**8**

votes

**2**answers

261 views

### Finiteness as a motivation for compactness

Another history question, and I am not sure if I will get any answers. (If anyone knows of a good history of math list to use for this question I would be happy for any tips. The one I used to post to ...

**11**

votes

**3**answers

1k views

### How were formulas / images added to books in post-printing-press / pre-digital times?

I have seen that Euclid's Elements was written 300 BC and first set in type in 1482. Are there scans of that old versions available?
How were formulas / images added to the books created with ...

**5**

votes

**2**answers

774 views

### Salvaging Leibnizian formalism?

Can one justify Leibniz's formalism in a suitable algebraic or topological context?
We have published some papers recently where we argue that Leibniz's formalism for the calculus wasn't ...

**32**

votes

**1**answer

3k views

### A topologist is not a mathematician - a small question

Years ago I read about a topologist who was to enter the states as an immigrant and was asked a question about his profession. He indicated he was a topologist, but as this was not included on the ...

**3**

votes

**0**answers

167 views

### History of limit point compact -/-> compact example

A standard example in elementary topology (e.g. Munkres) of a space which is limit-point compact (every infinite subset of the space has a limit point) but not compact is the minimal uncountable ...

**5**

votes

**1**answer

158 views

### Origins of the Jacobi matrix

I have several questions concerning history of Jacobi matrices.
Does anybody know why the Jacobi matrix (=symmetric tridiagonal matrix) is named by Carl Gustav Jacob Jacobi? What was his contribution ...

**1**

vote

**0**answers

107 views

### Default Orientation of Vectors [closed]

When I started studying math in 1982 in Germany, there seemed to have been a change in the choice of the default orientation of vectors; while it was row-vectors till then, it changed to ...

**12**

votes

**2**answers

655 views

### Maximal ideals are prime (history answer please!)

Please can someone tell me the history of the simple argument that any maximal ideal of a commutative ring or distributive lattice is prime? (It is understood that we have found the maximal one using ...

**10**

votes

**3**answers

952 views

### When did coordinate plane “as we know it” come into play?

This is a historical question that needs some background to make sense. Let me start with the longer version of the question:
When did negative numbers, algebra and coordinate plane come together?
...

**2**

votes

**2**answers

161 views

### Where can I find a translation of Caspar Wessel's “Om directionens analytiske betegning?”

I found a listing on Google books for a book containing the desired English translation, together with some biographical information on Wessel, and entitled On the Analytical Representation of ...

**17**

votes

**2**answers

921 views

### History of the connection between Riemann surfaces and complex algebraic curves

As noted in the question "Links between Riemann surfaces and algebraic geometry", there are strong connections between Riemann surfaces and algebraic geometry - for example, compact Riemann surfaces ...

**12**

votes

**2**answers

788 views

### Did Cauchy think that uniform and pointwise convergence were equivalent?

I've heard that Cauchy thought he'd proved that pointwise and uniform convergence are equivalent. Is this a historical fact? If it is indeed true, I was wondering if anyone had a reference.

**32**

votes

**6**answers

2k views

### Negative impact of wrong or non-rigorous proofs

The recent talks of Voevodsky (for example, http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/2014_IAS.pdf), which describe subtle errors in proofs by him as well as others, as well ...

**47**

votes

**19**answers

7k views

### Rediscovery of lost mathematics

Archimedes (ca. 287-212BC) described what are now known as the 13
Archimedean solids
in a lost work, later mentioned by Pappus.
But it awaited Kepler (1619) for the 13 semiregular polyhedra to be
...

**23**

votes

**0**answers

598 views

### History of the Proj construction in algebraic geometry

Projective geometry was introduced by fifteenth century Renaissance painters (like Alberti, da Vinci and Dürer) in the guise of perspective theory, although one could argue that Pappus was already ...

**37**

votes

**8**answers

4k views

### What recent programmes to alter highly-entrenched mathematical terminology have succeeded, and under what conditions do they tend to succeed or fail?

I think we all occasionally come across terminology that we'd like to see supplanted (e.g. by something more systematic). What I'd like to know is, under what circumstances is it reasonable to believe ...

**28**

votes

**26**answers

4k views

### Mathematicians who made important contributions outside their own field? [closed]

It is often said that scientists who cross disciplinary borders can make unexpected discoveries thanks to their fresh view of the problems at hand.
I am looking for mathematicians who did just that. ...

