✓On Proofs (2010-2019)

"A proof in mathematics is a compelling argument that a proposition holds without exception; a disproof requires only the demonstration of an exception. A mathematical proof does not, in general, establish the empirical truth of whatever is proved. What it establishes is that whatever is proved - usually a theorem - follows logically from the givens, or axioms." (Raymond S Nickerson, "Mathematical Reasoning", 2010)

"In the broad light of day mathematicians check their equations and their proofs, leaving no stone unturned in their search for rigour. But, at night, under the full moon, they dream, they float among the stars and wonder at the miracle of the heavens. They are inspired. Without dreams there is no art, no mathematics, no life." (Michael F Atiyah, "The Art of Mathematics", 2010)

"The difference between necessary and sufficient conditions seems an obvious one, yet they are surprisingly often confused in mathematical proofs. Formally, in graph theory, conditions are used to prove properties of graphs. When a condition C is said to be necessary, this means that a property P can hold only if C is met. When a condition C is said to be sufficient, this means that if C is met, then property P will hold true. And indeed, when property P is true if and only if condition C is met, indicates that C is a necessary and sufficient condition for property P to be valid." (Maarten van Steen, "Graph Theory and Complex Networks: An Introduction", 2010)

"A mathematical proof is a watertight argument which begins with information you are given, proceeds by logical argument, and ends with what you are asked to prove." (Sydney A Morris, "Topology without Tears", 2011)

"Just as music is not about reaching the final chord, mathematics is about more than just the result. It is the journey that excites the mathematician. I read and reread proofs in much the same way as I listen to a piece of music: understanding how themes are established, mutated, interwoven and transformed. What people don't realise about mathematics is that it involves a lot of choice: not about what is true or false (I can't make the Riemann hypothesis false if it's true), but from deciding what piece of mathematics is worth ‘listening to’." (Marcus du Sautoy, "Listen by Numbers: Music and Maths", 2011)

"Once a mathematical result is proven to the satisfaction of the discipline, it doesn’t need to be re-evaluated in the light of new evidence or refuted, unless it contains a mistake. If it was true for Archimedes, then it is true today." (Peter Rowlett, "The Unplanned Impact of Mathematics", Nature, 2011)

"To get a true understanding of the work of mathematicians, and the need for proof, it is important for you to experiment with your own intuitions, to see where they lead, and then to experience the same failures and sense of accomplishment that mathematicians experienced when they obtained the correct results. Through this, it should become clear that, when doing any level of mathematics, the roads to correct solutions are rarely straight, can be quite different, and take patience and persistence to explore." (Alan Sultan & Alice F Artzt, "The Mathematics that every Secondary School Math Teacher Needs to Know", 2011)

"Topology, like other branches of pure mathematics such as group theory, is an axiomatic subject. We start with a set of axioms and we use these axioms to prove propositions and theorems. It is extremely important to develop your skill at writing proofs." (Sydney A Morris, "Topology without Tears", 2011)

"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)

"Catastrophe theory can be thought of as a link between classical analysis, dynamical systems, differential topology" (including singularity theory), modern bifurcation theory and the theory of complex systems. [...] The name ‘catastrophe theory’ is used for a combination of singularity theory and its applications. [...] From the didactical point of view, there are two main positions for courses in catastrophe theory at university level: Trying to teach the theory as a perfect axiomatic system consisting of exact definitions, theorems and proofs or trying to teach mathematics as it can be developed from historical or from natural problems." (Werner Sanns, "Catastrophe Theory" [Mathematics of Complexity and Dynamical Systems, 2012])

"Proofs can cause dizziness or excessive drowsiness. Side effects of prolonged exposure may include night sweats, panic attacks, and, in rare cases, euphoria. Ask your doctor if proofs are right for you." (Steven Strogatz, "The Joy of X: A Guided Tour of Mathematics, from One to Infinity", 2012)

"The fact that some parts of mathematics are unexpected and others not, that some solutions are unique and others multiple, that some proofs are obvious and others take a huge amount of work to produce - all these have a bearing on how we describe the process of mathematical production, and all of them are entirely independent of one’s philosophical position." (Timothy Gowers, "Is Mathematics Discovered or Invented?",  ["The Best Writing of Mathematics: 2012"] 2012)

"The first thing that you should understand about science is that it is almost always uncertain. The scientific process allows science to move ahead without waiting for an elusive 'proof positive'. […] How can science afford to act on less than certainty? Because science is a continuing story - always retesting ideas. One scientific finding leads scientists to conduct more research, which may support and expand on the original finding." (Victor Cohn & Lewis Cope, "News & Numbers: A writer’s guide to statistics" 3rd Ed, 2012)

"The solution to a math problem is not a number; it’s an argument, a proof. We’re trying to create these little poems of pure reason. Of course, like any other form of poetry, we want our work to be beautiful as well as meaningful. Mathematics is the art of explanation, and consequently, it is difficult, frustrating, and deeply satisfying." (Paul Lockhart, "Measurement", 2012)

"[…] if one has a theory, one needs to be willing to try to prove it wrong as much as one tries to provide that it is right […]" (Lawrence M Krauss et al, A Universe from Nothing, 2013)

"If our machines are better at some things than we are, it makes sense to use machines. Proof techniques may change, but they do that all the time anyway: it’s called ‘research’. The concept of proof does not radically alter if some steps are done by a computer. A proof is a story; a computer-assisted proof is a story that’s too long to be told in full, so you have to settle for the executive summary and a huge automated appendix." (Ian Stewart, "Visions of Infinity", 2013)

"Often the key contribution of intuition is to make us aware of weak points in a problem, places where it may be vulnerable to attack. A mathematical proof is like a battle, or if you prefer a less warlike metaphor, a game of chess. Once a potential weak point has been identified, the mathematician’s technical grasp of the machinery of mathematics can be brought to bear to exploit it." (Ian Stewart, "Visions of Infinity", 2013)

