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Could someone list me the most important battles between mathematicians which happened in history, especially such that strong emotions played role in that time?

Perhaps the most known one is the battle between supporters of Newton and Leibniz on invention of calculus.. but..

What other similar campaigns are known in history of mathematics? Who was engaged and who finally won the battle if we can say that it was won? ( in the case of Newton and Leibniz I would say it was draw)

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    $\begingroup$ Not adding as a proper answer since I lack a historiographic source, but in the first half of the 20th century frequentists pushed against legitimacy of Bayesianism $\endgroup$
    – ain92
    Mar 5 at 12:38
  • $\begingroup$ Didn't Archimedes push someone off of a boat for insisting that pi was irrational? My old analysis professor told us that story to provide the motivation for the study of real analysis. $\endgroup$ Mar 5 at 23:06
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    $\begingroup$ @ForgottheJacobian There's a fairy story that Pythagoras pushed someone off a boat for discovering that the diagonal of a square is incommensurable to its side (i.e. $\sqrt{2}$ is irrational). That's probably what you or your analysis professor was misremembering, and it's made up anyway. $\endgroup$ Mar 6 at 12:14

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From Wikipedia:

From Great Feuds in Mathematics, you also have:

  • Tartaglia vs. Cardano (about the cubic equation priority, independent discovery, draw)
  • Fermat vs. Descartes (on tangent curves, at the time it was Descartes who was considered winner but now it is thought of as a draw)
  • Jakob Bernoulli vs. Johann Bernoulli (about the brachistochrone problem?, Johann had an incorrect proof and tried to pass Jakob’s solution as his own.)
  • Poincaré vs. Russell (on the nature of logic and Kant’s philosophy of mathematics, it seems it is either a draw or Russell was wrong, see this, specially since the development of Gödel's theorem)
  • Borel vs. Zermelo (on the axiom of choice, part of modern axioms of mathematics, but its validity is still discussed)

Book: Great Feuds in Mathematics: Ten of the Liveliest Disputes Ever, Hal Hellman, 2006

Note that all of these feuds have philosophical points of conflict that were not necessarily resolved, I just added a few remarks on the main issues.

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One battle that is somewhat overlooked is the battle over Bonaventura Cavalieri's indivisibles. His opponent was Paul (Habakkuk) Guldin, a Jesuit (and other Jesuits). Cavalieri was a Jesuat. Most engines today automatically correct "Jesuat" to "Jesuit", as the Jesuats are mostly unknown (for reasons that will become clear below). The Jesuat order was founded over a century before the Jesuits. Starting in the 1620s and 1630s, the two orders were lined up on opposite sides of a debate over indivisibles, with Cavalieri fiercely defending their legitimacy, and Guldin and others fiercely opposed to them.

The bottom line of the Jesuits' opposition to indivisibles were doctrinal: they were convinced that ideas such as atomism and indivisibles were contrary to Catholic dogma (obviously Cavalieri, who was a devout Catholic, did not think so). At issue is the rivalry between the Aristotelian doctrine of hylomorphism which since Aquinas became the theoretical "explanation" of the miracle of the Eucharist, and Democritus' atomism, thought of as subversive and heretical by the Jesuits.

The generals of the Jesuit order issued numerous bans against indivisibles. The Jesuits had far more influence in Rome than the Jesuats, and things came to a head in 1668 when the Jesuat order was suppressed abolished by papal bull ("bulla", not the animal). Cavalieri's student Degli Angeli, similarly a Jesuat, wrote about a dozen books about applications of the method of indivisibles, the last one I believe a year before the suppression of his order. After that, he didn't publish anything on indivisibles any more, though he remained active and published a book in mechanics, and continuing his teaching at the university for several decades after that. In the 1690s, Riccati attended his lectures at the university (obviously not about indivisibles).

