On Art: Poetry and Mathematics IV

"The belief that mathematics, because it is abstract, because it is static and cold and gray, is detached from life, is a mistaken belief. Mathematics, even in its purest and most abstract estate, is not detached from life. It is just the ideal handling of the problems of life, as sculpture may idealize a human figure or as poetry or painting may idealize a figure or a scene. Mathematics is precisely the ideal handling of the problems of life, and the central ideas of the science, the great concepts about which its stately doctrines have been built up, are precisely the chief ideas with which life must always deal and which, as it tumbles and rolls about them through time and space, give it its interests and problems, and its order and rationality." (Cassius J Keyser, "The Humanization of the Teaching of Mathematics", 1912)

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty - a beauty, cold and austere, like that of a sculpture. without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry." (Bertrand Russell, "The Study of Mathematics", 1919)

"Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature." (Albert Einstein, 1935)

"Mathematics is a form of poetry which transcends poetry in that it proclaims a truth; a form of reasoning which transcends reasoning in that it wants to bring about the truth it proclaims; a form of action, of ritual behavior, which does not find fulfilment in the act but must proclaim and elaborate a poetic form of truth." (Salomon Bochner, "Why Mathematics Grows", Journal of the History of Ideas, 1965)

"What binds us to space-time is our rest mass, which prevents us from flying at the speed of light, when time stops and space loses meaning. In a world of light there are neither points nor moments of time; beings woven from light would live ‘nowhere’ and ‘nowhen’; only poetry and mathematics are capable of speaking meaningfully about such things." (Yuri I Manin, "Space-Time as a Physical System", 1981)

 "Mathematical modeling is about rules - the rules of reality. What distinguishes a mathematical model from, say, a poem, a song, a portrait or any other kind of ‘model’, is that the mathematical model is an image or picture of reality painted with logical symbols instead of with words, sounds or watercolors." (John Casti, "Reality Rules", 1992)

"Whatever the ins and outs of poetry, one thing is clear: the manner of expression - notation - is fundamental. It is the same with mathematics - not in the aesthetic sense that the beauty of mathematics is tied up with how it is expressed - but in the sense that mathematical truths are revealed, exploited and developed by various notational innovations." (James R Brown, "Philosophy of Mathematics", 1999)

"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)

"Poetry and code - and mathematics - make us read differently from other forms of writing. Written poetry makes the silent reader read three kinds of pattern at once; code moves the reader from a static to an active, interactive and looped domain; while algebraic topology allows us to read qualitative forms and their transformations." (Stephanie Strickland & Cynthia L Jaramillo, "Dovetailing Details Fly Apart - All over, again, in code, in poetry, in chreods", 2007)

"Symmetry is a fundamental organizing principle of shape. It helps in classifying and understanding patterns in mathematics, nature, art, and, of course, poetry. And often the counterpoint to symmetry - the breaking or interruption of symmetry - is just as important in creative endeavors." (Marcia Birken & Anne C. Coon, "Discovering Patterns in Mathematics and Poetry", 2008)

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On Art: Poetry and Mathematics III

"Inspiration is needed in geometry, just as much as in poetry." (Aleksandr Pushkin)

"Mathematics is very much like poetry […] what makes a good poem, a great poem; is that there is a large amount of thought expressed in very few words.” (Lipman Bers)

"While most of us were just trying to learn to arrange logical statements into coherent arguments, Ted was quietly solving open problems and creating new mathematics. It was as if he could write poetry while the rest of us were trying to learn grammar." (Joel Shapiro)

“The poetry of science is in some sense embodied in its great equations.” (Graham Farmelo)

“Mathematics is, as it were, a sensuous logic, and relates to philosophy as do the arts, music, and plastic art to poetry.” (Friedrich von Schlegel)

“Proofs are to mathematics what spelling (or even calligraphy) is to poetry. Mathematical works do consist of proofs, just as poems do consist of words.” (Vladimir Arnold)

“We especially need imagination in science. It is not all mathematics, nor all logic, but it is somewhat beauty and poetry.” (Maria Mitchell)

“Mathematics and poetry are the two ways to drink the beauty of truth.” (Amit Ray)

“Poetry is a mystic, sensuous mathematics of fire, smoke-stacks, waffles, pansies, people, and purple sunsets.” (Carl Sandburg)

“The mathematics of rhythm are universal. They don't belong to any particular culture.” (John McLaughlin)

On Art: Poetry and Science II

"In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in poetry, it’s the exact opposite." (Paul A M Dirac)

"In the earliest ages science was poetry, as in the latter poetry has become science." (James Russell Lowell)

"Philosophy becomes poetry, and science imagination, in the enthusiasm of genius." (Isaac Disraeli)

"Science and art, or by the same token, poetry and prose differ from one another like a journey and an excursion. The purpose of the journey is its goal, the purpose of an excursion is the process." (Franz Grillparzer)

“Science and poetry are, in fact, inseparable. By providing a vision of life, of Earth, of the universe in all its splendor, science does not challenge human values; it can inspire human values. It does not negate faith; it celebrates faith.” (Jacques-Yves Cousteau)

“Science boasts of the distance of its stars; of the terrific remoteness of the things of which it has to speak. But poetry and religion always insist upon the proximity, the almost menacing closeness of the things with which they are concerned.” (Gilbert Keith Chesterton)

“Science is for those who learn; poetry for those who know.” (Joseph Roux)

"Science is the labor and handicraft of the mind; poetry can only be considered its recreation." (Sir Francis Bacon)

“Science sees signs: poetry, the thing signified.” (Julius Hare)

“There is poetry in science and the cultivation of the imagination is an essential prerequisite to the successful investigation of nature.” (Joseph Henry)

"We especially need imagination in science. It is not all mathematics, nor all logic, but it is somewhat beauty and poetry." (Maria Mitchell)

“You may translate books of science exactly. […] The beauties of poetry cannot be preserved in any language except that in which it was originally written.” (Samuel Johnson)

The Infinite and Its Difficulties II

“The existence of an actual infinite multitude is impossible. For any set of things one considers must be a specific set. And sets of things are specified by the number of things in them. Now no number is infinite, for number results from counting through a set of units. So no set of things can actually be inherently unlimited, nor can it happen to be unlimited.” (Thomas Aquinas, “Summa theologia”, 13th century) 

"[Paradoxes of the infinite arise] only when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited; but this […] is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another.” (Galileo Galilei, "Two New Sciences", 1638)

“Infinities and indivisibles transcend our finite understanding, the former on account of their magnitude, the latter because of their smallness; Imagine what they are when combined. In spite of this men cannot refrain from discussing them.” (Galileo Galilei, "Two New Sciences", 1638)

