# Tag Info

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One of the first application of mathematics and logic to biology was classification. This was done by Aristotle and his terms still apply today - ie genus, species. These are logical terms taken from his Organon which detail the Aristotelian understanding of logic. A good reference for this is the book by Rene Thom, Semiophysics where he also details the ...

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It is not that recent (apart from the name), before Fisher there were Bernoulli (1760), Malthus (1789) and Verhulst (1838), to name a few, although one might view them as pre-history. Here is a PowerPoint that has the main highlights with dates. Rashevsky is generally credited with establishing the field, see Hoffman, The Dawn of Mathematical Biology. For ...

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I'm not aware of any writings of Gauss that touched upon philosophical concerns. That doesn't mean that Gauss wasn't philosophically aware. He's been said to have read Kants Critique five times which probably isn't surprising given the esteem that Kant was held in the German consciousness of the time - rather like Hawking is held in ours. And to me, it's ...

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Negri and collaborators call "geometric" theories that can be axiomatized by FOL formulas of a special sort, namely universal closures of $A\to B$, where $A$ and $B$ do not contain $\to$ or $∀$, see e.g. Contraction-free sequent calculi for geometric theories with an application to Barr’s theorem. The terminology goes back to, at least, ...

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The geometry in geometric theories, as in topoi, comes from topology and hence also the name topos. They, that is the topoi, are seen as generalisation of a topology. This generalisation was by Grothendieck and generalises that of a topological covering which is, in his language, a sieve. And in this way we get elementary topoi, which was lawvere's ...

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This answer is CW, so I don't gain reputation for someone else's work. To quite Kevin Arlin's answer to the same question on math.stackexchange: Geometric logic constitutes the logic, models of whose theories are preserved by geometric morphisms between topoi. Geometric morphisms are those appropriate to toposes viewed as generalized spaces, for instance, ...

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This sounds like the physics equivalent of the Library of Babel by the Argentine author, Jorge Luis Borges and similarly, as fictive. Perhaps in the future we can imagine some such 'Encyclopadia Principae Physicae' hyperlinked to every journal ever printed and instantly accessible to everyone and running to millions of pages and many thousands of volumes and ...

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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 ...

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The essential part of the answer (page references) is contained in the comment of @Conifold. However his general conclusion is plain wrong, and I would like to put the things straight. A scientist makes ASSUMPTIONS. Then develops a theory. And then compares with observations/experiments. If this comparison works, this CONFIRMS his assumptions. For example, ...

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Here are the references to the original papers of Cantor: On a theorem concerning trigonometric series. (Ueber einen die trigonometrischen Reihen betreffenden Lehrsatz.) Borchardt J. LXXII, 130-138 (1870). Proof that a function given for every real value of by a trigonometric series has only one representation in this form. (Beweis, dass eine für jeden ...

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I think that the history of how we write fractions is helpful here. Although fractions were known in ancient times - the Babylonians and Egyptians used them - the modern notation for them began with the system of bhinnarasi by Aryabhatta around 5th Century AD and then Brahmagupta and (c. 626) and Bhaskara (c. 1150). In their works, they formed fractions by ...

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As @Andrei Kopylov noticed, epicycle theory is not the theory of Fourier series of a periodic function. Still this is called (generalized) Fourier analysis. Such functions are called almost periodic or quasi-periodic, and they expand into generalized Fourier series of the form $$\sum c_k e^{i\lambda_k t},$$ with arbitrary real $\lambda_k$, which is also ...

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This is very popular myth, but it is not true: Ptolemy's epicycles are not Fourier analysis! Fourier series can indeed approximate an arbitrary periodic function. And you can approximate an arbitrary motion of period $T$ by the series of epicycles. The first is just a circular motion of period $T$, the second is an epicycle with period $T/2$ and so on. But ...

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Yes. That was done by Giovanni Schiaparelli in a book called Scritti sulla storia dell'astronomia antica, published in 1926 and reprinted in 1997.

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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 ...

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paul garrett has the idea. $E \subseteq \mathbb R$ is a set of uniqueness if: given a trigonometric series $\sum_{n=-\infty}^\infty c_n e^{int}$, if it converges to $0$ except possibly on $E$, then $c_n = 0$ for all $n$. Here is a description. The empty set is a set of uniqueness. This is just a fancy way to say that if a trigonometric series converges to ...

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Unfortunately, I do not have a concrete reference... but I seem to recall that Cantor's earliest work was about "sets of uniqueness" for Fourier series (I think not Fourier transforms, but I could easily be mistaken). This would be similar to other late 19th century "constructive" analysis projects, where limits of limits of ... ...

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An old English translation is available on Google Books (or in transcribed form here). The Dictionary of Scientific Biography entry for Stevin lists several translations in its bibliography.

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The tensor product is actually a very simple concept. It goes back to Babylonian times when people realised that two edges describes an area. Intuitively they realised that geometric area was bilinear but all this wasn't formalised until the twentieth century. Mathematicians, being mathematicians, generalised by allowing the edges to take values in any ...

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Plato had a simple rule: learn geometry. This of course is well before the 1800s. This is tempered by the time of al-Jabr who taught us to learn algebra. Another useful heuristic is by Solomon Lefschetz. Unfortunately this is after your cut off date, being in the early to mid 20C. He had one simple rule: he said he wasn't interested in prettifying already ...

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Witten gave a spinorial proof of the positive energy theorem in GR. This was originally conjectured by Arnowitt, Deser and Misner in the early 60s. Special cases were then shown by a great many people with the general theorem finally established by Schoen and Yau. Witten also gave a super-symmetric physics proof of the Atiyah-Singer index theorem. This had ...

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The origins of differential homological algebra According to Weibel, who has written one of the standard texts on homological algebra: Homological algebra had its origins in the 19th century, via the work of Riemann (1857) and Betti (1871) on 'homology numbers' and the rigorous development of homology numbers by Poincare in 1895. An observation of Emmy ...

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I. Notes on notation and English terms in 1800s literature (primarily related to series) The following is from the introductory comments for something involving conditionally convergent series I was working on several years ago. The logarithm function is often denoted by a lower case L, sometimes italicized (e.g. Catalan's book) and sometimes not italicized ...

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Indeed they did, the OP story is told by Yau in The Shape of Inner Space, pp.169-70. Calculating the number of twisted cubics in the quintic hypersurface was a big early coup for mirror symmetry that attracted mathematicians' attention to string theory and revitalized enumerative geometry. In A pair of Calabi-Yau manifolds as an exactly soluble ...

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Nobody. Those who were first did not have a clear idea of real numbers or completeness, and by the time the concepts took shape those who used them were no longer first, see MacTutor, The real numbers: Stevin to Hilbert. The first to state completeness as an axiom, to back up his prior axiomatization of geometry, was Hilbert in Über den Zahlbegriff (1900), ...

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There is Betteridge's law of headlines: "Any headline that ends in a question mark can be answered by the word no." The article you quote (Euler's “Mistake”?) is not an exception. $\sqrt{ab}=\sqrt{a}\sqrt{b}$ is not a mistake. It is true if you understand square root as a multivalued function. Another common understanding of √ is positive square ...

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Bocher, the author of the 1892 article cited in Wikipedia, published a follow-up note: I find that in the Educational Times for March, 1864, Clifford refers incidentally, in the solution of a problem set by Prof. Sylvester, to the "nine-point conic;" thus showing that this conic, to which I called attention in the March number of the ANNALS, was ...

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