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Questions tagged [ho.history-overview]

History and philosophy of mathematics, biographies of mathematicians, mathematics education, recreational mathematics, communication of mathematics.

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Percentage of Ramanujan's conjectures that were proven correct

Today I read the following brief but insightful account of Ramanujan's approach to mathematics: https://www.imsc.res.in/~rao/ramanujan/images/KSRchap3.pdf and while reading this I wondered whether we ...
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2answers
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Complex analytic vs algebraic geometry

This is more of a philosophical or historical question, and I can be totally wrong in what I am about to write next. It looks to me, that complex-analytic geometry has lost its relative positions ...
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1answer
105 views

Semantics/Syntax distinction vs. Meta/Object language distinction

Distinguishing the semantics of a language from its syntax means (at least) distinguishing the meaning of the expressions (what is being represented) from the grammatical structure and formation rules ...
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11answers
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What definitions were crucial to further understanding?

Often the most difficult part of venturing into a field as a researcher is to come up with an appropriate definition. Sometimes definitions suggest themselves very naturally, as when you solve a ...
6
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1answer
564 views

Why did Voevodsky abandon his work on “singletons”?

In an interview (I link the Google translation), Voevodsky talks about how, in the late 2000s, he worked on the problem of "restoring the history of populations according to their modern genetic ...
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5answers
446 views

Mathematical phantoms, specifically but not exclusively in applied mathematics [duplicate]

A while ago over at our sister site, there was an interesting question [not by me] with next to no answers which I feel is, fleshed out in a more precise fashion, appropriate for MathOverflow. The ...
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0answers
361 views

Emanuel Lasker, Max Noether, and Emmy Noether

In 1900, Emanuel Lasker (world chess champion from 1894 to 1921) received his Ph.D. under Max Noether. In 1905, Lasker published a theorem that Emmy Noether generalized in 1921, now well known as the ...
110
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1answer
7k views

What happened to Suren Arakelov? [closed]

I heard that Professor Suren Arakelov got mental disorder and ceased research. However, a brief search on the Russian wikipedia page showed he was placed in a psychiatric hospital because of political ...
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2answers
432 views

Reference request: Oldest linear algebra books with exercises?

Inspired by the recent success of my "soft question" here, I also have to ask, what are some of the oldest linear algebra books out there with exercises? I'm fine with or without solutions, either way....
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5answers
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Reference request: Oldest calculus, real analysis books with exercises?

Per the title, what are some of the oldest calculus, real analysis books out there with exercises? Maybe there are some hidden gems from before the 20th century out there. Edit. Unsolved exercises ...
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0answers
261 views

Why does Loday call the permutohedra “zylchgons”?

Today I was reading Jean-Louis Loday's classic paper, "Realization of the Stasheff polytope", in which he produces a simple and very pretty realization of the associahedra as convex polytopes. He ...
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1answer
325 views

Young's natural representation of the symmetric group

The literature on the representation theory of the symmetric group contains some terminology that I find puzzling, and I am wondering if someone here knows the full story. One of the standard ways to ...
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0answers
187 views

English translation of Wilhelm Killing's 1889 paper

In the paper The greatest mathematical paper of all time, by A. J. Coleman (The Mathematical Intelligencer 11 (1989) pp29-38, https://doi.org/10.1007/BF03025189) the author argues that Wilhelm Killing'...
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1answer
147 views

Looking for an erratum (reference request)

Note: Since what I am asking about below touches on a potentially controversial subject, let me emphasize that I am only asking for a specific reference, and I am not asking for a discussion of the ...
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2answers
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Why did Dedekind claim that $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ hadn't been proved before?

In a letter to Lipschitz (1876) Dedekind doubts that $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ had been proved before: quoted from Leo Corry, Modern algebra, German original: Why did Dedekind doubt that $(\...
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5answers
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Early examples of mathematicians publishing (from home) in a foreign language?

