Mathematicians vs Scientists

"The man of science, who, forgetting the limits of philosophical inquiry, slides from these formulæ and symbols into what is commonly understood by materialism, seems to me to place himself on a level with the mathematician, who should mistake the x's and y's with which he works his problems for real entities - and with this further disadvantage, as compared with the mathematician, that the blunders of the latter are of no practical consequence, while the errors of systematic materialism may paralyse the energies and destroy the beauty of a life." (Thomas H Huxley, "Method and Results", 1893)

"Mathematicians and other scientists, however great they may be, do not know the future. Their genius may enable them to project their purpose ahead of them; it is as if they had a special lamp, unavailable to lesser men, illuminating their path; but even in the most favorable cases the lamp sends only a very small cone of light into the infinite darkness." (George Sarton, "The Study of the History of Mathematics", 1936)

"One of the difficulties which a mathematician has in describing his work to non-mathematicians is that the present day language of mathematics has become so esoteric that a well educated layman, or even a group of scientists, can comprehend essentially nothing of the discourse which mathematicians hold with each other, or of the accounts of their latest researches which are published in their professional journals." (Angus E Taylor," Some Aspects of Mathematical Research", American Scientist , Vol. 35, No. 2, 1947)

"A scientist worthy of the name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature. [...] we work not only to obtain the positive results which, according to the profane, constitute our one and only affection, as to experience this esthetic emotion and to convey it to others who are capable of experiencing it." (Henri Poincaré, "Notice sur Halphen", Journal de l'École Polytechnique, 1890)

"The entrepreneur's instinct is to exploit the natural world. The engineer's instinct is to change it. The scientist's instinct is to try to understand it - to work out what's really going on. The mathematician's instinct is to structure that process of understanding by seeking generalities that cut across the obvious subdivisions." (Ian Stewart, "Nature's Numbers", 1995)

"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)

"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)

Mathematical Experience II: Mathematicians

"A scientist worthy of the name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature. [...] we work not only to obtain the positive results which, according to the profane, constitute our one and only affection, as to experience this esthetic emotion and to convey it to others who are capable of experiencing it." (Henri Poincaré, "Notice sur Halphen", Journal de l'École Polytechnique, 1890)

"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", 1940)

"Nothing in our experience suggests the introduction of [complex numbers]. Indeed, if a mathematician is asked to justify his interest in complex numbers, he will point, with some indignation, to the many beautiful theorems in the theory of equations, of power series, and of analytic functions in general, which owe their origin to the introduction of complex numbers. The mathematician is not willing to give up his interest in these most beautiful accomplishments of his genius." (Eugene P Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", Communications in Pure and Applied Mathematics 13 (1), 1960)

"Any applied mathematicians - any engineer using mathematics - works sometimes more and sometimes less mathematically. When he is most mathematical he makes least appeal to experience." (Chandler Davis, "Materialist Mathematics", 1974)

"Every mathematician worthy of the name has experienced the state of lucid exaltation in which one thought succeeds another as if miraculously. This feeling may last for hours at a time, even for days. Once you have experienced it, you are eager to repeat it but unable to do it at will, unless perhaps by dogged work." (André Weil, "The Apprenticeship of a Mathematician", 1992)

"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)

"Mathematics is not a matter of ‘anything goes,’ and every mathematician is guided by explicit or unspoken assumptions as to what counts as legitimate – whether we choose to view these assumptions as the product of birth, experience, indoctrination, tradition, or philosophy. At the same time, mathematicians are primarily problem solvers and theory builders, and answer first and foremost to the internal exigencies of their subject." (Jeremy Avigad, "Methodology and Metaphysics in the Development of Dedekind’s Theory of Ideals", 2006)

"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)

"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)

"I think the thing which makes mathematics a pleasant occupation are those few minutes when suddenly something falls into place and you understand. Now a great mathematician may have such moments very often. Gauss, as his diaries show, had days when he had two or three important insights in the same day. Ordinary mortals have it very seldom. Some people experience it only once or twice in their lifetime. But the quality of this experience - those who have known it - is really joy comparable to no other joy." (Lipman Bers)

Mayme I Logsdon - Collected Quotes

"An incontestable claim of mathematics to importance in our civilization is that it is indispensable in a scientific explanation of what we observe in nature, i.e., the phenomena of nature. Of the several fields of elementary mathematics, the calculus may be called the motion-picture machine of mathematics which catches natural phenomena in the act of changing, or, as Newton called it, in a state of flux. Other fields of mathematics are to be likened to the camera which shows a still picture (of nature) as it appears at a given instant without regard to the possible appearance the following instant." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"In mathematics, as in the world about us, when one quantity depends on a second quantity, or when the value of one symbol depends on the value of another symbol, the first is said to be a function of the second. If the second quantity, or the second symbol, is thought of as taking on a number of arbitrary values (e.g., the angle A when it increases or decreases), it is called an independent variable and the function which depends on it is called a dependent variable. It may happen that a function depends on more than one independent variable." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"Mathematical theories have been of great service in many experimental sciences in correlating the results of observations and in predicting new data afterward verified by observation. This has happened particularly in geometry, physics, and astronomy. But the relationship between a mathematical theory and the data which it is designed to relate is often misunderstood. When such a theory has been successful as a correlating agent, the conviction is likely to become established that the theory has a unique relationship to nature as interpreted for us by the observations. Furthermore, it is sometimes inferred that nature behaves in precisely the way which the mathematics indicates. As a matter of fact, nature never does behave in this way, and there are always more mathematical theories than one whose results depart from a given set of data by less than the errors of observation." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"Neither the principle of cause and effect nor the principle of uncertainty can be precisely characteristic of the behavior of nature. They are merely most interesting theorems in two different theories by means of which we endeavor to correlate and interpret observed data. The ultimate choice between the two theories must be determined by convenience or by their relative accuracies of fit with observation, and not because of any supposedly precise correspondence with nature on the part of either one of them." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"[…] the purposes of an applied mathematical science are twofold: first to correlate and systematize data which may otherwise appear heterogeneous and unrelated in character, and second to predict by logical processes new results which might be difficult or impossible to discover by experimental methods alone." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"The purpose of a coordinate system is twofold. It enables one person to describe the position of points or objects in such a manner that others listening or reading may know exactly what points or objects are meant. And it is a device for linking algebra and geometry so that an algebraic equation corresponds to a geometric locus and, conversely, a geometric locus corresponds to one or more algebraic equations." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"The underlying notion of the integral calculus is also that of finding a limiting value, but this time it is the limiting value of a sum of terms when the number of terms increases without bound at the same time that the numerical value of each term approaches Zero. The area bounded by one or more curves is found as the limiting value of a sum of small rectangles; the length of an arc of a curve is found as the limiting value of a sum of lengths of straight lines (chords of the arc); the volume of a solid bounded by one or more curved surfaces is found as the limiting value of a sum of volumes of small solids bounded by planes; etc." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"The words 'maximum' and 'minimum' are used here in a technical sense. Maximum value of the function, for example, does not mean (as one might well suppose) the greatest value which the function attains for any value of x but, merely, the greatest value which it attains when, having been increasing, it ceases increasing and begins to decrease. In other words, the ordinate of a maximum point on a curve is greater than the ordinates of other nearby points. In a similar manner the ordinate of a minimum point is less than the ordinates of other nearby points." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"There is probably no one word which is more closely associated in everyone's mind with the mathematician than the word equation. The reason for this is easy to find. In the language of mathematics the word 'equation' plays the same role as that played by the word 'sentence' in a spoken and written language. Now the sentence is the unit for the expression of thought; the equation is the unit for the expression of a mathematical idea." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"To square a circle means to find a square whose area is equal to the area of a given circle. In its first form this problem asked for a rectangle whose dimensions have the same ratio as that of the circumference of a circle to its radius. The proof of the impossibility of solving this by use of ruler and compasses alone followed immediately from the proof, in very recent times, that π cannot be the root of a polynomial equation with rational coefficients." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"When an induction, based on observations, is made, it is not intended that it shall be accepted as a universal truth, but it is advanced as a hypothesis for further study. Additional observations are then made and the results compared with the results expected from the hypothesis. If there is more deviation between the experimental results and the computed results than can be expected from the inaccuracies of observation and measurement, the scientist discards the' hypothesis and tries to formulate another." (Mayme I Logsdon, "A Mathematician Explains", 1935)

