🥀On Aesthetic in Mathematics (2000-)

"Where we find certainty and truth in mathematics we also find beauty. Great mathematics is characterized by its aesthetics. Mathematicians delight in the elegance, economy of means, and logical inevitability of proof. It is as if the great mathematical truths can be no other way. This light of logic is also reflected back to us in the underlying structures of the physical world through the mathematics of theoretical physics." (F David Peat, "From Certainty to Uncertainty", 2002)

"Elegance and simplicity should remain important criteria in judging mathematics, but the applicability and consequences of a result are also important, and sometimes these criteria conflict. I believe that some fundamental theorems do not admit simple elegant treatments, and the proofs of such theorems may of necessity be long and complicated. Our standards of rigor and beauty must be sufficiently broad and realistic to allow us to accept and appreciate such results and their proofs. As mathematicians we will inevitably use such theorems when it is necessary in the practice our trade; our philosophy and aesthetics should reflect this reality." (Michael Aschbacher,"Highly complex proofs and implications", 2005)

"Metaphorizing is a manner of thinking, not a property of thinking. It is a capacity of thought, not its quality. It represents a mental operation by which a previously existing entity is described in the characteristics of another one on the basis of some similarity or by reasoning. When we say that something is" (like) something else, we have already performed a mental operation. This operation includes elements such as comparison, paralleling and shaping of the new image by ignoring its less satisfactory traits in order that this image obtains an aesthetic value. By this process, for an instant we invent a device, which serves as the pole vault for the comparison’s jump. Once the jump is made the pole vault is removed. This device could be a lightning-speed logical syllogism, or a momentary created term, which successfully merges the traits of the compared objects." (Ivan Mladenov,"Conceptualizing Metaphors: On Charles Peirce’s marginalia", 2006)

"Physicists have been drawn to elegant mathematical relationships that bind the subject together with economy and style, melding disparate qualities in subtle and harmonious ways. But this is to import a new factor into the argument - questions of aesthetics and taste. We are then on shaky ground indeed. It may be that M theory looks beautiful to its creators, but ugly to N theorists, who think that their theory is the most elegant. But then the O theorists disagree with both groups [...]" (Paul C W Davies, "Cosmic Jackpot: Why Our Universe Is Just Right for Life", 2007)

"[…] mathematics is not only an essential tool for science and technology, but also for humanities, in particular for art. And out of art we may say that mathematics gains one of its main reasons for developing and changing in time. Mathematics contributes to our way of conceiving and shaping the world we live in, while art develops the means to harmonize, describe, represent aesthetically - or even to transcend and transfigure - the world of our sensations and perception." (Mauro Francaviglia,"Art and Mathematics", 2008)

"A scale model represents, and yields information about what it is a model of, by selective resemblance. [...] Scale models can be produced for the sheer aesthetic pleasure of it, but more typically they serve in studies meant to design the very things of which they are meant to be the scaled down versions. " (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"Why, though, is symmetry so pervasive in nature? It is not just a matter of aesthetics. Just as it is for me and mathematics, symmetry in nature is about language. It provides a way for animals and plants to convey a multitude of messages, from genetic superiority to nutritional information. Symmetry is often a sign of meaning, and can therefore be interpreted as a very basic, almost primeval form of communication." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"If there is anything like a unifying aesthetic principle in mathematics, it is this: simple is beautiful. Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary." (Paul Lockhart,"A Mathematician's Lament", 2009)

"Mathematicians seek a certain kind of beauty. Perhaps mathematical beauty is a constant - as far as the contents of mathematics are concerned - and yet the forms this beauty takes are certainly cultural. And while the history of mathematics surely is many stranded, one of its most important strands is formed by such cultural forms of mathematical beauty." (Reviel Netz,"Ludic Proof: Greek Mathematics and the Alexandrian Aesthetic", 2009)

"As art, chess speaks to us of the personal decisions that are made in the course of a game. Looking at this facet of the game, the essential protagonist is the aesthetic sense rather than the capacity for calculation, which thus moves us closer to the human dimension and farther from mathematical algorithms." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Many terms that are used to comment on games are aesthetic allusions, indicating that among chess players it is hard to separate out the game’s creative and analytic aspects. Terms that are frequently used include subtlety, depth, beauty, surprise, vision, brilliance, elegance, harmony, and symmetry." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"There are three reasons for the study of inequalities: practical, theoretical and aesthetic. In many practical investigations, it is necessary to bound one quantity by another. The classical inequalities are very useful for this purpose. From the theoretical point of view, very simple questions give rise to entire theories. […] Finally, let us turn to the aesthetic aspects. As has been pointed out, beauty is in the eye of the beholder. However. it is generally agreed that certain pieces of music, art, or mathematics are beautiful. There is an elegance to inequalities that makes them very attractive." (Claudi Alsina & Roger B Nelsen, "When Less is More: Visualizing Basic Inequalities", 2009)

