Mathematical beauty

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An example of "beauty in method"—a simple and elegant proof of the Pythagorean theorem.

Many mathematicians derive aesthetic pleasure from their work, and from mathematics in general. They express this pleasure by describing mathematics (or, at least, some aspect of mathematics) as beautiful. Sometimes mathematicians describe mathematics as an art form or, at a minimum, as a creative activity. Comparisons are often made with music and poetry.

Bertrand Russell expressed his sense of mathematical beauty in these words:

Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.^[1]

Paul Erdős expressed his views on the ineffability of mathematics when he said, "Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is."

1 Beauty in method
2 Beauty in results
3 Beauty in experience
4 Beauty and philosophy
5 References
6 See also
7 External links

[edit] Beauty in method

Mathematicians describe an especially pleasing method of proof as elegant. Depending on context, this may mean:

A proof that uses a minimum of additional assumptions or previous results.
A proof that is unusually succinct.
A proof that derives a result in a surprising way (e.g., from an apparently unrelated theorem or collection of theorems.)
A proof that is based on new and original insights.
A method of proof that can be easily generalized to solve a family of similar problems.

In the search for an elegant proof, mathematicians often look for different independent ways to prove a result—the first proof that is found may not be the best. The theorem for which the greatest number of different proofs have been discovered is possibly the Pythagorean theorem, with hundreds of proofs having been published.^[2] Another theorem that has been proved in many different ways is the theorem of quadratic reciprocity—Carl Friedrich Gauss alone published eight different proofs of this theorem.

Conversely, results that are logically correct but involve laborious calculations, over-elaborate methods, very conventional approaches, or that rely on a large number of particularly powerful axioms or previous results are not usually considered to be elegant, and may be called ugly or clumsy.

[edit] Beauty in results

Starting at e0 = 1, travelling at the velocity i relative to one's position for the length of time π, and adding 1, one arrives at 0. (The diagram is an Argand diagram)

Starting at e⁰ = 1, travelling at the velocity i relative to one's position for the length of time π, and adding 1, one arrives at 0. (The diagram is an Argand diagram)

Some mathematicians (Rota 1977, p. 173) see beauty in mathematical results which establish connections between two areas of mathematics that at first sight appear to be totally unrelated. These results are often described as deep.

While it is difficult to find universal agreement on whether a result is deep, some examples are often cited. One is Euler's identity

$\displaystyle e^{i \pi} + 1 = 0.$

Richard Feynman called this "the most remarkable formula in mathematics". Modern examples include the modularity theorem, which establishes an important connection between elliptic curves and modular forms (work on which led to the awarding of the Wolf Prize to Andrew Wiles and Robert Langlands), and "monstrous moonshine," which connects the Monster group to modular functions via a string theory for which Richard Borcherds was awarded the Fields medal.

The opposite of deep is trivial. A trivial theorem may be a result that can be derived in an obvious and straightforward way from other known results, or which applies only to a specific set of particular objects such as the empty set. Sometimes, however, a statement of a theorem can be original enough to be considered deep, even though its proof is fairly obvious.

In his A Mathematician's Apology, Hardy suggests that mathematical beauty arises from an element of surprise. Rota, however, disagrees and proposes a counterexample:

"A great many theorems of mathematics, when ﬁrst published, appear to be surprising; thus for example some twenty years ago [from 1977] the proof of the existence of non-equivalent differentiable structures on spheres of high dimension was thought to be surprising, but it did not occur to anyone to call such a fact beautiful, then or now." (Rota 1977, p. 172)

Perhaps ironically, Monastyrsky (2001) writes:

"It is very diﬃcult to ﬁnd an analogous invention in the past to [Milnor's] beautiful construction of the different differential structures on the seven-dimensional sphere....The original proof of Milnor was not very constructive but later E. Briscorn showed that these differential structures can be described in an extremely explicit and beautiful form."

This disagreement illustrates both the subjective nature of mathematical beauty and its connection with mathematical results: in this case, not only the existence of exotic spheres, but also a particular realization of them.

[edit] Beauty in experience

Some degree of delight in the manipulation of numbers and symbols is probably required to engage in any mathematics. Given the utility of mathematics in science and engineering, it is likely that any technological society will actively cultivate these aesthetics, certainly in its philosophy of science if nowhere else.