"People think mathematics begins when you write down a theorem followed by a proof. That’s not the beginning, that’s the end. For me the creative place in mathematics comes before you start to put things down on paper, before you try to write a formula. You picture various things, you turn them over in your mind. You’re trying to create, just as a musician is trying to create music, or a poet. There are no rules laid down. You have to do it your own way. But at the end, just as a composer has to put it down on paper, you have to write things down. But the most important stage is understanding. A proof by itself doesn’t give you understanding." (Michael F Atiyah, [interview] 2013)

"What  brings  us  mathematical  knowledge?  The  carriers  of mathematical knowledge  are  proofs, more  generally  arguments  and constructions,  as embedded  in  larger  contexts. Mathematicians  and teachers  of higher mathematics  know  this,  but  it  should  be  said. Issues  about  competence  and intuition  can be  raised as  well  as  factors  of knowledge  involving  the general  dissemination  of analogical  or inductive  reasoning  or the  specific conveyance  of methods,  approaches  or ways  of thinking.  But  in the  end, what  can be  directly  conveyed  as  knowledge  are  proofs." (Akihiro Kanamori, "Mathematical  Knowledge:  Motley and  Complexity of Proof",  Annals  of  the  Japan Association  for  Philosophy  of  Science Vol. 21, 2013)

"When a mathematical conjecture eventually turns out to be correct, its history often follows a standard pattern. Over a period of time, various people prove the conjecture to be true provided special restrictions apply. Each such result improves on the previous one by relaxing some restrictions, but eventually this process runs out of steam. Finally, a new and much cleverer idea completes the proof." (Ian Stewart, "Visions of Infinity", 2013)

"Proof, in fact, is the requirement that makes great problems problematic. Anyone moderately competent can carry out a few calculations, spot an apparent pattern, and distil its essence into a pithy statement. Mathematicians demand more evidence than that: they insist on a complete, logically impeccable proof. Or, if the answer turns out to be negative, a disproof. It isn’t really possible to appreciate the seductive allure of a great problem without appreciating the vital role of proof in the mathematical enterprise. Anyone can make an educated guess. What’s hard is to prove it’s right. Or wrong." (Ian Stewart, "Visions of Infinity", 2013)

"It is still the case that mathematicians are most familiar with, and most comfortable with, a traditional, self-contained proof that consists of a sequence of logical steps recorded on a piece of paper. We still hope that some day there will be such a proof of the four-color theorem. After all, it is only a traditional, Euclidean-style proof that offers the understanding, the insight, and the sense of completion that all scholars seek. For now we live with the computer-aided proof of the four-color theorem." (Steven G Krantz & Harold R Parks, "A Mathematical Odyssey: Journey from the Real to the Complex", 2014)

"We are genetically predisposed to look for patterns and to believe that the patterns we observe are meaningful. […] Don’t be fooled into thinking that a pattern is proof. We need a logical, persuasive explanation and we need to test the explanation with fresh data." (Gary Smith, "Standard Deviations", 2014)

"The best way to think about mathematics is to include not only the content dimension of algorithms, procedures, theorems, and proofs but also the cognitive dimensions of learning, understanding, and creating mathematics." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"The human mind builds up theories by recognising familiar patterns and glossing over details that are well understood, so that it can concentrate on the new material. In fact it is limited by the amount of new information it can hold at any one time, and the suppression of familiar detail is often essential for a grasp of the total picture. In a written proof, the step-by-step logical deduction is therefore foreshortened where it is already a part of the reader’s basic technique, so that they can comprehend the overall structure more easily." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

"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)

"A theory is an organizational form of scientific knowledge about a certain set of objects, representing a system of interconnected assertions and proofs and containing methods of explanation and prediction of phenomena and processes in a given problem domain, i.e., of all phenomena and processes described by this theory." (Dmitry A Novikov, "Cybernetics: From Past to Future", 2016)

"A mathematician possesses a mental model of the mathematical entity she works on. This internal mental model is accessible to her direct observation and manipulation. At the same time, it is socially and culturally controlled, to conform to the mathematics community's collective model of the entity in question. The mathematician observes a property of her own internal model of that mathematical entity. Then she must find a recipe, a set of instructions, that enables other competent, qualified mathematicians to observe the corresponding property of their corresponding mental model. That recipe is the proof. It establishes that property of the mathematical entity." (Reuben Hersh, "Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)

"Imagine that each proof in this book is like a painting that one sees upon entering a gallery full of artwork, in which each work presents an artist’s unique vision of the same theme." (David Perkins, "φ, π, e & i", 2017) 

"Mathematics is a fascinating discipline that calls for creativity, imagination, and the mastery of rigorous standards of proof." (John Meier & Derek Smith, "Exploring Mathematics: An Engaging Introduction to Proof", 2017)

"Mathematicians don’t come up with the proofs first. First comes intuition. Rigor comes later. This essential role of in- tuition and imagination is often left out of high-school geometry courses, but it is essential to all creative mathematics." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

✓On Proofs (2000-2009)

"It is through proof that human mathematicians transcend the limitations of their humanity. Proofs link human mathematicians to truths of the universe. In the romance, proofs are discoveries of those truths." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"Mathematics is not placid, static and eternal. […] Most mathematicians are happy to make use of those axioms in their proofs, although others do not, exploring instead so-called intuitionist logic or constructivist mathematics. Mathematics is not a single monolithic structure of absolute truth!" (Gregory J Chaitin, "A century of controversy over the foundations of mathematics", 2000)

