Editing Mathematical problem (section)

==Abstract problems==
Abstract mathematical problems arise in all fields of mathematics. While mathematicians usually study them for their own sake, by doing so, results may be obtained that find application outside the realm of mathematics. [[Theoretical physics]] has historically been a rich source of [[artistic inspiration|inspiration]].

Some abstract problems have been rigorously proved to be unsolvable, such as [[squaring the circle]] and [[trisecting the angle]] using only the [[compass and straightedge constructions]] of classical geometry, and solving the general [[quintic equation]] algebraically. Also provably unsolvable are so-called [[undecidable problem]]s, such as the [[halting problem]] for [[Turing machine]]s.

Some well-known difficult abstract problems that have been solved relatively recently are the [[four-colour theorem]], [[Fermat's Last Theorem]], and the [[Poincaré conjecture]].

[[Computers]] do not need to have a sense of the motivations of mathematicians in order to do what they do.<ref>{{harv|Newby|Newby|2008}}, "The second test is, that although such machines might execute many things with equal or perhaps greater perfection than any of us, they would, without doubt, fail in certain others from which it could be discovered that they did not act from [[awareness#Self-awareness|knowledge]], but solely from the disposition of their organs: for while [[reason]] is an universal instrument that is alike available on every occasion, these organs, on the contrary, need a particular arrangement for each particular action; whence it must be morally impossible that there should exist in any machine a diversity of organs sufficient to enable it to act in all the occurrences of life, in the way in which our reason enable us to act." translated from <br/>{{harv|Descartes|1637}}, page =[https://gallica.bnf.fr/ark:/12148/btv1b86069594/f61.image 57], "Et le second est que, bien qu'elles fissent plusieurs choses aussy bien, ou peutestre mieux qu'aucun de nois, ells manqueroient infalliblement en quelques autres, par lesquelles on découuriroit quelles n'agiroient pas par connoissance, mais seulement par la disposition de leurs organs. Car, au lieu que la raison est un instrument univeersel, qui peut seruir en toutes sortes de rencontres, ces organs ont besoin de quelque particliere disposition pour chaque action particuliere; d'oǜ vient qu'il est moralement impossible qu'il y en ait assez de diuers en une machine, pour la faire agir en toutes les occurrences de la vie, de mesme façon que nostre raison nous fait agir."</ref> Formal definitions and computer-checkable [[deductive reasoning|deductions]] are absolutely central to [[mathematical science]].

{{see also |Logical positivism |de:Falsifikationismus}}