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The article has been recently proposed for deletion with the following motivation: "The article begins with 'In mathematics' and does not contain any mathematics except trivia. Most of the content consists of philosophical considerations that do not seem to be supported by the numerous philosophical sources. In summary, pure original research."

Another editor has endorsed this page's proposed deletion and added "Seconding, this is amateur pseudophilosophy which appears to be trying to derive an original theory of metaphysics."

As the PROD tag has been removed, I nominate the article for deletion.

Rational points

A quadric is defined over a field $F$ if the coefficients of its equation belong to $F.$ When $F$ is the field $\mathbb {Q}$ of the rational numbers, one can suppose that the coefficients are integers by clearing denominators.

A point of a quadric defined over a field $F$ is said rational over $F$ if its coordinates belong to $F.$ A rational point over the field $\mathbb {R}$ of the real numbers, is called a real point.

A rational point over $\mathbb {Q}$ is called simply a rational point. By clearing denominators, one can suppose and one supposes generally that the projective coordinates of a rational point (of a quadric defined over $\mathbb {Q}$) are integers. Also, by clearing denominators of the coefficients, one supposes generally that the all coefficients of the equation of the quadric, and of the polynomials occuring in the parametrization are integers.

Finding the rational points of a projective quadric amounts thus to solve a Diophantine equation.

Given a rational point A over a quadric over a field F, the parametrization described in the preceding section provides another rational point for every parameters in F, and, conversely, every rational point of the quadric can be obtained from parameters in F, if the point is not in the tangent hyperplane at A.

It follows that, if a quadric has a rational point, it has many other rational points (infinitely many if F is infinite), and these points can be algorithmically generated as soon one knows one of them.

As said above, in the case of projective quadrics defined over $\mathbb {Q} ,$ the parametrization takes the form

$X_{i}=F_{i}(T_{1},\ldots ,T_{n})\quad {\text{for }}i=0,\ldots ,n,$

where the $F_{i}$ are homogeneous polynomials of degree two with integer coefficients. Because of the homogeneity, one can consider only parameters that are setwise coprime integers. If $Q(X_{0},\ldots ,X_{n})=0$ is the equation of the quadric, a solution of this equation is said primitive if its components are setwise coprime integers, and the first nonzero element is positive. The primitive solutions are in one to one correspondence with the rational points of the quadric. The non-primitive integer solutions are obtained by multiplying primitive solutions by arbitrary integers; so they do not deserve a specific study. However, setwise coprime parameters can produce non-primitive solutions, and one may have to divide by a greatest common divisor for getting the associated primitive solution.

This is well illustrated by Pythagorean triples. A Pythagorean triple is a triple (a,b,c) of positive integers such that $a^{2}+b^{2}=c^{2}.$ A Pythagorean triple is primitive if $a,b,c$ are setwise coprime, or, equivalently, if any of the three pairs $(a,b),$ $(b,c)$ and $(a,c)$ is coprime.

By choosing $A=(-1,0,1),$ the above method provides the parametrization

${\begin{cases}a=m^{2}-n^{2}\\b=2mn\\c=m^{2}+m^{2}\end{cases}}$

for the quadric of equation $a^{2}+b^{2}-c^{2}=0.$ (The names of variables and parameters are been changed from the above ones to those that are common when considering Pythagorean triples).

If m and n are coprime integers such that $m>n>0,$ the resulting triple is a Pythagorean triple. If one of m and n is even and the other is even, this resulting triple is primitive; otherwise, m and n are both odd, and one gets a primitive triple by dividing by 2.

In summary, the primitive Pythagorean triples with $b$ even are obtained as

$a=m^{2}-n^{2},\quad b=2mn,\quad c=m^{2}+n^{2},$

with m and n coprime integers such that one is even and $m>n>0$ (this is Euclid's formula). The primitive Pythagorean triples with $b$ odd are obtained as

$a={\frac {m^{2}-n^{2}}{2}},\quad b=mn,\quad c={\frac {m^{2}+n^{2}}{2}},$

with m and n coprime odd integers such that $m>n>0.$

As the exchange of a and b transforms a Pythagorean triple into another Pythagorean triple, only one of the two cases is sufficient for getting all primitive Pythagorean triples.

