Amathematical problem is a problem that can berepresented, analyzed, and possibly solved, with the methods ofmathematics. This can be a real-world problem, such as computing theorbits of the planets in theSolar System, or a problem of a more abstract nature, such asHilbert's problems. It can also be a problem referring to thenature of mathematics itself, such asRussell's Paradox.
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 regularmathematical exercises like "5 − 3", even if one knows the mathematics required to solve the problem. Known asword problems, they are used inmathematics 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 amathematical 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, thesolution must be translated back into the context of the original problem.
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 ofinspiration.
Some abstract problems have been rigorously proved to be unsolvable, such assquaring the circle andtrisecting the angle using only thecompass and straightedge constructions of classical geometry, and solving the generalquintic equation algebraically. Also provably unsolvable are so-calledundecidable problems, such as thehalting problem forTuring machines.
Some well-known difficult abstract problems that have been solved relatively recently are thefour-colour theorem,Fermat's Last Theorem, and thePoincaré conjecture.
Computers do not need to have a sense of the motivations of mathematicians in order to do what they do.[1] Formal definitions and computer-checkabledeductions are absolutely central tomathematical science.
Mathematics educators usingproblem solving for evaluation have an issue phrased byAlan H. Schoenfeld:
The same issue was faced bySylvestre Lacroix almost two centuries earlier:
Such degradation of problems into exercises is characteristic of mathematics in history. For example, describing the preparations for theCambridge Mathematical Tripos in the 19th century, Andrew Warwick wrote:
