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Mathematical beauty

Mathematical beauty is the aesthetic pleasure derived from the abstractness, purity, simplicity, depth or orderliness of mathematics. Mathematicians may express this pleasure by describing mathematics (or, at least, some aspect of mathematics) as beautiful or describe mathematics as an art form, (a position taken by G. H. Hardy[1]) or, at a minimum, as a creative activity.

Comparisons are made with music and poetry.

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 or a collection of theorems).

theorem

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.

Mathematicians commonly describe an especially pleasing method of proof as elegant.[2] Depending on context, this may mean:


In the search for an elegant proof, mathematicians may search for multiple 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 being published up to date.[3] 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.[4]


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

In information theory[edit]

In the 1970s, Abraham Moles and Frieder Nake analyzed links between beauty, information processing, and information theory.[23][24] 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 (i.e., Kolmogorov complexity) relative to what the observer already knows.[25][26][27] 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 (possibly a predictive artificial neural network) 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.[28][29]

Cellucci, Carlo (2015), "Mathematical beauty, understanding, and discovery", , 20 (4): 339–355, doi:10.1007/s10699-014-9378-7, S2CID 120068870

Foundations of Science

Martin Gardner (April 1, 2007). . Scientific American.

"Is Beauty Truth and Truth Beauty?"

Stewart, Ian (2007). . New York: Basic Books, a member of the Perseus Books Group. ISBN 978-0-465-08236-0. OCLC 76481488.

Why beauty is truth : a history of symmetry

; Romaya, J. P.; Benincasa, D. M. T.; Atiyah, M. F. (2014), "The experience of mathematical beauty and its neural correlates", Frontiers in Human Neuroscience, 8: 68, doi:10.3389/fnhum.2014.00068, PMC 3923150, PMID 24592230

Zeki, S.

Mathematics, Poetry and Beauty

cut-the-knot.org

Is Mathematics Beautiful?

Justin Mullins.com

Edna St. Vincent Millay (poet): Euclid alone has looked on beauty bare

Mathbeauty Blog

The collection at the Internet Archive

Aesthetic Appeal

Jim Holt December 5, 2013 issue of The New York Review of Books review of Love and Math: The Heart of Hidden Reality by Edward Frenkel

A Mathematical Romance