The most intense experience of mathematical beauty for most mathematicians comes from actively engaging in mathematics. It is very difficult to enjoy or appreciate mathematics in a purely passive way—in mathematics there is no real analogy of the role of the spectator, audience, or viewer.^[3] Bertrand Russell referred to the austere beauty of mathematics.

[edit] Beauty and philosophy

Some mathematicians are of the opinion that the doing of mathematics is closer to discovery than invention. These mathematicians believe that the detailed and precise results of mathematics may be reasonably taken to be true without any dependence on the universe in which we live. For example, they would argue that the theory of the natural numbers is fundamentally valid, in a way that does not require any specific context. Some mathematicians have extrapolated this viewpoint that mathematical beauty is truth further, in some cases becoming mysticism.

Pythagoras (and his entire philosophical school of the Pythagoreans) believed in the literal reality of numbers. The discovery of the existence of irrational numbers was a shock to them—they considered the existence of numbers not expressible as the ratio of two natural numbers to be a flaw in nature. From the modern perspective, Pythagoras' mystical treatment of numbers was that of a numerologist rather than a mathematician. It turns out that what Pythagoras had missed in his insufficiently sophisticated world view was the limits of infinite sequences of ratios of natural numbers—the modern notion of a real number.

In Plato's philosophy there were two worlds, the physical one in which we live and another abstract world which contained unchanging truth, including mathematics. He believed that the physical world was a mere reflection of the more perfect abstract world.

Galileo Galilei is reported to have said, "Mathematics is the language with which God wrote the universe," a statement which (apart from the implicit theism) is consistent with the mathematical basis of all modern physics.

Hungarian mathematician Paul Erdős, although an atheist, spoke of an imaginary 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 exclaim "This one's from The Book!" This viewpoint expresses the idea that mathematics, as the intrinsically true foundation on which the laws of our universe are built, is a natural candidate for what has been personified as God by different religious mystics.

Twentieth-century French philosopher Alain Badiou claims that ontology is mathematics. Badiou also believes in deep connections between math, poetry and philosophy.

In some cases, natural philosophers and other scientists who have made extensive use of mathematics have made leaps of inference between beauty and physical truth in ways that turned out to be erroneous. For example, at one stage in his life, Johannes Kepler believed that the proportions of 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. As there are exactly five Platonic solids, Kepler's theory could only accommodate six planetary orbits and was disproved by the subsequent discovery of Uranus.

[edit] References

^ (The Study of Mathematics, in Mysticism and Logic, and Other Essays, ch. 4, London: Longmans, Green, 1918.)
^ Elisha Scott Loomis published over 360 proofs in his book Pythagorean Proposition (ISBN 0873530365).
^ "...there is nothing in the world of mathematics that corresponds to an audience in a concert hall, where the passive listen to the active. Happily, mathematicians are all doers, not spectators." Phillips, George M. (2005). Mathematics Is Not A Spectator Sport. Springer. ISBN 0387255281.

Aigner, Martin, and Ziegler, Gunter M. (2003), Proofs from THE BOOK, 3rd edition, Springer-Verlag.

Chandrasekhar, Subrahmanyan (1987), Truth and Beauty: Aesthetics and Motivations in Science, University of Chicago Press, Chicago, IL.

Hadamard, Jacques (1949), The Psychology of Invention in the Mathematical Field, 1st edition, Princeton University Press, Princeton, NJ. 2nd edition, 1949. Reprinted, Dover Publications, New York, NY, 1954.

Hardy, G.H. (1940), A Mathematician's Apology, 1st published, 1940. Reprinted, C.P. Snow (foreword), 1967. Reprinted, Cambridge University Press, Cambridge, UK, 1992.

Hoffman, Paul (1992), The Man Who Loved Only Numbers, Hyperion.

Huntley, H.E. (1970), The Divine Proportion: A Study in Mathematical Beauty, Dover Publications, New York, NY.

Loomis, Elisha Scott (1968), The Pythagorean Proposition, The National Council of Teachers of Mathematics. Contains 365 proofs of the Pythagorean Theorem.