"What does a rigorous proof consist of? The word ‘proof’ has a different meaning in different intellectual pursuits. A ‘proof’ in biology might consist of experimental data confirming a certain hypothesis; a ‘proof’ in sociology or psychology might consist of the results of a survey. What is common to all forms of proof is that they are arguments that convince experienced practitioners of the given field. So too for mathematical proofs. Such proofs are, ultimately, convincing arguments that show that the desired conclusions follow logically from the given hypotheses." (Ethan Bloch, "Proofs and Fundamentals", 2000)

"A felicitous but unproved conjecture may be of much more consequence for mathematics than the proof of many a respectable theorem." (Atle Selberg, 2001)

"By common consensus in the mathematical world, a good proof displays three essential characteristics: a good proof is (1) convincing, (2) surveyable, and (3) formalizable. The first requirement means simply that most mathematicians believe it when they see it. […] Most mathematicians and philosophers of mathematics demand more than mere plausibility, or even belief. A proof must be able to be understood, studied, communicated, and verified by rational analysis. In short, it must be surveyable. Finally, formalizability means we can always find a suitable formal system in which an informal proof can be embedded and fleshed out into a formal proof." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"Generally speaking, there are three grades of proof in mathematics. The first, or highest quality type of proof, is one that incorporates why and how the result is true, not simply that it is so. […] Second-grade proofs content themselves with showing that their   conclusion is true, by relying on the law of the excluded middle. Thus, they assume that the conclusion they want to demonstrate is false and then derive a contradiction from this assumption. In polite company, these are often termed 'nonconstructive proofs', since they lack the how and why. […] Finally, there is the third order, or lowest grade, of proof. In these situations, the idea of proof degenerates into mere verification, in which a (usually) large number of cases are considered separately and verified, one by one, very often by a computer." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"Somehow mathematicians seem to long for more than just results from their proofs; they want insight." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"That a proof must be convincing is part of the anthropology of mathematics, providing the key to understanding mathematics as a human activity. We invoke the logic of mathematics when we demand that every informal proof be capable of being formalized within the confines of a definite formal system. Finally, the epistemology of mathematics comes into play with the requirement that a proof be surveyable. We can't really say that we have created a genuine piece of knowledge unless it can be examined and verified by others; there are no private truths in mathematics." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"[…] a mathematician is more anonymous than an artist. While we may greatly admire a mathematician who discovers a beautiful proof, the human story behind the discovery eventually fades away and it is, in the end, the mathematics itself that delights us." (Timothy Gowers, "Mathematics: A Very Short Introduction", 2002)

"What a mathematical proof actually does is show that certain conclusions, such as the irrationality of, follow from certain premises, such as the principle of mathematical induction. The validity of these premises is an entirely independent matter which can safely be left to philosophers." (Timothy Gowers, "Mathematics: A Very Short Introduction", 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)

"We start from vague pictures or ideas […] which we encapsulate by rules, and then we discover that those rules persuade us to modify our mental images. We engage in a dialog between our mental images and our ability to justify them via equations. As we understand what we are investigating more clearly, the pictures become sharper and the equations more elaborate. Only at the end of the process does anything like a formal set of axioms followed by logical proofs" (E Brian Davies, "Science in the Looking Glass", 2003)

"When you’re trying to prove something, it helps to know it’s true. That gives you the confidence you need to keep searching for a rigorous proof." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"Proofs should be as short, transparent, elegant, and insightful as possible." (Burkard Polster, "Q.E.D.: Beauty in Mathematical Proof", 2004)

"The concept of proof perhaps marks the true beginning of mathematics as the art of deduction rather than just numerological observation, the point at which mathematical alchemy gave way to mathematical chemistry." (Marcus du Sautoy, "The Music of the Primes", 2004)

"There is a strong parallel between mountain climbing and mathematics research. When first attempts on a summit are made, the struggle is to find any route. Once on the top, other possible routes up may be discerned and sometimes a safer or shorter route can be chosen for the descent or for subsequent ascents. In mathematics the challenge is finding a proof in the first place. Once found, almost any competent mathematician can usually find an alternative often much better and shorter proof. At least in mountaineering we know that the mountain is there and that, if we can find a way up and reach the summit, we shall triumph. In mathematics we do not always know that there is a result, or if the proposition is only a figment of the imagination, let alone whether a proof can be found." (Kathleen Ollerenshaw, "To talk of many things: An autobiography", 2004)

"Conventional wisdom says the ideal proof should be short, simple, and elegant. However, there are now examples of very long, complicated proofs, and as mathematics continues to mature, more examples are likely to appear. Such proofs raise various issues. For example, it is impossible to write out a very long and complicated argument without error, so is such a ‘proof’ really a proof? What conditions make complex proofs necessary, possible, and of interest? Is the mathematics involved in dealing with information rich problems qualitatively different from more traditional mathematics?" (Michael Aschbacher, “Highly complex proofs and implications”, ‘Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences’ Vol. 363 (1835), 2005)

"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.", ‘Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences’ Vol. 363 (1835), 2005)

"Mathematical truth is not totally objective. If a mathematical statement is false, there will be no proofs, but if it is true, there will be an endless variety of proofs, not just one! Proofs are not impersonal, they express the personality of their creator/discoverer just as much as literary efforts do. If something important is true, there will be many reasons that it is true, many proofs of that fact. [...] each proof will emphasize different aspects of the problem, each proof will lead in a different direction. Each one will have different corollaries, different generalizations. [...] the world of mathematical truth has infinite complexity […]" (Gregory Chaitin, "Meta Math: The Quest for Omega", 2005) 

"The charm of our studies, the enchantment of science, is that, everywhere and always, we can give the justification of our principles and the proof of our discoveries." (Mordechai Ben-Ari, "Just a Theory: Exploring the Nature of Science", 2005)