Today this is common, but how exactly did it start? I am looking for examples in various languages, and suggest: Exclude Latin (as more “ancient” or “international” than “foreign”) Exclude French ...
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1answer
575 views

Quantum functional analysis

Can one explain some philosophy behind "quantum functional analysis" (or "quantized functional analysis") which was initiated and developed by such researchers as: Ruan Z.-J., Pisier J., Effros E.G., ...
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0answers
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History of the definition of completeness of algebraic varieties (and properness of morphisms)

An algebraic variety (say over an algebraically closed field) $X$ is complete if for every variety $Y$, the projection $X \times Y \to Y$ is closed (with respect to the Zariski topology). As it is ...
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4answers
707 views

History of ODE and PDE reference request

Is there any reference (book or articles) which made the history (up to the modern times) and the conceptual development of Ordinary Differential Equations and Partial Differential Equations? It will ...
6
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0answers
117 views

What (if any) was known about null sets before Lebesgue?

The notion af a null set, i. e., a set of Lebesgue measure zero, does not require a full blown construction of Lebesgue measure: A set is $E\subset \mathbb{R}$ is called a null-set if it can be ...
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3answers
1k views

“Mächtigkeit” versus “Kardinalität”?

In Cantor's set theory, is there any difference between the terms Mächtigkeit and Kardinalität ?
8
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1answer
483 views

Whence “Durchschnitt” and “Vereinigung”?

Today the set-theoretic operations of intersection $\cap$ [German: Durchschnitt] and union $\cup$ [German: Vereinigung] are standard. The modern notations are present in the first edition of van der ...
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5answers
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Has incorrect notation ever led to a mistaken proof?

In mathematics we introduce many different kinds of notation, and sometimes even a single object or construction can be represented by many different notations. To take two very different examples, ...
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2answers
562 views

What did the Intuitionists want to do with applied mathematics?

Oversimplification: Newton & Leibnitz &c build the calculus and other methods that solve a vast number of practical problems. Weierstrass, Dedekind, Cantor &c build a foundation under it ...
9
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1answer
190 views

Origin of the concept of “homomorphism”? [duplicate]

When was the concept of a "homomorphism" of algebraic structures first introduced? Steinitz' 1910 paper Algebraic Theory of Fields is often pointed to as the first true work of abstract algebra, yet ...
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6answers
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“Long-standing conjectures in analysis … often turn out to be false”

The title is a quote from a Jim Holt article entitled, "The Riemann zeta conjecture and the laughter of the primes" (p. 47).1 His example of a "long-standing conjecture" is the Riemann hypothesis,...
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2answers
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Who first chose the names Alice and Bob for players A and B? [closed]

Who first chose the names Alice and Bob for the players (or observers) A and B?
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0answers
475 views

Eckmann-Hilton argument / Grothendieck

In terse terms, the “Eckmann-Hilton argument” says that a monoid in the category of monoids is commutative. It is essentially used, for example, to prove that homotopy groups of topological groups are ...
4
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1answer
361 views

Constructibility of the regular 17-gon [closed]

There is a standard construction of a regular heptadecagon by H.W. Richmond (1893) (https://en.wikipedia.org/wiki/Heptadecagon ) (As anyone knows, it was Gauss who found out that it is possible to do ...
26
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1answer
3k views

Why did Euler consider the zeta function?

Many zeta functions and L-functions which are generalizations of the Riemann zeta function play very important roles in modern mathematics (Kummer criterion, class number formula, Weil conjecture, BSD ...
4
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2answers
435 views

Priority for lemniscate of Gerono?

The Lemniscate of Gerono is a special case of the Lissajous curves. The dates for the two mathematicians are fairly close: Gerono (1799-1891) and Lissajous (1822-1880). There seems to have been ...
3
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0answers
142 views

In history of algebra, who was the first to add one equation to another equation?

In history of algebra, who was the first to add one equation to another equation? Someone gave me the name of an Italian mathematician of Renaissance period, but I lost the email.
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374 views

Kodaira's Fields medal citation

In 1954, Kodaira won the Fields medal. His citation was Achieved major results in the theory of harmonic integrals and numerous applications to Kählerian and more specifically to algebraic ...
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2answers
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How were modular forms discovered?

