Mathematical beauty
Mathematical beauty is a type of aesthetic value that is experienced in doing or contemplating mathematics. The testimonies of mathematicians indicate that various aspects of mathematics—including results, formulae, proofs and theories—can trigger subjective responses similar to the beauty of art, music, or nature. The pleasure in this experience can serve as a motivation for doing mathematics, and some mathematicians, such as G.H. Hardy, have characterized mathematics as an art form that seeks beauty.
Beauty in mathematics has been subject to examination by mathematicians themselves and by philosophers, psychologists, and neuroscientists. Understanding beauty in general can be difficult because it is a subjective response to sense-experience but is perceived as a property of an external object, and because it may be shaped by cultural influence or personal experience. Mathematical beauty presents additional problems, since the aesthetic response is evoked by abstract ideas which can be communicated symbolically, and which may only be available to a minority of people with mathematical ability and training. The appreciation of mathematics may also be less passive than listening to music. Furthermore, beauty in mathematics may be connected to other aesthetic or non-aesthetic values. Some authors identify mathematical elegance with mathematical beauty; others distinguish elegance as a separate aesthetic value, or as being, for instance, limited to the form mathematical exposition. Beauty itself is often linked to, or thought to be dependent on, the abstractness, purity, simplicity, depth or order of mathematics.
Examples of beautiful mathematics
Results
is often given as an example of a beautiful result:This expression ties together arguably the five most important mathematical constants with the two most common mathematical symbols. Euler's identity is a special case of Euler's formula, which the physicist Richard Feynman called "our jewel" and "the most remarkable formula in mathematics".
Another example is Fermat's theorem on sums of two squares, which says that any prime number such that can be written as a sum of two square numbers, which both G.H. Hardy and E.T. Bell thought was a beautiful result.
In a survey in which mathematicians were asked to evaluate 24 theorems for their beauty, the top-rated three theorems were: Euler's equation; Euler's polyhedron formula, which asserts that for a polyhedron with V vertices, E edges, and F faces, ; and
Euclid's theorem that there are infinitely many prime numbers, which was also given by Hardy as an example of a beautiful theorem.
Proofs
, which establishes that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers, has been cited by both mathematicians and philosophers as an example of a beautiful proof.Image:Proofwithoutwords.svg|left|thumb|A proof without words for the sum of odd numbers theorem
Visual proofs, such as the illustrated proof of the Pythagorean theorem, and other proofs without words generally, such as the shown proof that the sum of all positive odd numbers up to 2n − 1 is a perfect square, have been thought beautiful.
The mathematician Paul Erdős spoke of The Book, an imaginary infinite book in which God has written down all the most beautiful mathematical proofs. When Erdős wanted to express particular appreciation of a proof, he would proclaim it "straight from The Book!". His rhetorical device inspired the creation of Proofs from THE BOOK, a collection of such proofs, including many suggested by Erdős himself.
Objects
In Plato's Timaeus, the five regular convex polyhedra, called the Platonic solids for their role in this dialogue, are called the "most beautiful" bodies. In the Timaeus, they are described as having been used by the demiurge, or creator-craftsman who builds the cosmos, for the four classical elements plus the heavens, because of their beauty.In his 1596 book Mysterium Cosmographicum, Johannes Kepler argued that the orbits of the then-known planets in the Solar System have been arranged by God to correspond to a concentric arrangement of the five Platonic solids, each orbit lying on the circumsphere of one polyhedron and the insphere of another. For Kepler, God had wanted to shape the universe according to the five regular solids because of their beauty, and this explained why there were six planets.
A more modern example is the exceptional simple Lie group, which has been called "perhaps the most beautiful structure in all of mathematics".
Scientific theories
The mathematical statements of scientific theories, especially in physics, are sometimes considered to be mathematically beautiful. For example, Roger Penrose thought there was a "special beauty" in Maxwell's equations of electromagnetism:Einstein's theory of general relativity has been characterized as a work of art, and, among other aesthetic praise, was described by Paul Dirac as having "great mathematical beauty" and by Penrose as having "supreme mathematical beauty".
