**64**

votes

**59**answers

15k views

### Pseudonyms of famous mathematicians

Many mathematicians know that Lewis Carroll was quite a good mathematician, who wrote about logic (paradoxes) and determinants. He found an expansion formula, which bears his real name (Charles ...

**0**

votes

**0**answers

42 views

### Reference request: Uniformly totally bounded classes of compact metric spaces are Gromov-Hausdorff precompact

The following Theorem can be found for instance here (Theorem 7.4.15):
Theorem. (author ?) Any uniformly totally bounded class $\mathfrak X$ of compact metric spaces is pre-compact in the ...

**50**

votes

**1**answer

2k views

### Did Bourbaki write a text on algebraic geometry?

Certainly Bourbaki never wrote an introduction to algebraic geometry: we would have heard about it, right?

**13**

votes

**2**answers

2k views

### What are examples of theorems which were once “valid”, then became “invalid” as standard definitions shifted?

That is, results established by correct proofs within some framework, yet the manner in which their author or the general mathematical community at the time would describe these results would, in ...

**2**

votes

**2**answers

348 views

### What are the sense and reference of the propositions $R \notin R$, $R \in R$, where $R=\{x \mid x \notin x\}$ in Frege's Grundgesetze?

In the paper,
Aldo Antonelli and Robert May, Frege's new science, Notre Dame J. Form. Log. 41 (2000), no. 3, 242–-270, MR 1943495.
the authors give the following quote of Frege, from his paper ...

**2**

votes

**6**answers

2k views

### Looking for a source for Intended Interpretation

Hao Wang writes: "The originally intended, or standard, interpretation takes the ordinary nonnegative integers $\{0, 1, 2, \ldots \}$ as the domain, the symbols $0$ and $1$ as denoting zero and one, ...

**20**

votes

**2**answers

2k views

### Euler's mathematics in terms of modern theories?

Some aspects of Euler's work were formalized in terms of modern infinitesimal theories by Laugwitz, McKinzie, Tuckey, and others. Referring to the latter, G. Ferraro claims that "one can see in ...

**8**

votes

**3**answers

739 views

### English or French translation of Gauss' “Summatio Quarumdam Serierum Singularium”

I'm interested in looking at the details of Gauss' method of determining the sign of the Gauss sum in his "Summatio Quarumdam Serierum Singularium", and I was wondering if anyone knew if there was an ...

**24**

votes

**1**answer

2k views

### More on “Transalgebraic Theories” (a 19th century yoga)?

Among the talks at occasion of the Galois Bicentennial, one is about "Transalgebraic Theories". Unfortunately I found only this article describing that fascinating idea as " an extremely powerful ...

**-4**

votes

**1**answer

277 views

### Proof of formula for $\pi$ [closed]

The number $\pi$ can be expressed as $\pi=\lim_{n\to\infty} \frac{n\sqrt[n]{-1}-n}{\sqrt{-1}}$ or more poetically $\pi=\frac{\infty\sqrt[\infty]{-1}-\infty}{\sqrt{-1}}$. Here we choose the principal ...

**1**

vote

**0**answers

53 views

### History of Cauchy-Euler Equations

As I teach a class in ODE, and following this post and Rota's paper, I wandered what is the history of the research of -
$\sum\limits_{k=0}^n a_k x^k y^{(k)}(x) = g(x),\quad \forall ...

**42**

votes

**15**answers

6k views

### Abstract Thought vs Calculation

Jeremy Avigad and Erich Reck in their remarkable historical paper "Clarifying the nature of the infinite: the development of metamathematics and proof theory" claim that one of the factors of becoming ...

**10**

votes

**1**answer

495 views

### Why did Alonzo Church choose the letter $\lambda$ as the “binding operator”?

Is there any known reason why Alonzo Church chose Greek $\lambda$ as the "binding operator" for the Lambda Calculus?

**0**

votes

**1**answer

199 views

### Pasch axiom and Pythagorean field condition?

I am looking for a reference for the claim that the Pasch axiom is equivalent to the Pythagorean field condition, and with respect to what base theory this should be true.
Since posting the question, ...

**49**

votes

**9**answers

6k views

### When have we lost a body of mathematics because errors were found?

The history of mathematics over the last 200 years has many occasions when the fundamental assumptions of an area have been shown to be flawed, or even wrong. Yet I cannot think of any examples where, ...

**63**

votes

**10**answers

7k views

### Have you solved problems in your sleep?

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 ...