"Mathematicians often get bored by a problem after they have fully understood it and have given proofs of their conjectures. Sometimes they even forget the precise details of what they have done after the lapse of years, having refocused their interest in another area. The common notion of the mathematician contemplating timeless truths, thinking over the same proof again and again - Euclid looking on beauty bare - is rarely true in any static sense." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Nowadays, however, we are much more aware of the fact that the best proof in the world is worth no more than its premises: every scientific theory is transitory and provisional, in wait for a better one, and accepted only as long as the experimental results conform to its predictions." (Ivar Ekeland, "The Best of All Possible Worlds", 2006)

"A proof is a series of steps based on the (adopted) axioms and deduction rules which reaches a desired conclusion." (Cristian S Calude et al, "Proving and Programming", 2007)

"In mathematics, the first principles are called axioms, and the rules are referred to as deduction/inference rules. A proof is a series of steps based on the (adopted) axioms and deduction rules which reaches a desired conclusion. Every step in a proof can be checked for correctness by examining it to ensure that it is logically sound." (Cristian S Calude et al, "Proving and Programming", 2007)

"It thus stands to reason that mathematical structures have a dual origin: in part human, in part purely logical. Human mathematics requires short formulations (because of our poor memory, etc.). But mathematical logic dictates that theorems with a short formulation may have very long proofs, as shown by Gödel. Clearly you don't want to go through the same long proof again and again. You will try instead to use repeatedly the short theorem that you have obtained. And an important tool to obtain short formulations is to give short names to mathematical objects that occur repeatedly. These short names describe new concepts. So we see how concept creation arises in the practice of mathematics as a consequence of the inherent logic of the sub ject and of the nature of human mathematicians." (David Ruelle, "The Mathematician's Brain", 2007)

"Mathematicians attempt to justify their claims by proofs. The quest for cast iron rational arguments is the driving force of pure mathematics. Chains of correct deduction from what is known or assumed, lead the mathematician to a conclusion which then enters the established mathematical storehouse." (Tony Crilly, "50 Mathematical Ideas You Really Need to Know", 2007)

"The goal of a definition is to introduce a mathematical object. The goal of a theorem is to state some of its properties, or interrelations between various objects. The goal of a proof is to make such a statement convincing by presenting a reasoning subdivided into small steps each of which is justified as an "elementary" convincing argument." (Yuri I Manin, "Mathematics as Metaphor: Selected Essays of Yuri I. Manin", 2007)

 "The ever-present rigorous proof is both a science and an art." (Edward B. Burger, Zero To Infinity: A History of Numbers", 2007)

"Popular accounts of mathematics often stress the discipline’s obsession with certainty, with proof. And mathematicians often tell jokes poking fun at their own insistence on precision. However, the quest for precision is far more than an end in itself. Precision allows one to reason sensibly about objects outside of ordinary experience. It is a tool for exploring possibility: about what might be, as well as what is." (Donal O’Shea, "The Poincaré Conjecture", 2007)

"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)

"We speak of mathematical reality as we speak of physical reality. They are different but both quite real. Mathematical reality is of logical nature, while physical reality is tied to the universe in which we live and which we perceive through our senses. This is not to say that we can readily define mathematical or physical reality, but we can relate to them by making mathematical proofs or physical experiments." (David Ruelle, "The Mathematician's Brain", 2007)

"Why are proofs so important? Suppose our task were to construct a building. We would start with the foundations. In our case these are the axioms or definitions - everything else is built upon them. Each theorem or proposition represents a new level of knowledge and must be firmly anchored to the previous level. We attach the new level to the previous one using a proof. So the theorems and propositions are the new heights of knowledge we achieve, while the proofs are essential as they are the mortar which attaches them to the level below. Without proofs the structure would collapse." (Sidney A Morris, "Topology without Tears", 2007)

"A proof is like a piece of theatre or music, with moments of high drama where some major shift takes the audience into a new realm." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"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)

"As students, we learned mathematics from textbooks. In textbooks, mathematics is presented in a rigorous and logical way: definition, theorem, proof, example. But it is not discovered that way. It took many years for a mathematical subject to be understood well enough that a cohesive textbook could be written. Mathematics is created through slow, incremental progress, large leaps, missteps, corrections, and connections." (Richard S Richeson, "Eulers Gem: The Polyhedron Formula and the birth of Topology", 2008)

"A proof in mathematics is a compelling argument that a proposition holds without exception; a disproof requires only the demonstration of an exception. A mathematical proof does not, in general, establish the empirical truth of whatever is proved. What it establishes is that whatever is proved - usually a theorem - follows logically from the givens, or axioms." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures and Proofs", 2009)

"It can be asserted that a 'proof' [...] is a psychological device for convincing the reader that an assertionis true. However our view in this book is more rigid: a proof is a sequence

of applications of the rules of logic to derive the assertion from the axioms. There is no room for opinion here. The axioms are plain. The rules are rigid. A proof is like a sequence of moves in a game of chess. If the rules are followed, then the proof is correct, otherwise it is not." (Steven G Krantz, "Essentials of Topology with Applications”, 2009)

"[…] proof is the key ingredient of the emotional side of mathematics; proof is the ultimate explanation of why something is true, and a good proof often has a powerful emotional impact, boosting confidence and encouraging further questions ‘why’." (Alexandre V Borovik, "Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice", 2009)

"The characterization of mathematics as a deductive discipline is accurate but incomplete. It represents the finished and polished consequences of the work of mathematicians, but it does not adequately represent the doing of mathematics. It describes theorem proofs but not theorem proving. Moreover, the history of mathematics is not the emotionless chronology of inventions of evermore esoteric formalisms that some people imagine it to be. It has its full share of color, mystery, and intrigue." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2009)

