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27

I will start by answering why matrix algebra became important, and then discuss approximately when. "Matrices" underpin what is often called operations research. That is, the theory of decision making. They are particularly useful in computer science, which features strings, arrays, etc., with machines substituting for human beings in (mechanical) decision ...

24

First hand testimony and insightful thoughts on Ramanujan's background and way of doing mathematics can be found in Hardy's lecture Indian Mathematician Ramanujan. Hardy is the British mathematician who first appreciated the full extent of Ramanujan's talent, knew him well personally, and had a fruitful mathematical collaboration with him. Here is Hardy's ...

18

The general question is difficult to answer. Why were the British and Dutch scientists of 17-th century so impressive? Why was French science of the 18-th and 19-th century so impressive, especially in the early part of 19-th century ? The only thing which is clearly seen, is the correlation of these periods with vigorous economic development of these ...

18

The "movement" against relativistic mass was started by Adler in 1987 with Does Mass Really Depend on Velocity, Dad? (his answer, "actually no, but don't tell your teacher"). It got a boost from Okun's two 1989 papers. In 1990 American Journal of Physics solicited Okun's contribution on the relativistic mass to be published alongside Sandin's, who defended ...

13

Mauro Allegranza's comment pretty much says it all but to elaborate a bit one could mention that Leibniz came to mathematics rather late in his intellectual career and was essentially a self-educated scholar. His older colleague Huygens encouraged him to pursue mathematics, and his encouragement (on many occasions) was instrumental in Leibniz's development. ...

12

Well Ramanujan was such a mathematician. He was not so poor in math exams but he scored somewhat unbelievable marks in mathematics. In $1907$, he appeared in FA Examination at Pachaiyappa College, after studying privately. He got $85$ out of $150$ in mathematics and failed in English, Sanskrit, Physiology and History. Now one would expect that Ramanujan ...

10

I can only talk about math books. Problems in the end of the chapters is a British-American custom. German, French and Russian books do not have them. Instead they publish separate problem books, usually by a different author. Problem books contain only problems, and in the introduction they mention which textbook(s) are recommended. British/American ...

8

The most famous case of this sort is Galois failing the entrance exam to Ecole Polytechnique.

8

Given the historical circumstances the German education system might or might not have played a significant role in the formation of these scientists. In his collection of essays entitled "Brocas Brain", Carl Sagan gives some excellent insight into Einsteins biography: (I think this an accurate reprint of the original piece) Einstein dropped out of school ...

8

In 1969 Lang wrote an article for the Columbia Daily Spectator, Don't Blame Us if You Flunk Math (Volume III, Number 4, December 8, 1969). The phrasing of the subline illustrates how much the times have changed:"A fifteen minute quiz raises questions what kind of people should be taught what kinds of math at Columbia". He provides a basic introductory test ...

7

A similar question was asked to the theoretical physicist Gerald 't Hooft on his page 'Theoretical Physics as a Challenge'. It is mentioned at the bottom of the linked page: Mr. Hisham Kotry came with an important question: "... Two years ago I decided to self-study theoretical physics by following the syllabus of a renown university and the advice from ...

7

Here is one data point. (It does not contradict Alexandre's assertion that this is an American innovation, but does provide an earlier starting date.) Day's Algebra (I picked this volume up in a used book store once.) This textbook does have lists of exercises at the end of each section.

7

"Examples of the Catholic church commenting on, or intervening in, the development of mathematics:" Counter-reformation issues in the 17th century around transubstantiation/consubstantiation interpretation of the eucharist directly influenced attitudes toward the techniques of indivisibles that were seen as closely related to atomism and therefore opposed on ...

7

A relevant source to find answers to this questions is the book (out of print in Germany): [Ferber 1956] Christian von Ferber: Die Entwicklung des Lehrkörpers der deutschen Universitäten und Hochschulen 1864-1954. Band 3 von Untersuchungen zur Lage der deutschen Hochschullehrer. Vandenhoeck & Ruprecht, 1956, 244 pages which is the declared source ...

6

Cauchy, who gave calculus its modern formalization (cf. Grabiner's The Origins of Cauchy's Rigorous Calculus), was a "significant mathematician who was also a practicing Catholic." From Belhoste's biography of him, p. viii: "Truth," he wrote in 1842, "is a priceless treasure which, whenever we manage to acquire it, cannot bring us remorse and sorrow; ...