**34**

votes

**2**answers

4k views

### How did “normal” come to mean “perpendicular”?

How and when did the word "normal" acquire this meaning? When I first thought of this, I couldn't really come up with any explanation that wasn't complete speculation -- pretty much all I was able to ...

**10**

votes

**1**answer

320 views

### Elements of the method of forcing in some papers of N. N. Luzin

In the paper
Eléments de la méthode de forcing dans quelques travaux de N. N. Lousin. (French) [Elements of the method of forcing in some papers of N. N. Luzin] Amphora, 469–479, Birkhäuser, ...

**7**

votes

**2**answers

564 views

### Survey of the history of calculus?

Boyer 1939 is a nice readable survey of the history of the calculus, but it's showing its age. Discussing the notion of instantaneous velocity, he has:
Mathematics knows no minimum interval of ...

**9**

votes

**2**answers

558 views

### Origin of the term “generic” in set theory

In set theory, in particular the context of forcing, if $M$ is a model of $\sf ZFC$ and $P\in M$ is a partial order, we say that $G\subseteq P$ is a generic filter (or $M$-generic or generic over $M$) ...

**25**

votes

**3**answers

2k views

### Who invented diagrammatic algebra?

There is a strong and growing trend to do mathematics via diagrammatic algebra, which involves constructing and manipulating equations whose elements are diagrams drawn in the plane. The manipulations ...

**13**

votes

**1**answer

701 views

### Who first dubbed them “expander graphs”?

Expander graphs
("sparse graphs that have strong connectivity properties")
burst onto the mathematical scene around the millennium, but I have not
been successful in tracing the origin of
(a) the ...

**3**

votes

**1**answer

387 views

### Levi's book on Leibnizian calculus

Raphael Levi learned from Leibniz at a late stage in Leibniz's career. This might be a definite advantage for understanding Leibniz. Leibniz did not elaborate some of the philosophical principles ...

**6**

votes

**0**answers

148 views

### Origin of Lie Product Formula

I'm interested in where Lie wrote down the Lie Product formula (for finite matrices)
(the precursor of the Trotter product formula; see http://en.wikipedia.org/wiki/Lie_product_formula). With a ...

**5**

votes

**1**answer

416 views

### What was Seifert's contribution to the Seifert-van Kampen theorem?

The Seifert-van Kampen theorem is the classical theorem of algebraic topology that the fundamental group functor $\pi_1$ preserves pushouts; more often than not this is referred to simply as the van ...

**6**

votes

**1**answer

838 views

### Is there a “big program” in mathematics at the moment? [closed]

I apologize in the event that you should find this question off topic. Please feel free to delete it if that is the case.
Years ago, I studied undergrad mathematics at university. The understanding ...

**6**

votes

**1**answer

262 views

### When was the word “stable” first used to describe stable homotopy theory?

The word "stable" has many uses in mathematics, but in the context of stable homotopy theory, one might take it to mean one of two things:
Homotopy groups stabilize after taking suspensions ...

**43**

votes

**4**answers

3k views

### History of “without loss of generality”

"Without loss of generality" is a standard in the mathematical lexicon, and I am writing to ask if anyone knows where the expression was popularized. (The idea has been around since antiquity, I'm ...

**11**

votes

**3**answers

2k views

### Why are smooth numbers called “smooth”?

"Adleman refers to integers which factor completely into small primes as “smooth” numbers." (ME Hellman, JM Reyneri. Advances in Cryptology, 1983: citation link.)
Does anyone know what is the ...

**15**

votes

**4**answers

848 views

### Who first used the multiplication operator version of spectral theory

This is another history question.
Hilbert phrased the spectral theorem in terms of resolutions of the identity.
While this remained the form of Stone and von Neumann, they did also have the ...

**11**

votes

**3**answers

310 views

### Origin of number theoretic invariants associated to hyperbolic 3-manifolds

I've been studying number theoretic methods of classifying hyperbolic 3-manifolds for over a year now. In particular, there is are the trace field, invariant trace field, quaternion algebra, and ...

**52**

votes

**8**answers

5k views

### Have you solved problems in your sleep? [closed]

I have hit upon major (for me—relative to my trivial accomplishments)
insights in my research
in various sleep-deprived altered states of consciousness,
e.g., long solo car-drives extending ...

**17**

votes

**1**answer

866 views

### Reference for Diagonalization Trick

There is a standard trick in analysis, where one chooses a subsequence, then a subsequence of that... and wants to get an eventual subsubsequence of all of them and you take the diagonal. I've always ...