"A proof is simply a story. The characters are the elements of the problem, and the plot is up to you. The goal, as in any literary fiction, is to write a story that is compelling as a narrative. In the case of mathematics, this means that the plot not only has to make logical sense but also be simple and elegant. No one likes a meandering, complicated quagmire of a proof. We want to follow along rationally to be sure, but we also want to be charmed and swept off our feet aesthetically. A proof should be lovely as well as logical.(Paul Lockhart, "Measurement", 2012)

"The need for overall understanding is not just aesthetic or educational. The human mind tends to make errors: errors of fact, errors of judgement, errors of interpretation. In the step-by-step method we might not notice that one line is not a logical consequence of preceding ones. Within the overall framework, however, if an error leads to a conclusion that does not fit into the total picture, the conflict will alert us to the possibility of a mistake." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

"The true foundations of mathematics do not lie in axioms, definitions, and logical inference, which are the foundational elements of formal mathematics. The true foundations of mathematics lie in the minds of mathematicians as they interact with and try to make sense of their world - in their ideas, their intuitions, and their aesthetic sensibility." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

🥀On Aesthetic in Mathematics (1970-1979)

"If some great mathematicians have known how to give lyrical expression to their enthusiasm for the beauty of their science, nobody has suggested examining it as if it were the object of an art - mathematical art - and consequently the subject of a theory of aesthetics, the aesthetics of mathematics" (François Le Lionnais, Great Currents of Mathematical Thought", 1971)

"The evolutionary vision is agnostic in regard to systems in the universe of greater complexity than those of which human beings have clear knowledge. It recognizes aesthetic, moral, and religious ideas and experiences as a species, in this case of mental structures or of images, which clearly interacts with other species in the world's great' ecosystem." (Kenneth Boulding," Ecodynamics: A New Theory of Societal Evolution", 1978)

[...] 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)

"Geometry is the study of shapes. Although true, this definition is so broad that it is almost meaningless. The judge of a beauty contest is, in a sense, a geometrician because he is judging […] shapes, but this is not quite what we want the word to mean. It has been said that a curved line is the most beautiful distance between two points. Even though this statement is about curves, a proper element of geometry, the assertion seems more to be in the domain of aesthetics rather than mathematics." (Martin Gardner, "Aha! Insight", 1978)

"The evolutionary vision is agnostic in regard to systems in the universe of greater complexity than those of which human beings have clear knowledge. It recognizes aesthetic, moral, and religious ideas and experiences as a species, in this case of mental structures or of images, which clearly interacts with other species in the world's great' ecosystem." (Kenneth Boulding," Ecodynamics: A New Theory of Societal Evolution", 1978)

"The motivation and standards of creative mathematics are more like those of art than like those 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 Arthur Steen, "Mathematics Today: Twelve Informal Essays", 1978)

"There are three reasons for the study of inequalities: practical, theoretical and aesthetic. On the aesthetic aspects, as has been pointed out, beauty is in the eyes of the beholder. However, it is generally agreed that certain pieces of music, art, or mathematics are beautiful. There is an elegance to inequalities that makes them very attractive." (Richard E Bellman, 1978)

"Mathematics is not a branch of aesthetics. The mistake, which is common enough, probably stems from the requirement of aesthetic unity but is not identical with that unity." (J K Feibleman,"Assumptions of Grand Logics", 1979)

🥀On Aesthetic in Mathematics (1950-1969)

 "It would seem that more than function itself, simplicity is the deciding factor in the aesthetic equation. One might call the process beauty through function and simplification." (Raymond Loewy, "Never Leave Well Enough Alone", 1951) 

"[...] sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work - that is, correctly to describe phenomena from a reasonably wide area. Furthermore, it must satisfy certain aesthetic criteria - that is, in relation to how much it describes, it must be rather simple." (John von Neumann, "Method in the Physical Sciences", 1955)

"When, for instance, I see a symmetrical object, I feel its pleasurable quality, but do not need to assert explicitly to myself, ‘How symmetrical!’. This characteristic feature may be explained as follows. In the course of individual experience it is found generally that symmetrical objects possess exceptional and desirable qualities. Thus our own bodies are not regarded as perfectly formed unless they are symmetrical. Furthermore, the visual and tactual technique by which we perceive the symmetry of various objects is uniform, highly developed, and almost instantaneously applied. It is this technique which forms the associative 'pointer.' In consequence of it, the perception of any symmetrical object is accompanied by an intuitive aesthetic feeling of positive tone." (George D Birkhoff, "Mathematics of Aesthetics", 1956)

"It is sometimes said of two expositions of one and the same mathematical proof that the one is simpler or more elegant than the other. This is a distinction which has little interest from the point of view of the theory of knowledge; it does not fall within the province of logic, but merely indicates a preference of an aesthetic or pragmatic character." (Karl Popper,"The Logic of Scientific Discovery", 1959)

"[...] sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work - that is, correctly to describe phenomena from a reasonably wide area. Furthermore, it must satisfy certain aesthetic criteria - that is, in relation to how much it describes, it must be rather simple." (John von Neumann, "Method in the physical sciences", 1961)

