39

Commutativity is over-rated: in fact, it holds back bicomplex numbers: It prevents your number system characterising non-commuting operations, e.g. rotations in $3$-dimensional space, Hamilton's original focus. Since $0=i^2-j^2=(i-j)(i+j)$ in bicomplex numbers, you have zero divisors, so it's not a normed division algebra; no convenient conjugates, no ...


33

Yes, it is true. Fermat's own copy was used in the publication of Diophantus by Fermat's son Samuel, and he included Fermat's notes. The original with Fermat's handwriting is lost. https://www.joh.cam.ac.uk/library/special_collections/early_books/fermat.htm#:~:text=When%20reviewing%20his%20copy%20of,to%20fit%20in%20the%20margin. (The page with his most ...


33

Physics cannot help giving physical significance to things. But, yes, the first item on your list should have been Lie Groups. Developed as a mathematical "sudoku" game generalizing rotations, in the late 19th century, by Sophus Lie, Felix Klein, Friedrich Engel, Henri Poincare, and, as associated applied mathematical structures by James Joseph ...


32

Giovanni Schiaparelli ... He wrote in 1877 about his telescopic observations of Mars. He described some features using the Italian word canali. English translation would be channels. But the term was mistranslated as English canals, which was taken to mean they were artificial constructions. This led to much controversy about whether Mars was inhabited!


31

That's John von Neumann, about whom George Pólya wrote: There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn’t say anything but after five minutes he raised his hand. When I called on him he went to the ...


24

Before 1991 - nowhere, there were no platforms to "post" preprints on. Some were distributed by mail, and even collected and catalogued by large libraries. US National Institutes of Health even briefly supported Information Exchange Groups that circulated preprints, from 1961 to 1967. It was shut down because journals refused to publish shared ...


23

Your description "total uselessness of quaternions" in a comment above is poorly chosen, and reflects more on your interests than on the real state of knowledge of mathematics. The Hamilton quaternions are the simplest nontrivial example of a quaternion algebra, which has turned out to be a really important concept in mathematics. It is useful to ...


22

Sylvester's line problem (1893) was to prove that there exists no finite configuration of points in real projective plane such that every line through two points actually contains at least 3 points, unless all points are on the same line. Sylvester himself found such a configuration in the complex projective plane. This problem was quite famous, and was ...


21

I remember those days. (I am in mathematics.) When I completed a paper, I would mail (not email) photocopies (called "preprints") to researchers I knew that I hoped would be interested in it. Then I might attend a conference or visit another university, where I would give a talk on my work, and bring a stack of preprints for interested ...


21

fractal The so-called Cantor set was described by Georg Cantor, 1884 (or H. J. S. Smith, 1875?) Sets with "fractional" dimension were described by Felix Hausdorff, 1918 Investigated thoroughly by A. S. Besicovitch, 1930s - 1950s But these were only mathematical abstractions. Benoit B. Mandelbrot, 1960s and later claimed relevance of these sets in ...


18

It was Fourier series rather than Fourier transform. Considering that the sets where Fourier series converge can be very intricate it is not that surprising that they led Cantor to develop set theory for subsets of real numbers. But at some point he took a turn into the abstract (for which he is best known today) that was not really motivated by the initial ...


15

The following quote is from C. N. Yang, delivered at a 1979 symposium dedicated to the geometer Chern: "When I met Chern, I told him that I finally understood the beauty of the theory of fibre bundles and the elegant Chern​–Weil theorem. I was struck that gauge fields, in particular connections on fibre bundles, were studied by mathematicians without ...


12

Yes, Euler wrote letters in Russian: (Euler's letter to Nartov, June 18, 1743) Interestingly, he signed papers as "Леонгардъ Эулеръ", a direct transliteration of his name, not how he is called in Russian today (imitating pronunciation).


11

Hamilton expected that the quaternions would be of physical interest. In this, he was right. But he was too early. He had discovered them in 1843, it was almost a century later, in 1928, when Dirac discovered his equation involving the Pauli matrices, that it was seen that the quaternions were naturally implicated in quantum field theory. (Here, the Pauli ...


11

I have not been able to find any primary sources that confirm that Gerwien's first name was Paul. Gerwien published three papers in Crelle's "Journal für die reine und angewandte Mathematik", where he is listed in the table of contents as P. Gerwien. High-quality scans of these publications are available from the Göttingen Digitization Center. P. ...


