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There have been several versions of the memoir. The last version was sent by Galois to his friend Auguste Chevallier. just before the duel and was published by Camille Jordan. For details see the book.


It is the opposite: for Euclid there is nothing but the (2- or 3-)space. All the other things (points, lines, planes, etc.) live there.


I do not completely agree with Alexander Eremenko's answer (though I understand his point - just hope the following does not sound too provocative). Was Italian algebraic geometry at the beginning of the XXth century providing proofs of their statement? Yes, certainly, though a number of them turned out to be wrong, therefore not rigorous. And yet their ...


The reason is that there is no analogue of Church–Turing thesis in logic. Let me explain. In computational theory there is well-known Church–Turing thesis that states that all reasonable definitions of computation are equivalent to each other. That is, all possible definitions of computation (like Turing machine, lambda-calculus, Minsky machine) turn out to ...


Riemann geometry: Einstein had to look around for the mathematics that could describe General Relativity. Before that nobody would have guessed that curved geometry has anything to do with space-time and gravity.


I think one of the best example is the binary system, the representing of numbers in 0 and 1 by diving them by 2. This is how every processor, ram, HDD, SSD,.. etc keep data and how digital communications work. Also the whole Boole algebra is the basics of how computer processors calculate numbers. In 1605 Francis Bacon discussed a system whereby letters of ...


There are many examples of a physical significance given to abstract mathematical ideas. One of the most prominent is the spacetime as introduced in general relativity. In general relativity a mathematical framework is given that informs you about the structure of space. It even relates space with time, thereby introducing new physical stuff, i.e., spacetime....


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.


Spinors were introduced by E.Cartan and found applications in quantum theory 15 years later.


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


Not really physics, but: I think that number theory (edit: specifically research on primes) was an entirely idle mathematical pastime until cryptography became a thing.


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 ...


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 ...


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 ...


First of all, Slovic is a psychologist, not a mathematician. Most likely, his knowledge of modern mathematics is quite limited. (Modern here means, roughly, developed in the last 100 years.) I would be very surprived if he were able to read any papers published, say, by Annals of Mathematics (which is widely regarded as the best math journal) in the last 80 ...


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 ...


The closest thing I can think of are symbols that mean "(the absolute value of) the difference between", a sort of "commutative subtraction" operator: $4 \sim 5 = 5 \sim 4 = 1$ I have access to a copy of Webster's New International Dictionary, Second Edition with a (publication?) date of 1949 on the title page, copyright date of 1934, and ...


Diophantus wrote one of the first number theory books "Arithmetica".


Quoting from A History of Mathematics (3rd edition), by Victor J. Katz: Although there were earlier versions of Elements before that of Euclid, his is the only one to survive, perhaps because it was the first one written after both the foundations of proportion theory and the theory of irrationals had been developed and the careful distinctions always to be ...

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