Properties of beautiful mathematics
Many mathematicians and philosophers who have written about mathematical beauty have tried to identify properties or criteria that are conducive to the perception of beauty in a piece of mathematics. It is debated whether beauty can be clarified or explained by such properties: Paul Erdős thought that it was no more possible to convince someone of the beauty of a piece of mathematics than to convince them of the beauty of Beethoven's Ninth Symphony, if they couldn't see it for themselves.Results
In his 1940 essay A Mathematician's Apology, G. H. Hardy said that a beautiful result, including its proof, possesses three "purely aesthetic qualities", namely "inevitability", "unexpectedness", and "economy". He particularly excluded enumeration of cases as "one of the duller formsof mathematical argument".
In 1997, Gian-Carlo Rota disagreed with unexpectedness as a sufficient condition for beauty and proposed a counterexample:
In contrast, Monastyrsky wrote in 2001:
This disagreement illustrates both the subjective nature of mathematical beauty, like other forms of beauty in general, and its connection with mathematical results: in this case, not only the existence of exotic spheres, but also a particular realization of them.
Proofs
Besides Hardy's properties of "unexpectedness", "inevitability", "economy", which he applied to proofs as well as results, mathematicians have customarily thought beautiful proofs that are short and simple.In the search for an elegant proof, mathematicians often look for different independent ways to prove a result—as the first proof that is found can often be improved. The theorem for which the greatest number of different proofs have been discovered is possibly the Pythagorean theorem, with hundreds of proofs having being published. Another theorem that has been proved in many different ways is the theorem of quadratic reciprocity. In fact, Carl Friedrich Gauss alone had eight different proofs of this theorem, six of which he published.
In contrast, results that are logically correct but involve laborious calculations or consideration of many cases, are not usually considered beautiful, and may be even referred to as ugly or clumsy. For example, Kenneth Appel and Wolfgang Haken's proof of the four color theorem made use of computer checking of over a thousand cases. Philip J. Davis and Reuben Hersh said that when they first heard that about the proof, they hoped it contained a new insight "whose beauty would transform my day", and were disheartened when informed the proof was by case enumeration and computer verification. Paul Erdős said it was "not beautiful" because it gave no insight into why the theorem was true.
Philosophical analysis
thought that beauty was found especially in mathematics, writing in the Metaphysics thatThe logician and philosopher Bertrand Russell made a now-famous statement characterizing mathematical beauty in terms of purity and austerity:
In the twentieth century, some philosophers questioned whether there was genuinely beauty in mathematics. The philosopher of science Rom Harré argued that there were no true aesthetic appraisals of mathematics, but only quasi-aesthetic appraisals. Any mathematical success described by an aesthetic term was a second-order success besides understanding and correctness. In contrast, aesthetic appraisal of a work of art was first-order. Harré considered this to be the difference between a quasi-aesthetic and a genuinely aesthetic appraisal.
Nick Zangwill thought that there were no true aesthetic experiences of mathematics and that a proofs or theories could only be metaphorically beautiful. His argument had two bases. First, he thought that aesthetic properties depended on sensory properties, and so abstract entities could not have aesthetic properties. Second, he thought that proofs, theorems, theories, and so on had purposes such as demonstrating correctness or granting understanding, and that any praise of them reflected only how well they achieved their purpose.
Scientific analysis
Information-theory model
In the 1970s, Abraham Moles and Frieder Nake analyzed links between beauty, information processing, and information theory. In the 1990s, Jürgen Schmidhuber formulated a mathematical theory of observer-dependent subjective beauty based on algorithmic information theory: the most beautiful objects among subjectively comparable objects have short algorithmic descriptions relative to what the observer already knows. Schmidhuber explicitly distinguishes between beautiful and interesting. The latter corresponds to the first derivative of subjectively perceived beauty:the observer continually tries to improve the predictability and compressibility of the observations by discovering regularities such as repetitions and symmetries and fractal self-similarity. Whenever the observer's learning process leads to improved data compression such that the observation sequence can be described by fewer bits than before, the temporary interesting-ness of the data corresponds to the compression progress, and is proportional to the observer's internal curiosity reward.