A mathematical problem is a problem that is
amenable to being 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.
== 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 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.
== 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,
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.
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)
maked 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.
Computer 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 ==
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.
== See also ==
List of unsolved problems in mathematics
Problem solving
Mathematical game
List of mathematical concepts named after
places
