"[...] despite an objectivity about mathematical results that has no parallel in the world of art, the motivation and standards of creative mathematics are more like those of art than of science. Aesthetic judgments transcend both logic and applicability in the ranking of mathematical theorems: beauty and elegance have more to do with the value of a mathematical idea than does either strict truth or possible utility." (Lynn A Steen, „Mathematics Today: Twelve Informal Essays", 1978)
"The test of the intelligibility of any statement that overwhelms us with its air of profundity is its translatability into language that lacks the elevation and verve of the original statement but can pass muster as a simple and clear statement in ordinary, everyday speech. Most of what has been written about beauty will not survive this test. In the presence of many of the most eloquent statements about beauty, we are left speechless - speechless in the sense that we cannot find other words for expressing what we think or hope we understand." (Mortimer J Adler, Six Great Ideas, 1981)
"[...] despite an objectivity about mathematical results that has no parallel in the world of art, the motivation and standards of creative mathematics are more like those of art than of science. Aesthetic judgments transcend both logic and applicability in the ranking of mathematical theorems: beauty and elegance have more to do with the value of a mathematical idea than does either strict truth or possible utility." (Lynn A Steen," Mathematics Today: Twelve Informal Essays", 1978)
"Perhaps the best way to approach the question of what mathematics is, is to start at the beginning. In the far distant prehistoric past, where we must look for the beginnings of mathematics, there were already four major faces of mathematics. First, there was the ability to carry on the long chains of close reasoning that to this day characterize much of mathematics. Second, there was geometry, leading through the concept of continuity to topology and beyond. Third, there was number, leading to arithmetic, algebra, and beyond. Finally there was artistic taste, which plays so large a role in modern mathematics. There are, of course, many different kinds of beauty in mathematics. In number theory it seems to be mainly the beauty of the almost infinite detail; in abstract algebra the beauty is mainly in the generality. Various areas of mathematics thus have various standards of aesthetics." (Richard Hamming, "The Unreasonable Effectiveness of Mathematics", The American Mathematical Monthly Vol. 87 (2), 1980)
"In lieu of the traditional confrontation between theory and experiment, superstring theorists pursue an inner harmony where elegance, uniqueness and beauty define truth. The theory depends for its existence upon magical coincidences, miraculous cancellations and relations among seemingly unrelated (and possibly undiscovered) fields of mathematics." (Sheldon L Glashow, "Desperately Seeking Superstrings?", Physics Today, 1986)
"The principle of mathematical beauty, like related aesthetic principles, is problematical. The main problem is that beauty is essentially subjective and hence cannot serve as a commonly defined tool for guiding or evaluating science. It is, to say the least, difficult to justify aesthetic judgment by rational arguments. Within literary and art criticism there is, indeed, a long tradition of analyzing the idea of beauty, including many attempts to give the concept an objective meaning. Objectivist and subjectivist theories of aesthetic judgment have been discussed for centuries without much progress, and today the problem seems as muddled as ever. Apart from the confused state of art in aesthetic theory, it is uncertain to what degree this discussion is relevant to the problem of scientific beauty. I, at any rate, can see no escape from the conclusion that aesthetic judgment in science is rooted in subjective and social factors. The sense of aesthetic standards is pan of the socialization that scientists acquire; but scientists, as well as scientific communities, may have widely different ideas of how to judge the aesthetic merit of a particular theory. No wonder that eminent physicists do not agree on which theories are beautiful and which are ugly." (Helge Kragh, 1990)
"Order wherever it reigns, brings beauty with it. Theory not only renders the group of physical laws it represents easier to handle, more convenient, and more useful, but also more beautiful." (Pierre Maurice Marie Duhem, "The Aim and Structure of Physical Theory", 1991)
"The material world begins to seem so trivial, so arbitrary, so ephemeral when contrasted with the timeless beauty of mathematics." (William Dunham, "The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems, and Personalities", 1994)
"Order wherever it reigns, brings beauty with it. Theory not only renders the group of physical laws it represents easier to handle, more convenient, and more useful, but also more beautiful." (Pierre Maurice Marie Duhem, "The Aim and Structure of Physical Theory", 1991)
"The material world begins to seem so trivial, so arbitrary, so ephemeral when contrasted with the timeless beauty of mathematics." (William Dunham, "The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems, and Personalities", 1994)
"Mathematics-as-science naturally starts with mysterious phenomena to be explained, and leads (if you are successful) to powerful and harmonious patterns. Mathematics-as-a-game not only starts with simple objects and rules, but involves all the attractions of games like chess: neat tactics, deep strategy, beautiful combinations, elegant and surprising ideas. Mathematics-as-perception displays the beauty and mystery of art in parallel with the delight of illumination, and the satisfaction of feeling that now you understand.