“Whatever we imagine is finite. Therefore, there is no idea or conception of anything we call finite. No man can have in his mind an image of infinite magnitude; nor conceive infinite swiftness, infinite time, or infinite force, or inmate power.” (Thomas Hobbes, "Of Man", 1658)

“Man is equally incapable of seeing the nothingness from which he emerges and the infinity in which he is engulfed.” (Blaise Pascal, "Pensées", 1670)

“Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. From this it follows not only that they remain on the fringes, but in addition they entertain strange ideas about the concept of the infinite, which they must try to use." (Leonhard Euler, "Introduction to Analysis of the Infinite", 1748)

 “A great deal of misunderstanding is avoided if it be remembered that the terms infinity, infinite, zero, infinitesimal must be interpreted in connexion with their context, and admit a variety of meanings according to the way in which they are defined.” (George B Mathews, “Theory of Numbers”, 1892)

“Like children who are not permitted to do certain things, we are not permitted by nature to think in terms of infinity.” (Robert Tuttle Morris, “Microbes and Men”, 1916)

 "The infinite in mathematics is always unruly unless it is properly treated."  (Edward Kasner & James Newman, “Mathematics and the Imagination”, 1940)

“I am incapable of conceiving infinity, and yet I do not accept finity.” (Simone de Beauvoir, “La Vieillesse”, 1970)

On Problem Solving X: Mathematicians II

“More than any other science, mathematics develops through a sequence of consecutive abstractions. A desire to avoid mistakes forces mathematicians to find and isolate the essence of problems and the entities considered. Carried to an extreme, this procedure justifies the well-known joke that a mathematician is a scientist who knows neither what he is talking about nor whether whatever he is talking about exists or not.” (Élie Cartan) 

"Mathematics consists of content and know-how. What is know-how in mathematics? The ability to solve problems." (George Polya) 

"The science of physics does not only give us [mathematicians] an oportunity to solve problems, but helps us also to discover the means of solving them, and it does this in two ways: it leads us to anticipate the solution and suggests suitable lines of argument." (Henri Poincaré)
 
“Solving problems can be regarded as the most characteristically human activity.” (George Polya, 1981)

 “Solving problems is the specific achievement of intelligence.” (George Polya, 1957)

"Mystery is found as much in mathematics as in detective stories. Indeed, the mathematician could well be described as a detective, brilliantly exploiting a few initial clues to solve the problem and reveal its innermost secrets. An especially mathematical mystery is that you can often search for some mathematical object, and actually know a lot about it, if it exists, only to discover that in fact it does not exist at all - you knew a lot about something which cannot be." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Mental acuity of any kind comes from solving problems yourself, not from being told how to solve them.” (Paul Lockhart, “A Mathematician's Lament”, 2009)
 
“To learn mathematics is to learn mathematical problem solving.” (Patrik W Thompson, 1985)

„[...] mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It’s the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation.“ (Ivars Peterson, „Islands of Truth: A Mathematical Mystery Cruise“, 1990)

"The real raison d'etre for the mathematician's existence is simply to solve problems. So what mathematics really consists of is problems and solutions. And it is the 'good' problems, the ones that challenge the greatest minds for decades, if not centuries, that eventually become enshrined as mathematical mountaintops." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"For solving problems in the real world, not only a knowledge of mathematics is useful, but also the mathematician is meaningful: the mathematician uses specific ways of thinking such as abstraction, generalization, and extraction of truth from various realities to crystallize to a simple statement as a theorem. Moreover, in various applications, mathematicians do not merely apply knowledge in an existing specific area of mathematics, but can build up a new theoretical system of mathematics for solving concrete problems." (Masahiro Yamamoto, "Mathematics for Industry: Principle, Reality and Practice, from the Point of View of a Mathematician", [in "What Mathematics Can Do for You"] 2013)

"What I really am is a mathematician. Rather than being remembered as the first woman this or that, I would prefer to be remembered, as a mathematician should, simply for the theorems I have proved and the problems I have solved.“ (Julia Robinson)

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On Theorems (until 1699)

"How many theorems in geometry which have seemed at first impracticable are in time successfully worked out!" (Archimedes, "On Spirals", cca. 225 BC)

"I am persuaded that this method [for calculating the volume of a sphere] will be of no little service to mathematics. For I foresee that once it is understood and established, it will be used to discover other theorems which have not yet occurred to me, by other mathematicians, now living or yet unborn." (Archimedes, "On Spirals", cca. 225 BC)

“It is essential that the treatment [of geometry] should be rid of everything superfluous, for the superfluous is an obstacle to the acquisition of knowledge; it should select everything that embraces the subject and brings it to a focus, for this is of the highest service to science; it must have great regard both to clearness and to conciseness, for their opposites trouble our understanding; it must aim to generalize its theorems, for the division of knowledge into small elements renders it difficult of comprehension.” (Proclus, cca. 5th century)

"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)

“The Excellence of Modern Geometry is in nothing more evident, than in those full and adequate Solutions it gives to Problems; representing all possible Cases in one view, and in one general Theorem many times comprehending whole Sciences; which deduced at length into Propositions, and demonstrated after the manner of the Ancients, might well become the subjects of large Treatises: For whatsoever Theorem solves the most complicated Problem of the kind, does with a due Reduction reach all the subordinate Cases.” (Edmund Halley, “An Instance of the Excellence of Modern Algebra in the resolution of the problem of finding the foci of optic glasses universally”, Philosophical Transactions, 1694)

On Theorems (1700-1799)

“[…] for the saving the long progression of the thoughts to remote and first principles in every case, the mind should provide itself several stages; that is to say, intermediate principles, which it might have recourse to in the examining those positions that come in its way. These, though they are not self-evident principles, yet, if they have been made out from them by a wary and unquestionable deduction, may be depended on as certain and infallible truths, and serve as unquestionable truths to prove other points depending upon them, by a nearer and shorter view than remote and general maxims. […] And thus mathematicians do, who do not in every new problem run it back to the first axioms through all the whole train of intermediate propositions. Certain theorems that they have settled to themselves upon sure demonstration, serve to resolve to them multitudes of propositions which depend on them, and are as firmly made out from thence as if the mind went afresh over every link of the whole chain that tie them to first self-evident principles.” John Locke, “The Conduct of the Understanding”, 1706)
 
„It may be observed of mathematicians that they only meddle with such things as are certain, passing by those that are doubtful and unknown. They profess not to know all things, neither do they affect to speak of all things. What they know to be true, and can make good by invincible arguments, that they publish and insert among their theorems. Of other things they are silent and pass no judgment at all, choosing rather to acknowledge their ignorance, than affirm anything rashly.“ (Isaac Barrow, „Mathematical Lecture“, 1734)

“One should not be deceived by philosophical works that pretend to be mathematical, but are merely dubious and murky metaphysics. Just because a philosopher can recite the words lemma, theorem and corollary doesn't mean that his work has the certainty of mathematics. That certainty does not derive from big words, or even from the method used by geometers, but rather from the utter simplicity of the objects considered by mathematics.” (Pierre L Maupertuis, "Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique", 1746)

“The advantages which mathematics derives from the peculiar nature of those relations about which it is conversant, from its simple and definite phraseology, and from the severe logic so admirably displayed in the concatenation of its innumerable theorems, are indeed immense, and well entitled to separate and ample illustration.” (Dugald Stewart, “Philosophy of the Human Mind”, 1792)

See also:
Theorems I, III, IV, V, VI, VII, VIII, IX, X
Proofs I, II, III, IV, V,. VI, VII, VIII, IX

On Theorems (1925-1949)

“How many properties were there of which the compass knew nothing, how many cunning laws lay contained in embryo within an equation, the mysterious nut which must be artistically cracked to extract the rich kernel, the theorem!” (Jean-Henri Fabre, “The Life of the Fly”, 1925)

„Any mathematical science is a body of theorems deduced from a set of axioms. A geometry is a mathematical science. The question then arises why the name geometry is given to some mathematical sciences and not to others. It is likely that there is no definite answer to this question, but that a branch of mathematics is called a geometry because the name seems good, on emotional and  people.“ (John H C Whitehead, „The Foundation of Differential Geometry“, 1932)

„[...] the abstract mathematical theory has an independent, if lonely existence of its own. But when a sufficient number of its terms are given physical definitions it becomes a part of a vital organism concerning itself at every instant with matters full of human significance. Every theorem can be given the form ‘if you do so and so, such and such will happen'.“ (Oswald Veblen, “Remarks on the Foundation of Geometry”, Bulletin of the American Mathematical Society, Vol. 35, 1935)

“All the theories and hypotheses of empirical science share this provisional character of being established and accepted ‘until further notice’, whereas a mathematical theorem, once proved, is established once and for all; it holds with that particular certainty which no subsequent empirical discoveries, however unexpected and extraordinary, can ever affect to the slightest extent.” (Carl G Hempel, "Geometry and Empirical Science”, 1935)

 „The mathematician is still regarded as the hermit who knows little of the ways of life outside his cell, who spends his time compounding incredible and incomprehensible theorems in a strange, clipped, unintelligible jargon.“ (James R Newman, „Mathematics and the Imagination“, 1940)

 “Perhaps the greatest paradox of all is that there are paradoxes in mathematics […] because mathematics builds on the old but does not discard it, because its theorems are deduced from postulates by the methods of logic, in spite of its having undergone revolutionary changes we do not suspect it of being a discipline capable of engendering paradoxes.” (James R Newman, “Mathematics and the Imagination”, 1940)

„It is a melancholic experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done [...] there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.“  (Godfrey H Hardy, „A Mathematician's Apology“, 1941)

“The fact that the proof of a theorem consists in the application of certain simple rules of logic does not dispose of the creative element in mathematics, which lies in the choice of the possibilities to be examined.” (Richard Courant & Herbert Robbins, “What Is Mathematics?: An Elementary Approach to Ideas and Methods”, 1941)

"Exact figures have, in principle, the same role in geometry as exact measurements in physics; but, in practice, exact figures are less important than exact measurements because the theorems of geometry are much more extensively verified than the laws of physics. The beginner, however, should construct many figures as exactly as he can in order to acquire a good experimental basis; and exact figures may suggest geometric theorems also to the more advanced. Yet, for the purpose of reasoning, carefully drawn free-hand figures are usually good enough, and they are much more quickly done." (George Pólya, "How to solve it", 1945)

"The materials necessary for solving a mathematical problem are certain relevant items of our formerly acquired mathematical knowledge, as formerly solved problems, or formerly proved theorems. Thus, it is often appropriate to start the work with the question; Do you know a related problem?" (George Pólya, "How to Solve It", 1945)

See also:
Theorems I, II, III, IV, VI, VII, VIII, IX, X
Proofs I, II, III, IV, V,. VI, VII, VIII, IX

On Theorems (2000-2009)

"Theorems are fun especially when you are the prover, but then the pleasure fades. What keeps us going are the unsolved problems." (Carl Pomerance, 2000)

"Despite the unworldly nature of mathematics, mathematicians still have egos that need massaging. Nothing acts as a better drive to the creative process than the thought of the immortality bestowed by having your name attached to a theorem.” (Marcus du Sautoy, "The Music of the Primes”, 2003)

"A theorem is never arrived at in the way that logical thought would lead you to believe or that posterity thinks. It is usually much more accidental, some chance discovery in answer to some kind of question. Eventually you can rationalize it and say that this is how it fits. Discoveries never happen as neatly as that. You can rewrite history and make it look much more logical, but actually it happens quite differently.” (Sir Michael Atiyah, 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.", ‘Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences’ Vol. 363 (1835), 2005)

"Some number patterns, like even and odd numbers, lie on the surface. But the more you learn about numbers, both experimentally and theoretically, the more you discover patterns that are not so obvious. […] After a hidden pattern is exposed, it can be used to find more hidden patterns. At the end of a long chain of patterned reasoning, you can get to very difficult theorems, exploring facts about numbers that you otherwise would not know were true." (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)

"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)

"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)

"Obviously, the final goal of scientists and mathematicians is not simply the accumulation of facts and lists of formulas, but rather they seek to understand the patterns, organizing principles, and relationships between these facts to form theorems and entirely new branches of human thought.” (Clifford A Pickover, "The Math Book”, 2009)

On Theorems (1970-1979)

“In many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance.” (Martin Gardner, “Mathematical Games”, Scientific American, 1973)

“The world is anxious to admire that apex and culmination of modern mathematics: a theorem so perfectly general that no particular application of it is feasible.” (George Pólya, “A Story With a Moral”, Mathematical Gazette 57 (400), 1973) 

 „For hundreds of pages the closely-reasoned arguments unroll, axioms and theorems interlock. And what remains with us in the end? A general sense that the world can be expressed in closely-reasoned arguments, in interlocking axioms and theorems.“ (Michael Frayn, „Constructions“, 1974)

“Mathematics does not grow through a monotonous increase of the number of indubitably established theorems, but through the incessant improvement of guesses by speculation and criticism.” (Imre Lakatos, "Proofs and Refutations", 1976)

“The esthetic side of mathematics has been of overwhelming importance throughout its growth. It is not so much whether a theorem is useful that matters, but how elegant it is.” (Stanislaw Ulam “Adventures of a Mathematician”, 1976)

“At the heart of mathematics is a constant search for simpler and simpler ways to prove theorems  and solve problems.” (Martin Gardner, “Aha! Insight”, 1978)

 “A good theorem will almost always have a wide-ranging influence on later mathematics, simply by virtue of the fact that it is true. Since it is true, it must be true for some reason; and if that reason lies deep, then the uncovering of it will usually require a deeper understanding of neighboring facts and principles.” (Ian Richards, “Number theory”, 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)

