"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)
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