Questions tagged [ho.history-overview]
History and philosophy of mathematics, biographies of mathematicians, mathematics education, recreational mathematics, communication of mathematics.
1,407
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What is the shortest Ph.D. thesis? [closed]
The question is self-explanatory, but I want to make some remarks in order to prevent the responses from going off into undesirable directions.
It seems that every few years I hear someone ask this ...
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1
answer
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Is there evidence whether undergraduate math courses improve problem-solving?
The most commonly stated reason for why mathematics should be a required condition for graduating is }to improve problem-solving skills". Usually it's taken for granted that taking a mathematics ...
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Fulfilling Pythagoras' Dream using Nonstandard Models of Arithmetic and/or Surreal Numbers
Pythagoras and his followers believed that the Universe was made of numbers. Specifically, they thought that if you compared any magnitudes of the same kind, say the lengths of two objects, you would ...
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The story about Milnor proving the Fáry-Milnor theorem
This question is similar to a previous one about "urban legends", but not the same. It is established that Milnor proved the Fáry-Milnor theorem as an undergraduate at Princeton. For the record, ...
421
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91
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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 ...
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History of the Lagrange Inversion Theorem
I'm doing research on the history of the Lagrange inversion theorem. The earliest predecessor I've found is the one referenced by De Morgan; viz. Jo. H. Lambert's construction in Observationes Variae ...
3
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1
answer
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Whence the k-tuple conjecture?
What is the source of the $k$-tuple conjecture, that every integer tuple $(k_1,\ldots,k_n)$ either contains all members of a congruence class mod a prime or has infinitely many primes amongst $(k_1+c,\...
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14
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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 ...
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The different Branches of Arithmetic
... "and then the
different branches of Arithmetic--
Ambition, Distraction, Uglification,
and Derision."
(Alice in Wonderland, chapter IX: the Mock Turtle's story)
As a child I wondered for ...
134
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69
answers
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Mathematical "urban legends"
When I was a young and impressionable graduate student at Princeton, we scared each other with the story of a Final Public Oral, where Jack Milnor was dragged in against his will to sit on a committee,...
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Where can you find Grothendieck's "Récoltes et Semailles"?
Where can you find Grothendieck's "Récoltes et Semailles"?
Is it available anywhere?
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2
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What is the shortest proof of the existence of a prime between $p$ and $p^2$ ? other examples? [closed]
1) It is well known that between a prime $p$ and $p^2$ always exist a prime, but what is the shortest proof of that (by elementary methods or not)?
(One can say that we can have it as a collorary of ...
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7
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History of classifying spaces
Where did the idea and formal definition of the "classifying space of a (small) category" first appear?
Added: As Andy Putman noted below, the "classical" early reference for this is G. Segal's ...
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4
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How did Bernoulli prove L'Hôpital's rule?
To prove L'Hôpital's rule, the standard method is to use use Cauchy's Mean Value Theorem (and note that once you have Cauchy's MVT, you don't need an $\epsilon$-$\delta$ definition of limit to ...
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Why the name 'separable' space?
It is well known that a separable space is a topological space that has a countable dense subset. I am wondering how is this related to the name 'separable'? Any intuition where the name come from?
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7
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How might M.C. Escher have designed his patterns?
I realize this question isn't strictly mathematical, and if it doesn't fit with the content on this site then feel free (moderators/high-rep users) to close it. But when I thought up the question it ...
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3
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Did André Weil predict that the Riemann Hypothesis would be settled by prime number theory rather than by analysis?
Did André Weil predict that the Riemann Hypothesis would be settled by prime number theory rather than by analysis? If so, what are a reference and/or a quotation?
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What would you want to see at the Museum of Mathematics? [closed]
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 ...
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Historical and terminological questions about Dan Kan's Ex functor and its relation to the classical case of simplicial complexes
Recall that we may define a functor $\xi:\Delta\to \operatorname{Poset}$ sending a simplex $[n]$ to the set of monotone injections $[k]\hookrightarrow [n]$ for $k\geq 0$ (effectively, $k\leq n$ as ...
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2
answers
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History of Irrationality results
The Greeks knew that numbers of the form $\sqrt{n}$ for nonsquare
integers $n$ are not rational. Much later, Lambert (1768) proved that
the values of $e^x$ and $\tan x$ are irrational for nonzero ...
40
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1
answer
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First correct proof of FLT for exponent 3?
It is well known that Euler gave the first proof of FLT ($x^n + y^n = z^n$ has
no nontrivial integral solutions for $n > 2$) for exponent $n=3$, but that his proof
had gaps (which are not as easily ...
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4
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Did Grothendieck write about modular forms?
This question might be astoundingly naive, because my understanding of modular forms is so meek. It occurred to me that the reason I was never able to penetrate into the field of modular forms, ...
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Proofs that require fundamentally new ways of thinking
I do not know exactly how to characterize the class of proofs that interests me, so let me give some examples and say why I would be interested in more. Perhaps what the examples have in common is ...
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Has the mathematical content of Grothendieck's "Récoltes et Semailles" been used?