Relationship with science

Carl Friedrich Gauss, known as the prince of mathematicians

There is still a philosophical debate whether mathematics is a science. However, in practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences. Like them, it is falsifiable, which means in mathematics that, if a result or a theory is wrong, this can be proved by providing a counterexample. Similarly as in science, theories and results (theorems) are often obtained from experimentation. In mathematics, the experimentation may consist of computation on selected examples or of the study of figures or other representations of mathematical objects (often mind representations without physical support). For example, when asked how he came about his theorems, Gauss (one of the greatest mathematicians of the 19th century) once replied "durch planmässiges Tattonieren" (through systematic experimentation). However, some authors emphasize that mathematics differs from the modern notion of science by not relying on empirical evidence.

What precedes is only one aspect of the relationship between mathematics and other sciences. Other aspects are considered in the next subsections.

Pure and applied mathematics

Main articles: Applied mathematics and Pure mathematics

Isaac Newton (left) and Gottfried Wilhelm Leibniz developed infinitesimal calculus.

Until the end of the 19th century, the development of mathematics was mainly motivated by the needs of technology and science, and there was no clear distinction between pure and applied mathematics. For example, the natural numbers and arithmetic were introduced for the need of counting, and geometry was motivated by surveying, architecture and astronomy. Later, Isaac Newton introduced infinitesimal calculus for explaining the movement of the planets with his law of gravitation. Moreover, most mathematicians were also scientists, and many scientists were also mathematicians. However, a notable exception occurred in Ancient Greece; see Pure mathematics § Ancient Greece.

In the second half ot the 19th century, new mathematical theories were introduced which were not related with the physical world (at least at that time), in particular, non-Euclidean geometries and Cantor's theory of transfinite numbers. This was one of the starting points of the foundational crisis of mathematics, which was eventually solved by the systematization of the axiomatic method for defining mathematical structures.

So, many mathematicians focused their research on internal problems, that is pure mathematics, and this led to split mathematics into pure mathematics and applied mathematics, the latter being often considered as having a lower value.

During the second half of the 20th century, it appeared that many theories issued from applications are also interesting from the point of view of pure mathematics, and that many results of pure mathematics have applications outside mathematics (see next section), and that, in turn, the study of these applications may give new insights on the "pure theory". An example of the first case is the theory of distributions, introduced by Laurent Schwartz for validating computations done in quantum mechanics, which became immediately an important tool of (pure) mathematical analysis. An example of the second case is the decidability of the first-order theory of the real numbers, a problem of pure mathematics that was proved true by Alfred Tarski, with an algorithm that is definitely impossible to implement, because of a computational complexity that was much too high. For getting an algorithm that can be implemented and can solve systems of polynomial equations and inequalities, George Collins introduced the cylindrical algebraic decomposition that became a fundamental tool in real algebraic geometry.

So, the distinction between pure and applied mathematics is presently more a question of personal research aim of mathematicians than a division of mathematics into broad areas. The Mathematics Subject Classification does not mention "pure mathematics" nor "applied mathematics". However, these terms are still used in names of some university departments, such as at the Faculty of Mathematics at Cambridge University.

Unreasonable effectiveness

The unreasonable effectiveness of mathematics is a phenomenon that has been named and first made explicit by physicist Eugene Wigner. It is the fact that many mathematical theories, even the "purest" have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced.

A famous example is the prime factorization of natural numbers that has been discovered more than 2,000 years before its common use for secure internet communications through the RSA cryptosystem.

Another historical example is the theory of ellipses. They were studied by the ancient Greek mathematicians as conic sections (that is, intersections of cones with planes). It is almost 2,000 years later that Johannes Kepler discovered that the trajectories of the planets are ellipses.

In the 19th century, the internal development of geometry (pure mathematics) lead to define and study non-Euclidean geometries, spaces of dimension higher than three and manifolds. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, Albert Einstein developed the theory of relativity that uses fundamentally these concepts. In particular, spacetime of the special relativity is a non-Euclidean space of dimension four, and spacetime of the general relativity is a (curved) manifold of dimension four.

Similar examples of unexpected applications of mathematical theories can be found in many areas of mathematics.'

The connection between abstract mathematics and material reality has led to philosophical debates since at least the time of Pythagoras. The ancient philosopher Plato argued this was possible because material reality reflects abstract objects that exist outside time. As a result, the view that mathematical objects somehow exist on their own in abstraction is often referred to as Platonism. Although most mathematicians don't typically concern themselves with philosophical questions, they may be generally considered as Platonists, since think and talk of their objects of study as real objects. Nevertheless, Platonism and the concurrent views on abstraction do not explain the unreasonable effectiveness of mathematics.