"The reasoning of the mathematician and that of the scientist are similar to a point. Both make conjectures often prompted by particular observations. Both advance tentative generalizations and look for supporting evidence of their validity. Both consider specific implications of their generalizations and put those implications to the test. Both attempt to understand their generalizations in the sense of finding explanations for them in terms of concepts with which they are already familiar. Both notice fragmentary regularities and - through a process that may include false starts and blind alleys - attempt to put the scattered details together into what appears to be a meaningful whole. At some point, however, the mathematician’s quest and that of the scientist diverge. For scientists, observation is the highest authority, whereas what mathematicians seek ultimately for their conjectures is deductive proof." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures and Proofs", 2009)

"Without denying the usefulness of the distinction between intuition and proof, I believe it can be drawn too sharply; intuition plays an essential role in the making and evaluating of proofs and is sometimes changed as a consequence of these processes. In this respect, the distinction is like that between creative and critical thinking; while this distinction too is a useful one, it is not possible to have either in any very satisfactory sense without the other." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2009)

✓On Proofs (1950-1959)

"We could compare mathematics so formalized to a game of chess in which the symbols correspond to the chessmen; the formulae, to definite positions of the men on the board; the axioms, to the initial positions of the chessmen; the directions for drawing conclusions, to the rules of movement; a proof, to a series of moves which leads from the initial position to a definite configuration of the men." (Friedrich Waismann, "Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics", 1951)

"Many proofs in mathematics have been actually found by extremely roundabout processes. A man starts to prove this theorem and he finds that he wanders all over the map. He starts off and prove a good many results which don’t seem to be leading anywhere and then eventually ends up by the back door on the solution of the given problem." (Claude E Shannon, "Creative Thinking", 1952)

"The diagrams incorporate a large amount of information. Their use provides extensive savings in space and in mental effort. In the case of many theorems, the setting up of the correct diagram is the major part of the proof. We therefore urge that the reader stop at the end of each theorem and attempt to construct for himself the relevant diagram before examining the one which is given in the text. Once this is done, the subsequent demonstration can be followed more readily; in fact, the reader can usually supply it himself." (Samuel Eilenberg & Norman E. Steenrod, "Foundations of Algebraic Topology", 1952)

"The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference." (George Pólya, "Mathematics and plausible reasoning" Vol. 1, 1954)

"We secure our mathematical knowledge by demonstrative reasoning, but we support our conjectures by plausible reasoning. A mathematical proof is demonstrative reasoning, but the inductive evidence of the physicist, the circumstantial evidence of the lawyer, the documentary evidence of the historian, and the statistical evidence of the economist belong to plausible reasoning." (George Pólya, "Mathematics and Plausible Reasoning", 1954)

"You have to guess the mathematical theorem before you prove it: you have to guess the idea of the proof before you carry through the details. You have to combine observations and follow analogies: you have to try and try again. The result of the mathematician’s creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing." (George Polya, "Mathematics and plausible reasoning" Vol. 1, 1954)

"[…] no branch of mathematics competes with projective geometry in originality of ideas, coordination of intuition in discovery and rigor in proof, purity of thought, logical finish, elegance of proofs and comprehensiveness of concepts. The science born of art proved to be an art." (Morris Kline, "Projective Geometry", Scientific America Vol. 192 (1), 1955)

"We speak in terms of ‘acceptance’, ‘confidence’, and ‘probability’, not ‘proof’. If by proof it is meant the establishment of eternal and absolute truth, open to no possible exception or modification, then proof has no place in the natural sciences." (George G Simpson, “Life: An Introduction to Biology”, 1957)

"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)

✓On Proofs (1850-1874)

"It is easily seen from a consideration of the nature of demonstration and analysis that there can and must be truths which cannot be reduced by any analysis to identities or to the principle of contradiction but which involve an infinite series of reasons which only God can see through." (Gottfried W Leibniz, "Nouvelles lettres et opuscules inédits", 1857)

"One now finds indeed approximately this number of real roots within these limits, and it is very probable that all roots are real. Certainly one would wish for a stricter proof here; I have meanwhile temporarily put aside the search for this after some fleeting futile attempts, as it appears unnecessary for the next objective of my investigation." (Bernhard Riemann, "On the Number of Prime Numbers less than a Given Quantity" ["Ueber die Anzahl der Primzahlen unter einer gegebenen Gröosse"], Monatsberichte der Berliner Akademie, 1859) 

"We must never assume that which is incapable of proof.”  (George H Lewes, “The Physiology of Common Life” Vol. 2, 1860)

"Few will deny that even in the first scientific instruction in mathematics the most rigorous method is to be given preference over all others. Especially will every teacher prefer a consistent proof to one which is based on fallacies or proceeds in a vicious circle, indeed it will be morally impossible for the teacher to present a proof of the latter kind consciously and thus in a sense deceive his pupils. Notwithstanding these objectionable so-called proofs, so far as the foundation and the development of the system is concerned, predominate in our textbooks to the present time. Perhaps it will be answered, that rigorous proof is found too difficult for the pupil’s power of comprehension. Should this be anywhere the case, - which would only indicate some defect in the plan or treatment of the whole, - the only remedy would be to merely state the theorem in a historic way, and forego a proof with the frank confession that no proof has been found which could be comprehended by the pupil; a remedy which is ever doubtful and should only be applied in the case of extreme necessity. But this remedy is to be preferred to a proof which is no proof, and is therefore either wholly unintelligible to the pupil, or deceives him with an appearance of knowledge which opens the door to all superficiality and lack of scientific method." (Hermann G Grassmann, "Stücke aus dem Lehrbuche der Arithmetik", 1861)

"Besides accustoming the student to demand complete proof, and to know when he has not obtained it, mathematical studies are of immense benefit to his education by habituating him to precision. It is one of the peculiar excellencies of mathematical discipline, that the mathematician is never satisfied with à peu près. He requires the exact truth." (John S Mill, "An Examination of Sir William Hamilton's Philosophy", 1865)