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    $\begingroup$ Do you have any reference where we can read more about this? Also, I think the right term in English is Jesuati or Jesuate but I might be wrong as these were Italian terms. $\endgroup$
    – Mauricio
    Mar 4 at 12:52
  • $\begingroup$ Amir Alexander uses the spelling "jesuat" in his book on infinitesimals, but you are right that the other two spellings are more commonly used. Another spelling found in the literature is "gesuati". I have a forthcoming paper on this that I prefer not to publicize for now, but if you have any specific questions I would be glad to provide more details. @Mauricio $\endgroup$ Mar 4 at 13:46
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In computable algebraic geometry there is the "elimination of elimination theory" controversy.

It starts with Kronecker in 1880 providing an approach towards the solution of polynomial systems that eliminates one variable after the other using resultants. This way fundamental theorems like the Nullstellensatz are proven constructively, but with great machinery. At the time no complexity analysis was done.

Hilbert in the 1890's provided a shorter, basically non-constructive proof of the Nullstellensatz and related results. The further development of algebraic theory followed mainly Hilberts approach, to the point that in editions after 1930 of vander Wardens influential book any mention of elimination theory was eliminated. Previously it contained a nice elementary exposition in the first chapters.

The development of the Gröbner basis algorithm in the 1960's (?) was taken as confirmation of the Hilbert approach. Such algorithms are widely implemented in CAS, but contain a large heuristic part.

Only starting in the 1990's were efficient elimination procedures developed. This is still a small fringe of the community, but has some advantages: there is a theoretically accessible complexity theory for it and the complexity magnitudes are better than all what is available for Gröbner bases. and their complexity proven

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  • $\begingroup$ Any source to read on this? $\endgroup$
    – Mauricio
    Mar 5 at 9:57
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    $\begingroup$ Generally the group TERA around Marc Giusti, Joos Heintz, Louis Pardo et al. Gregoire Lecerf for the implementation of these ideas as the "Kronecker" library. A historical overview was formulated in "Kroneckers smart little black boxes" PDF. Honorable mention to Sommese, Verschelde et al. from the PHCpack project. $\endgroup$ Mar 5 at 12:58
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I would say Mochizuki vs Scholze fits the bill, though this is very recent history:-)

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Dorian invasion. Unlike native Achaeans, the Dorians viewed the world in terms of Euclidean geometry (not yet called that) in which "large and small can be the same" (similarities are automorphisms), didn't use intentional perspective and destroyed almost all works of art they could find which used it (and other artifacts as well; they were aware of cultural assimilation of invaders being the usual outcome and adamant in avoiding it, even if it meant lower quality of life). They clearly won – nowadays we have no problem using maps, technical drawings in scale, or copying, without specific instruction, large letters from a blackboard into a notebook, unconsciously scaling them down, even as children.

Akousmatics vs mathematicians (first group of people who called themselves that), the factions into which Pythagoreans split upon discovery of irrational numbers. Result: mathematicians are a (fairly) respected profession, akousmatics quickly became an esoteric fringe subculture.

Linear algebra (with linear equations), preferred on the continent, particularly among Germans, vs matrix theory, developed and favoured on the British Isles. Result: draw IMO; emotions subsided and theorems showing relevant equivalences became common knowledge in the field; both tools are often reached for, depending on the problem and other factors.

Analysts vs algebraists, logicians and theoretical computer scientists. Ongoing. Sometimes with heightened emotions.

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    $\begingroup$ Can you provide some sources for your claims? $\endgroup$ Mar 5 at 8:53
  • $\begingroup$ On akousmatics almost any source describing them will do, including Wikipedia and Stanford Encyclopedia of Philosophy. A single source which handles all of them, I believe, to various degrees, as well as many mentioned above by others, plus mathematical matches popular in Renaissance Italy, is "Wykłady z historii matematyki" (in Polish) by Marek Kordos. $\endgroup$
    – ByteEater
    Mar 5 at 12:47
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    $\begingroup$ Kordos would be good for the Polish StackExchange but you would do the users here a favor by linking the relevant passages in your answer to the appropriate sources. $\endgroup$ Mar 5 at 12:54

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