See also:
Theorems I, II, III, IV, V, VI, VIII, IX, X
Proofs I, II, III, IV, V,. VI, VII, VIII, IX

On Theorems (1800-1899)

"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 Friedrich Gauss, 1817)

 "Every theorem in geometry is a law of external nature, and might have been ascertained by generalizing from observation and experiment, which in this case resolve themselves into comparisons and measurements. But it was found practicable, and being practicable was desirable, to deduce these truths by ratiocination from a small number of general laws of nature, the certainty and universality of which was obvious to the most careless observer, and which compose the first principles and ultimate premises of the science." (John S Mill, "System of Logic", 1843)

"Mathematics is peculiarly and preeminently the science of relations, and whether quantity or direction may severally form its object, these are never contemplated in characters purely absolute, but invariably in comparison with other objects like themselves; and it is hence that relations once established by the unerring theorems of the science, we are enabled, disregarding magnitude in itself, to pass indifferently from the finite to the infinite, from the limited regions of sense to those of conception, and with all the assurance and all the certainty that even the geometry of the ancients could confer." (John H W Waugh, Mathematical Essays", 1854)

"In abstract mathematical theorems the approximation to absolute truth is perfect, because we can treat of infinitesimals. In physical science, on the contrary, we treat of the least quantities which are perceptible." (William S Jevons, „The Principles of Science: A Treatise on Logic and Scientific Method", 1887)

"I compare arithmetic with a tree that unfolds upwards in a multitude of techniques and theorems while the root drives into the depths." (Gottlob Frege, "Grundgesetze der Arithmetik", 1893) 

"Many theorems are obvious upon looking at a moderately-sized figure; but the reasoning must be such as to convince the mind of their truth when, from excessive increase or diminution of the scale, the figures themselves have past the boundary even of imagination." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1898)

"In mathematics we see the conscious logical activity of our mind in its purest and most perfect form; here is made manifest to us all the labor and the great care with which it progresses, the precision which is necessary to determine exactly the source of the established general theorems, and the difficulty with which we form and comprehend abstract conceptions; but we also learn here to have confidence in the certainty, breadth, and fruitfulness of such intellectual labor." (Hermann von Helmholtz, "Vorträge und Reden", 1896)

See also:
Theorems I, II, IV, V, VI, VII, VIII, IX, X
Proofs I, II, III, IV, V,. VI, VII, VIII, IX

On Theorems (1990-1999)

"[…] mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It’s the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation." (Ivars Peterson, "Islands of Truth", 1990)

"A distinctive feature of mathematics, that feature in virtue of which it stands as a paradigmatically rational discipline, is that assertions are not accepted without proof. […] By proof is meant a deductively valid, rationally compelling argument which shows why this must be so, given what it is to be a triangle. But arguments always have premises so that if there are to be any proofs there must also be starting points, premises which are agreed to be necessarily true, self-evident, neither capable of, nor standing in need of, further justification. The conception of mathematics as a discipline in which proofs are required must therefore also be a conception of a discipline in which a systematic and hierarchical order is imposed on its various branches. Some propositions appear as first principles, accepted without proof, and others are ordered on the basis of how directly they can be proved from these first principle. Basic theorems, once proved, are then used to prove further results, and so on. Thus there is a sense in which, so long as mathematicians demand and provide proofs, they must necessarily organize their discipline along lines approximating to the pattern to be found in Euclid's Elements." (Mary Tiles, "Mathematics and the Image of Reason", 1991)

"[...] there is no criterion for appreciation which does not vary from one epoch to another and from one mathematician to another. [...] These divergences in taste recall the quarrels aroused by works of art, and it is a fact that mathematicians often discuss among themselves whether a theorem is more or less 'beautiful'. This never fails to surprise practitioners of other sciences: for them the sole criterion is the 'truth' of a theory or formula." (Jean Dieudonné, "Mathematics – The Music of Reason", 1992)

"An intuitive proof allows you to understand why the theorem must be true; the logic merely provides firm grounds to show that it is true." (Ian Stewart, "Concepts of Modern Mathematics",  1995)

"Mathematics is about theorems: how to find them; how to prove them; how to generalize them; how to use them; how to understand them. […] But great theorems do not stand in isolation; they lead to great theories. […] And great theories in mathematics are like great poems, great paintings, or great literature: it takes time for them to mature and be recognized as being 'great'." (John L Casti, "Five Golden Rules", 1995)

"To be an engineer, and build a marvelous machine, and to see the beauty of its operation is as valid an experience of beauty as a mathematician's absorption in a wondrous theorem. One is not ‘more’ beautiful than the other. To see a space shuttle standing on the launch pad, the vented gases escaping, and witness the thunderous blast-off as it climbs heavenward on a pillar of flame - this is beauty. Yet it is a prime example of applied mathematics.” (Calvin C Clawson, “Mathematical Mysteries”, 1996)

"The lack of beauty in a piece of mathematics is of frequent occurrence, and it is a strong motivation for further mathematical research. Lack of beauty is associated with lack of definitiveness. A beautiful proof is more often than not the definitive proof (though a definitive proof need not be beautiful); a beautiful theorem is not likely to be improved upon or generalized." (Gian-Carlo Rota, "The phenomenology of mathematical proof", Synthese, 111(2), 1997)

"The most common instance of beauty in mathematics is a brilliant step in an otherwise undistinguished proof. […] A beautiful theorem may not be blessed with an equally beautiful proof; beautiful theorems with ugly proofs frequently occur. When a beautiful theorem is missing a beautiful proof, attempts are made by mathematicians to provide new proofs that will match the beauty of the theorem, with varying success. It is, however, impossible to find beautiful proofs of theorems that are not beautiful." (Gian-Carlo Rota, "The Phenomenology of Mathematical Beauty", 1997)

"Mathematical truth is found to exceed the proving of theorems and to elude total capture in the confining meshes of any logical net." (John Polkinghorne, "Belief in God in an Age of Science", 1998)

"A mathematician experiments, amasses information, makes a conjecture, finds out that it does not work, gets confused and then tries to recover. A good mathematician eventually does so – and proves a theorem." (Steven Krantz, "Conformal Mappings", American Scientist, Sept.–Oct. 1999)

"Let us regard a proof of an assertion as a purely mechanical procedure using precise rules of inference starting with a few unassailable axioms. This means that an algorithm can be devised for testing the validity of an alleged proof simply by checking the successive steps of the argument; the rules of inference constitute an algorithm for generating all the statements that can be deduced in a finite number of steps from the axioms." (Edward Beltrami, "What is Random?: Chaos and Order in Mathematics and Life", 1999)