11

Hanna Neumann conjecture was open for 40+ years and was solved in 2011 by relatively elementary methods independently and almost simultaneously by Friedman and Mineyev. The existing proofs actually proved the strengthened Hanna Neumann conjecture formulated in 1990 by Walter Neumann. All tools used in Mineyev's proof certainly existed in 1990 and much ...


11

While some sources claim it is a modern invention, Encyclopedia Brittanica claims that Romans already differentiated right from left footwear, see shoe. Look also at these sandals from Ancient Egypt. Looking on the web, it is a mess as is full of blog/news articles without sources pointing to somebody in the 1800s. For English sources it is sometimes claimed ...


10

I will skip the pre-history of solving polynomial equations and factoring polynomials. Let me mention that the analogy between long division of numbers and polynomials goes back to medieval Islamic mathematician al-Samawal, see Who invented short and long division?, and the Euclidean algorithm for polynomials was optimized by Hudde, a younger contemporary of ...


10

Using planimeters to illustrate Green's theorem is a relatively recent didactic development. Neither Green, nor Cauchy, nor Riemann had any interest in the instruments, and vice versa, planimeter developers did not involve something so abstract in explaining their "principle" mathematically. The kinds of mathematical proofs given can be seen in ...


10

You might be looking for the Italian School of Algebraic Geometry. It has become the canonical example of problems with a lack of rigour. The short summary is that the school started with some unfounded postulates, that they used to derive a wide number of results. This must be understood in the context of an incipient field, where no rigorous foundation was ...


10

Depends on what "introduced" means. If we skip arithmetic and geometric series as too simple to suggest something general the idea goes back to Fibonacci and his rabbit breeding problem that sets up a recurrence in Liber Abaci (1202). Jacob (1564) and Kepler (1618) pointed out that the limit ratios of consecutive terms approach the golden ratio, ...


9

The Delian problem (doubling the cube) and the angle trisection problem are probably the oldest ones, though their exact age is not known. (Same applies to all problems of constructions with ruler and compass). They were certainly around in 4 bc, and existing sources show that by that time they were already old. Since both problems were solved only in 19th ...


9

The statement you cite is somewhat misleading. If you compare Euclid or Archimedes with modern mathematical publications, there is almost no difference in the standard of rigor. When they say that "standard of rigor changes", they usually mean the epoch from 17 to 19 centuries when calculus was invented, and its practitioners were impatient to ...


9

"Are they mathematical curiosities instead of the heroes they were supposed to be by Hamilton?" Neither. It is true that there was a great hype around them, after their discovery. The hype passed. And they took their modest place among the rest of mathematics. Quaternions indeed give a very convenient way to calculate with rotations, and widely ...


8

This is a case where it seems that the symbol should be old, from Euler's or Gauss's time at least, but it is not. It does not appear in Dickson's History of the theory of numbers (1919), whose entire first volume is dedicated to divisibility, nor in Cajori's comprehensive History Of Mathematical Notations (1928), and not even in van der Waerden's Moderne ...


8

The following information is found HERE According to Daniel L. Klaasen in Historical Topics for the Mathematical Classroom: Isaac Newton was the first to think of using polar coordinates. In a treatise Method of Fluxions (written about 1671), which dealt with curves defined analytically, Newton showed ten types of coordinate systems that could be used; one ...


8

The closest match I could find is in H. Weyl, "Harmonics on homogeneous manifolds." Annals of Mathematics, Second Series, Vol. 35, No. 3, July 1934, pp. 486-499, as reproduced in K. Chandrasekharan (ed.), "Hermann Weyl: Gesammelte Abhandlungen Band III," Springer 1968. On page 389 of "Gesammelte Abhandlungen": A "finite&...


8

Zariski introduced his topology in this paper: The compactness of the Riemann manifold of an abstract field of algebraic functions, Bull. Amer. Math. Soc., 50 (1944), 683-691. You can read it online here. See also p. 79 of his talk at the 1950 ICM: The fundamental ideas of abstract algebraic geometry, pp. 77-89 in "Proceedings of the International ...


8

Abelian and non-abelian group theory -> quantum chromodynamics Noneuclidean geometry -> general relativity Sorry, I cannot write you any equations as examples.


8

I don't know if this can be counted for as physics, but to my knowledge the radon transformation was mostly something mathematicians thought about without any application. Now, it is widely used (and necessary) to transform images in tomography. 1917 - Radon transform introduced by Johann Radon 1970s - First Computer Tomograph Scan https://en.wikipedia.org/...


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