"The process of preparing programs for a digital computer is especially attractive, not only because it can economically and scientifically rewarding, but also because it can be an aesthetic experience much like composing poetry or music." (Donald E Knuth, "The Art of Computer Programming: Fundamental algorithms", 1968)

🥀On Aesthetic in Mathematics (1925-1949)

"Nature is probably quite indifferent to the aesthetic preferences of mathematicians." (Alfred N Whitehead, "Science and the Modern World", 1925)

"A system of laws may be beautiful, or a mathematical proof may be elegant, although no auditory or visual experience is directly involved in either case. It would seem indeed that all feeling of desirability which is more than mere appetite has some claim to be regarded as aesthetic feeling." (George D Birkhoff, "Aesthetic Measure", 1933

"When, for instance, I see a symmetrical object, I feel its pleasurable quality, but do not need to assert explicitly to myself, ‘How symmetrical!’. This characteristic feature may be explained as follows. In the course of individual experience it is found generally that symmetrical objects possess exceptional and desirable qualities. Thus our own bodies are not regarded as perfectly formed unless they are symmetrical. Furthermore, the visual and tactual technique by which we perceive the symmetry of various objects is uniform, highly developed, and almost instantaneously applied. It is this technique which forms the associative 'pointer.' In consequence of it, the perception of any symmetrical object is accompanied by an intuitive aesthetic feeling of positive tone." (George D Birkhoff, "Mathematics of Aesthetics", 1933)

"It is sometimes said of two expositions of one and the same mathematical proof that the one is simpler or more elegant than the other. This is a distinction which has little interest from the point of view of the theory of knowledge; it does not fall within the province of logic, but merely indicates a preference of an aesthetic or pragmatic character." (Karl R Popper,"The Logic of Scientific Discovery", 1934)

"Mathematics as an expression of the human mind reflects the active will, the contemplative reason. and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science." (Richard Courant ‎& Herbert Robbins,"What is Mathematics?", 1941)

"It is to be hoped that in the future more and more theoretical physicists will command a deep knowledge of mathematical principles; and also that mathematicians will no longer limit themselves so exclusively to the aesthetic development of mathematical abstractions." (George D Birkhoff, "Mathematical Nature of Physical Theories" American Scientific Vol. 31" (4), 1943)

"I think that it is a relatively good approximation to truth - which is much too complicated to allow anything but approximations - that mathematical ideas originate in empirics. But, once they are conceived, the subject begins to live a peculiar life of its own and is […] governed by almost entirely aesthetical motivations. In other words, at a great distance from its empirical source, or after much ‘abstract’ inbreeding, a mathematical subject is in danger of degeneration. Whenever this stage is reached the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas." (John von Neumann, "The Mathematician", The Works of the Mind Vol. I" (1), 1947)

"True, the universe is more than a collection of objective experimental data; more than the complexus of theories, abstractions, and special assumptions devised to hold the data together; more, indeed, than any construct modeled on this cold objectivity. For there is a deeper, more subjective world, a world of sensation and emotion, of aesthetic, moral, and religious values as yet beyond the grasp of objective science. And towering majestically over all, inscrutable and inescapable, is the awful mystery of Existence itself, to confound the mind with an eternal enigma." (Banesh Hoffmann, "The Strange Story of the Quantum", 1947)

🥀On Aesthetic in Mathematics XV

"In addition to this it [mathematics] provides its disciples with pleasures similar to painting and music. They admire the delicate harmony of the numbers and the forms; they marvel when a new discovery opens up to them an unexpected vista; and does the joy that they feel not have an aesthetic character even if the senses are not involved at all? […] For this reason I do not hesitate to say that mathematics deserves to be cultivated for its own sake, and I mean the theories which cannot be applied to physics just as much as the others." (Henri Poincaré, 1897)

"What are the mathematical entities to which we attribute this character of beauty and elegance, which are capable of developing in us a kind of aesthetic emotion? Those whose elements are harmoniously arranged so that the mind can, without effort, take in the whole without neglecting the details. This harmony is at once a satisfaction to out esthetic requirements, and an assistance to the mind which supports and guides." (Henri Poincaré, 1908)

"Mathematics has beauties of its own - a symmetry and proportion in its results, a lack of superfluity, an exact adaptation of means to ends, which is exceedingly remarkable and to be found elsewhere only in the works of the greatest beauty." (Jacob W A Young,"The Teaching of Mathematics", 1907) 

"What are the mathematical entities to which we attribute this character of beauty and elegance, which are capable of developing in us a kind of aesthetic emotion? Those whose elements are harmoniously arranged so that the mind can, without effort, take in the whole without neglecting the details. This harmony is at once a satisfaction to out esthetic requirements, and an assistance to the mind which supports and guides." (Henri Poincaré, 1908) 

"Since the primary object of the scientific theory is to express the harmonies which are found to exist in nature, we see at once that these theories must have an aesthetic value. The measure of the success of a scientific theory is, in fact, a measure of its aesthetic value, since it is a measure of the extent to which it has introduced harmony in what was before chaos." (John W N Sullivan, "The Justification of the Scientific Method", The Athenaeum" (4644), 1919)