"Prime numbers belong to an exclusive world of intellectual conceptions. We speak of those marvelous notions that enjoy simple, elegant description, yet lead to extreme - one might say unthinkable - complexity in the details. The basic notion of primality can be accessible to a child, yet no human mind harbors anything like a complete picture. In modern times, while theoreticians continue to grapple with the profundity of the prime numbers, vast toil and resources have been directed toward the computational aspect, the task of finding, characterizing, and applying the primes in other domains." (Richard Crandall & Carl Pomerance,"Prime Numbers: A Computational Perspective", 2001)
"A good poem has a unified structure, each word fits perfectly, there is nothing arbitrary about it, metaphors hold together and interlock, the sound of a word and its reflections of meaning complement each other. Likewise postmodern physics asks: How well does everything fit together in a theory? How inevitable are its arguments? Are the assumptions well founded or somewhat arbitrary? Is its overall mathematical form particularly elegant?" (F David Peat, "From Certainty to Uncertainty", 2002)
"Complex numbers are really not as complex as you might expect from their name, particularly if we think of them in terms of the underlying two dimensional geometry which they describe. Perhaps it would have been better to call them 'nature's numbers'. Behind complex numbers is a wonderful synthesis between two dimensional geometry and an elegant arithmetic in which every polynomial equation has a solution." (David Mumford, Caroline Series & David Wright, "Indra’s Pearls: The Vision of Felix Klein", 2002)
'"There is an old debate', Erdos liked to say, 'about whether you create mathematics or just discover it. In other words, are the truths already there, even if we don't yet know them?' Erdos had a clear answer to this question: Mathematical truths are there among the list of absolute truths, and we just rediscover them. Random graph theory, so elegant and simple, seemed to him to belong to the eternal truths. Yet today we know that random networks played little role in assembling our universe. Instead, nature resorted to a few fundamental laws [...]. Erdos himself created mathematical truths and an alternative view of our world by developing random graph theory. (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)
"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)
"[…] admiration for elegant symmetry never dies […]" (Robert Kaplan & Ellen Kaplan, "The Art of the Infinite: The Pleasures of Mathematics", 2003)
"Pure mathematics was characterized by an obsession with proof, rigor, beauty, and elegance, and sought its foundations in the disembodied worlds of logic or intuition. Far from being coextensive with physics, pure mathematics could be ‘applied’ only after it had been made foundationally secure by the purists." (Andrew Warwick,"Masters of Theory: Cambridge and the rise of mathematical physics", 2003)
"Proofs should be as short, transparent, elegant, and insightful as possible." (Burkard Polster,"Q.E.D.: Beauty in Mathematical Proof", 2004)
"Theoretical physics borrows from mathematics (or, if there's none to borrow, they invent new mathematics) in order to create a mathematical roadmap of things that can happen in the real world, in nature. It strives to explain all of the many different phenomena observed in the universe, perhaps ultimately seeking one elegant and economical logical system. However, physicists usually settle for lesser triumphs, in which many physical systems with common and comprehensible behaviors are successfully described. This description is always created in the abstract language of mathematics." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)
"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)
"Evolution moves towards greater complexity, greater elegance, greater knowledge, greater intelligence, greater beauty, greater creativity, and greater levels of subtle attributes such as love. […] Of course, even the accelerating growth of evolution never achieves an infinite level, but as it explodes exponentially it certainly moves rapidly in that direction." (Ray Kurzweil, "The Singularity is Near", 2005)
"We all know what we like in music, painting or poetry, but it is much harder to explain why we like it. The same is true in mathematics, which is, in part, an art form. We can identify a long list of desirable qualities: beauty, elegance, importance, originality, usefulness, depth, breadth, brevity, simplicity, clarity. However, a single work can hardly embody them all; in fact, some are mutually incompatible. Just as different qualities are appropriate in sonatas, quartets or symphonies, so mathematical compositions of varying types require different treatment." (Michael Atiyah,"Mathematics: Art and Science" Bulletin of the AMS 43, 2006)
"Mathematical problems, or puzzles, are important to real mathematics (like solving real-life problems), just as fables, stories, and anecdotes are important to the young in understanding real life. Mathematical problems are ‘sanitized’ mathematics, where an elegant solution has already been found (by someone else, of course), the question is stripped of all superfluousness and posed in an interesting and (hopefully) thought-provoking way. If mathematics is likened to prospecting for gold, solving a good mathematical problem is akin to a ‘hide-and-seek’ course in gold-prospecting: you are given a nugget to find, and you know what it looks like, that it is out there somewhere, that it is not too hard to reach, that it is unearthing within your capabilities, and you have conveniently been given the right equipment (i.e. data) to get it. It may be hidden in a cunning place, but it will require ingenuity rather than digging to reach it." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)
"We all know what we like in music, painting or poetry, but it is much harder to explain why we like it. The same is true in mathematics, which is, in part, an art form. We can identify a long list of desirable qualities: beauty, elegance, importance, originality, usefulness, depth, breadth, brevity, simplicity, clarity. However, a single work can hardly embody them all; in fact, some are mutually incompatible. Just as different qualities are appropriate in sonatas, quartets or symphonies, so mathematical compositions of varying types require different treatment." (Michael Atiyah, "Mathematics: Art and Science" Bulletin of the AMS 43, 2006)
"Mathematicians, then, do not just care about proving theorems: they care about proving interesting, deep, fruitful theorems, by means of elegant, ingenious, explanatory, memorable, or even amusing proofs. If we wish to understand more about the character of mathematical knowledge, we ought to investigate these kinds of evaluative claims made by mathematicians." (Mary Leng ["Mathematical Knowledge", Ed. by Mary Leng, Alexander Paseau and Michael Potter], 2007)
"A theory appears to be beautiful or elegant (or simple, if you prefer) when it can be expressed concisely in terms of mathematics we already have." (Murray Gell-Mann, "Beauty and Truth in Physics", 2007)
"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)
"I enjoy mathematics so much because it has a strange kind of unearthly beauty. There is a strong feeling of pleasure, hard to describe, in thinking through an elegant proof, and even greater pleasure in discovering a proof not previously known." (Martin Gardner, 2008)
"In mathematics, it’s the limitations of a reasoned argument with the tools you have available, and with magic it’s to use your tools and sleight of hand to bring about a certain effect without the audience knowing what you’re doing. [...]When you’re inventing a trick, it’s always possible to have an elephant walk on stage, and while the elephant is in front of you, sneak something under your coat, but that’s not a good trick. Similarly with mathematical proof, it is always possible to bring out the big guns, but then you lose elegance, or your conclusions aren’t very different from your hypotheses, and it’s not a very interesting theorem." (Persi Diaconis, 2008)
"We can describe general relativity using either of two mathematically equivalent ideas: curved space-time or metric field. Mathematicians, mystics and specialists in general relativity tend to like the geometric view because of its elegance. Physicists trained in the more empirical tradition of high-energy physics and quantum field theory tend to prefer the field view, because it corresponds better to how we (or our computers) do concrete calculations." (Frank Wilczek, "The Lightness of Being: Mass, Ether, and the Unification of Forces", 2008)
"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)
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