Elegant mathematics

Summary: A mathematical accomplishment may be considered elegant because of its conceptual depth, its aesthetic appeal, its importance and implications, its rigorousness, or the surprise of its results.

Elegant mathematics is an elusive idea, often being an aesthetical judgment determined subjectively as a reflection of one’s knowledge and understanding of mathematics. That is, an intricate proof in number theory or analysis may be deemed “elegant” by mathematicians, but it would be mere nonsense to a struggling high school student. In turn, the visual “completing of the square” as a proof of the quadratic formula may be deemed elegant by high school students, but it would be too simplistic and inefficient to mathematicians. Thus, it is necessary to dig deeper into the meaning of “elegant mathematics,” trying to focus on the many forms of mathematics—its methods, its visual aspects, and its role as a language.

The word “elegance” often is defined as an attribute that is effective and simple. Elegant mathematics can then be defined as mathematics that is effective and simple. However, this definition can be deceptive because one of the primary roles of mathematics is as a language, capitalizing on its ability to effectively record and model ideas and situations using a symbolic notation that is both effective and simple. Thus, since mathematics is broader than a mere language, the view of aspects of mathematics as elegant should include other attributes, such as surprise, nontriviality, consistency, power, conceptual depth, and even beauty.

When discussing elegant mathematics, mathematicians usually refer to proofs as prime examples, shifting the focus from the correctness of the proof’s logical structure to its effectiveness and simplicity. Specific elements that suggest elegance are the following:

• Uses a minimal number of necessary assumptions
• Is unusually succinct yet understandable
• Avoids complex or laborious calculations
• Offers a surprising path from assumptions to conclusion
• Models “out-of-the-box” thinking
• Achieves a difficult result with a minimum of work
• Includes original conceptual insights that clarify both the “how” and the “why”
• Can be generalized to a broader context or set of problems
• Displays the power of mathematics as both a method and a language

An example of an elegant mathematical proof is Euclid’s proof that an infinite number of primes exist. Using the process of “reductio ad absurdum,” assume that the number of primes is finite, which may be written as p₁, p₂ , p₃,… , p_n. Let

N = p₁p₂p₃… p_n-1p_{n + 1}

which cannot be prime because N>p_n. Thus, N must be composite and have a prime factor. However, all of the known primes p₁, p₂ , p₃,… , p_n are not factors of N because on division they leave a remainder of 1. Thus, there must exist another prime q>p_i for all i, such that q is a factor of N. But, this is a contradiction of the original assumption, and the number of primes is infinite. Euclid’s proof is mathematically elegant because it is effective, simple, powerful, and surprising. Many other mathematical proofs are regarded as elegant, such as these examples:

• Archimedes’s use of mechanical concepts to prove that the volume of a sphere is two-thirds the volume of its circumscribing cylinder
• The Chinese “Behold!” proof of the Pythagorean Theorem
• Fourier’s use of series to prove that the number e is irrational
• Euler’s proofs involving infinite series
• Cantor’s diagonal proof of the countability of the rationals, as well as his related proof that the reals are not countable

Paul Erdös, a Hungarian mathematician, often referred to an imaginary book in which God had included all the most beautiful or elegant proofs in mathematics. Then, when he came across a proof that he felt was elegant, Erdös would suggest, “This one’s from The Book!” In the 1990s, Martin Aigner and Günter Ziegler capitalized on Erdös’s ideas and published Proofs From THE BOOK. The most recent edition (2009) includes 30 sections involving elegant proofs from number theory, geometry, combinatorics, analysis, and graph theory.

Inelegant Proofs

A mathematics proof that is not elegant is viewed as ugly, laborious, awkward, or pedantic. Inelegant mathematical proofs often involve computer-based computations that cannot be easily replicated by mathematicians within a reasonable time frame. These inelegant yet effective proofs are akin to proofs by exhaustion involving a great number of cases, thereby disguising any elements of brevity or simplicity. A primary example of such a proof is Kenneth Appel and Wolfgang Haken’s proof of the Four Color Theorem in 1976. Despite their use of some clever categorizing techniques, the final steps in the proof required more than 1000 hours of computer time to check 1,936 maps of reducible configurations as possible counterexamples. In fact, some mathematicians do not accept the proof because of its reliance on computers. Yet, the Four Color Theorem as a conceptual statement is itself considered to be elegant.