When modular forms are usually introduced, it is by: "We have the standard action of $SL(2,\mathbb Z)$ on the upper half-plane, so let us study functions which are (almost) invariant under such ...
4
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1answer
141 views

Citations graphs what is known?

There have been much research related to webgraphs and social graphs. They can be thought of a kind of random graphs, but the point is that they are different from the well-known Erdős–Rényi model. ...
50
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5answers
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The Logic of Buddha: A Formal Approach

Buddhist logic is a branch of Indian logic (see also Nyaya), one of the three original traditions of logic, alongside the Greek and the Chinese logic. It seems Buddha himself used some of the features ...
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1answer
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Some clarifications on Connes' approach to RH

How serious/promising is Connes' work on the Riemann Hypothesis? Connes is mostly known these days for his work in non-commutative geometry, having previously earned a Fields medal* for his work on ...
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0answers
87 views

History of the relation between $p$-adic measures and power series

In 1964, Kubota and Leopoldt defined the $p$-adic $L$-function by means of some $p$-adic sums (now called the Volkenborn integral which is a $p$-adic distribution). Later, Mazur (in his secret ...
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0answers
512 views

On the basis of a finite dimensional vector space (revised)

Revision in response to the comments to earlier version: To introduce the notion of a basis of a finite dimensional vector space over an arbitrary field $\Lambda$, without performing any computation ...
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1answer
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New articles by Errett Bishop on constructive type theory?

Recently two formerly unknown articles by Errett Bishop (1928-1983) were posted online by Martín Escardó. One is entitled "A general language", deals with constructive type theory, and ...
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2answers
146 views

History of an open problem on partial tilting modules

The following is an open problem: Given a partial tilting module $T$ over a finite dimensional algebra $A$ (that is $Ext_A^i(T,T)=0$ for all $i \geq 1$ and $pd(T) < \infty$), then $T$ is a tilting ...
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2answers
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“This category obviously leads to paradoxes of set theory.” What is the paradox?

Eilenberg and Mac Lane formally defined categories in their 1945 paper General Theory of Natural Equivalences. Their definition of a category starts as follows: "A category {A,a} is an aggregate of ...
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1answer
446 views

Ehresmann's approach to differential geometry

I have come accross this brief description of Charles Ehresmann's life given by his wife: http://www.cs.le.ac.uk/people/ah83/cat-myths/myth0002.html I quote the part from the text relevant to my ...
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1answer
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Who was the first to capitalize Real?

For example in Atiyah's $KR$-theory there is the notion of a Real vector bundle in contrast to complex or real vector bundles. I am also familiar with the notion of a Real $C^*$-algebra and there are ...
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1answer
499 views

Who was in the Fields committee for ICM 1962 (the first appointed by IMU)?

Traditionally, at the presentation of the Fields medals at the ICM opening ceremony, the composition of the Fields medal committee is disclosed. This information can be found in the first volume of ...
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4answers
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What “metatheory” did early set theory/logic researchers use to prove semantic results?

Things like the first-order completeness theorem and the Löwenheim-Skolem theorem are considered foundational in mathematical logic. The modern approach seems to be, usually, to interpret a "model" ...
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1answer
871 views

Historically, which came first: the Lie algebras or their classification?

The classification of the complex simple Lie algebras by their Dynkin diagrams gives rise to five exceptional complex simple Lie algebras: $F_4, G_2, E_6, E_7$ and $E_8$. I am trying to find out ...
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1answer
355 views

Different definitions of category

It appears that there are two different definitions of category. Some authors require the Hom-sets to be pairwise disjoint. Eilenberg and Mac Lane in their original definition require each identity ...
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0answers
484 views

What is your favourite wrong proof of RH? [closed]

Some of the users here receive claimed proofs of the Riemann hypotheses on a regular bases. As fas as we know all of them have been wrong. But sometimes failure is also interesting. So for all cases ...
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2answers
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When did people start thinking of elliptic curves as groups?

I have been reading some old papers of Cassels and Selmer from around 1950, and they talk about generators of rational solutions to elliptic curves, in the sense of Mordell–Weil, but do not appear to ...