23

Hamilton and Klein, Klein was more explicit about it. Hamilton in Lectures on Quaternions (1853) realized that his representation of rotations of rigid bodies by the unit quaternions was not $1$-$1$, but $2$-$1$. Klein in Lectures on the Ikosahedron and the Solution of Equations of the Fifth Degree (1888) replaced the unit quaternions by $2 × 2$ unitary ...


19

It is not random. These names are of Greek origin, and -ic or -ics are Anglicizations of the Greek suffix -ikos, which meant "pertaining to". In other languages it can be rendered as -ika or -ica, Wolfram's "Mathematica" uses such a version. From the Online Etymology Dictionary: "-ics in the names of sciences or disciplines (acoustics, aerobics, ...


17

The letter Ш (sha) of the Cyrillic alphabet is widely accepted in theoretical computer science as the symbol for the shuffle product, which gives the shuffle algebra. The same letter is also used to denote the Tate-Shafarevich group, but I'm not sure if it's really a standard (the letter was introduced by Cassels only in 1990 in 1962 instead of TS, see below ...


16

Before Hamilton (1847) one should cite Euler (1771), Gauss (1819), Rodrigues (1840), and Cayley (1845). Detailed references in e.g. Pujol, J., Hamilton, Rodrigues, Gauss, quaternions, and rotations: a historical reassessment, Commun. Math. Anal. 13, No. 2, 1-14 (2012). ZBL1268.01010. Specifically, to four numbers $p,q,r,s$ with $pp+qq+rr+ss=u$, Euler (1771,...


16

The same person who introduced it, Cayley. Sylvester first used the term "matrix" (womb in Latin) for an array of numbers in 1848, but did not do much with it. Cayley started developing matrix algebra in 1855 and summarized his theory in A Memoir on the Theory of Matrices (1858). In the opening paragraphs he writes: "It will be, seen that matrices (...


15

Archimedes' books, Stomachion and The Method of Mechanical Theorems were lost until rediscovered in 2006. The only known copy is the Archimedes Palimpsest. These two texts comprise many theorems. The Method describes Archimedes' very early use of Riemann sums to compute areas and a variation of Dedekind cuts (via a pair, one a strictly monotonically ...


14

Kolmogorov was not exactly free to express his views considering the situation in the Soviet Union. Philosophical issues, even concerning mathematics, were ideologically sensitive, and everyone had to express allegiance, in one form or another, to the dialectical materialism of Marx and Engels. It went beyond that, as the only grand philosophy available it ...


13

The Grothendieck Circle site suggests a more innocent explanation for the loss these letters. Having left Montpellier in 1984.... In May of that year [1985] a secretary informed him that his office on the fourth floor of the institute had been cleared out. Seeing this incident as an egregious example of the general decline of mores, an outraged ...


12

I assume this refers to Lagrange's 1768 proof of the Diophantine approximation theorem. The proof was simplified by Dirichlet in 1842, using the idea twice. He named it Schubfachprinzip (drawer principle), and it is with Dirichlet that the principle came to be most commonly associated. Many authors date Dirichlet's use back to 1834, but without any reference....


12

It is discussed in multiple manuscripts, letters and publications from 1675 to 1701. According to Fracois Ziegler's post on MO Did Leibniz really get the Leibniz rule wrong?, Leibniz originally thought $d(uv)=du\,dv$ in a special case, but corrected his mistake the same month in the manuscript Methodi tangentium inversae exempla (November 11, 1675). Later ...


11

It is sometimes asserted that $\varnothing$ for the empty set was introduced by Bourbaki using a Danish and Norwegian letter. EDIT: The source is the Weil autobiography, cited in Jeff Miller's collection of the origins of mathematical expressions: André Weil (1906-1998) says in his autobiography that he was responsible for the symbol: Wisely, we had ...


11

TL;DR Dijkstra basically got it right. The primary means for computing with Roman numerals were the abacus and the reckoning board. The use of small pebbles in this manner of computation is the origin of our word calculus. Roger Cooke, "The History of Mathematics. A Brief Course 2nd. ed.", Wiley 2005, p. 144, gives a brief but useful overview how and when ...


10

The nickname appears to be a creation of the New Math movement, and spread from the math education literature. The notation itself in its modern form can be traced back to Lefschetz's Algebraic Topology (1942), and variants appear already in Principia (1910) and von Neumann's Zur Einführung der transfiniten Zahlen (1923). See Who first discovered the ...


10

There are several non-alphabetic symbols, the best known is the integral sign $\int$ and the Weierstrass $P$-function $\wp$. To be sure their origins are letters of Latin alphabet, but they are special stylized symbols, and as far as I know there is no computer code for them in the standard sets of computer characters. Strictly speaking they do not belong to ...


10

Vinogradov likely adapted $\ll$ from Poincare and Borel, who used it for asymptotic series in 1890s (Cajori cites Borel, *Lecons sur les series divergentes", 1901). Physicists used it for vague "much less than" as early as 1918 (Heurlinger's doctoral dissertation Untersuchungen über die Struktur der Bandenspektra). Whether they reinterpreted Poincare's ...


