Solution (math)

In mathematics, a solution is a mathematical object , for example a number or a function , that meets the requirements of a well-defined mathematical problem . If one regards a task as a set of formalized statements, for example of equations or inequalities , which contain free variables , then one solution is an assignment of the variables with elements from a well-defined domain that fulfills all statements at the same time : If the free occurrences of the variables are replaced by the values assigned in the assignment, then all of these statements must be true at the same time . The set of all such assignments is the solution set of the problem.

More generally, one speaks of the "solution" of the problem not only for equations, but also for any mathematical problem. Such a solution can be a construction or a proof, for example.

On the other hand, the metamathematic literature frequently addresses an antagonism between "problem solvers" (e.g. Paul Erdős ) and "theory builders" (e.g. Alexander Grothendieck ). A well-known linguistic picture is the comparison of problem solvers and theory builders with frogs and birds, which goes back to John von Neumann (see quote below).

The systematic investigation of problem-solving strategies is known as heuristics , and the Hungarian mathematician George Pólya in particular has made extensive contributions in this area. A standard work is his book " Schule des Denkens ".

Special solutions

Depending on the nature of the problem, solutions often have special names:

Zero : solving an equation of the form. In the case of polynomials , one speaks of the "roots" of the equation. ${\ displaystyle f (x) = 0}$

Fixed point : solving an equation of the form.

{\ displaystyle f (x) = x}

optimal solution , especially as an extreme solution : solution of an equation under constraints of the form or.

{\ displaystyle f (x) = \ max \ {f (t) \}}

{\ displaystyle f (x) = \ min \ {f (t) \}}

Examples

The equation is in the real numbers , the solution , in the complex numbers , the amount of solution in the finite field the solution . ${\ displaystyle x ^ {3} -1 = 0}$ ${\ displaystyle x = 1}$ ${\ displaystyle \ left \ {1, - {\ frac {1} {2}} + i {\ frac {\ sqrt {3}} {2}}, - {\ frac {1} {2}} - i {\ frac {\ sqrt {3}} {2}} \ right \}}$ ${\ displaystyle GF (2)}$ ${\ displaystyle x = 1}$

The differential equation has (in the differentiable functions ) the solution set . ${\ displaystyle f '(x) = f (x)}$ ${\ displaystyle f \ colon \ mathbb {R} \ to \ mathbb {R}}$ ${\ displaystyle \ left \ {ce ^ {x} \ colon c \ in \ mathbb {R} \ right \}}$

The optimization problem has the solution for .

{\ displaystyle \ left \ {x + y = 1, f (x, y) = xy = \ max! \ right \}}

{\ displaystyle f (x, y) = {\ frac {1} {4}}}

{\ displaystyle x = y = {\ frac {1} {2}}}

Quotes

"Some mathematicians are birds, others are frogs. Birds fly high in the air and survey broad vistas of mathematics out to the far horizon. They delight in concepts that unify our thinking and bring together various problems from different parts of the landscape. Frogs live in the mud below and see only the flowers that grow nearby. They delight in the details of particular objects, and they solve problems one at a time. (...) "

"The landlady hurried into the yard and put the mousetrap on the floor (it was an old-fashioned trap, a cage with a trap door) and called the daughter to fetch the cat. The mouse in the trap seemed to be the target of these operations feel that she was running around in the cage and throwing herself violently against the bars, once on this side and then on that, and at the last moment she managed to squeeze through the bars and disappear into the neighbor's field have given this point a slightly larger space between the bars ... I quietly congratulated the mouse, for it had solved a significant problem and set an important example.

This is the right way to solve problems. We have to try again and again until we finally see the small differences in the openings on which everything depends. We must vary our attempts to explore all sides of the problem. In fact, we cannot know in advance which side is the only possible opening through which we can escape.

The basic method of mice and humans is the same: try, try again, and the attempts vary so that we don't miss the few cheap opportunities.

It's true that humans are usually better problem solvers than mice. A person does not have to physically throw himself against an obstacle, he can do so mentally; a person can vary his attempts better and he can learn more from his unsuccessful attempts. "(George Pólya:" Die Maus ")

literature

George Pólya : On solving mathematical problems. [On solving mathematical problems.] Insight and discovery, learning and teaching. [Insight and discovery, learning and teaching] Second edition. Translated from English by Lulu Bechtolsheim. Science and Culture, 20. Birkhauser Verlag, Basel-Boston, Mass., 1979. ISBN 3-7643-1101-0 (Volume I), ISBN 3-7643-0298-4 (Volume II).
George Pólya: School of Thought. About solving math problems (“How to solve it”). 4th edition Francke Verlag, Tübingen 1995, ISBN 3-7720-0608-6 ( Dalp collection ).

Web links

ZUM-Wiki:

Individual evidence

^ Freeman Dyson: Birds and Frogs Notices Amer. Math. Soc. 56 (2009), no. 2, 212-223.

[1] Freeman Dyson: Birds and Frogs Notices Amer. Math. Soc. 56 (2009), no. 2, 212-223.