David Wells - Collected Quotes

"All the [mathematical] models that we have considered […] have been rough and ready. They have all been obviously crude approximations, and no one supposes that they are anything more. This does not mean that they are useless - far from it - but it does mean that the answers they give to practical questions are also approximations. There is a pragmatic payoff here between the use of simple models which give good-enough answers which are good value for money, and the use of much more sophisticated models which are more powerful, but also more complex to use, perhaps requiring more advanced mathematics and the use of computers." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"By practice and experience, much of it vicarious, through studying the achievements of others, we develop the strength to tackle novel and unfamiliar situations." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Chess players recognize and applaud good play, from a single smart move to a brilliant combination to an entire game which is considered a masterpiece. Mathematicians recognize and applaud good mathematics, from clever tricks to brilliant proofs, and from beautiful conceptions to grand and deep ideas which advance our understanding of mathematics as a whole. It takes imagination and insight to discover the best moves, at chess or mathematics, and the more difficult the position, the harder they are to find. Chess players learn by experience to recognize types of positions and situations and to know what kind of moves are likely to be successful; they exploit brilliant local tactics as well as deep stategical ideas. So do mathematicians. Neither games nor mathematics play themselves - they both need a player with understanding, good ideas, judgement and discrimination to play them. To develop these essential attributes, the player must explore the game by playing it, thinking about it and analysing it. For the chess player and the mathematician, this process is scientific: you test ideas, experiment with new possibilities, develop the ones that work and discard the ones that fail. This is how chess players and mathematicians develop their tactical and strategical understanding; it is how they give meaning to chess and mathematics." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Do mathematicians know exactly what they are talking about, or don't they? This is the first bridge that mathematicians have to cross in their search for certainty, and it has to be admitted that they cannot cross it with complete confidence, so their search for absolute certainty is already compromised. They can certainly rely on the long history of their most basic ideas, and the universal agreement that these concepts 'work'. However, this is another way of saying that, if a concept is new and untried, then even mathematicians should be wary of it. They may well discover with experience that their intuitive expectations of it are false. Indeed, this is one of the great values of experimental mathematics - experiments provide the mathematician with data against which mathematicians' natural expectations can be continually tested." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Graph theory is typical of much modern mathematics. Its subject matter is not traditional, and it is not a development from traditional theories. Its applications are not traditional either. […] Graph theory is not concerned with continuous quantities. It often involves counting, but in integers, not measuring using fractions. Graph theory is an example of discrete mathematics. Graphs are put together in pieces, in chunks, rather like Meccano or Lego, or a jigsaw puzzle." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"However mathematics starts, whether it is in counting and measuring in everyday life, or in puzzles and riddles, or in scientific queries about projectiles, floating bodies, levers and balances, or magnetic lines of force, it eventually becomes detached from its roots and develops a life of its own. It becomes more powerful, because it can be applied not just to the situations in which it originated but to all other comparable situations. It also becomes more abstract, and more game-like." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Ironically, mathematicians often infer all sorts of properties about objects which they only suspect, or hope, actually exist. If their suspicions turn out to be unfounded, then they seem to end up by knowing rather a lot about something which does not exist and which might seem, therefore, not to have any properties at all." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"It is typical that there is more than one way of looking at a geometrical figure, just as there are many ways of looking at lines of algebra. Perception, 'seeing', is an essential feature of mathematics. This is obvious when we are looking for patterns - how can you possibly 'spot' a pattern if you cannot in some sense 'see' it? But it is just as true when the mathematician is looking for hidden connections, or studying a position in a mathematical game, searching for a tactical sequence, or trying to 'see' the possibilities clearly. Superficially, it might seem that it is only geometry (and related fields of mathematics) that depends on perception, but this is not so. Perception is everywhere in mathematics." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Mathematical models are continually invoking ideas of infinitely smooth surfaces, weightless strings, weightless beams, perfectly spherical balls, projectiles flying through airless space, gases which are perfectly compressible and liquids which are perfectly incompressible, and so on. The purpose of such simplifications is, in theory, to understand the world better despite the oversimplification, which you hope either will not matter or will be corrected when you construct a second (better) model." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Mathematicians get a different kind of pleasure from the illumination of solving a problem, when what was once mysterious and obscure is made plain. Revealing the hidden connections in a situation is delightful - like reaching the top of a mountain after a hard climb, and seeing the landscape spread out before you. All of a sudden, everything is clear! If the result is extremely simple, so much the better . To start with confusing complexity and transform it into revealing simplicity is a marvellous reward for hard work. It really does give the mathematician a 'kick'!" (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Mathematics-as-science naturally starts with mysterious phenomena to be explained, and leads (if you are successful) to powerful and harmonious patterns. Mathematics-as-a-game not only starts with simple objects and rules, but involves all the attractions of games like chess: neat tactics, deep strategy, beautiful combinations, elegant and surprising ideas. Mathematics-as-perception displays the beauty and mystery of art in parallel with the delight of illumination, and the satisfaction of feeling that now you understand." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Mathematics depends on the ability to 'see', literally and abstractly, in novel and unexpected ways, which is another way of saying that it depends on the brilliance and subtlety of the human brain which just happens to be wonderfully adapted to this very purpose. [...] It is one of the marvels of mathematics that everything can be seen in different ways from different points of view, sometimes literally, sometimes metaphorically speaking." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"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)

"Playing the game of mathematics is much harder than investigating scientifically! To jot down some numbers, a few differences, and spot a pattern is child's play compared to playing the game of algebra." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Puzzle composers share another feature with mathematicians. They know that, generally speaking, the simpler a puzzle is to express, the more attractive it is likely to be found: similarly, simplicity is for both a desirable feature of the solution. Especially satisfying solutions are often described as 'elegant', a word that - no surprise here - is also used by scientists, engineers and designers, indeed by anyone with a problem to solve. However, simplicity is by no means the only reward of success. Far from it! Mathematicians (and scientists and others) can reasonably expect two further returns: they are (in no particular order) firstly the power to do things, and secondly the perception of connections which were never before suspected, leading in turn to the insight and illumination that mathematicians expect from their best arguments." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995) 

"'Technique' is a term used equally by chess players and mathematicians to describe sequences of moves which are standard, familiar and unoriginal. Once upon a time, the particular technique was an invention, a new discovery, but no longer. The precise sequence of moves required may never have been played before in the history of the world, yet no new ideas, no originality and no imagination are demanded, at least of the experienced player. (To the learner, of course, the most mundane sequences will appear novel and require original thought.)" (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"The easiest way to discover mathematical facts and theorems is to treat mathematical objects just as if they were objects in the real world, and to make observations and do experiments, by drawing and measuring on geometrical figures, and making calculations with numbers (or in algebra) […]" (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"[…] the most important techniques are the final results of some of the most significant breakthroughs in the history of mathematics, beautifully simplified and explained for the benefit of players everywhere. As such, technique is used by pure mathematicians and applied mathematicians alike, but in rather different ways. Pure mathematicians use familiar techniques on the way to discovering or proving new results; applied mathematicians use techniques to model phenomena in the real world […]" (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"There is a difficulty, however, which has to be borne in mind. As long as you are behaving like a scientist, you can never be quite certain that your mathematical results are true. Therefore, scientific practice, although it is an essential tool of the mathematician, is also a source of danger. The mathematician who is fast asleep will not see the danger lurking, and will fall straight into the traps and snares that await. The mathematician who is wide awake, however, will gain the double benefit from scientific experiment and observation, of discovering all sorts of supposed facts, and gaining some idea why they might be true." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"There is no sharp dividing line between scientific theories and models, and mathematics is used similarly in both. The important thing is to possess a delicate judgement of the accuracy of your model or theory. An apparently crude model can often be surprisingly effective, in which case its plain dress should not mislead. In contrast, some apparently very good models can be hiding dangerous weaknesses." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"This process, of answering deep questions and so discovering what you really meant by what you've been talking about all the time, is very common in mathematics. It is also very important and very scientific. It has to be scientific, because you cannot prove that one definition is superior to another by logic: you can only make judgements that this definition is preferable to that. You could say that, until mathematicians have examined very deeply and scientifically the objects they talk about so glibly, then they do not really know what they are talking about." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Yet, Einstein's theories are also not the last word: quantum theory and relativity are inconsistent, and Einstein himself, proclaiming that 'God does not play dice!', rejected the basic reliance of quantum theory on chance events, and looked forward to a theory which would be deterministic. Recent experiments suggest that this view of Einstein's conflicts with his other deeply held beliefs about the nature of the physical universe. Certain it is that somewhere, beyond physicists' current horizons, are even more powerful theories of how the world is." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