"It has long been a complaint against mathematicians that they are hard to convince: but it is a far greater disqualification both for philosophy, and for the affairs of life, to be too easily convinced; to have too low a standard of proof. The only sound intellects are those which, in the first instance, set their standards of proof high. Practice in concrete affairs soon teaches them to make the necessary abatement: but they retain the consciousness, without which there is no sound practical reasoning, that in accepting inferior evidence because there is no better to be had, they do not by that acceptance raise it to completeness." (John S Mill, "An Examination of Sir William Hamilton's Philosophy", 1865)

"Mathematical training is almost purely deductive. The mathematician starts with a few simple propositions, the proof of which is so obvious that they are called self-evident, and the rest of his work consists of subtle deductions from them." (Thomas H Huxley, "Scientific Education: Notes of an After Dinner Speech", Macmillan’s Magazine Vol. XX, 1869)

"The mathematician starts with a few propositions, the proof of which is so obvious that they are called self-evident, and the rest of his work consists of subtle deductions from them. The teaching of languages, at any rate as ordinarily practised, is of the same general nature: authority and tradition furnish the data, and the mental operations are deductive." (Thomas H Huxley, 1869)

"Proof requires a person who can give and a person who can receive." (Augustus De Morgan, "A Budget of Paradoxes", 1872)

"Simplification of modes of proof is not merely an indication of advance in our knowledge of a subject, but is also the surest guarantee of readiness for farther progress." (Lord Kelvin, "Elements of Natural Philosophy", 1873)

✓On Proofs (1800-1824)

"The integral along two different paths will always have the same value if it is never the case that φ(x) = ∞ in the space between the curves representing the paths. This is a beautiful theorem, whose not-too-difficult proof I will give at a suitable opportunity." (Carl F Gauss, [letter to Bessel] 1811)

"It is characteristic of higher arithmetic that many of its most beautiful theorems can be discovered by induction with the greatest of ease but have proofs that lie anywhere but near at hand and are often found only after many fruitless investigations with the aid of deep analysis and lucky combinations." (Carl F Gauss, 1817)

"Up to now, one as has not succeeded to find a rigourous proof of that truth [Euclid’s axiom on parallels]. Those which were given may be named only explanations, but do not deserve to be considered, in the full sense, mathematical proofs. " (Nikolai I Lobachevsky, 1823)

"Mathematical proofs, like diamonds, are hard and clear, and will be touched with nothing but strict reasoning." (John Locke, 1824)

"Mathematics in gross, it is plain, are a grievance in natural philosophy, and with reason. […] Mathematical proofs are out of the reach of topical arguments, and are not to be attacked by the equivocal use of words or declamation, that make so great a part of other discourses; nay, even of controversies." (John Locke, 1824)

✓On Proofs (1600-1699)

"It is very difficult to write mathematics books today. If one does not take pains with the fine points of theorems, explanations, proofs and corollaries, then it won’t be a mathematics book; but if one does these things, then the reading of it will be extremely boring." (Johannes Kepler, "Astronomia Nova", 1609)

"For, Mathematical Demonstrations being built upon the impregnable Foundations of Geometry and Arithmetick, are the only Truths, that can sink into the Mind of Man, void of all Uncertainty; and all other Discourses participate more or less of Truth, according as their Subjects are more or less capable of Mathematical Demonstration." (Christopher Wren, [lecture at Gresham College] 1657)

"There is scarcely any one who states purely arithmetical questions, scarcely any who understands them. Is this not because arithmetic has been treated up to this time geometrically rather than arithmetically? This certainly is indicated by many works ancient and modern. Diophantus himself also indicates this. But he has freed himself from geometry a little more than others have, in that he limits his analysis to rational numbers only; nevertheless the Zetcica of Vieta, in which the methods of Diophantus are extended to continuous magnitude and therefore to geometry, witness the insufficient separation of arithmetic from geometry. Now arithmetic has a special domain of its own, the theory of numbers. This was touched upon but only to a slight degree by Euclid in his Elements, and by those who followed him it has not been sufficiently extended, unless perchance it lies hid in those books of Diophantus which the ravages of time have destroyed. Arithmeticians have now to develop or restore it. To these, that I may lead the way, I propose this theorem to be proved or problem to be solved. If they succeed in discovering the proof or solution, they will acknowledge that questions of this kind are not inferior to the more celebrated ones from geometry either for depth or difficulty or method of proof: Given any number which is not a square, there also exists an infinite number of squares such that when multiplied into the given number and unity is added to the product, the result is a square." (Pierre de Fermat, [Letter to Frénicle] 1657)

"I am not content with algebra, in that it yields neither the shortest proofs nor the most beautiful constructions of geometry. Consequently, in view of this, I consider that we need yet another kind of analysis, geometric or linear, which deals directly with position, as algebra deals with magnitude." (Gottfried Leibniz, [letter to Christiaan Huygens] 1670)

"Algebra is nothing but the characteristic of undetermined numbers or magnitudes. But it does not directly express the place, angles and motions, from which it follows that it is often difficult to reduce, in a computation, what is in a figure, and that it is even more difficult to find geometrical proofs and constructions which are enough practical even when the Algebraic calculus is all done." (Gottfried W Leibniz, [letter to Christiaan Huygens] 1679)

"I found the elements of a new characteristic, completely different from Algebra and which will have great advantages for the exact and natural mental representation, although without figures, of everything that depends on the imagination. Algebra is nothing but the characteristic of undetermined numbers or magnitudes. But it does not directly express the place, angles and motions, from which it follows that it is often difficult to reduce, in a computation, what is in a figure, and that it is even more difficult to find geometrical proofs and constructions which are enough practical even when the Algebraic calculus is all done." (Gottfried W Leibniz, [letter to Christiaan Huygens] 1679)

