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)
From Wikipedia:
From Great Feuds in Mathematics, you also have:
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.
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).
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
I would say Mochizuki vs Scholze fits the bill, though this is very recent history:-)
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.