**2**

votes

**3**answers

669 views

### The Hidden Aspect of Set Theory [closed]

This question is inspired by a similar question at the beginning of Kunen's new book, "Set Theory".
Many mathematicians believe they are exploring a "real" universe. In such a Platonic point of ...

**1**

vote

**1**answer

109 views

### Original sources for two theorems by Bass, Matlis and Papp

It is an interesting fact that a commutative ring $R$ is noetherian if and only if direct sums of injective $R$-modules are injective, and if and only if every injective $R$-module is a direct sum of ...

**13**

votes

**2**answers

578 views

### What was the Question that led Euler to his Investigations on Polyhedra?

The question that led Euler to his investigations on graphs is the well-known question related to the seven bridges of Königsberg, and that story is a must in every introduction to graph theory.
...

**4**

votes

**1**answer

180 views

### What is the early history of the concepts of probabilistic independence and conditional probability/expectation?

In the 1738 second edition of The Doctrine of Chances, de Moivre writes,
Two Events are independent, when they have no connexion one with the other, and that the happening of one neither forwards ...

**7**

votes

**2**answers

569 views

### $\aleph$ looks like $\mathbb N$?

We all know the notation $\aleph_\lambda$ for the $\lambda$th (or, I guess, $\lambda+1$st) infinite cardinal number; in particular $\aleph_0$ is the cardinality of the the set of natural numbers ...

**1**

vote

**0**answers

448 views

### Is it possible to give a fair assessment of the influence of Bourbaki's “Eléments de mathématique”? [closed]

Well, I apologize if this "soft-question" (related to the "Arnold-Serre" debate) is considered as irrelevant for MO, and for possible misunderstandings in the two earlier versions of this post (which ...

**1**

vote

**1**answer

162 views

### First Parameterized Subset of Primes that was Related to a Mathematical Result

To my knowledge, Fermat primes, i.e. primes of the form $2^{2^n}+1$ were the first to play a role in a mathematical result, namely in the characterization of constructible regular n-gons. Gauss ...

**23**

votes

**2**answers

2k views

### Was Vinogradov's 1937 proof of the three-prime theorem effective?

Was Vinogradov's first proof of the three-prime theorem effective?
Reasons for my question: Vinogradov presented his proof in 1937 in a monograph; the English translation by K.F. Roth and A. ...

**10**

votes

**1**answer

408 views

### Who first resolved Hilbert's 20th problem?

Hilbert's 20th problem concerns the existence of solutions to the fundamental problem in the calculus of variations. I understand that Hilbert, Lesbesgue and Tonelli were pioneers in this area.
In ...

**6**

votes

**2**answers

1k views

### Ramanujan's tau function

Why was Ramanujan interested in the his tau function before the advent of modular forms? The machinery of modular forms used by Mordel to solve the multiplicative property seems out of context until I ...

**13**

votes

**2**answers

859 views

### Who first defined _simply connected_, reference?

The following definition is due to Donald J. Newman:
A connected open subset $D$ of the plane $\mathbb C$
is simply connected
if and only if its complement $\widetilde D = \mathbb C \setminus D$
...

**73**

votes

**9**answers

8k views

### Analogues of P vs. NP in the history of mathematics

Recently I wrote a blog post entitled "The Scientific Case for P≠NP". The argument I tried to articulate there is that there seems to be an "invisible electric fence" separating the problems in P ...

**6**

votes

**1**answer

808 views

### Source for Derogatory Quote About Graph Theory

(Edited in accordance with suggestions in comments.)
I remember once I read a quote that sounded like "graph theory is the scum of topology" (please approximate). I can not find it on the web, and I ...

**0**

votes

**3**answers

593 views

### Definition of Prime Numbers [duplicate]

The first time I heard of prime numbers, they were defined as natural numbers $n$ that can only be divided by 1 and themselves without remainder; later, when prime factorization was introduced, I ...

**25**

votes

**1**answer

5k views

### Who made the famous error in calculation that 'wasted' the final years of his life?

Sorry, I am merely a Middle School maths teacher at an Australian secondary school. I remember reading years ago about a famous mathematician (18th or 19th Century?) who calculated table upon table of ...

**10**

votes

**2**answers

416 views

### Historical quotation search: Equations/formulae in (Latin?) prose, before modern symbolic notation

I have been trying, without success, to find a vaguely-remembered quotation: the quadratic equation (or perhaps the quadratic formula), given in (Latin?) prose, along lines like “Consider that ...