6

I would add, as example, that can be a bit investigated the work of Padre Girolamo Saccheri, a Jesuit priest, and his famous "Euclides ab omni naevo vindicatus" (Euclid Vindicated from Every Blemish) in which he first discovered many theorems of what will then be called hyperbolic geometry. He however ended his books by saying that such theorems were ...

6

I would say that in Germany there was a gradual development towards the matrix notation of linear equation systems from the 1920s onwards. Courant certainly was a pioneer in this development as he tells in this interview. This textbook from 1927 on Statik im Eisenbetonbau, i.e. statics of concrete structures, features the term "matrix" 65 times and was ...

6

There is a large list of links to biographies here. Most mathematicians on the list are not as well known as Gauss and Euler, and they link to MacTutor, which is more reliable than Wikipedia.

6

I recommend the authoritative volume "Das Studium der Mathematik an den deutschen Universitäten seit Anfang des 19. Jahrhunderts" by Wilhelm Lorey, published in 1916. This digitalized edition can be accessed from inside the US. As for the number of professors: to the best of my knowledge, German universities had two ordinary and one extraordinary professor, ...

6

Mathematicians rarely describe the process which led them to their discoveries. One notable exception was Euler. Some books on the subject written by great mathematicians are: J. Hadamard, The Mathematician's Mind. The Psychology of Invention in the Mathematical Field (English translation). G. Polya, a) Mathematics and plausible reasoning. b) Mathematical ...

5

Newton somewhat infamously failed his examinations at Cambridge: he was questioned orally about the proofs in Euclid, and since he had looked at Euclid once and thought it a total waste of time, and had just rederived all the results himself, he had no idea how to answer things like "How does Euclid prove X.1?" Somehow he was awarded the scholarship anyway -...

5

You must tell them about the book Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, by Amir Alexander. It's about, among other things, how the Jesuit priests stopped the progress of Infinitesimal Calculus in Italy because the foundations of the subject were shaky at the time (to say the least). Perhaps that you might also mention ...

5

This question comes up often, but there is no up to date scholarly study of it that I know of. The most comprehensive recent accounts (and they are rather brief) seem to be Jeff Miller's Earliest Uses of Symbols of Operation and Peterson's post on Math Forum, parts of which are aped on various pop-sites, often without attribution. But Peterson admits:"I have ...

5

Students learned theorems, propositions and their proofs. A teacher would call a student to the blackboard and ask to reproduce a proof. A shorter test would be just to state the theorem. Many students memorized theorems and proofs by heart. A good (but not popular) teacher would make his own picture on the blackboard, with his own notation, different from ...

4

An interesting story is the one of Matteo Ricci, a missionary jesuit in China. He wrote the first Chinese translation of Euclid's Elements. He was also a cartographer. Regarding real analysis, Pietro Mengoli proposed one of the most influential problem in the early days of calculus: the Basel problem, famously solved by Euler. Also, Francesco Faà di Bruno ...

4

G.H. Hardy. From http://www-history.mcs.st-and.ac.uk/Biographies/Hardy.html "While at Winchester Hardy won an open scholarship to Trinity College, Cambridge, which he entered in 1896. At Cambridge Hardy was assigned to the most famous coach R R Webb. He quickly realised that the point of the training was simply to achieve the best possible marks in the ...

4

Cédric Villani's Living theorem does exactly that. A great read for scientists, even non-mathematicians.

4

The first mathematician who did that was Archimedes, in his The Method of Mechanical Theorems (usually known as The Method).

3

The best example I can think of is George Green. George Green quit his studies at the age of 9 to work with his father and eventually was owner of a mill. He spent the majority of his life as a miller but found time to work on advanced math and developed concepts far ahead of his own time. He published an essay on electromagnetism in a little known journal ...

3

The German/Prussian education system was based on two pillars: The first was a mandatory comprehensive primary public school education with the aim to provide literacy, numeracy and good educational background for everybody (primary school = Volksschule). Even the most deserted villages got their “Volksschule”. The follow up secondary and “Gymnasium” ...

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