"But the most powerful proof of the reality of phenomena (a proof which is, indeed, sufficient by itself) is success in predicting future phenomena from those which are past and present, whether the prediction be founded upon the success, so far, of a reason or hypothesis, or upon custom so far observed.” (Gottfried W Leibniz, "De Modo Distinguendi phenomena realia ab imaginariis" ["On the Method of Distinguishing Real from Imaginary Phenomena”], cca. 1684)

"Mathematics in gross, it is plain, are a grievance in natural philosophy, and with reason: for mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning. Mathematical proofs are out of the reach of topical arguments; and are not to be attacked by the equivocal use of words or declaration, that make so great a part of other discourses, - nay, even of controversies." (John Locke, 1690)

✓On Proofs (500-1599)

"Geometry enlightens the intellect and sets one's mind right. All of its proofs are very clear and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In this convenient way, the person who knows geometry acquires intelligence." (Ibn Khaldun, cca. 14th century)

"Among the various branches of philosophy, logic has two prerogatives: it has both the honor of coming first and the distinction of serving as an efficacious instrument throughout the whole body [of philosophy]. Natural and moral philosophers can construct their principles only by the forms of proof supplied by logicians. Also, in order to define and divide correctly, they must borrow and employ the art of the logicians. And if, perchance, they succeed in this without logic, their success is due to luck, rather than to science. Logic is 'rational' [philosophy], and we may readily see from the very name, what progress in philosophy can be expected from one who [since he lacks logic] lacks reason." (John of Salisbury, "Metalogicon", 1159)

"But every art has its own special methods, which we may figuratively characterize as its 'approaches' or 'keys'. Seeking is a necessary preliminary to finding, and one who cannot endure the hardship of inquiry cannot expect to harvest the fruit of knowledge. Demonstrative logic, however, seeks methods [of proof] involving necessity, and arguments which establish the essential identification of terms that cannot be thrust asunder. Only that which cannot possibly be otherwise is necessary." (John of Salisbury, "Metalogicon", 1159)

"The method of demonstration is therefore generally feeble and ineffective with regard to facts of nature (I refer to corporeal and changeable things). But it quickly recovers its strength when applied to the field of mathematics. For whatever it concludes in regard to such things as numbers, proportions and figures is indubitably true, and cannot be otherwise. One who wishes to become a master of the science of demonstration should first obtain a good grasp of probabilities. Whereas the principles of demonstrative logic are necessary; those of dialectic are probable." (John of Salisbury, "Metalogicon", 1159)

"Reason may be employed in two ways to establish a point: first for the purpose of furnishing sufficient proof of some principle, as in natural science, where sufficient proof can be brought to show that the movement of the heavens is always of uniform velocity. Reason is employed in another way, not as furnishing a sufficient proof of a principle, but as confirming an already established principle, by showing the congruity of its results […]" (St. Thomas Aquinas, "Summa Theologica", cca. 1266-1273)

"Geometry enlightens the intellect and sets one’s mind right. All of its proofs are very clear and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In this convenient way, the person who knows geometry acquires intelligence." (Ibn Khaldun, "The Muqaddimah: An Introduction to History", 1377)

"Reason may be employed in two ways to establish a point: first for the purpose of furnishing sufficient proof of some principle, as in natural science, where sufficient proof can be brought to show that the movement of the heavens is always of uniform velocity. Reason is employed in another way, not as furnishing a sufficient proof of a principle, but as confirming an already established principle, by showing the congruity of its results [...]" (Saint Thomas Aquinas, "Summa Theologica", 1485)

"At first, the thing seemed to me to be based more on sophism than on truth, but I searched until I found a proof." (Rafael Bombelli. "L'algebra", 1550 [1572]) [comment on the casus irreducibilis]

Proofs ( - 499 AD)

"It is evidently equally foolish to accept probable reasoning from a mathematician and to demand from a rhetorician demonstrative proofs." (Aristotle, cca. 4th century BC)

"Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth; synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it." (Eudoxus, cca. 4th century BC)

"Analysis is a method where one assumes that which is sought, and from this, through a series of implications, arrives at something which is agreed upon on the basis of synthesis; because in analysis, one assumes that which is sought to be known, proved, or constructed, and examines what this is a consequence of and from what this latter follows, so that by backtracking we end up with something that is already known or is part of the starting points of the theory; we call such a method analysis; it is, in a sense, a solution in reversed direction. In synthesis we work in the opposite direction: we assume the last result of the analysis to be true. Then we put the causes from analysis in their natural order, as consequences, and by putting these together we obtain the proof or the construction of that which is sought. We call this synthesis." (Pappus of Alexandria, cca. 4th century BC)

"In analysis, we start from what is required, we take it for granted, and we draw consequences from it, and consequences from the consequences, till we reach a point that we can use as starting point in synthesis. For in analysis we assume what is required to be done as already done (what is sought as already found, what we have to prove as true). We inquire from what antecedent the desired result could be derived; then we inquire again what could be the antecedent of that antecedent, and so on, until passing from antecedent to antecedent, we come eventually upon something already known or admittedly true. This procedure we call analysis, or solution backwards, or regressive reasoning." (Pappus of Alexandria, cca. 4th century BC)

"Now analysis is of two kinds; the one is the analysis of the ‘problems to prove’ and aims at establishing true theorems; the other is the analysis of the ‘problems to find’ and aims at finding the unknown."  (Pappus of Alexandria, cca. 4th century BC)

"It is of course easier to supply the proof when we have previously acquired some knowledge of the questions by the method, than it is to find it without any previous knowledge." (Archimedes, cca. 3rd century BC)

"It is not possible to find in all geometry more difficult and more intricate questions or more simple and lucid explanations [than those given by Archimedes]. Some ascribe this to his natural genius; while others think that incredible effort and toil produced these, to all appearance, easy and unlaboured results. No amount of investigation of yours would succeed in attaining the proof, and yet, once seen, you immediately believe you would have discovered it; by so smooth and so rapid a path he leads you to the conclusion required." (Plutarch, cca. 1st century)