"There is much beauty in nature's clues, and we can all recognize it without any mathematical training. There is beauty, too, in the mathematical stories that start from the clues and deduce the underlying rules and regularities, but it is a different kind of beauty, applying to ideas rather than things." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)
"The lack of beauty in a piece of mathematics is of frequent occurrence, and it is a strong motivation for further mathematical research. Lack of beauty is associated with lack of definitiveness. A beautiful proof is more often than not the definitive proof (though a definitive proof need not be beautiful); a beautiful theorem is not likely to be improved upon or generalized." (Gian-Carlo Rota, "The phenomenology of mathematical proof", Synthese, 111(2), 1997)
"The most common instance of beauty in mathematics is a brilliant step in an otherwise undistinguished proof. […] A beautiful theorem may not be blessed with an equally beautiful proof; beautiful theorems with ugly proofs frequently occur. When a beautiful theorem is missing a beautiful proof, attempts are made by mathematicians to provide new proofs that will match the beauty of the theorem, with varying success. It is, however, impossible to find beautiful proofs of theorems that are not beautiful." (Gian-Carlo Rota, "The Phenomenology of Mathematical Beauty", 1997)
"Whatever the ins and outs of poetry, one thing is clear: the manner of expression - notation - is fundamental. It is the same with mathematics - not in the aesthetic sense that the beauty of mathematics is tied up with how it is expressed - but in the sense that mathematical truths are revealed, exploited and developed by various notational innovations." (James R Brown, “Philosophy of Mathematics”, 1999)
"The spirit of mathematics and the essence of its beauty is remarkably fragile, because mathematics is about ideas and about thought. Mathematics takes place in the mind, and no two minds are the same. After many years of study and work, a mathematician may stumble on a vast and beautiful vista that unifies and simplifies many things that once appeared disparate and complicated. Mathematicians can share a beautiful mathematical vista with one another, but there is no camera that can easily capture an image of such a vista to convey it in full to people who have not trudged along many of the same trails." (Silvio Levy, "The Eightfold Way: The Beauty of Klein’s Quartic Curve", 1999)
"Whatever the ins and outs of poetry, one thing is clear: the manner of expression - notation - is fundamental. It is the same with mathematics - not in the aesthetic sense that the beauty of mathematics is tied up with how it is expressed - but in the sense that mathematical truths are revealed, exploited and developed by various notational innovations." (James R Brown, “Philosophy of Mathematics”, 1999)
"The spirit of mathematics and the essence of its beauty is remarkably fragile, because mathematics is about ideas and about thought. Mathematics takes place in the mind, and no two minds are the same. After many years of study and work, a mathematician may stumble on a vast and beautiful vista that unifies and simplifies many things that once appeared disparate and complicated. Mathematicians can share a beautiful mathematical vista with one another, but there is no camera that can easily capture an image of such a vista to convey it in full to people who have not trudged along many of the same trails." (Silvio Levy, "The Eightfold Way: The Beauty of Klein’s Quartic Curve", 1999)
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