently reading Jeremy Gray's "Plato's Ghost", and I run into the following passage (Chapter 5, page 332). The point is, it seems to me that it contains two very elementary mistakes that feel ...
6
votes
6answers
1k views

Michael Spivak in “Calculus” asserts that $\sqrt2$ cannot be proven to exist, and that such a proof is impossible. What does he mean by “exist”?

Michael Spivak in "Calculus" asserts that $\sqrt2$ cannot be proven to exist, and that such a proof is impossible. What does he mean by "exist"? How are you to prove that any number "exists"? Why ...
5
votes
4answers
412 views

From continuity to differentiability and analyticity- what's next?

Continuity is an intuitive concept. I will not dwell on the precise definitions of continuity and the rest here. Note that differentiability is a more restrictive condition than continuity, while ...