**3**

votes

**1**answer

207 views

### Have there been any claims of mathematical breakthroughs while in altered states of consciousness?

This certainly is a related question:
Have you solved problems in your sleep?
Has anyone seriously attempted to make a similar claim for other altered states, besides dreaming?
I know the claim has ...

**11**

votes

**5**answers

1k views

### Texts on the General History of Contemporary Combinatorics

I am looking for some core texts (books, book chapters, papers) about the general history of contemporary combinatorics, starting, say, from the interwar period up to today.
Texts about the history ...

**12**

votes

**5**answers

2k views

### What are hypergroups and hyperrings good for?

I came across the concept of a hyperring in two recent papers by Connes and Consani (From monoids to hyperstructures: in search of an absolute arithmetic and The hyperring of adèle classes). It's a ...

**5**

votes

**0**answers

93 views

### Historical perspectives on CAT(0) spaces

Does there exist a survey on the early developments of CAT(k) spaces, with the first motivations and the first problems considered? I looked at Bridson and Haefliger's book On metric spaces of ...

**9**

votes

**0**answers

300 views

### Why do we study symplectic geometry? [closed]

What is the motivation behind studying smooth manifolds with a non-degenerate closed two-form?
The subject certainly originated from physics, but is there a deeper reason for why it is still an ...

**3**

votes

**1**answer

181 views

### Early examples of problems that are easier in high dimension

In many areas of mathematics, there are problems that admit a natural formulation in any dimension. It often happens that such a problem is easier to solve in dimension $n>k$ as compared to ...

**83**

votes

**29**answers

8k views

### Examples of theorems misapplied to non-mathematical contexts

For something I'm writing -- I'm interested in examples of bad arguments which involve the application of mathematical theorems in non-mathematical contexts. E.G. folks who make theological arguments ...

**22**

votes

**3**answers

2k views

### Nelson's proof of Liouville's theorem

The paper "A proof of Liouville's theorem" by E. Nelson, published in 1961 in Proceedings of AMS, contains just one paragraph, giving a (now) standard proof that every bounded harmonic function in ...

**3**

votes

**1**answer

180 views

### Fréchet L-Spaces

According to the paper The emergence of open sets, closed sets, and limit points
in analysis and topology famous mathematician Maurice Fréchet who introduced the concept of metric spaces has also ...

**3**

votes

**1**answer

421 views

### origin of analogy “primes as the atoms of number theory/ arithmetic”

a math student recently challenged me on the old comparison/ analogy of prime numbers to "the atoms of number theory or arithmetic" and then was wondering the origin of the phrase.
where does this ...

**12**

votes

**1**answer

537 views

### Have Grothendieck's notes in Montpellier already been investigated?

Grothendieck, who passed away on November 13, 2014, left a huge amount (around 20.000 sheets) of personal notes in the University of Montpellier that he thought he was the only one to be able to ...

**0**

votes

**2**answers

276 views

### The Zeta Function Before Riemann [duplicate]

Leonhard Euler studied the function that is now known as the Riemann zeta function. I have not found the notation $\zeta$ in any of the works of any mathematicians prior to Bernhard Riemann's paper On ...

**1**

vote

**1**answer

203 views

### Have some works by Émile Borel ever been translated from French to English or another foreign language?

I plan to submit a couple of questions around Émile Borel's works in probability theory to MO.
In this scope, I'd like to know if the following works have ever been translated from French to English ...

**42**

votes

**31**answers

7k views

### Trichotomies in mathematics

Added. Thanks to all who participated! Let me humbly apologize to those who were annoyed (quite understandably) by this thread, deeming it nothing more than an exercise in futility. If you thought the ...

**24**

votes

**1**answer

631 views

### Biholomophic non-Algebraically Isomorphic Varieties

Recently, when writing a review for MathSciNet, the following question arose:
Is it true that two smooth complex varieties that are biholomorphic are algebraically isomorphic? The converse is true ...

**26**

votes

**16**answers

3k views

### What are some examples of narrowly missed discoveries in the history of mathematics?

What are the examples of some mathematicians coming very close to a very promising theory or a correct proof of a big conjecture but not making or missing the last step?

**7**

votes

**1**answer

472 views

### Where was the arithmetic zeta function of a scheme first defined?

Let $X$ be an arithmetic scheme, that is, a scheme of finite type over the integers. We denote the set of closed points of $X$ by $|X|$. For every $x\in|X|$, write $N(x)$ for the cardinality of the ...