"Mathematicians, like the rest of us, cherish clever ideas; in particular they delight in an ingenious picture. But this appreciation does not overwhelm a prevailing skepticism. After all, a diagram is - at best - just a special case and so can't establish a general theorem. Even worse, it can be downright misleading. Though not universal, the prevailing attitude is that pictures are really no more than heuristic devices; they are psychologically suggestive and pedagogically important - but they prove nothing. I want to oppose this view and to make a case for pictures having a legitimate role to play as evidence and justification - a role well beyond the heuristic.  In short, pictures can prove theorems." (James R Brown, "Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures, 1999)

See also:
Theorems I, II, III, IV, V, VI, VII, VIII, X
Proofs I, II, III, IV, V,. VI, VII, VIII, IX

On Theorems (1950-1969)

“On the basis of what has been proved so far, it remains possible that there may exist (and even be empirically discoverable) a theorem-proving machine which in fact is equivalent to mathematical intuition, but cannot be proved to be so, nor even be proved to yield only correct theorems of finitary number theory.” (Kurt Gödel, 1951)

"Mathematicians do not know what they are talking about because pure mathematics is not concerned with physical meaning. Mathematicians never know whether what they are saying is true because, as pure mathematicians, they make no effort to ascertain whether their theorems are true assertions about the physical world." (Morris Kline, “Mathematics in Western Culture”, 1953)

"The construction of hypotheses is a creative act of inspiration, intuition, invention; its essence is the vision of something new in familiar material. The process must be discussed in psychological, not logical, categories; studied in autobiographies and biographies, not treatises on scientific method; and promoted by maxim and example, not syllogism or theorem." (Milton Friedman, "Essays in Positive Economics", 1953)

“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)

“Mathematics, springing from the soil of basic human experience with numbers and data and space and motion, builds up a far-flung architectural structure composed of theorems which reveal insights into the reasons behind appearances and of concepts which relate totally disparate concrete ideas.” (Saunders MacLane, “Of Course and Courses”, The American Mathematical Monthly, Vol. 61, No. 3, March, 1954)

”Mathematics is a creation of the mind. To begin with, there is a collection of things, which exist only in the mind, assumed to be distinguishable from one another; and there is a collection of statements about these things, which are taken for granted. Starting with the assumed statements concerning these invented or imagined things, the mathematician discovers other statements, called theorems, and proves them as necessary consequences. This, in brief, is the pattern of mathematics. The mathematician is an artist whose medium is the mind and whose creations are ideas.” (Hubert Stanley Wall, “Creative Mathematics”, 1963)

“So the first thing we have to accept is that even in mathematics you can start in different places. If all these various theorems are interconnected by reasoning there is no real way to say ‘These are the most fundamental axioms’, because if you were told something different instead you could also run the reasoning the other way. It is like a bridge with lots of members, and it is over-connected; if pieces have dropped out you can reconnect it another way.” (Richard Feynman, “The Character of Physical Law”, 1965)

"A mathematical proof, as usually written down, is a sequence of expressions in the state space. But we may also think of the proof as consisting of the sequence of justifications of consecutive proof steps - i.e., the references to axioms, previously-proved theorems, and rules of inference that legitimize the writing down of the proof steps. From this point of view, the proof is a sequence of actions (applications of rules of inference) that, operating initially on the axioms, transform them into the desired theorem." (Herbert A Simon, "The Logic of Heuristic Decision Making", [in "The Logic of Decision and Action"], 1966)

“A theorem is no more proved by logic and computation than a sonnet is written by grammar and rhetoric, or than a sonata is composed by harmony and counterpoint, or a picture painted by balance and perspective.” (George Spencer-Brown, “Laws of Form”, 1969)

See also:
Theorems I, II, III, IV, V, VII, VIII, IX, X
Proofs I, II, III, IV, V,. VI, VII, VIII, IX

On Theorems (1980-1989)

“Some people believe that a theorem is proved when a logically correct proof is given; but some people believe a theorem is proved only when the student sees why it is inevitably true.” (Wesley R Hamming, “Coding and Information Theory”, 1980)

“We become quite convinced that a theorem is correct if we prove it on the basis of reasonably sound statements about numbers or geometrical figures which are intuitively more acceptable than the one we prove.” (Morris Kline, “Mathematics: The loss of certainty”, 1980)

“For what is important when we give children a theorem to use is not that they should memorize it. What matters most is that by growing up with a few very powerful theorems one comes to appreciate how certain ideas can be used as tools to think with over a lifetime. One learns to enjoy and to respect the power of powerful ideas. One learns that the most powerful idea of all is the idea of powerful ideas.” (Seymour Papert, “Mindstorms: Children, Computers and Powerful Ideas”, 1980)

"The elegance of a mathematical theorem is directly proportional to the number of independent ideas one can see in the theorem and inversely proportional to the effort it takes to see them." (George Pólya, "Mathematical Discovery", 1981)

"To many, mathematics is a collection of theorems. For me, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples [...]" (John B Conway, “Subnormal Operators”, 1981)

“Proof serves many purposes simultaneously […] Proof is respectability. Proof is the seal of authority. Proof, in its best instance, increases understanding by revealing the heart of the matter. Proof suggests new mathematics […] Proof is mathematical power, the electric voltage of the subject which vitalizes the static assertions of the theorems.” (Reuben Hersh, “The Mathematical Experience”, 1981)

“There are no deep theorems - only theorems that we have not understood very well.” (Nicholas P Goodman, “Reflections on Bishops Philosophy of Mathematics”, 1983)

“Mathematics is not a deductive science - that's a cliche. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork." (Paul Halmos, “I Want to Be a Mathematician”, 1985)

„The pursuit of pretty formulas and neat theorems can no doubt quickly degenerate into a silly vice, but so can the quest for austere generalities which are so very general indeed that they are incapable of application to any particular.“ (Eric T Bell, „Men of Mathematics“, 1986)

“Mathematics is more than doing calculations, more than solving equations, more than proving theorems, more than doing algebra, geometry or calculus, more than a way of thinking. Mathematics is the design of a snowflake, the curve of a palm frond, the shape of a building, the joy of a game, the frustration of a puzzle, the crest of a wave, the spiral of a spider's web. It is ancient and yet new. Mathematics is linked to so many ideas and aspects of the universe.” (Theoni Pappas, “More Joy of Mathematics: Exploring Mathematics All Around You”, 1986)

“Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems - general and specific statements - can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences.” (Sir Erik C Zeeman, “Private Games”, 1988)

On Theorems (1900-1924)

"No theorem can be new unless a new axiom intervenes in its demonstration; reasoning can only give us immediately evident truths borrowed from direct intuition; it would only be an intermediary parasite." (Henri Poincaré, "Science and Hypothesis", 1902)

