the phenomenology of mathematical beauty

gian-carlo rota

beauty is applied first to theorems, then to proofs

then to “a brilliant step” in an otherwise undistinguished proof

“An example of mathematical beauty upon which all mathematiciansagree is Picard’s theorem, asserting that an entire function of a complex variable takes all values with at most one exception. The limpid statement of this theorem is matched by the beauty of the five-line proof providedby Picard”

he gives a lot of examples of theorems/proofs being beautiful

“This agreement is notmerely the perception of an aesthetic quality superimposed on the contentof a piece of mathematics. A piece of mathematics that is agreed to be beau-tiful is more likely to be included in school curricula; the discoverer of abeautiful theorem is rewarded by promotions and awards; a beautiful argu-ment will be imitated. In other words, the beauty of a piece of mathematicsdoes not consist merely of the subjective feelings experienced by anobserver. The beauty of a theorem is an objective property on a par withits truth. The truth of a theorem does not differ from its beauty by a greaterdegree of objectivity.”

mathematics are very aware that beauty is context-dependent

“ne can characterize a mathematical paper as “cre-ative” only after the paper has been understood. It is, however, impossibleto produce on commission a “creatively” written mathematical paper”

“AAppreciation of mathematical beauty requiresfamiliarity with a mathematical theory, which is arrived at at the cost of time, effort, exercise, and Sitzfleischrather than by training in beautyappreciation.”

“There is a difference between mathematical beauty and mathematicalelegance. Although one cannot strive for mathematical beauty, one canachieve elegance in the presentation of mathematics. In preparing to delivera mathematics lecture, mathematicians often choose to stress elegance andsucceed in recasting the material in a fashion that everyone will agree iselegant. Mathematical elegance has to do with the presentation of mathe-matics, and only tangentially does it relate to its content. A beautifulproof can be presented elegantly or not”

lack of beauty is related to lack of definitiveness

Mathematicians seldom use the word “ugly.” In its place are such dis-paraging terms as “clumsy,” “awkward,” “obscure,” “redundant,” and, inthe case of proofs, “technical,” “auxiliary,” and “pointless.” But the mostfrequent expression of condemnation is the rhetorical question, “What isthis good for?”

“The beauty of a piece of mathematics is frequently associated with short-ness of statement or of proof”

“A proof is viewedas beautiful only after one is made aware of previous, clumsier proofs.”

“But when mathematicaltheorems from disparate areas are strung together and presented as “pearls,”they are likely to be appreciated only by those who are already familiarwith them.” -> programming pearls

“The mathematician who is baffled and asks “What is thisgood for?” is missing the senseof the statement that has been verified to betrue. Verification alone does not give us a clue as to the role of a statementwithin the theory; it does not explain the relevanceof the statement. Inshort, the logical truth of a statement does not enlighten us as to the senseof the statement. Enlightenment, not truth, is what the mathematician seekswhen asking, “What is this good for?” Enlightenment is a feature of math-ematics about which very little has been written.”

ENLIGHTENMENT -> understanding