Robert Musil - Collected Quotes

"It is only when one looks not towards the outside, at their utility, but within mathematics itself at the relationship among the unused parts that one sees the other, real face of this science. It is not goal-oriented but uneconomical and passionate… (The mathematician) believes that what he is doing will probably eventually lead to some practical cash value, but this is not what spurs him on; he serves the truth, which is to say his destiny, not its purpose. The result may be economical a thousand times over; what is immanent is a total surrender and a passionate devotion." (Robert Musil, "The Mathematical Man", 1913)

"Let no one object that outside their field mathematicians have banal or silly minds, […] but in their field they do what we ought to be doing in ours. Therein lies the significant lesson and model of their existence; they are an analogy for the intellectual of the future." (Robert Musil, "The Mathematical Man", 1913)

"Mathematics is the bold luxury of pure reason, one of the few that remain today." (Robert Musil, "The Mathematical Man", 1913)

"In their field they [mathematicians] do what we ought to be doing in ours. Therein lies the significant lesson … of their existence. They are an analogy for the intellectual of the future. (Robert Musil," The Mathematical Man", 1913)

"A metaphor holds a truth and an untruth, felt as inextricably bound up with each other. If one takes it as it is and gives it some sensual form, in the shape of reality, one gets dreams and art; but between these two and real, full-scale life there is a glass partition. If one analyzes it for its rational content and separates the unverifiable from the verifiable, one gets truth and knowledge but kills the feeling." (Robert Musil, "Man Without Qualities", 1930)

"Mathematics is the source of a wicked intellect that, while making man the lord of the earth, also makes him the slave of the machine." (Robert Musil, "The Man Without Qualities", 1930)

"What is done for science must also be done for art: accepting undesirable side effects for the sake of the main goal, and moreover diminishing their importance by making this main goal more magnificent. For one should reform forward, not backward: social illnesses, revolutions, are evolutions inhibited by a conserving stupidity. (Robert Musil", "Precision and Soul: Essays and Addresses", 1990)

On Patterns (1940-1949)

"A mathematician, like a painter or a poet, is a maker of patterns. [...]. The mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics." (Godfrey H Hardy, "A Mathematician's Apology", 1941)

"It will probably be the new mathematical discoveries which are suggested through physics that will always be the most important, for, from the beginning Nature has led the way and established the pattern which mathematics, the Language of Nature, must follow." (George D Birkhoff, "Mathematical Nature of Physical Theories" American Scientific Vol. 31 (4), 1943)

"One may generalize upon these processes in terms of group equilibrium. The group may be said to be in equilibrium when the interactions of its members fall into the customary pattern through which group activities are and have been organized. The pattern of interactions may undergo certain modifications without upsetting the group equilibrium, but abrupt and drastic changes destroy the equilibrium." (William F Whyte, "Street Corner Society", 1943)

"Those who are content with a positivist conception of the aims of science will feel that he is in an entirely satisfactory position; he has discovered the pattern of events, and so can predict accurately; what more can he want? A mental picture would be an added luxury, but also a useless luxury. For if the picture did not bear any resemblance at all to the reality it would be valueless, and if it did it would be unintelligible […]" (James H Jeans," Physics and Philosophy" 3rd Ed., 1943)

"Without falling into the trap of attempting a precise definition, we may suggest a theory as to the general nature of symbolism, viz. that it is the ability of processes to parallel or imitate each other, or the fact that they can do so since there are recurrent patterns in reality." (Kenneth Craik, "The Nature of Explanation", 1943)

"Science in general […] does not consist in collecting what we already know and arranging it in this or that kind of pattern. It consists in fastening upon something we do not know, and trying to discover it. (Robin G Collingwood, "The Idea of History", 1946)

Out of Context: On Mathematician (Definitions)

"The mathematician is perfect only in so far as he is a perfect man, in so far as he senses in himself the beauty of truth; only then will his work be thorough, transparent, prudent, pure, clear, graceful, indeed elegant." (Plato, "Republic", cca. 375 BC)

"A mathematician is only perfect insofar as he is a perfect man, sensitive to the beauty of truth." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1833)

"[…] 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", 1917)

"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 mathematician is not a man who can readily manipulate figures; often he cannot. He is not even a man who can readily perform the transformations of equations by the use of calculus. He is primarily an individual who is skilled in the use of symbolic logic on a high plane, and especially he is a man of intuitive judgment in the choice of the manipulative processes he employs." (Vannevar Bush, "As We May Think", 1945)

"The mathematician is not willing to give up his interest in these most beautiful accomplishments of his genius." (Eugene P Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", Communications in Pure and Applied Mathematics 13 (1), 1960)

"The mathematician is an artist whose medium is the mind and whose creations are ideas." (Hubert S Wall, "Creative Mathematics", 1963)

"Mathematicians are there to find the constraints and to eliminate those things that aren't constraints [...]" (Robert E Machol, Mathematicians are useful, 1971)

"Mathematicians are more like classical composers, typically working within a much tighter framework, reluctant to go to the next step until all previous ones have been established with due rigor." (Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory", 1999)

"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." (Gregory J Chaitin, "A century of controversy over the foundations of mathematics", 2000)

"[…] a mathematician is more anonymous than an artist." (Timothy Gowers, "Mathematics: A Very Short Introduction", 2002)

"[…] mathematicians are much more concerned for example with the structure behind something or with the whole edifice. Mathematicians are not really puzzlers." (Carlo Beenakker, [interview] 2006)

"[...] mathematicians are primarily problem solvers and theory builders, and answer first and foremost to the internal exigencies of their subject." (Jeremy Avigad, "Methodology and Metaphysics in the Development of Dedekind’s Theory of Ideals", 2006)

"Mathematicians are used to game-playing according to a set of rules they lay down in advance, despite the fact that nature always writes her own." (Philip W Anderson, "More and Different: Notes from a Thoughtful Curmudgeon", 2011)

"The mathematician is entirely free, within the limits of his imagination, to construct what worlds he pleases." (John W N Sullivan)

"The mathematician is the tamer who tamed infinity." (Lucian Blaga)

"The real mathematician is an enthusiast per se. ." (Friederich von Hardenberg [Novalis])

On Knowledge (-1699)

"In all disciplines in which there is systematic knowledge of things with principles, causes, or elements, it arises from a grasp of those: we think we have knowledge of a thing when we have found its primary causes and principles, and followed it back to its elements." (Aristotle, "Physics", cca. 350 BC)

"Thinking is different from perceiving and is held to be in part imagination, in part judgement: we must therefore first mark off the sphere of imagination and then speak of judgement. If then imagination is that in virtue of which an image arises for us, excluding metaphorical uses of the term, is it a single faculty or disposition relative to images, in virtue of which we discriminate and are either in error or not? The faculties in virtue of which we do this are sense, opinion, knowledge, thought." (Aristotle, "De Anima", cca. 350 BC)

"Knowledge, then, is a state of capacity to demonstrate, and has the other limiting characteristics which we specify in the Analytics; for it is when one believes in a certain way and the principles are known to him that he has knowledge, since if they are not better known to him than the conclusion, he will have his knowledge only on the basis of some concomitant." (Aristotle," Nicomachean Ethics", cca. 340 BC)

"What we know is not capable of being otherwise; of things capable of being otherwise we do not know, when they have passed outsideour observation, whether they exist or not. Therefore the object of knowledge is of necessity. Therefore it is eternal; for things that are of necessity in the unqualified sense are all eternal; and things that are eternal are ungenerated and imperishable. " (Aristotle, "Nicomachean Ethics", cca. 340 BC)

"We can get some idea of a whole from a part, but never knowledge or exact opinion. Special histories therefore contribute very little to the knowledge of the whole and conviction of its truth. It is only indeed by study of the interconnexion of all the particulars, their resemblances and differences, that we are enabled at least to make a general survey, and thus derive both benefit and pleasure from history." (Polybius, "The Histories", cca. 150 BC)