Elegant Versus Ugly

Famous mathematicians such as Bertrand Russell, G. H. Hardy, Richard Feynman, and Paul Erdös also have shared their opinions relative to the distinctions between elegant and inelegant proofs (or mathematics, in general), often taking strong stands. For example, in a letter to Max Wertheimer, Albert Einstein even discussed the distinctions between elegant and ugly proofs. For him, a proof was ugly if it depended on the artificial introduction of additional elements, such as constructing auxiliary lines, which distracted the reader from the flow and “symmetry” of a proof. In his letter, Einstein provides both elegant and ugly examples of proofs of Menelaus’s Theorem on Colinearity.

Authors have jumped on this “elegant versus ugly” bandwagon, extending it by their evaluations of both the proof and the conceptual claims associated with a mathematical theorem. The result is published resources such as The Most Beautiful Mathematical Formulas (1992) and An Introduction to the World’s Most Elegant Mathematics (2006).

Unfortunately, the sorting process is not as straight-forward as these authors suggest. Often, mathematicians vacillate, being unsure in the classification of a proof as either elegant or inelegant. A current example of this indecision is Andrew Wiles’s proof of Fermat’s Last Theorem, which conjectures that no three whole numbers a, b, and c can satisfy the equation aⁿ+bⁿ=cⁿ for any integral value of n>2. On one level, Wiles’s approach was ingenious (and thereby elegant) in his use of elliptic curve theory and modular forms to solve this famous extension of the Pythagorean Theorem. And on the other hand, Wiles’s final proof is inelegant because it involves more than 100 pages of very difficult mathematics that deters both mathematicians and non-mathematicians. The same can be said for the proof of the Monster Group.

Elegance

Moving the focus beyond proofs alone, mathematicians tend to classify mathematical ideas, such as theorems and concepts, as “elegant” if they establish insightful connections between two areas of mathematics that were assumed to be unrelated. The most famous example perhaps is Leonard Euler’s identity that relates special mathematical constants: e^iπ+1=0.

Framed copies of this fascinating identity often will be found hanging on the walls of mathematicians’ offices. It exudes simplicity and explains unexpected connections of several different mathematical ideas.

The symbolic simplicity of the above identity also illustrates the elegance of mathematics as a language. In fact, combinations of mathematical symbols with words can convey mathematical ideas that are simultaneously complex and powerful. Combined further with mathematical graphics, the elegance of mathematics as a language is enhanced by the ability to convey complex ideas efficiently, consistently, and with economy.

And, partially because of its elegance in form, mathematics also is the preferred language of the sciences and many other disciplines that involve quantitative models. Awareness of this elegance led physicist Eugene Wigner to write his famous essay, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” in 1960. He concludes his paper with the statement that “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”

It is expected that the idea of elegant mathematics will remain an elusive one, because it is a subjective judgment of the aesthetics of proof and ideas within mathematics. Though the quandary will lead to arguments, it should not have any impact on the continuing development of mathematics. That is, because of the nature of both elegance and mathematics, it is not possible to merge them as a thinking strategy. Rather, as history has demonstrated, the mathematics is first developed and proven and only then can the aesthetic judgments (elegant versus ugly) begin. And one cannot ignore the quality of the considerable mathematics that lies between these two extremes.

Bibliography

Aigner, Martin, and Günter Ziegler. Proofs From THE BOOK. 4th ed. New York: Springer, 2009.

Clements, Ken, and Nerida Ellerton. “Historical Perspectives on Mathematical Elegance.” Conference Proceedings, Mathematics Education Group of Australia, Adelaide, Australia: MERGA, 2006.

Hardy, G. H. A Mathematician’s Apology. Cambridge, England: Cambridge University Press, 1941.

Salem, Lionel, Frédéric Testard, and Coralie Salem. The Most Beautiful Mathematical Formulas. Hoboken, NJ: Wiley, 1992.

Wigner, Eugene. “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Communications in Pure and Applied Mathematics 13, no. 1 (February 1960).