"A mathematical theorem and its demonstration are prose. But if the mathematician is overwhelmed with the grandeur and wondrous harmony of geometrical forms, of the importance and universal application of mathematical maxims, or, of the mysterious simplicity of its manifold laws which are so self-evident and plain and at the same time so complicated and profound, he is touched by the poetry of his science; and if he but understands how to give expression to his feelings, the mathematician turns poet, drawing inspiration from the most abstract domain of scientific thought." (Paul Carus," Friedrich Schiller: A Sketch of His Life and an Appreciation of His Poetry", 1905)

"Generally speaking, mathematical theorems are no analytic judgements yet, but we can reduce them to analytic ones through the hypothetical addition of synthetic premises. The logically reduced mathematical theorems emerging in this way are analytically hypothetical judgements which constitute the logical skeleton of a mathematical theory." (Ernst Zermelo, "Mathematische Logik. Vorlesungen gehalten von Prof. Dr. E. Zermelo zu Göttingen im S.S.", 1908)

"The beautiful has its place in mathematics as elsewhere. The prose of ordinary intercourse and of business correspondence might be held to be the most practical use to which language is put, but we should be poor indeed without the literature of imagination. Mathematics too has its triumphs of the Creative imagination, its beautiful theorems, its proofs and processes whose perfection of form has made them classic. He must be a 'practical' man who can see no poetry in mathematics." (Wiliam F White, "A Scrap-book of Elementary Mathematics: Notes, Recreations, Essays", 1908)

"Theorems valid 'in the small' are those which affirm a statement about a certain neighborhood of a point without making any statement about the size of that neighborhood." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"[...] the mathematician is always walking upon the brink of a precipice, for, no matter how many theorems he deduces, he cannot tell that some contradiction will not await him in the infinity of consequences." (Richard A Arms, "The Notion of Number and the Notion of Class Mathematical Usage", 1917)

"The axioms and provable theorems (i.e. the formulas that arise in this alternating game [namely formal deduction and the adjunction of new axioms]) are images of the thoughts that make up the usual procedure of traditional mathematics; but they are not themselves the truths in the absolute sense. Rather, the absolute truths are the insights (Einsichten) that my proof theory furnishes into the provability and the consistency of these formal systems. (David Hilbert; "Die logischen Grundlagen der Mathematik", Mathematische Annalen 88 (1), 1923)

On Numeracy III

“To be numerate means to be competent, confident, and comfortable with one’s judgements on whether to use mathematics in a particular situation and if so, what mathematics to use, how to do it, what degree of accuracy is appropriate, and what the answer means in relation to the context.” (Diana Coben, “Numeracy, mathematics and adult learning”, 2000)

"Numeracy is the ability to process, interpret and communicate numerical, quantitative, spatial, statistical, even mathematical information, in ways that are appropriate for a variety of contexts, and that will enable a typical member of the culture or subculture to participate effectively in activities that they value." (Jeff Evans, “Adults´ Mathematical Thinking and Emotion”, 2000)

"Ignorance of relevant risks and miscommunication of those risks are two aspects of innumeracy. A third aspect of innumeracy concerns the problem of drawing incorrect inferences from statistics. This third type of innumeracy occurs when inferences go wrong because they are clouded by certain risk representations. Such clouded thinking becomes possible only once the risks have been communicated." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"In my view, the problem of innumeracy is not essentially 'inside' our minds as some have argued, allegedly because the innate architecture of our minds has not evolved to deal with uncertainties. Instead, I suggest that innumeracy can be traced to external representations of uncertainties that do not match our mind’s design - just as the breakdown of color constancy can be traced to artificial illumination. This argument applies to the two kinds of innumeracy that involve numbers: miscommunication of risks and clouded thinking. The treatment for these ills is to restore the external representation of uncertainties to a form that the human mind is adapted to." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"Overcoming innumeracy is like completing a three-step program to statistical literacy. The first step is to defeat the illusion of certainty. The second step is to learn about the actual risks of relevant events and actions. The third step is to communicate the risks in an understandable way and to draw inferences without falling prey to clouded thinking. The general point is this: Innumeracy does not simply reside in our minds but in the representations of risk that we choose." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"Statistical innumeracy is the inability to think with numbers that represent uncertainties. Ignorance of risk, miscommunication of risk, and clouded thinking are forms of innumeracy. Like illiteracy, innumeracy is curable. Innumeracy is not simply a mental defect 'inside' an unfortunate mind, but is in part produced by inadequate 'outside' representations of numbers. Innumeracy can be cured from the outside." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

“Mathematics is often thought to be difficult and dull. Many people avoid it as much as they can and as a result much of the population is mathematically illiterate. This is in part due to the relative lack of importance given to numeracy in our culture, and to the way that the subject has been presented to students.“ (Julian Havil , “Gamma: Exploring Euler's Constant”, 2003)

“One can be highly functionally numerate without being a mathematician or a quantitative analyst. It is not the mathematical manipulation of numbers (or symbols representing numbers) that is central to the notion of numeracy. Rather, it is the ability to draw correct meaning from a logical argument couched in numbers. When such a logical argument relates to events in our uncertain real world, the element of uncertainty makes it, in fact, a statistical argument.” (Eric R Sowey, “The Getting of Wisdom: Educating Statisticians to Enhance Their Clients' Numeracy”, The American Statistician 57(2), 2003)

“Mathematics and numeracy are not congruent. Nor is numeracy an accidental or automatic by-product of mathematics education at any level. When the goal is numeracy some mathematics will be involved but mathematical skills alone do not constitute numeracy.” (Theresa Maguire & John O'Donoghue, “Numeracy concept sophistication - an organizing framework, a useful thinking tool”, 2003)

“Mathematical literacy is an individual’s capacity to identify and understand the role that mathematics plays in the world, to make well-founded judgements and to use and engage with mathematics in ways that meet the needs of that individual’s life as a constructive, concerned and reflective citizen.” (OECD, “Assessing scientific, reading and mathematical literacy: a framework for PISA 2006”, 2006)

“Statistical literacy is more than numeracy. It includes the ability to read and communicate the meaning of data. This quality makes people literate as opposed to just numerate. Wherever words (and pictures) are added to numbers and data in your communication, people need to be able to understand them correctly.” (United Nations, “Making Data Meaningful” Part 4: “A guide to improving statistical literacy”, 2012)

“When a culture is founded on the principle of immediacy of experience, there is no need for numeracy. It is impossible to consume more than one thing at a time, so differentiating between 'a small amount', 'a larger amount' and 'many' is enough for survival.” (The Open University, “Understanding the environment: learning and communication”, 2016)

See also: Numeracy I & II 

On Numeracy II

“People often feel inept when faced with numerical data. Many of us think that we lack numeracy, the ability to cope with numbers. […] The fault is not in ourselves, but in our data. Most data are badly presented and so the cure lies with the producers of the data. To draw an analogy with literacy, we do not need to learn to read better, but writers need to be taught to write better.” (Andrew Ehrenberg, “The problem of numeracy”, American Statistician 35(2), 1981)