"The mathematician speculates the causes of a certain sensible effect, without considering its actual existence; for the contemplation of universals excludes the knowledge of particulars; and he whose intellectual eye is fixed on that which is general and comprehensive, will think but little of that which is sensible and singular." (Proclus Lycaeus, cca 5th century)

"All knowledge or cognition possessed by creatures is limited. Infinite knowledge belongs solely to God, because of His infinite nature." (John of Salisbury, "Metalogicon", 1159)

"All things have a way of adding up together, so that one will become more proficient in any proposed branch of learning to the extent that he has mastered neighboring and related departments of knowledge." (John of Salisbury, "Metalogicon", 1159)

"In our acquisition of [scientific] knowledge, investigation is the first step, and comes before comprehension, analysis, and retention. Innate ability, although it proceeds from nature, is fostered by study and exercise. What is difficult when we first try it, becomes easier after assiduous practice, and once the rules for doing it are mastered, very easy, unless languor creeps in, through lapse of use or carelessness, and impedes our efficiency. This, in short, is how all the arts have originated: Nature, the first fundamental, begets the habit and practice of study, which proceeds to provide an art, and the latter, in turn, finally furnishes the faculty whereof we speak. Natural ability is accordingly effective. So, too, is exercise. And memory likewise, is effective, when employed by the two aforesaid. With the help of the foregoing, reason waxes strong, and produces the arts, which are proportionate to [man’s] natural talents." (John of Salisbury, "Metalogicon", 1159)

"There are four great sciences, without which the other sciences cannot be known nor a knowledge of things secured […] Of these sciences the gate and key is mathematics […] He who is ignorant of this [mathematics] cannot know the other sciences nor the affairs of this world." (Roger Bacon, "Opus Majus", 1267)

"There are two modes of acquiring knowledge, namely, by reasoning and experience. Reasoning draws a conclusion and makes us grant the conclusion, but does not make the conclusion certain, nor does it remove doubt so that the mind may rest on the intuition of truth unless the mind discovers it by the path of experience." (Roger Bacon, "Opus Majus", 1267)

"That faculty which perceives and recognizes the noble proportions in what is given to the senses, and in other things situated outside itself, must be ascribed to the soul. It lies very close to the faculty which supplies formal schemata to the senses, or deeper still, and thus adjacent to the purely vital power of the soul, which does not think discursively […] Now it might be asked how this faculty of the soul, which does not engage in conceptual thinking, and can therefore have no proper knowledge of harmonic relations, should be capable of recognizing what is given in the outside world. For to recognize is to compare the sense perception outside with the original pictures inside, and to judge that it conforms to them." (Johannes Kepler, "Harmonices Mundi" ["Harmony of the World"] , 1619)

"Knowledge being to be had only of visible and certain truth, error is not a fault of our knowledge, but a mistake of our judgment, giving assent to that which is not true." (John Locke, "An Essay Concerning Human Understanding", 1689)

"[…] the highest probability amounts not to certainty, without which there can be no true knowledge." (John Locke, "An Essay Concerning Human Understanding", 1689)

On Knowledge (1775-1799)

"Cultivate that kind of knowledge which enables us to discover for ourselves in case of need that which others have to read or be told of." (Georg C Lichtenberg, Notebook D, 1773-1775)

"Knowledge is of two kinds. We know a subject ourselves, or we know where we can find information upon it." (Samuel Johnson, 1775)

"Our knowledge springs from two fundamental sources of the mind; the first is the capacity of receiving representations (receptivity for impressions), the second is the power of knowing an object through these representations (spontaneity [in the production] of concepts)." (Immanuel Kant, "Critique of Pure Reason", 1781)

"Philosophical knowledge is the knowledge gained by reason from concepts; mathematical knowledge is the knowledge gained by reason from the construction of concepts." (Immanuel Kant, "Critique of Pure Reason", 1781)

"Thoughts without content are empty, intuitions without concepts are blind. The understanding can intuit nothing, the senses can think nothing. Only through their unison can knowledge arise." (Immanuel Kant, "Critique of Pure Reason", 1781)

"The word ‘chance’ then expresses only our ignorance of the causes of the phenomena that we observe to occur and to succeed one another in no apparent order. Probability is relative in part to this ignorance, and in part to our knowledge." (Pierre-Simon Laplace, "Mémoire sur les Approximations des Formules qui sont Fonctions de Très Grands Nombres", 1783)

"The mathematician pays not the least regard either to testimony or conjecture, but deduces everything by demonstrative reasoning, from his definitions and axioms. Indeed, whatever is built upon conjecture, is improperly called science; for conjecture may beget opinion, but cannot produce knowledge." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"On completing one discovery we never fail to get an imperfect knowledge of others of which you could have no idea before […]" (Joseph Priestley, 1786)

"As there is no study which may be so advantageously entered upon with a less stock of preparatory knowledge than mathematics, so there is none in which a greater number of uneducated men have raised themselves, by their own exertions, to distinction and eminence. […] Many of the intellectual defects which, in such cases, are commonly placed to the account of mathematical studies, ought to be ascribed to the want of a liberal education in early youth." (Dugald Stewart, "Elements of the Philosophy of the Human Mind", 1792)

"The power of Reason […] is unquestionably the most important by far of those which are comprehended under the general title of Intellectual. It is on the right use of this power that our success in the pursuit of both knowledge and of  happiness depends; and it is by the exclusive possession of it that man is distinguished, in the most essential respects, from the lower animals. It is, indeed, from their subserviency to its operations, that the other faculties […] derive their chief value." (Dugald Stewart, "Elements of the Philosophy of the Human Mind", 1792)

"Conjecture may lead you to form opinions, but it cannot produce knowledge. Natural philosophy must be built upon the phenomena of nature discovered by observation and experiment." (George Adams, "Lectures on Natural and Experimental Philosophy" Vol. 1, 1794)

On Real Numbers I

"Because all conceivable numbers are either greater than zero or less than 0 or equal to 0, then it is clear that the square roots of negative numbers cannot be included among the possible numbers [real numbers]. Consequently we must say that these are impossible numbers. And this circumstance leads us to the concept of such numbers, which by their nature are impossible, and ordinarily are called imaginary or fancied numbers, because they exist only in the imagination." (Leonhard Euler, "Vollständige Anleitung zur Algebra", 1768-69)

"[…] with few exceptions all the operations and concepts that occur in the case of real numbers can indeed be carried over unchanged to complex ones. However, the concept of being greater cannot very well be applied to complex numbers. In the case of integration, too, there appear differences which rest on the multplicity of possible paths of integration when we are dealing with complex variables. Nevertheless, the large extent to which imaginary forms conform to the same laws as real ones justifies the introduction of imaginary forms into geometry." (Gottlob Frege, "On a Geometrical Representation of Imaginary forms in the Plane", 1873)

"Mathematics is a study which, when we start from its most familiar portions, may be pursued in either of two opposite directions. The more familiar direction is constructive, towards gradually increasing complexity: from integers to fractions, real numbers, complex numbers; from addition and multiplication to differentiation and integration, and on to higher mathematics. The other direction, which is less familiar, proceeds, by analyzing, to greater and greater abstractness and logical simplicity." (Bertrand Russell, "Introduction to Mathematical Philosophy", 1919)

"There is more to the calculation of π to a large number of decimal places than just the challenge involved. One reason for doing it is to secure statistical information concerning the 'normalcy' of π. A real number is said to be simply normal if in its decimal expansion all digits occur with equal frequency, and it is said to be normal if all blocks of digits of the same length occur with equal frequency. It is not known if π (or even √2, for that matter) is normal or even simply normal." (Howard Eves, "Mathematical Circles Revisited", 1971)

"Surreal numbers are an astonishing feat of legerdemain. An empty hat rests on a table made of a few axioms of standard set theory. Conway waves two simple rules in the air, then reaches into almost nothing and pulls out an infinitely rich tapestry of numbers that form a real and closed field. Every real number is surrounded by a host of new numbers that lie closer to it than any other 'real' value does. The system is truly 'surreal.'" (Martin Gardner, "Mathematical Magic Show", 1977)