🥀On Beauty: Beauty & Mathematics (2000-2019)

"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)

“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)

“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)

"Mathematicians often get bored by a problem after they have fully understood it and have given proofs of their conjectures. Sometimes they even forget the precise details of what they have done after the lapse of years, having refocused their interest in another area. The common notion of the mathematician contemplating timeless truths, thinking over the same proof again and again - Euclid looking on beauty bare - is rarely true in any static sense." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Still, in the end, we find ourselves drawn to the beauty of the patterns themselves, and the amazing fact that we humans are smart enough to prove even a feeble fraction of all possible theorems about them. Often, greater than the contemplation of this beauty for the active mathematician is the excitement of the chase. Trying to discover first what patterns actually do or do not occur, then finding the correct statement of a conjecture, and finally proving it - these things are exhilarating when accomplished successfully. Like all risk-takers, mathematicians labor months or years for these moments of success." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Arguments for the Riemann hypothesis often include its widespread ramifications and appeals to mathematical beauty; however, we also have a large corpus of hard facts. With the advent of powerful computational tools over the last century, mathematicians have increasingly turned to computational evidence to support conjectures, and the Riemann hypothesis is no exception." (Peter Borwein et al, "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", 2007)

"I think that the beauty of mathematics lies in uncovering the hidden simplicity and complexity that coexist in the rigid logical framework that the subject imposes." (David Ruelle, "The Mathematician's Brain", 2007)

"[...] it is while doing mathematical research that one truly comes to see the beauty of mathematics. It faces you in those moments when the underlying simplicity of a question appears and its meaningless complications can be forgotten. In those moments a piece of a colossal logical structure is illumi￾nated, and some of the meaning hidden in the nature of things is finally revealed." (David Ruelle, "The Mathematician's Brain", 2007)

“The immediate evidence from the natural world may seem to be chaotic and without any inner regularity, but mathematics reveals that under the surface the world of nature has an unexpected simplicity - an extraordinary beauty and order.” (William Byers, “How Mathematicians Think”, 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, beauty is a very important ingredient. Beauty exists in mathematics as in architecture and other things. It is a difficult thing to define but it is something you recognise when you see it. It certainly has to have elegance, simplicity, structure and form. All sorts of things make up real beauty. There are many different kinds of beauty and the same is true of mathematical theorems. Beauty is an important criterion in mathematics because basically there is a lot of choice in what you can do in mathematics and science. It determines what you regard as important and what is not.” (Michael Atiyah, 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)

“Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion - not because it makes no sense to you, but because you gave it sense and you still don't understand what your creation is up to; to have a break-through idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it.” (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)

“Mathematicians are sometimes described as living in an ideal world of beauty and harmony. Instead, our world is torn apart by inconsistencies, plagued by non sequiturs and, worst of all, made desolate and empty by missing links between words, and between symbols and their referents; we spend our lives patching and repairing it. Only when the last crack disappears are we rewarded by brief moments of harmony and joy.” (Alexandre V Borovik, “Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice”, 2009)

“What is the basis of this interest in beauty? Is it the same in both mathematics and science? Is it rational, in either case, to expect or demand that the products of the discipline satisfy such a criterion? Is there an underlying assumption that the proper business of mathematics and science is to discover what can be discovered about reality and that truth - mathematical and physical - when seen as clearly as possible, must be beautiful? If the demand for beauty stems from some such assumption, is the assumption itself an article of blind faith? If such an assumption is not its basis, what is?” (Raymond S Nickerson, “Mathematical Reasoning:  Patterns, Problems, Conjectures, and Proofs”, 2010)

"You do not study mathematics because it helps you build a bridge. You study mathematics because it is the poetry of the universe. Its beauty transcends mere things." (Jonathan D Farley, 2011)

"Mathematics leads to simple and beautiful conclusions by ignoring and idealizing various factors in order to abstract. However, reality is both more complex and discrete, and in many cases mathematical conclusions are only approximately valid. There is also the risk that the assumptions used for valuation models differ from the market (model risk), as well as issues with the reliability of the parameters used in the model (parameter risk)." (Kenichi Watanabe, "The Role of Mathematics in Finance: Applied Mathematics and Risk", 2013)

🥀On Beauty: Beauty & Mathematics VI (Unsourced)

“The beauty in mathematics is seeing the truth without effort.” (George Pólya)

"Mathematics has beauty and romance. It's not a boring place to be, the mathematical world. It's an extraordinary place; it's worth spending time there." (Marcus Du Sautoy)

"When I am working on a problem I never think about beauty. I only think about how to solve the problem. But when I have finished, if the solution is not beautiful, I know it is wrong." (Buckminster Fuller)

"[…] the feeling of mathematical beauty, of the harmony of numbers and of forms, of geometric elegance. It is a genuinely esthetic feeling, which all mathematicians know. And this is sensitivity." (Henri Poincaré)

“The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living. Of course I do not here speak of that beauty that strikes the senses, the beauty of qualities and appearances; not that I undervalue such beauty, far from it, but it has nothing to do with science; I mean that profounder beauty which comes from the harmonious order of the parts, and which a pure intelligence can grasp.” (Henri Poincaré)

"All mathematicians share ... a sense of amazement over the infinite depth and the mysterious beauty and usefulness of mathematics." (Martin Gardner)

"There are some traits all mathematicians share. An obvious one is a sense of amazement over the infinite depth and the mysterious beauty and usefulness of mathematics." (Martin Gardner)

"I was struck by the art with which mathematicians remove, reject, and little by little eliminate everything that is not necessary for expressing the absolute with the least possible number of terms, while preserving in the arrangement of these terms a discrimination, a parallelism, a symmetry which seems to be the visible elegance and beauty of an eternal idea. (Edgar Quinet)