“We would wish ‘numerate’ to imply the possession of two attributes. The first of these is an ‘at-homeness’ with numbers and an ability to make use of mathematical skills which enable an individual to cope with the practical mathematical demands of his everyday life. The second is ability to have some appreciation and understanding of information which is presented in mathematical terms, for instance in graphs, charts or tables or by reference to percentage increase or decrease.” (Cockcroft Committee, “Mathematics Counts: A Report into the Teaching of Mathematics in Schools”, 1982)

“To function in today's society, mathematical literacy - what the British call ‘numeracy' - is as essential as verbal literacy […] Numeracy requires more than just familiarity with numbers. To cope confidently with the demands of today's society, one must be able to grasp the implications of many mathematical concepts - for example, change, logic, and graphs - that permeate daily news and routine decisions - mathematical, scientific, and cultural - provide a common fabric of communication indispensable for modern civilized society. Mathematical literacy is especially crucial because mathematics is the language of science and technology.” (National Research Council, “Everybody counts: A report to the nation on the future of mathematics education”, 1989)

“Illiteracy and innumeracy are social ills created in part by increased demand for words and numbers. As printing brought words to the masses and made literacy a prerequisite for productive life, so now computing has made numeracy an essential feature of today's society. But it is innumeracy, not numeracy, that dominates the headlines: ignorance of basic quantitative tools is endemic […] and is approaching epidemic levels […].” (Lynn A Steen, “Numeracy”, Daedalus Vol. 119 No. 2, 1990)

“[…] data analysis in the context of basic mathematical concepts and skills. The ability to use and interpret simple graphical and numerical descriptions of data is the foundation of numeracy […] Meaningful data aid in replacing an emphasis on calculation by the exercise of judgement and a stress on interpreting and communicating results.” (David S Moore, “Statistics for All: Why, What and How?”, 1990)

“To be numerate is more than being able to manipulate numbers, or even being able to ‘succeed’ in school or university mathematics. Numeracy is a critical awareness which builds bridges between mathematics and the real world, with all its diversity. […] in this sense […] there is no particular ‘level’ of Mathematics associated with it: it is as important for an engineer to be numerate as it is for a primary school child, a parent, a car driver or gardener. The different contexts will require different Mathematics to be activated and engaged in […] ”(Betty Johnston, “Critical Numeracy”, 1994)

“We believe that numeracy is about making meaning in mathematics and being critical about maths. This view of numeracy is very different from numeracy just being about numbers, and it is a big step from numeracy or everyday maths that meant doing some functional maths. It is about using mathematics in all its guises - space and shape, measurement, data and statistics, algebra, and of course, number - to make sense of the real world, and using maths critically and being critical of maths itself. It acknowledges that numeracy is a social activity. That is why we can say that numeracy is not less than maths but more. It is why we don’t need to call it critical numeracy being numerate is being critical.” (Dave Tout & Beth Marr, “Changing practice: Adult numeracy professional development”, 1997)

"We need literacy in all sense of the word: linguistic, visual, and symbolic. Mathematics is a multisemiotic enterprise." (Kay O’Halloran)

See also:Numeracy I & III

On Numeracy I

“The great body of physical science, a great deal of the essential fact of financial science, and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of one of the new great complex world-wide States that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and write.” (Herbert G Wells, „Mankind in the Making”,  1903)

“[…] statistical literacy. That is, the ability to read diagrams and maps; a “consumer” understanding of common statistical terms, as average, per cent, dispersion, correlation, and index number.” (Douglas Scates, “Statistics: The Mathematics for Social Problems”, 1943)

„Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write.” (Samuel S Wilks, 1951 [paraphrasing Herber Wells] )

“Just as by ‘literacy’, in this context, we mean much more than its dictionary sense of the ability to read and write, so by ‘numeracy’ we mean more than mere ability to manipulate the rule of three. When we say that a scientist is ‘illiterate’, we mean that he is not well enough read to be able to communicate effectively with those who have had a literary education. When we say that a historian or a linguist is ‘innumerate’ we mean that he cannot even begin to understand what scientists and mathematicians are talking about.” (Sir Geoffrey Crowther, “A Report of the Central Advisory Committee for Education”, 1959)

“It is perhaps possible to distinguish two different aspects of numeracy […]. On the one hand is an understanding of the scientific approach to the study of phenomena - observation, hypothesis, experiment, verification. On the other hand, there is the need in the modern world to think quantitatively, to realise how far our problems are problems of degree even when they appear as problems of kind.” (Sir Geoffrey Crowther, “A Report of the Central Advisory Committee for Education”, 1959)

“Numeracy has two facets-reading and writing, or extracting numerical information and presenting it. The skills of data presentation may at first seem ad hoc and judgemental, a matter of style rather than of technology, but certain aspects can be formalized into explicit rules, the equivalent of elementary syntax.” (Andrew Ehrenberg, “Rudiments of Numeracy”, Journal of Royal Statistical Society, 1977)

See also:
Numeracy II & III

On Proofs (1900 - 1924)

"[…] it is an error to believe that rigor in the proof is the enemy of simplicity." (David Hilbert, Paris International Congress, 1900)

“Besides it is an error to believe that rigour is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigor forces us to find out simpler methods of proof.” (David Hilbert, “Mathematical Problems”, Bulletin of the American Mathematical Society, 1902)

"It is one of the chief merits of proofs that they instill a certain skepticism as to the result proved." (Bertrand Russell, "The Principles of Mathematics", 1903)

"It is by logic that we prove, but by intuition that we discover. [...] Every definition implies an axiom, since it asserts the existence of the object defined. The definition then will not be justified, from the purely logical point of view, until we have proved that it involves no contradiction either in its terms or with the truths previously admitted." (Henri Poincaré, "Science and Method", 1908)

"Banishing fundamental facts or problems from science merely because they cannot be dealt with by means of certain prescribed principles would be like forbidding the further extension of the theory of parallels in geometry because the axiom upon which this theory rests has been shown to be unprovable. Actually, principles must be judged from the point of view of science, and not science from the point of view of principles fixed once and for all." (Ernst Zermelo, "Neuer Beweis für die Möglichkeit einer Wohlordnung", Mathematische Annalen 65, 1908)

"Now even in mathematics unprovability, as is well known, is in no way equivalent to nonvalidity, since, after all, not everything can be proved, but every proof in turn presupposes unproved principles. Thus, in order to reject such a fundamental principle, one would have to ascertain that in some particular case it did not hold or to derive contradictory consequences from it; but none of my opponents has made any attempt to do this." (Ernst Zermelo, "Neuer Beweis für die Möglichkeit einer Wohlordnung", Mathematische Annalen 65, 1908)