"If explaining minds seems harder than explaining songs, we should remember that sometimes enlarging problems makes them simpler! The theory of the roots of equations seemed hard for centuries within its little world of real numbers, but it suddenly seemed simple once Gauss exposed the larger world of so-called complex numbers. Similarly, music should make more sense once seen through listeners' minds." (Marvin Minsky, "Music, Mind, and Meaning", 1981)

“The letter ‘i’ originally was meant to suggest the imaginary nature of this number, but with the greater abstraction of mathematics, it came to be realized that it was no more imaginary than many other mathematical constructs. True, it is not suitable for measuring quantities, but it obeys the same laws of arithmetic as do the real numbers, and, surprisingly enough, it makes the statement of various physical laws very natural.” (John A Paulos, “Beyond Numeracy”, 1991)

"A real number that satisfies (is a solution of) a polynomial equation with integer coefficients is called algebraic. […] A real number that is not algebraic is called transcendental. There is nothing mystic about this word; it merely indicates that these numbers transcend (go beyond) the realm of algebraic numbers."  (Eli Maor, "e: The Story of a Number", 1994)

"The real numbers are one of the most audacious idealizations made by the human mind, but they were used happily for centuries before anybody worried about the logic behind them. Paradoxically, people worried a great deal about the next enlargement of the number system, even though it was entirely harmless. That was the introduction of square roots for negative numbers, and it led to the 'imaginary' and 'complex' numbers. A professional mathematican should never leave home without them […]" (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

On Imagination (1750-1799)

"The imagination in a mathematician who creates makes no less difference than in a poet who invents […]." (Jean Le Rond d'Alembert, "Discours Preliminaire de L'Encyclopedie", 1751)

"Thus, metaphysics and mathematics are, among all the sciences that belong to reason, those in which imagination has the greatest role." (Jean Le Rond d'Alembert, "Discours Preliminaire de L'Encyclopedie", 1751)

"Things which do not now exist in the mind itself, can only be perceived, remembered, or imagined, by means of the ideas or images in the mind, which are the immediate objects of perception, remembrance, and imagination." (Thomas Reid, "An Inquiry into the Human Mind on the Principles", 1764)

"Men always fool themselves when they give up experience for systems born of the imagination. Man is the work of nature, he exists in nature, he is subject to its laws, he can not break free, he can not leave even in thought; it is in vain that his spirit wants to soar beyond the bounds of the visible world, he is always forced to return." (Paul-Henri T d’ Holbach, "Système de la Nature", 1770)

"Psychologists have hitherto failed to realize that imagination is a necessary ingredient of perception itself." (Immanuel Kant, "Critique of Pure Reason", 1781)

"The schema is in itself always a product of imagination. Since, however, the synthesis of imagination aims at no special intuition, but only at unity in the determination of sensibility, the schema has to be distinguished from the image." (Immanuel Kant," Critique of Pure Reason", 1781)

"There are conceptions which may be called fancy pictures. They are commonly called creatures of fancy, or of imagination. They are not the copies of any original that exists, but are originals themselves […]. They were conceived by their creators, and may be conceived by others, but they never existed. We do not ascribe the qualities of true or false to them, because they are not accompanied with any belief, nor do they imply any affirmation or negation." (Thomas Reid,"Essays on the Intellectual Powers of Man", 1785)

"The moment a person forms a theory, his imagination sees, in every object, only the traits which favor that theory." (Thomas Jefferson, [letter to Charles Thompson] 1787)

"Conjectures in philosophy are termed hypotheses or theories; and the investigation of an hypothesis founded on some slight probability, which accounts for many appearances in nature, has too often been considered as the highest attainment of a philosopher. If the hypothesis (sic) hangs well together, is embellished with a lively imagination, and serves to account for common appearances - it is considered by many, as having all the qualities that should recommend it to our belief, and all that ought to be required in a philosophical system." (George Adams, "Lectures on Natural and Experimental Philosophy" Vol. 1, 1794)

"Wit is the appearance, the external flash of imagination. Thus its divinity, and the witty character of mysticism." (K W Friedrich von Schlegel, "Dialogue on Poetry and Literary Aphorisms", [Aphorism 26] 1797) 

"The imagination is an eye where images remain forever." (Joseph Joubert, [Letter to Revd. Dr. Trusler] 1799)

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On Patterns (1990-1999)

"Mathematics is an exploratory science that seeks to understand every kind of pattern - patterns that occur in nature, patterns invented by the human mind, and even patterns created by other patterns." (Lynn A Steen, "The Future of Mathematics Education", 1990)

"Phenomena having uncertain individual outcomes but a regular pattern of outcomes in many repetitions are called random. 'Random' is not a synonym for 'haphazard' but a description of a kind of order different from the deterministic one that is popularly associated with science and mathematics. Probability is the branch of mathematics that describes randomness." (David S Moore, "Uncertainty", 1990)

"Systems thinking is a framework for seeing interrelationships rather than things, for seeing patterns rather than static snapshots. It is a set of general principles spanning fields as diverse as physical and social sciences, engineering and management." (Peter Senge, "The Fifth Discipline", 1990)

"The term chaos is used in a specific sense where it is an inherently random pattern of behaviour generated by fixed inputs into deterministic (that is fixed) rules (relationships). The rules take the form of non-linear feedback loops. Although the specific path followed by the behaviour so generated is random and hence unpredictable in the long-term, it always has an underlying pattern to it, a 'hidden' pattern, a global pattern or rhythm. That pattern is self-similarity, that is a constant degree of variation, consistent variability, regular irregularity, or more precisely, a constant fractal dimension. Chaos is therefore order (a pattern) within disorder (random behaviour)." (Ralph D Stacey, "The Chaos Frontier: Creative Strategic Control for Business", 1991)

"Chaos demonstrates that deterministic causes can have random effects […] There's a similar surprise regarding symmetry: symmetric causes can have asymmetric effects. […] This paradox, that symmetry can get lost between cause and effect, is called symmetry-breaking. […] From the smallest scales to the largest, many of nature's patterns are a result of broken symmetry; […]" (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"In everyday language, the words 'pattern' and 'symmetry' are used almost interchangeably, to indicate a property possessed by a regular arrangement of more-or-less identical units […]” (Ian Stewart & Martin Golubitsky, “Fearful Symmetry: Is God a Geometer?”, 1992)

"Scientists have discovered many peculiar things, and many beautiful things. But perhaps the most beautiful and the most peculiar thing that they have discovered is the pattern of science itself. Our scientific discoveries are not independent isolated facts; one scientific generalization finds its explanation in another, which is itself explained by yet another. By tracing these arrows of explanation back toward their source we have discovered a striking convergent pattern - perhaps the deepest thing we have yet learned about the universe." (Steven Weinberg, "Dreams of a Final Theory: The Scientist’s Search for the Ultimate Laws of Nature", 1992)

"Searching for patterns is a way of thinking that is essential for making generalizations, seeing relationships, and understanding the logic and order of mathematics. Functions evolve from the investigation of patterns and unify the various aspects of mathematics." (Marilyn Burns, "About Teaching Mathematics: A K–8 Resource", 1992)

"Symmetry is bound up in many of the deepest patterns of Nature, and nowadays it is fundamental to our scientific understanding of the universe. Conservation principles, such as those for energy or momentum, express a symmetry that (we believe) is possessed by the entire space-time continuum: the laws of physics are the same everywhere." (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"World view, a concept borrowed from cultural anthropology, refers to the culturally dependent, generally subconscious, fundamental organization of the mind. This conceptual organization manifests itself as a set of presuppositions that predispose one to feel, think, and act in predictable patterns." (Kenneth G Tobin, "The practice of constructivism in science education", 1993)

"[For] us to be able to speak and understand novel sentences, we have to store in our heads not just the words of our language but also the patterns of sentences possible in our language. These patterns, in turn, describe not just patterns of words but also patterns of patterns. Linguists refer to these patterns as the rules of language stored in memory; they refer to the complete collection of rules as the mental grammar of the language, or grammar for short." (Ray Jackendoff, "Patterns in the Mind", 1994)

"A neural network is characterized by A) its pattern of connections between the neurons (called its architecture), B) its method of determining the weights on the connections (called its training, or learning, algorithm), and C) its activation function." (Laurene Fausett, "Fundamentals of Neural Networks", 1994)