9

This is a collection of open problems concerning various areas in function theory, functional analysis, theory of linear and nonlinear partial differential equations. Seventy Five (Thousand) Unsolved Problems in Analysis and Partial Differential Equations


9

The earliest journals were multidisciplinary, they were published by academies since the second half of 17th century. Before that time, communication was only by books and letters. Letters sometimes were copied and circulated. The first French, English and German journals were: Journal des sçavans, 1665 (published papers in all sciences and humanities) ...


9

Special cases were handled algebraically even before the "l'Hopital's" rule, which appears in l'Hopital's 1696 transcription of tips on calculus he purchased (literally) from Johann Bernoulli in 1694, see Indeterminate Forms Revisited, by Boas. For example, Descartes's method of finding tangents involved resolving "indeterminate forms" like $0/0$, see Is ...


9

Indeed, funding of mathematics, together with other science and engineering fields experienced a sharp increase between 1957 and 1970. The number of PhD awarded in these fields in USA tripled during this period, and this was the fastest growth for any period after WWII. https://www.nsf.gov/statistics/2018/nsf18304/static/report/nsf18304-report.pdf But the "...


9

Kolmogovov expressed his views in this paper: MR2278817 Kolmogorov, A. N. Modern debates on the nature of mathematics. (Russian). With a commentary by V. A. Uspenskiĭ. Reprinted from Nauchnoe Slovo 1929, no. 6, 41–54. Problemy Peredachi Informatsii 42 (2006), no. 4, 129–141; translation in Probl. Inf. Transm. 42 (2006), no. 4, 379–389. It is a reprint of ...


9

Newton studied at school and at the university, but he mostly taught himself by reading. (At his secondary school he certainly learned Latin, Greek, the Bible and some arithmetic. In the universities, they mostly studied Aristotle at that time, which has nothing to do with mathematics). Besides textbooks that existed at that time he mastered Euclid, and ...


9

Such terms as “given in species” are defined in Euclid’s Data (Greek, English): III. Rectilineal figures are said to be given in species, which have each of their angles given, and the ratios of their sides given. (English version, R. Simpson, 1810, p. 367) [Species is the translation of eidos, shape or form; see LSJ, εἶδος, def. A.2.b.]


9

That formula was stated (albeit in a rather different notation) and derived in section 149 of Galloway (1839, A treatise on probability, Adam and Charles Black), of which Google Books has the full text available. That work appears to be a republication as a book of an article from the 7th edition of Encyclopaedia Britannica, which was published in 1827. I ...


8

The answer depends somewhat on what "explicitly" means. The appearance of continuous distributions, and their densities, is generally attributed to Simpson. Namely, his 1757 response to Bayes's critique of his 1755 letter on de Moivre's theory of errors. As Stigler writes in The History of Statistics: "Simpson's 1757 republication of the letter ...


8

Gauss also came up with the more discrete $n/\ln (n)$- in volume 10 of his collected works appears a short (5-6 pages) fragment entitled "asymptotic laws of arithmetics", which is dated to the year 1791. In [1] of this fragment Gauss states this approximation of the primes counting function, as well as additional conjecture on the asymptotics of k-prime ...


7

Yes, he did, multiple times. Singular double integrals (1814) In Mémoire sur les intégrales définies (1814) Cauchy studied why switching the order of integration in a double integral can sometimes lead to different results. This led him to introduce the notion of "singular integral". For $K = \frac{z}{x^2 + z^2}$ he showed that $\int_0^1\int_0^1\frac{\...


7

Feynman is being... liberally creative. What he says is his own interpolation that "makes sense" from the perch of today. "Must have been psychologically wonderful", perhaps, but "freeing of man from the intimidation of the ancients" is not how the men of Renaissance generally felt. The intimidation they sought the freeing from was not of the ancients, but ...


7

It appears to be from the following broadcast (source): ARTE (FRANCE TÉLÉVISION - LA CINQUIÈME) Émission « Archimède » du 14 novembre 2000 consacrée à Bourbaki, réalisée avec la collaboration de Maurice Boulanger et Pierre Samuel. On a présenté de courts extraits de films d’amateur tournés par Pierre Samuel, ancien membre de Bourbaki, durant divers ...


7

The Rogers-Ramanujan identities? The formulas have a curious history, having been proved by Rogers (1894) in a paper that was completely ignored, then rediscovered (without proof) by Ramanujan sometime before 1913. The formulas were communicated to MacMahon, who published them in his famous text, still without proof. Then, in 1917, Ramanujan accidentally ...


7

I did once write an email to Robert Geroch, because an unpublished paper of his was listed as a bibliographical reference in a paper of Hajicek, which contained a few theorems regarding non-Hausdorff spacetimes. He apparently had no recollection of this paper, and it was never published. I assume there are many other such papers, the myriad of unpublished ...


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