Some Mathématicien S seek in their work or mathematics in general, an esthetic pleasure. They express this pleasure by describing “beautiful” parts of mathematics.
They go even sometimes until regarding mathematics a Art or as a creative activity. Comparisons are often made with the Musique and the Poésie.
Bertrand Russell gave his direction of the mathematical beauty in these terms: “The mathematics, considered with their right measurement, has not only the truth, but the supreme beauty, a beauty cold and austere, like that of a sculpture, without reference to part of our fragile nature, without the splendid effects of illusion of painting or the music, however pure and sublimates, capable of a severe perfection such as only greatest arts can show it. The true spirit of the pleasure, the exaltation, the impression to be more than one man, who is the stone of key of the most raised excellence, must as surely be found in mathematics as poetry. ”.
Paul Erdős evoked the unutterable character of the beauty of mathematics by declaring “why the numbers are beautiful? That amounts wondering why the ninth Symphonie of Beethoven is beautiful. If you do not see why, nobody will be able to explain it to you. I know that the numbers are beautiful. If they are not beautiful, nothing is it”.
The beauty in the formulas
A formula is regarded as “beautiful” if it produces a significant result and produces a result surprising by its simplicity compared to complexity thus connect (in particular an equality of which one of the members is very simple whereas the other member is very complicated.Certain mathematicians consider that most beautiful of the formulas of Léonard Euler is that, whose Euler itself said that it showed the presence of the hand of God.
In the novel Enigma , the fictitious mathematician Tom Jericho describes as “crystalline” the beauty of the formula (formula of Leibniz).
The beauty in the methods
The mathematicians can qualify a method in a demonstration of “elegant” when:- It uses little of results preliminary,
- It is exceptionally short,
- It establishes result of way surprising (for example starting from Théorème S which is apparently not in connection with this one),
- It is based on new original concepts,
- It calls upon a method which can be generalized to solve a family of similar problems easily.
In the search of an elegant demonstration, the mathematicians seek often various independent manners to establish a theorem; the first found demonstration can not be the best. The theorem for which the greatest number of different demonstrations was found is probably the Théorème of Pythagore since hundreds of evidence were published. Another theorem which was shown of many ways is the quadratic theorem of reciprocity of Karl Friedrich Gauss from which at least eight demonstrations different were published.
Conversely, of the logically correct methods but which imply hard calculations, too moderate methods, very conventional approaches, or which are based on a great number of particularly powerful axioms or on preliminary results themselves usually considered as not very elegant, can be described of ugly or awkward . This is related to the principle of the Rasoir of Occam.
The beauty in the theorems
The mathematicians see the beauty in the mathematical theorems which make it possible to establish the link between two fields of mathematics which seems completely independent at first sight. These results are often regarded as “deep”.Whereas it is often difficult to find an agreement universal to decide if a theorem must be regarded as deep, certain examples are often quoted in the scientific literature. It is the case of the Identité of Euler which was called “the most remarkable formula in mathematics” by Richard Feynman. The modern examples include the Théorème of Taniyama-Shimura which establishes an important bond between the elliptic curved and the modular forms (work for which its authors Andrew Wiles and Robert Langlands accepted the Prix Wolf), and the “Conjecture Monstrous Moonshine” which establishes a bond between the Groupe Monster and the modular functions via the Théorie of the cords for which Richard Borcherds was seen decreeing the Médaille Fields.
The opposite of “deep” is “commonplace”. A commonplace theorem can be a proposal which can be deduced in a way obvious and immediate of other known theorems, or which applies only to one specific whole of particular objects such as the empty set. However, it happens that a theorem is sufficiently original to be regarded as deep, although its demonstration is rather obvious.
The beauty in the experiment
A certain pleasure in the handling of the numbers and the symbols is probably necessary to begin in mathematics. Being given the utility of mathematics in the sciences and the Technologie, it is probable that any technological company cultivates actively its needs for Esthétique.The most intense experiment of the mathematical beauty for the majority of the mathematicians comes from their active engagement in mathematics. It is very difficult to like or appreciate mathematics in a purely passive way. In mathematics, there is no really analogy between the role of spectator, assistant or televiewer.
Bertrand Russell evoked the austere beauty mathematics.
The beauty and philosophy
Certain mathematicians agree to say that to make mathematics is closer to the discovery than of the invention. These mathematicians believe that the detailed and precise theorems of mathematics can be reasonably regarded as truths independently of the universe in which we live. For example, some claim that the theory of the natural numbers Entiers is basically valid, in a manner which does not require any specific context. Mathematicians extrapolated this point of view by regarding the mathematical beauty as a truth, approaching in certain cases the Mysticisme. Pythagore (and its whole philosophical school) believed in the literal reality of the numbers. The discovery of the existence of irrational numbers caused a great distress within the school; they regarded the existence of these natural numbers not exprimables as ratio of two whole, like a dust in the universe. From the modern point of view, the mystical consideration of Pythagore of the numbers would be that of a Numérologiste rather than that of a mathematician.In the philosophy of Plato there are two worlds, the physical world in which we live and an abstract world different which contains the invariable truth, including that from mathematics. He thought that the physical world was a simple reflection of the world abstracts more perfect.
It is reported that Galileo claimed that “mathematics is the language with which God wrote the universe”.
The mathematician Hungarian Paul Erdős, although atheist, spoke about an imaginary book, in which God foot-note all most beautiful mathematical demonstrations. When Erdős wanted to express its particular satisfaction of a demonstration, he would have exclaimed “this one comes from the book! ”. This point of view expresses the idea that mathematics, being the intrinsically true base on which the laws of our universe are established, is a natural candidate to be personified as a God, according to various mystical religions.
The French philosopher Alain Badiou of the twentieth century affirms that the ontology is mathematics. Badiou also believes in the major bonds between mathematics, poetry and philosophy.
In certain cases, the philosophers and the scientists who used much mathematics established bonds between the beauty and the physical truth in manners which proved to be false. For example, a stage in its life, Johannes Kepler accepted that the proportions of the orbits of planets known hitherto in the Solar system had been arranged by God to make them correspond to a concentric arrangement of the five solid Platonic, each orbit being on the circumference of a polyhedron and the insphere of the others. As there are five Platonic solids exactly, the theory of Kepler could only apply only to six planetary orbits, and was refuted later on by the discovery of Uranus. James Watson made a similar error when he postulated that each of the four bases of DNA is connected to a base of the same type being on the other hand (Thymine connected to the thymine, etc) while being based on the belief that “what is beautiful must be true”.
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