This question is partly motivated by Never appeared forthcoming papers.
Motivation
Grothendieck's "Récoltes et Semailles" has been cited on various occasions on this forum. See for instance ...
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Why are differential forms called closed and exact?
It seems to me that "exact" relates to exact differential equation. So, why are they called exact?
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William Rowan Hamilton and Algebra as Time
This question ended up longer than I intended (though most of the bulk is interesting remarks by Hamilton), so I thought it might be good to include my question at the beginning before the admittedly-...
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16
answers
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What do named "tricks" share?
There are a number of theorems or lemmas or mathematical ideas that come to be known as eponymous
tricks, a term which in this context is in no sense derogatory.
Here is a list of 11 such tricks (the ...
17
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1
answer
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A synopsis of Adyan’s solution to the general Burnside problem?
Where can I find a high-level overview
of Adyan’s original proof of the existence of finitely generated infinite groups with finite exponent?
Additionally:
If possible, would an expert please ...
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4
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The history of Proper Forcing
What were the initial motivations of the use of the proper forcing.?
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5
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"Let $x \in A$", beginning a proof of "$\forall x \in A$ ...", if A were empty [closed]
I work at a four-year teaching school, where we pride ourselves on teaching pure math, proof, and a rather obsessive carefulness of work. Recently I have been criticized for saying that "Let $x \in A$...
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12
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What are some early examples of creation of lists / catalogues of (particularly) combinatorial objects?
A lot of effort in discrete maths / combinatorics is expended in the construction of lists, catalogues or census [sic] of combinatorial objects such as groups, graphs, designs etc. These catalogues ...
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Countable connected Hausdorff space
Let me start by reminding two constructions of topological spaces with such exotic combination of properties:
1) The elements are non-zero integers; base of topology are (infinite) arithmetic ...
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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 ...
136
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7
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Is the boundary $\partial S$ analogous to a derivative?
Without prethought, I mentioned in class once that the reason the symbol $\partial$
is used to represent the boundary operator in topology is
that its behavior is akin to a derivative.
But after ...
6
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2
answers
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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 ...
92
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74
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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 ...
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Why is "abelian" infrequently capitalized?
Posted with input from meta for improvement. I usually read, e.g. "Gaussian integers" and "Riemannian metrics", and occasionally "euclidean" or "cartesian" or even "lorentzian space", but the latter ...
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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 ...
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Most memorable titles
Given the vast number of new papers / preprints that hit the internet everyday, one factor that may help papers stand out for a broader, though possibly more casual, audience is their title. This view ...
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What are Central Limit Theorems and why are they called so?
I know two opinions:
1) "Central" means "very important" (as it was central problem in probability for many decades), and CLT is a statement about Gaussian limit distribution. If the limit ...
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What's coherent about coherent sheaves?
In a recent answer to a recent question, BCnrd wrote
[...] beyond the coherent case one cannot expect information about a fiber (e.g., vanishing, 6 generators, etc.) to "propogate" to information ...
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Who is Kirszbraun?
Kirszbraun's theorem is one of my favorite theorems in mathematics.
I always wanted to know something about Kirszbraun, or at least to see his picture.
Do you have any information about him?
(I know ...
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Extremely messy proofs
Currently in my undergraduate courses I am being taught how to set up various machinery using slick, short proofs and then how to apply that machinery. What I am not being taught, largely, is what ...
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Birkhoff's theorem about doubly stochastic matrices
Birkhoff's theorem states:
The set of $n \times n$ doubly stochastic matrices is a convex set whose extreme points are the permutation matrices
This theorem seems to be commonly attributed to ...
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Missing exposes in SGA 5, and the composition of the SGA's
Over the past couple of years I had to look in SGA for various results, and I can't but marvel at how poorly constructed it is. In SGA1 expose VII "n'existe pas", SGA 1 references higher SGA's, and so ...
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What was Weierstrass's counterexample to the Dirichlet Principle?
Shortly after his work on the foundations of geometry David Hilbert turned his attention to finding a suitable statement of the Dirichlet principle, from which to prove the Riemann mapping theorem and ...
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Where does the Chebyshev polynomial notation come from?
The $k$th Chebyshev polynomial is denoted by $T_k$ where
$T_k(x) = \cos(k\cos^{-1}(x))$
I was wondering where this notation came from. It has been suggested that it comes from Tschebyscheff (the ...
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4
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who fixed the topology on ideles?
I am teaching a course leading up to Tate's thesis and I told the students last week, when defining ideles, that the first topology that was put on the ideles was not so good (e.g., it was not ...
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What was Casimir's precise role in describing the center of the universal enveloping algebra of a semisimple Lie algebra?
This question is prompted by a recent MO question on explicit computations of Weyl group invariants for certain exceptional simple Lie algebras:
37602. Like some others who started graduate study in ...
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What if Current Foundations of Mathematics are Inconsistent? [closed]
The title of the question is also the title of a talk by Vladimir Voevodsky, available here.
Had this kind of opinion been expressed before?
EDIT. Thanks to all answerers, commentators, voters, ...