**40**

votes

**14**answers

6k views

### Does any research mathematics involve solving functional equations?

This is a somewhat frivolous question, so I won't mind if it gets closed. One of the categories of Olympiad-style problems (e.g. at the IMO) is solving various functional equations, such as those ...

**6**

votes

**1**answer

624 views

### Fibonacci = Leonardo Pisano?

Leonardo of Pisa is best known as Fibonacci; various stories found in books and on the web claim that the name Fibonacci was invented by Edouard Lucas or Guillaume Libri in the 19th century, and that ...

**6**

votes

**2**answers

448 views

### English translation of Lambert's Theorie der Parallellinien?

Does anyone know if there is an available (published or unpublished) English translation of Johann Lambert's Theorie der Parallellinien? I was able to find it online in German by way of the ...

**60**

votes

**15**answers

7k views

### Mathematical research published in the form of poems

The article
Friedrich Wille: Galerkins Lösungsnäherungen bei monotonen Abbildungen,
Math. Z. 127 (1972), no. 1, 10-16
is written in the form of a lengthy poem, in a style similar to that
of the ...

**9**

votes

**0**answers

673 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" ...

**11**

votes

**2**answers

742 views

### Gauss proof of fundamental theorem of algebra

My question concerns the argument given by Gauss in his "geometric proof" of the fundamental theorem of Algebra. At one point he says (I am reformulating) :
A branch (a component) of any algebraic ...

**256**

votes

**72**answers

95k views

### Video lectures of mathematics courses available online for free

It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some ...

**180**

votes

**36**answers

49k views

### Widely accepted mathematical results that were later shown wrong?

I wonder if there are any examples in the history of mathematics of a mathematical proof that was initially reviewed and widely accepted as valid, only to be disproved a significant amount of time ...

**5**

votes

**1**answer

373 views

### First mention of the fundamental bigroupoid of a space?

The fundamental bigroupoid $\Pi_2(X)$ of a space $X$ was independently described by Hardie, Kamps and Kieboom (paywall) and Stevenson (arXiv) around the year 2000. HKK cite Baez-Dolan's seminal HDA0, ...

**23**

votes

**4**answers

2k views

### In what ways did Leibniz's philosophy foresee modern mathematics?

Leibniz was a noted polymath who was deeply interested in philosophy as well as mathematics, among other things. From my mathematical readings I have the impression that Leibniz's stature as a ...

**3**

votes

**2**answers

349 views

### Archive of the Work of J Sutherland Frame

Does anyone know of the existence of an archive of the work of J Sutherland Frame?
The Briscoe Center for American History maintains about 100 archives of American mathematics and I have found the ...

**94**

votes

**94**answers

12k views

### What would you want to see at the Museum of Mathematics?

EDIT (30 Nov 2012): MoMath is opening in a couple of weeks, so this seems like it might be a good time for any last-minute additions to this question before I vote to close my own question as "no ...

**2**

votes

**1**answer

653 views

### Reference for Connes Bourbaki membership or otherwise

Alain Connes being a leading French mathematician today one could ask whether he is a member of the Bourbaki group. Is there a published reference that would either refute or confirm this?

**10**

votes

**1**answer

381 views

### What was a cusp to Hurwitz in 1892?

Let $d\in\mathbb{N}$
be squarefree.
Let $\mathcal{O}_d$
be the ring of integers of $\mathbb{Q}(\sqrt{-d})$.
Let $\Gamma_d=\mathrm{PSL}_2(\mathcal{O}_d)$.
Let $\mathcal{H}^3$
be the upper half-space ...

**9**

votes

**1**answer

350 views

### Analogy between Lagrange's Theorem and Rank-Nullity Theorem?

One can view view Lagrange's Theorem $$|G/H|=|G|/|H|$$ and the Rank-Nullity Theorem $$\dim(V/U)=\dim(V)-\dim(U)$$ as directly analogous. Does anyone know a high-level explanation of this analogy? I ...

**241**

votes

**21**answers

30k views

### Thinking and Explaining

How big a gap is there between how you think about mathematics and what you say to others? Do you say what you're thinking? Please give either personal examples of how your thoughts and words ...

**102**

votes

**6**answers

8k views

### Source and context of $\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)$?

Possibly the most striking proof of Archimedes's inequality $\pi < 22/7$ is an integral formula for the difference:
$$
\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2},
$$
where the ...