"To reach our goal [of proving consistency], we must make the proofs as such the object of our investigation; we are thus compelled to a sort of proof theory which studies operations with the proofs themselves." (David Hilbert, 1922)

"Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this is so only because mathematics does not attempt to draw absolute conclusions. All mathematical truths are relative, conditional." (Charles P Steinmetz, 1923)

On Proofs (1925-1949)

"As the objects of abstract geometry cannot be totally grasped by space intuition, a rigorous proof in abstract geometry can never be based only on intuition, but it must be founded on logical deduction from valid and precise axioms. Nevertheless intuition maintains, also in precision geometry, its irreplaceable value that cannot be substituted by logical considerations. Intuition helps us to construct a proof and to gain an overview, it is, moreover, a source of inventions and new mental connections." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"Proof is an idol before whom the pure mathematician tortures himself. In physics we are generally content to sacrifice before the lesser shrine of Plausibility." (Sir Arthur S Eddington, "The Nature of the Physical World", 1928)

"A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories." (Stefan Banach, cca. 1930)

“A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details.” (Hermann Weyl, "Unterrichtsblätter für Mathematik und Naturwissenschaften", 1932)

"The search for the most general conditions of validity of a determined statement, if it is ready to reveal its causal proof, doesn’t succeed without a constant reworking of the implemented notions. (Georges Bouligand, "La causalite des theories mathématiques", Actualités Scientifiques et Industrielles 184, 1935)

“A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.“ (Godfrey H Hardy, “A Mathematician’s Apology”, 1940)

“Without the strictest deductive proof from admitted assumptions, explicitly stated as such, mathematics does not exist.” (Eric T Bell, “The Development of Mathematics”, 1940)

"Heuristic reasoning is reasoning not regarded as final and strict but as provisional and plausible only, whose purpose is to discover the solution of the present problem. We are often obliged to use heuristic reasoning. We shall attain complete certainty when we shall have obtained the complete solution, but before obtaining certainty we must often be satisfied with a more or less plausible guess. We may need the provisional before we attain the final. We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building." (George Pólya, "How to solve it", 1945)

"In mathematics as in the physical sciences we may use observation and induction to discover general laws. But there is a difference. In the physical sciences, there is no higher authority than observation and induction but In mathematics there is such an authority: rigorous proof." (George Pólya, "How to solve it", 1945)

"The cookbook gives a detailed description of ingredients and procedures but no proofs for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. [...] Mathematics cannot be tested in exactly the same manner as a pudding; if all sorts of reasoning are debarred, a course of calculus may easily become an incoherent inventory of indigestible information." (George Pólya, "How to solve it", 1945)

"Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions.” (Eric T Bell, Mathematics Magazine, 1949)

On Proofs (1975 - 1989)

“The conception of the mental construction which is the fully analysed proof as being an infinite structure must, of course, be interpreted in the light of the intuitionist view that all infinity is potential infinity: the mental construction consists of a grasp of general principles according to which any finite segment of the proof could be explicitly constructed.” (Michael Dummett, “The philosophical basis of intuitionistic logic”, 1975)

“No theory ever agrees with all the facts in its domain, yet it is not always the theory that is to blame. Facts are constituted by older ideologies, and a clash between facts and theories may be proof of progress. It is also a first step in our attempt to find the principles implicit in familiar observational notions.”  (Paul K Feyerabend, “Against Method: Outline of an Anarchistic Theory of Knowledge”, 1975)

“There is an infinite regress in proofs; therefore proofs do not prove. You should realize that proving is a game, to be played while you enjoy it and stopped when you get tired of it.” (Imre Lakatos, “Proofs and Refutations”, 1976)

“On the face of it there should be no disagreement about mathematical proof. Everybody looks enviously at the alleged unanimity of mathematicians; but in fact there is a considerable amount of controversy in mathematics. Pure mathematicians disown the proofs of applied mathematicians, while logicians in turn disavow those of pure mathematicians. Logicists disdain the proofs of formalists and some intuitionists dismiss with contempt the proofs of logicists and formalists.” (Imre Lakatos, “Mathematics, Science and Epistemology” Vol. 2, 1978)

"Mathematics is a way of finding out, step by step, facts which are inherent in the statement of the problem but which are not immediately obvious. Usually, in applying mathematics one must first hit on the facts and then verify them by proof. Here we come upon a knotty problem, for the proofs which satisfied mathematicians of an earlier day do not satisfy modem mathematicians." (John R Pierce, "An Introduction to Information Theory: Symbols, Signals & Noise" 2nd Ed., 1980)

"Mathematicians start out with certain assumptions and definitions, and then by means of mathematical arguments or proofs they are able to show that certain statements or theorems are true." (John R Pierce, "An Introduction to Information Theory: Symbols, Signals & Noise" 2nd Ed., 1980)

“When a mathematician asks himself why some result should hold, the answer he seeks is some intuitive understanding. In fact, a rigorous proof means nothing to him if the result doesn’t make sense intuitively.” (Morris Kline, “Mathematics: The Loss of Certainty”, 1980)

“No proof is final. New counterexamples undermine old proofs. The proofs are then revised and mistakenly considered proven for all time. But history tells us that this merely means that the time has not yet come for a critical examination of the proof” (Morris Kline, “Mathematics: The Loss of Certainty”, 1980)

“If the proof starts from axioms, distinguishes several cases, and takes thirteen lines in the text book […] it may give the youngsters the impression that mathematics consists in proving the most obvious things in the least obvious way.” (George Pólya, “Mathematical Discovery: on Understanding, Learning, and Teaching Problem Solving”, 1981)

“A proof transmits conviction from its premises down to its conclusion, so it must start with premises […] for which there already is conviction; otherwise, there will be nothing to transmit.” (Robert Nozick, Philosophical Explanations, 1981)

“We often hear that mathematics consists mainly in ‘proving theorems’. Is a writer’s job mainly that of ‘writing sentences’? A mathematician’s work is mostly a tangle of guesswork, analogy, wishful thinking and frustration, and proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks.” (Gian-Carlo Rota, “Complicating Mathematics” in “Discrete Thoughts”, 1981)

“People might suppose that a mathematical proof is conceived as a logical progression, where each step follows upon the ones that have preceded it. Yet the conception of a new argument is hardly likely actually to proceed in this way. There is a globality and seemingly vague conceptual content that is necessary in the construction of a mathematical argument; and this can bear little relation to the time that it would seem to take in order fully to appreciate a serially presented proof” (Roger Penrose, “The Emperor’s New Mind”, 1989)

See also:

Proofs I, II, III, IV, V, VI, VIII, IX
Theorems I, II, III, IV, V, VI, VII, VIII, IX, X

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)

"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)

"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)

"[…] 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)

"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)

"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)

"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)

"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)