"At the other far extreme, we find many systems ordered as a patchwork of parallel operations, very much as in the neural network of a brain or in a colony of ants. Action in these systems proceeds in a messy cascade of interdependent events. Instead of the discrete ticks of cause and effect that run a clock, a thousand clock springs try to simultaneously run a parallel system. Since there is no chain of command, the particular action of any single spring diffuses into the whole, making it easier for the sum of the whole to overwhelm the parts of the whole. What emerges from the collective is not a series of critical individual actions but a multitude of simultaneous actions whose collective pattern is far more important. This is the swarm model." (Kevin Kelly, "Out of Control: The New Biology of Machines, Social Systems and the Economic World", 1995)

"Each of nature's patterns is a puzzle, nearly always a deep one. Mathematics is brilliant at helping us to solve puzzles. It is a more or less systematic way of digging out the rules and structures that lie behind some observed pattern or regularity, and then using those rules and structures to explain what's going on." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Human mind and culture have developed a formal system of thought for recognizing, classifying, and exploiting patterns. We call it mathematics. By using mathematics to organize and systematize our ideas about patterns, we have discovered a great secret: nature's patterns are not just there to be admired, they are vital clues to the rules that govern natural processes." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Patterns possess utility as well as beauty. Once we have learned to recognize a background pattern, exceptions suddenly stand out." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Self-organization refers to the spontaneous formation of patterns and pattern change in open, nonequilibrium systems. […] Self-organization provides a paradigm for behavior and cognition, as well as the structure and function of the nervous system. In contrast to a computer, which requires particular programs to produce particular results, the tendency for self-organization is intrinsic to natural systems under certain conditions." (J A Scott Kelso, "Dynamic Patterns : The Self-organization of Brain and Behavior", 1995)

"Symmetry is basically a geometrical concept. Mathematically it can be defined as the invariance of geometrical patterns under certain operations. But when abstracted, the concept applies to all sorts of situations. It is one of the ways by which the human mind recognizes order in nature. In this sense symmetry need not be perfect to be meaningful. Even an approximate symmetry attracts one's attention, and makes one wonder if there is some deep reason behind it." (Eguchi Tohru & ?K Nishijima , "Broken Symmetry: Selected Papers Of Y Nambu", 1995)

"Whatever the reasons, mathematics definitely is a useful way to think about nature. What do we want it to tell us about the patterns we observe? There are many answers. We want to understand how they happen; to understand why they happen, which is different; to organize the underlying patterns and regularities in the most satisfying way; to predict how nature will behave; to control nature for our own ends; and to make practical use of what we have learned about our world. Mathematics helps us to do all these things, and often it is indispensable." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"If we are to have meaningful, connected experiences; ones that we can comprehend and reason about; we must be able to discern patterns to our actions, perceptions, and conceptions. Underlying our vast network of interrelated literal meanings (all of those words about objects and actions) are those imaginative structures of understanding such as schema and metaphor, such as the mental imagery that allows us to extrapolate a path, or zoom in on one part of the whole, or zoom out until the trees merge into a forest." (William H Calvin, "The Cerebral Code", 1996)

"The methods of science include controlled experiments, classification, pattern recognition, analysis, and deduction. In the humanities we apply analogy, metaphor, criticism, and (e)valuation. In design we devise alternatives, form patterns, synthesize, use conjecture, and model solutions." (Béla H Bánáthy, "Designing Social Systems in a Changing World", 1996)

"The more complex the network is, the more complex its pattern of interconnections, the more resilient it will be." (Fritjof Capra, "The Web of Life: A New Scientific Understanding of Living Systems", 1996)

"The role of science, like that of art, is to blend proximate imagery with more distant meaning, the parts we already understand with those given as new into larger patterns that are coherent enough to be acceptable as truth. Biologists know this relation by intuition during the course of fieldwork, as they struggle to make order out of the infinitely varying patterns of nature." (Edward O Wilson, "In Search of Nature", 1996)

"Mathematics can function as a telescope, a microscope, a sieve for sorting out the signal from the noise, a template for pattern perception, a way of seeking and validating truth. […] A knowledge of the mathematics behind our ideas can help us to fool ourselves a little less often, with less drastic consequences." (K C Cole, "The Universe and the Teacup: The Mathematics of Truth and Beauty", 1997)

"Mathematics is a way of thinking that can help make muddy relationships clear. It is a language that allows us to translate the complexity of the world into manageable patterns. In a sense, it works like turning off the houselights in a theater the better to see a movie. Certainly, something is lost when the lights go down; you can no longer see the faces of those around you or the inlaid patterns on the ceiling. But you gain a far better view of the subject at hand." (K C Cole, "The Universe and the Teacup: The Mathematics of Truth and Beauty", 1997)

"A formal system consists of a number of tokens or symbols, like pieces in a game. These symbols can be combined into patterns by means of a set of rules which defines what is or is not permissible (e.g. the rules of chess). These rules are strictly formal, i.e. they conform to a precise logic. The configuration of the symbols at any specific moment constitutes a ‘state’ of the system. A specific state will activate the applicable rules which then transform the system from one state to another. If the set of rules governing the behaviour of the system are exact and complete, one could test whether various possible states of the system are or are not permissible." (Paul Cilliers, "Complexity and Postmodernism: Understanding Complex Systems", 1998)

"Mathematics, in the common lay view, is a static discipline based on formulas taught in the school subjects of arithmetic, geometry, algebra, and calculus. But outside public view, mathematics continues to grow at a rapid rate, spreading into new fields and spawning new applications. The guide to this growth is not calculation and formulas but an open-ended search for pattern." (Lynn A Steen, "The Future of Mathematics Education", 1998)

"A neural network consists of large numbers of simple neurons that are richly interconnected. The weights associated with the connections between neurons determine the characteristics of the network. During a training period, the network adjusts the values of the interconnecting weights. The value of any specific weight has no significance; it is the patterns of weight values in the whole system that bear information. Since these patterns are complex, and are generated by the network itself (by means of a general learning strategy applicable to the whole network), there is no abstract procedure available to describe the process used by the network to solve the problem. There are only complex patterns of relationships." (Paul Cilliers, "Complexity and Postmodernism: Understanding Complex Systems", 1998)

"Mathematics has traditionally been described as the science of number and shape. […] When viewed in this broader context, we see that mathematics is not just about number and shape but about pattern and order of all sorts. Number and shape - arithmetic and geometry - are but two of many media in which mathematicians work. Active mathematicians seek patterns wherever they arise." (Lynn A Steen, "The Future of Mathematics Education", 1998)

"Often, we use the word random loosely to describe something that is disordered, irregular, patternless, or unpredictable. We link it with chance, probability, luck, and coincidence. However, when we examine what we mean by random in various contexts, ambiguities and uncertainties inevitably arise. Tackling the subtleties of randomness allows us to go to the root of what we can understand of the universe we inhabit and helps us to define the limits of what we can know with certainty." (Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari", 1998)

"Sequences of random numbers also inevitably display certain regularities. […] The trouble is, just as no real die, coin, or roulette wheel is ever likely to be perfectly fair, no numerical recipe produces truly random numbers. The mere existence of a formula suggests some sort of predictability or pattern." (Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari", 1998)

"We use mathematics and statistics to describe the diverse realms of randomness. From these descriptions, we attempt to glean insights into the workings of chance and to search for hidden causes. With such tools in hand, we seek patterns and relationships and propose predictions that help us make sense of the world."  (Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari", 1998)

"Complexity is looking at interacting elements and asking how they form patterns and how the patterns unfold. It’s important to point out that the patterns may never be finished. They’re open-ended. In standard science this hit some things that most scientists have a negative reaction to. Science doesn’t like perpetual novelty." (W Brian Arthur, 1999)

"Randomness is the very stuff of life, looming large in our everyday experience. […] The fascination of randomness is that it is pervasive, providing the surprising coincidences, bizarre luck, and unexpected twists that color our perception of everyday events." (Edward Beltrami, "Chaos and Order in Mathematics and Life", 1999)

