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{{Short description|Problem that can be possibly solved via mathematics}}
A '''mathematical problem''' is a problem that can be [[Representation (mathematics)|represented]], analyzed, and possibly solved, with the methods of [[mathematics]]. This can be a real-world problem, such as computing the [[Orbit#Planetary orbits|orbit]]s of the planets in the solar system, or a problem of a more abstract nature, such as [[Hilbert's problems]]. It can also be a problem referring to the [[Foundations of mathematics|nature of mathematics]] itself, such as [[Russell's Paradox]].
==Real-world problems==
Informal "real-world" mathematical problems are questions related to a concrete setting, such as "Adam has five apples and gives John three. How many has he left?".  Such questions are usually more difficult to solve than regular [[mathematical exercise]]s like "5&nbsp;&minus;&nbsp;3", even if one knows the mathematics required to solve the problem. Known as [[word problem (mathematics education)|word problem]]s, they are used in [[mathematics education]] to teach students to connect real-world situations to the abstract language of mathematics.
In general, to use mathematics for solving a real-world problem, the first step is to construct a [[mathematical model]] of the problem. This involves abstraction from the details of the problem, and the modeller has to be careful not to lose essential aspects in translating the original problem into a mathematical one. After the problem has been solved in the world of mathematics, the [[solution (mathematics)|solution]] must be translated back into the context of the original problem.
==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}}
==Degradation of problems to exercises==
Mathematics educators using [[problem solving]] for evaluation have an issue phrased by [[Alan H. Schoenfeld]]:
:How can one compare test scores from year to year, when very different problems are used? (If similar problems are used year after year, teachers and students will learn what they are, students will practice them: problems become [[exercise (mathematics)|exercise]]s, and the test no longer assesses problem solving).<ref>[[Alan H. Schoenfeld]] (editor) (2007) ''Assessing mathematical proficiency'',  preface pages x, xi, Mathematical Sciences Research Institute, [[Cambridge University Press]] {{isbn|978-0-521-87492-2}}</ref>
The same issue was faced by [[Sylvestre Lacroix]] almost two centuries earlier:
:... it is necessary to vary the questions that students might communicate with each other. Though they may fail the exam, they might pass later. Thus distribution of questions, the variety of topics, or the answers, risks losing the opportunity to compare, with precision, the candidates one-to-another.<ref>[[S. F. Lacroix]] (1816) ''Essais sur l’enseignement en general, et sur celui des mathematiques en particulier'', page 201</ref>
Such degradation of problems into exercises is characteristic of mathematics in history. For example, describing the preparations for the [[Cambridge Mathematical Tripos]] in the 19th century, Andrew Warwick wrote:
:... many families of the then standard problems had originally taxed the abilities of the greatest mathematicians of the 18th century.<ref>Andrew Warwick (2003) ''Masters of Theory: Cambridge and the Rise of Mathematical Physics'', page 145, [[University of Chicago Press]] {{isbn|0-226-87375-7}}</ref>
==See also==
*[[List of unsolved problems in mathematics]]
*[[Problem solving]]
*[[Mathematical game]]
==References==
{{Reflist}}
* {{cite web |last1 =Newby |first1 =Ilana |last2 =Newby |first2 =Greg |date =2008-07-01 |language =en |website =[[Project Gutenberg]] |title =Discourse on the Method of rightly conducting the reason, and seeking truth in the sciences by Rene Descartes <!--chapter =Part V (paragraph 7)--> |url =http://www.gutenberg.org/files/59/59-h/59-h.htm |access-date =2019-02-13 }}, translated from
**{{cite book |last=Descartes |first=René |author-link =René Descartes |year =1637 |title =Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les scienses, plus la dioptrique, les météores et la géométrie qui sont des essais de cette method |language =fr |url =https://gallica.bnf.fr/ark:/12148/btv1b86069594/f5.image |publisher=Gallica - The [[BnF]] digital library }}
{{commons category|Mathematical problems}}
[[Category:Mathematics education]]
[[Category:Elementary mathematics]]
[[Category:Mathematical problems| ]]