A mathematical problem is a problem that can be represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits 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 nature of mathematics itself, such as Russell's Paradox.
The result of mathematical problem solved is demonstrated and examined formally.
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 exercises like "5 − 3", even if one knows the mathematics required to solve the problem. Known as word problems, 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 must be translated back into the context of the original problem.
By outward seeing, there is various phenomenon from simple to complex in the real world. The some of that have also the complex mechanism with microscopic observation whereas they have the simple outward look. It depend to the scale of the observation and the stability of the mechanism. There is not only the case that simple phenomenon explained by the simple model, but also the case that simple model might be able to explain the complex phenomenon. One of example model is a model by the chaos theory.
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, and remains, a rich source of 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 problems, such as the halting problem for Turing machines.
Many abstract problems can be solved routinely, others have been solved with great effort, for some significant inroads have been made without having led yet to a full solution, and yet others have withstood all attempts, such as Goldbach's conjecture and the Collatz conjecture. 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.
The all of mathematical new ideas which develop a new horizon on our imagination not correspond to the real world. Science is a way of seeking only new mathematics, if all of that correspond. On the view of modern mathematics, It have thought that to solve a mathematical problem be able to reduced formally to an operation of symbol that restricted by the certain rules like chess (or shogi, or go). On this meaning, Wittgenstein interpret the mathematics to a language game (de:Sprachspiel). So a mathematical problem that not relation to real problem is proposed or attempted to solve by mathematician. And it may be that interest of studying mathematics for the mathematician himself (or herself) made much than newness or difference on the value judgment of the mathematical work, if mathematics is a game. Popper criticize such viewpoint that is able to accepted in the mathematics but not in other science subjects.
Computers do not need to have a sense of the motivations of mathematicians in order to do what they do. Formal definitions and computer-checkable deductions are absolutely central to mathematical science. The vitality of computer-checkable, symbol-based methodologies is not inherent to the rules alone, but rather depends on our imagination.
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 exercises, and the test no longer assesses problem solving).
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.
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.
- 斉藤, 隆央 (2008-02-15). 超ひも理論を疑う：「見えない次元」はどこまで物理学か? (in Japanese) (1st ed.). Tokyo: 早川書房. p. 17. ISBN 978-4-15-208892-5, translated fromCS1 maint: postscript (link)
Krauss, Lawrence M. (2005). Hiding in the Mirror: The Quest for Alternative Realities, from Plato to String Theory by way of Alice in Wonderland, Einstein, and The Twilight Zone. USA: Penguin Group.
- 前原, 昭二 (1968-09-30). 集合論1. ブルバキ数学原論 (in Japanese) (1st. ed.). Tokyo: 東京図書. pp. 1–4. translated from
Bourbaki, Nicolas (1966). Théorie des ensembles. ÉLÉMENTS DE MATHÉMATIQUE (3 ed.). Paris: Hermann.
- (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 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
(Descartes 1637) harv error: no target: CITEREFDescartes1637 (help), page =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."
- Heaton, Luke (2015). "Lived Experience and the Nature of Facts". A Brief History of Mathematical Thought. Great Britain: Robinson. p. 305. ISBN 978-1-4721-1711-3.
- 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
- S. F. Lacroix (1816) Essais sur l’enseignement en general, et sur celui des mathematiques en particulier, page 201
- Andrew Warwick (2003) Masters of Theory: Cambridge and the Rise of Mathematical Physics, page 145, University of Chicago Press ISBN 0-226-87375-7
- Newby, Ilana; Newby, Greg (2008-07-01). "Discourse on the Method of rightly conducting the reason, and seeking truth in the sciences by Rene Descartes". Project Gutenberg. Retrieved 2019-02-13., translated from
- René, Descartes (1637). 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. Gallica - The BnF digital library (in French).
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