"The first view of randomness is of clutter bred by complicated entanglements. Even though we know there are rules, the outcome is uncertain. Lotteries and card games are generally perceived to belong to this category. More troublesome is that nature's design itself is known imperfectly, and worse, the rules may be hidden from us, and therefore we cannot specify a cause or discern any pattern of order. When, for instance, an outcome takes place as the confluence of totally unrelated events, it may appear to be so surprising and bizarre that we say that it is due to blind chance." (Edward Beltrami. "What is Random?: Chance and Order in Mathematics and Life", 1999)

Calculus I: Differential Calculus I

"Thus, differential calculus has all the exactitude of other algebraic operations." (Pierre-Simon Laplace, "A Philosophical Essay on Probabilities", 1814)

"The invention of a new symbol is a step in the advancement of civilisation. Why were the Greeks, in spite of their penetrating intelligence and their passionate pursuit of Science, unable to carry Mathematics farther than they did? and why, having formed the conception of the Method of Exhaustions, did they stop short of that of the Differential Calculus? It was because they had not the requisite symbols as means of expression. They had no Algebra. Nor was the place of this supplied by any other symbolical language sufficiently general and flexible; so that they were without the logical instruments necessary to construct the great instrument of the Calculus." (George H Lewes "Problems of Life and Mind", 1873)

"Everyone who understands the subject will agree that even the basis on which the scientific explanation of nature rests is intelligible only to those who have learned at least the elements of the differential and integral calculus, as well as analytical geometry." (Felix Klein, Jahresbericht der Deutsche Mathematiker Vereinigung Vol. 1, 1902)

"The chief difficulty of modern theoretical physics resides not in the fact that it expresses itself almost exclusively in mathematical symbols, but in the psychological difficulty of supposing that complete nonsense can be seriously promulgated and transmitted by persons who have sufficient intelligence of some kind to perform operations in differential and integral calculus […]" (Celia Green, "The Decline and Fall of Science", 1976)

"The invention of the differential calculus was based on the recognition that an instantaneous rate is the asymptotic limit of averages in which the time interval involved is systematically shrunk. This is a concept that mathematicians recognized long before they had the skill to actually compute such an asymptotic limit." (Michael Guillen,"Bridges to Infinity: The Human Side of Mathematics", 1983)

"The acceptance of complex numbers into the realm of algebra had an impact on analysis as well. The great success of the differential and integral calculus raised the possibility of extending it to functions of complex variables. Formally, we can extend Euler's definition of a function to complex variables without changing a single word; we merely allow the constants and variables to assume complex values. But from a geometric point of view, such a function cannot be plotted as a graph in a two-dimensional coordinate system because each of the variables now requires for its representation a two-dimensional coordinate system, that is, a plane. To interpret such a function geometrically, we must think of it as a mapping, or transformation, from one plane to another." (Eli Maor, "e: The Story of a Number", 1994)

"By studying analytic functions using power series, the algebra of the Middle Ages was connected to infinite operations (various algebraic operations with infinite series). The relation of algebra with infinite operations was later merged with the newly developed differential and integral calculus. These developments gave impetus to early stages of the development of analysis. In a way, we can say that analyticity is the notion that first crossed the boundary from finite to infinite by passing from polynomials to infinite series. However, algebraic properties of polynomial functions still are strongly present in analytic functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996) 

"In fact the complex numbers form a field. [...] So however strange you may feel the very notion of a complex number to be, it does turn out to provide a 'normal' type of arithmetic. In fact it gives you a tremendous bonus not available with any of the other number systems. [...] The fundamental theorem of algebra is just one of several reasons why the complex-number system is such a 'nice' one. Another important reason is that the field of complex numbers supports the development of a powerful differential calculus, leading to the rich theory of functions of a complex variable." (Keith Devlin, "Mathematics: The New Golden Age", 1998)

"Thus, calculus proceeds in two phases: cutting and rebuilding. In mathematical terms, the cutting process always involves infinitely fine subtraction, which is used to quantify the differences between the parts. Accordingly, this half of the subject is called differential calculus. The reassembly process always involves infinite addition, which integrates the parts back into the original whole. This half of the subject is called integral calculus." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"Nothing has afforded me so convincing a proof of the unity of the Deity as these purely mental conceptions of numerical and mathematical science which have been by slow degrees vouchsafed to man, and are still granted in these latter times by the Differential Calculus, now superseded by the Higher Algebra, all of which must have existed in that sublimely omniscient Mind from eternity." (Mary Somerville)

On Differential Equations IV

"Problems relative to the uniform propagation, or to the varied movements of heat in the interior of solids, are reduced […] to problems of pure analysis, and the progress of this part of physics will depend in consequence upon the advance which may be made in the art of analysis. The differential equations […] contain the chief results of the theory; they express, in the most general and concise manner, the necessary relations of numerical analysis to a very extensive class of phenomena; and they connect forever with mathematical science one of the most important branches of natural  philosophy." (Jean-Baptiste-Joseph Fourier, "The Analytical Theory of Heat", 1822)

"It is well known that the central problem of the whole of modern mathematics is the study of the transcendental functions defined by differential equations." (Felix Klein, "Lectures on Mathematics", 1911)

"Men have fallen in love with statues and pictures. I find it easier to imagine a man falling in love with a differential equation, and I am inclined to think that some mathematicians have done so. Even in a nonmathematician like myself, some differential equations evoke fairly violent physical sensations to those described by Sappho and Catallus when viewing their mistresses. Personally, I obtain an even greater 'kick' from finite difference equations, which are perhaps more like those which an up-to-date materialist would use to describe human behavior." (John B S Haldane, "The Inequality of Man and Other Essays", 1932)

"The method of successive approximations is often applied to proving existence of solutions to various classes of functional equations; moreover, the proof of convergence of these approximations leans on the fact that the equation under study may be majorised by another equation of a simple kind. Similar proofs may be encountered in the theory of infinitely many simultaneous linear equations and in the theory of integral and differential equations. Consideration of semiordered spaces and operations between them enables us to easily develop a complete theory of such functional equations in abstract form." (Leonid V Kantorovich, "On one class of functional equations", 1936)

"The emphasis on mathematical methods seems to be shifted more towards combinatorics and set theory - and away from the algorithm of differential equations which dominates mathematical physics." (John von Neumann & Oskar Morgenstern, "Theory of Games and Economic Behavior", 1944)

"The study of changes in the qualitative structure of the flow of a differential equation as parameters are varied is called bifurcation theory. At a given parameter value, a differential equation is said to have stable orbit structure if the qualitative structure of the flow does not change for sufficiently small variations of the parameter. A parameter value for which the flow does not have stable orbit structure is called a bifurcation value, and the equation is said to be at a bifurcation point." (Jack K Hale & Hüseyin Kocak, "Dynamics and Bifurcations", 1991)

"Dynamical systems that vary in discrete steps […] are technically known as mappings. The mathematical tool for handling a mapping is the difference equation. A system of difference equations amounts to a set of formulas that together express the values of all of the variables at the next step in terms of the values at the current step. […] For mappings, the difference equations directly express future states in terms of present ones, and obtaining chronological sequences of points poses no problems. For flows, the differential equations must first be solved. General solutions of equations whose particular solutions are chaotic cannot ordinarily be found, and approximations to the latter are usually determined by numerical methods." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Faced with the overwhelming complexity of the real world, time pressure, and limited cognitive capabilities, we are forced to fall back on rote procedures, habits, rules of thumb, and simple mental models to make decisions. Though we sometimes strive to make the best decisions we can, bounded rationality means we often systematically fall short, limiting our ability to learn from experience." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"Following the traditional classification in the field of control systems, a system that describes the input-output behavior in a way similar to a mathematical mapping without involving a differential operator or equation is called a static system. In contrast, a system described by a differential operator or equation is called a dynamic system." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)

"The standard view among most theoretical physicists, engineers and economists is that mathematical models are syntactic (linguistic) items, identified with particular systems of equations or relational statements. From this perspective, the process of solving a designated system of (algebraic, difference, differential, stochastic, etc.) equations of the target system, and interpreting the particular solutions directly in the context of predictions and explanations are primary, while the mathematical structures of associated state and orbit spaces, and quantity algebras – although conceptually important, are secondary." (Zoltan Domotor, "Mathematical Models in Philosophy of Science" [Mathematics of Complexity and Dynamical Systems, 2012])

