All Questions
Tagged with foundations ho.history-overview
12 questions
18
votes
3
answers
3k
views
What's the earliest result (outside of logic) that cannot be proven constructively?
Although mathematicians usually do not work in constructive mathematics per se, their results often are constructively valid (even if the original proof isn't).
An obvious counter-example is the law ...
- 6,353
14
votes
5
answers
1k
views
Emergence of the discrete from the continuum
An almost eternal theme in Mathematics is the approximation of the Continuum by the Discrete. This core idea goes back at least to Archimedes, and remains active to these very days (and quite likely ...
- 7,853
6
votes
1
answer
545
views
Historical origin of the empty set
The question is in the title:
Who first claimed the existence / necessity of the empty set ? When did this happen ?
Of course I know that the notation $\emptyset$ goes back to André Weil, and that ...
- 52.3k
3
votes
1
answer
610
views
Problem Understanding Euclid Book 10 Proposition 1 [closed]
this is embarrassing, but I am having trouble reading through Proposition 1 of Book 10 of Euclid's elements. I'm struggling with Euclid's terminology and don't have a clear picture of what divisions ...
- 595
13
votes
3
answers
2k
views
History of the abstract method in mathematics
Recently I have "finished" a 13-year on and off research on the history of the mathematical notion of equivalence. At the end of which, I learned that we owe the nowadays rather elementary process of "...
- 2,437
9
votes
0
answers
965
views
Has anyone pursued Frege's idea of numbers as second-order concepts?
Gottlob Frege was a pivotal figure in the history of mathematical logic. He gave an analysis of numbers that proceeded along roughly the following lines, in his books "The Foundations of Arithmetic" (...
- 4,587
27
votes
4
answers
4k
views
Who introduced the terms "equivalence relation" and "equivalence class"?
Consider that the question does not concern the origin of the ideas of equivalence relation and equivalence class. It exactly concerns the origin of the terms "equivalence relation" and "equivalence ...
- 2,437
11
votes
4
answers
2k
views
Why do mathematicians prefer one definition over the other when they both define the same concept?
Here is a basic, though very important, example:
Hilbert takes as primary the notion of “congruence” (or “equal”) between segments. His first axiom of congruence “requires the possibility of ...
- 2,437
7
votes
2
answers
1k
views
Sine and Archimedes' derivation of the area of the circle
The elementary "opposite over hypotenuse" definition of the sine function defines the sine of an angle, not a real number. As discussed in the article "A Circular Argument" [Fred Richman, The College ...
- 313
3
votes
2
answers
473
views
Evolution of the Mapping/Function Concept
Hello! I'm looking for a survey (of the history) of the concept of mapping/function. How the concept was evolving. Especially I'm interested in what it turned into during the last 50 years.
So ...
- 441
12
votes
1
answer
3k
views
Up-to-date version of Principia Mathematica?
Background: I found this interesting translation of Godel's On formally undecidable propositions of Principia Mathematica and related systems I that, along with translating it into English, uses more ...
- 1,052
4
votes
2
answers
2k
views
Dedekind's theorem
In "Was sind und was sollen die Zahlen?" Dedekind gives a noncircular
proof of the statement that a set is finite if and only if it cannot be
put in bijective correspondence with a proper subset. By "...
- 18.6k