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On Topology V

"In mathematics, logic, linguistics, and other abstract disciplines, the systems are not assigned to objects. They are defined by an enumeration of the variables, their admissible values, and their algebraic, topological, grammatical, and other properties which, in the given case, determine the relations between the variables under consideration." (George Klir, "An approach to general systems theory", 1969)

"Because of its foundation in topology, catastrophe theory is qualitative, not quantitative. Just as geometry treated the properties of a triangle without regard to its size, so topology deals with properties that have no magnitude, for example, the property of a given point being inside or outside a closed curve or surface. This property is what topologists call 'invariant' -it does not change even when the curve is distorted. A topologist may work with seven-dimensional space, but he does not and cannot measure (in the ordinary sense) along any of those dimensions. The ability to classify and manipulate all types of form is achieved only by giving up concepts such as size, distance, and rate. So while catastrophe theory is well suited to describe and even to predict the shape of processes, its descriptions and predictions are not quantitative like those of theories built upon calculus. Instead, they are rather like maps without a scale: they tell us that there are mountains to the left, a river to the right, and a cliff somewhere ahead, but not how far away each is, or how large." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

"Geometry and topology most often deal with geometrical figures, objects realized as a set of points in a Euclidean space (maybe of many dimensions). It is useful to view these objects not as rigid (solid) bodies, but as figures that admit continuous deformation preserving some qualitative properties of the object. Recall that the mapping of one object onto another is called continuous if it can be determined by means of continuous functions in a Cartesian coordinate system in space. The mapping of one figure onto another is called homeomorphism if it is continuous and one-to-one, i.e. establishes a one-to-one correspondence between points of both figures." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"Homeomorphism is one of the basic concepts in topology. Homeomorphism, along with the whole topology, is in a sense the basis of spatial perception. When we look at an object, we see, say, a telephone receiver or a ring-shaped roll and first of all pay attention to the geometrical shape (although we do not concentrate on it specially) - an oblong figure thickened at the ends or a round rim with a large hole in the middle. Even if we deliberately concentrate on the shape of the object and forget about its practical application, we do not yet 'see' the essence of the shape. The point is that oblongness, roundness, etc. are metric properties of the object. The topology of the form lies 'beyond them'." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"Since geometry is the mathematical idealization of space, a natural way to organize its study is by dimension. First we have points, objects of dimension O. Then come lines and curves, which are one-dimensional objects, followed by two-dimensional surfaces, and so on. A collection of such objects from a given dimension forms what mathematicians call a 'space'. And if there is some notion enabling us to say when two objects are 'nearby' in such a space, then it's called a topological space." (John L Casti, "Five Golden Rules", 1995)

"One of the basic tasks of topology is to learn to distinguish nonhomeomorphic figures. To this end one introduces the class of invariant quantities that do not change under homeomorphic transformations of a given figure. The study of the invariance of topological spaces is connected with the solution of a whole series of complex questions: Can one describe a class of invariants of a given manifold? Is there a set of integral invariants that fully characterizes the topological type of a manifold? and so forth." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

"Topology studies those characteristics of figures which are preserved under a certain class of continuous transformations. Imagine two figures, a square and a circular disk, made of rubber. Deformations can convert the square into the disk, but without tearing the figure it is impossible to convert the disk by any deformation into an annulus. In topology, this intuitively obvious distinction is formalized." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

"[...] there is no area of mathematics where thinking abstractly has paid more handsome dividends than in topology, the study of those properties of geometrical objects that remain unchanged when we deform or distort them in a continuous fashion without tearing, cutting, or breaking them." (John L Casti, "Five Golden Rules", 1995)

"At first, topology can seem like an unusually imprecise branch of mathematics. It’s the study of squishy play-dough shapes capable of bending, stretching and compressing without limit. But topologists do have some restrictions: They cannot create or destroy holes within shapes. […] While this might seem like a far cry from the rigors of algebra, a powerful idea called homology helps mathematicians connect these two worlds. […] homology infers an object’s holes from its boundaries, a more precise mathematical concept. To study the holes in an object, mathematicians only need information about its boundaries." (Kelsey Houston-Edwards, "How Mathematicians Use Homology to Make Sense of Topology", Quanta Magazine, 2021) [source]

"In geometry, shapes like circles and polyhedra are rigid objects; the tools of the trade are lengths, angles and areas. But in topology, shapes are flexible things, as if made from rubber. A topologist is free to stretch and twist a shape. Even cutting and gluing are allowed, as long as the cut is precisely reglued. A sphere and a cube are distinct geometric objects, but to a topologist, they’re indistinguishable." (David E Richeson, "Topology 101: The Hole Truth", 2021) [source]

On Heuristics I

"The materialistic point of view in psychology can claim, at best, only the value of an heuristic hypothesis." (Wilhelm Wundt, "Principles of Physiological Psychology", 1874)

"Heuristic reasoning is good in itself. What is bad is to mix up heuristic reasoning with rigorous proof. What is worse is to sell heuristic reasoning for rigorous proof." (George Pólya, "How to Solve It", 1945)

"Heuristic, or heuretic, or 'ars inveniendi' was the name of a certain branch of study, not very clearly circumscribed, belonging to logic, or to philosophy, or to psychology, often outlined, seldom presented in detail, and as good as forgotten today. The aim of heuristic is to study the methods and rules of discovery and invention. [...] Heuristic, as an adjective, means 'serving to discover'." (George Pólya, "How to Solve It", 1945)

"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)

"The attempt to characterize exactly models of an empirical theory almost inevitably yields a more precise and clearer understanding of the exact character of a theory. The emptiness and shallowness of many classical theories in the social sciences is well brought out by the attempt to formulate in any exact fashion what constitutes a model of the theory. The kind of theory which mainly consists of insightful remarks and heuristic slogans will not be amenable to this treatment. The effort to make it exact will at the same time reveal the weakness of the theory." (Patrick Suppes," A Comparison of the Meaning and Uses of Models in Mathematics and the Empirical Sciences", Synthese  Vol. 12 (2/3), 1960)

"Factoring big numbers is a strange kind of mathematics that closely resembles the experimental sciences, where nature has the last and definitive word. […] as with the experimental sciences, both rigorous and heuristic analyses can be valuable in understanding the subject and moving it forward. And, as with the experimental sciences, there is sometimes a tension between pure and applied practitioners." (Carl B Pomerance, "A Tale of Two Sieves", The Notices of the American Mathematical Society 43, 1996)

"[…] mathematics does not come to us written indelibly on Nature’s Tablets, but rather is the product of a controlled search governed by metaphorical considerations, the premier instance being the heuristics of the conservation principles." (Philip Mirowski, "More Heat than Light: Economics as Social Physics: Physics as Nature’s Economics", 1989)

"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)

"In the language of mental models, such past experience provided the default assumptions necessary to fill the gaps in the emerging and necessarily incomplete framework of a relativistic theory of gravitation. It was precisely the nature of these default assumptions that allowed them to be discarded again in the light of novel information - provided, for instance, by the further elaboration of the mathematical formalism - without, however, having to abandon the underlying mental models which could thus continue to function as heuristic orientations." (Jürgen Renn, "Before the Riemann Tensor: The Emergence of Einstein’s Double Strategy", [in "The Universe of General Relativity"] 2000)

"You can often hear from non-mathematicians, especially from philosophers, that mathematics consists exclusively in drawing conclusions from clearly stated premises; and that in this process, it makes no difference what these premises signify, whether they are true or fa1se, provided only that they do not contradict one another. But a per. son who has done productive mathematical work will talk quite differently. In fact these people [the non-mathematicians] are thinking only of the crystallized form into which finished mathematica1 theories are finally cast. However, the investigator himself, in mathematics as in every other science, does not work in this rigorous deductive fashion. On the contrary, he makes essential use of his imagination and proceeds inductively aided by heuristic expedients. One can give numerous examples of mathematicians who have discovered theorems of the greatest importance which they were unable to prove. Should one then refuse to recognize this as a great accomplishment and in deference to the above definition insist that this is not mathematics? After all it is an arbitrary thing how the word is to be used, but no judgment of value can deny that the inductive work of the person who first announces the theorem is at least as valuable as the deductive work. of the one who proves it. For both are equally necessary and the discovery is the presupposition of the later conclusion." (Felix Klein)