On Theories (1915-1929)

"Theory is the best guide for experiment - that were it not for theory and the problems and hypotheses that come out of it, we would not know the points we wanted to verify, and hence would experiment aimlessly" (Henry Hazlitt, "Thinking as a Science", 1916)

"As soon as science has emerged from its initial stages, theoretical advances are no longer achieved merely by a process of arrangement. Guided by empirical data, the investigator rather develops a system of thought which, in general, is built up logically from a small number of fundamental assumptions, the so-called axioms. We call such a system of thought a theory. The theory finds the justification for its existence in the fact that it correlates a large number of single observations, and it is just here that the 'truth' of the theory lies. " (Albert Einstein: "Relativity: The Special and General Theory", 1916)

"No fairer destiny could be allotted to any physical theory, than that it should of itself point out the way to the introduction of a more comprehensive theory, in which it lives on as a limiting case." (Albert Einstein: "Relativity, The Special and General Theory", 1916)

"To come very near to a true theory, and to grasp its precise application, are two very different things, as the history of a science teaches us. Everything of importance has been said before by somebody who did not discover it." (Alfred N Whitehead, "The Organization of Thought", 1917)

"Facts are carpet-tacks under the pneumatic tires of theory." (Austin O’Malley, "Keystones of Thought", 1918)

"[…] analogies are not ‘aids’ to the establishment of theories; they are an utterly essential part of theories, without which theories would be completely valueless and unworthy of the name. It is often suggested that the analogy leads to the formulation of the theory, but that once the theory is formulated the analogy has served its purpose and may be removed or forgotten. Such a suggestion is absolutely false and perniciously misleading." (Norman R Campbell, "Physics, the Elements", 1920) 

"Nothing is more interesting to the true theorist than a fact which directly contradicts a theory generally accepted up to that time, for this is his particular work." (Max Planck, "A Survey of Physics", 1925)

"[…] the mere collection of facts, without some basis of theory for guidance and elucidation, is foolish and profitless." (Gamaliel Bradford, "Darwin", 1926)

"In scientific thought we adopt the simplest theory which will explain all the facts under consideration and enable us to predict facts of the same kind. The  catch in this criterion lies in the world 'simplest'." (John B S Haldane, "Possible Worlds and Other Essays", 1928)

"It is characteristic of a good scientific theory that it makes no more assumptions than are needed to explain the facts under consideration and predict a few more." (John B S Haldane, "Possible Worlds and Other Essays", 1928)

"Often a liberal antidote of experience supplies a sovereign cure for a paralyzing abstraction built upon a theory." (Benjamin N Cardozo, "Paradoxes of Legal Science", 1928)

"[…] facts are too bulky to be lugged about conveniently except on the wheels of theory." (Julian Huxley, "Essays of a Biologist", 1929)

 "We can invent as many theories we like, and any one of them can be made to fit the facts. But that theory is always preferred which makes the fewest number of assumptions." (Albert Einstein [interview] 1929)

Immanuel Kant - Collected Quotes

"God put a secret art into the forces of Nature so as to enable it to fashion itself out of chaos into a perfect world system." (Immanuel Kant, "Universal Natural History and Theory of the Heavens", 1755)

"It falls into this difficulty without any fault of its own. It begins with principles, which cannot be dispensed with in the field of experience, and the truth and sufficiency of which are, at the same time, insured by experience. With these principles it rises, in obedience to the laws of its own nature, to ever higher and more remote conditions. But it quickly discovers that, in this way, its labours must remain ever incomplete, because new questions never cease to present themselves; and thus it finds itself compelled to have recourse to principles which transcend the region of experience, while they are regarded by common sense without distrust. It thus falls into confusion and contradictions, from which it conjectures the presence of latent errors, which, however, it is unable to discover, because the principles it employs, transcending the limits of experience, cannot be tested by that criterion. The arena of these endless contests is called Metaphysic.” (Immanuel Kant, “The Critique of Pure Reason”, 1781)

"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)

"That metaphysics has hitherto remained in so vacillating a state of uncertainty and contradiction, is only to be attributed to the fact, that this great problem, and perhaps even the difference between analytical and synthetical judgements, did not sooner suggest itself to philosophers. Upon the solution of this problem, or upon sufficient proof of the impossibility of synthetical knowledge a priori, depends the existence or downfall of metaphysics. (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)

"Natural processes should be judged different from mechanical ones because they are self-organizing." (Immanuel Kant, "Critique of Judgment", 1790)

"We must think of each part as an organ, that produces the other parts (so that each reciprocally produces the other) […] Because of this, [the organism] will be both an organized and self-organizing being." (Immanuel Kant, "Critique  of  Judgment", 1790)

“We construct concepts when we represent them in intuition a priori, without experience, or when we represent in intuition  the object which corresponds to our concept of it. - The mathematician can never apply his reason to mere concepts, nor the philosopher to the construction of concepts. - In mathematics  the reason is employed in concreto, however, the intuition is not  empirical, but the object of contemplation is something a priori.” (Immanuel Kant, “Logic”, 1800)

"With the synthesis of every new concept in the aggregation of coordinate characteristics the extensive or complex distinctness is increased; with the further analysis of concepts in the series of subordinate characteristics the intensive or deep distinctness is increased. The latter kind of distinctness, as it necessarily serves the thoroughness and conclusiveness of cognition, is therefore mainly the business of philosophy and is carried farthest especially in metaphysical investigations." (Immanuel Kant, “Logic”, 1800)

“One says of a person who has travelled much, that he has seen the world. to the knowledge of the world than just seeing it. Whoever wants to must draw up a plan beforehand and must not just regard the world senses.” (Immanuel Kant “Physische Geographie” [Physical Geography], 1802)

"[…] all mathematical cognition has this pecularity: that it must first exhibit its concept in intuitional form. […] Without this, mathematics cannot take a single step. Its judgements are therefore always intuitional, whereas philosophy must make do with discursive judgements from mere concepts. It may illustrate its judgements by means of a visual form, but it can never derive them from such a form.” (Immanuel Kant)

"[…] there is a God precisely because Nature itself, even in chaos, cannot proceed except in an orderly and regular manner." (Immanuel Kant)

“God has put a secret art into the forces of Nature so as to enable it to fashion itself out of chaos into a perfect world system.” (Immanuel Kant)

"Mathematics is pure poetry." (Immanuel Kant)

Stephen J Gould - Collected Quotes

"Facts do not ‘speak for themselves’; they are read in the light of theory. Creative thought, in science as much as in the arts, is the motor of changing opinion. Science is a quintessentially human activity, not a mechanized, robot-like accumulation of objective information, leading by laws of logic to inescapable interpretation." (Stephen J Gould, "Ever Since Darwin: Reflections in Natural History", 1977)

"Science is not a heartless pursuit of objective information. It is a creative human activity, its geniuses acting more as artists than information processors. Changes in theory are not simply the derivative results of the new discoveries but the work of creative imagination influenced by contemporary social and political forces." (Stephen J Gould, "Ever Since Darwin: Reflections in Natural History", 1977)

"[…] science must be understood as a social phenomenon, a gutsy, human enterprise, not the work of robots programmed to collect pure information. […] Science, since people must do it, is a socially embedded activity. It progresses by hunch, vision, and intuition." (Stephen J Gould, "The Mismeasure of Man", 1980)

"Science, since people must do it, is a socially embedded activity. It progresses by hunch, vision, and intuition. Much of its change through time does not record a closer approach to absolute truth, but the alteration of cultural contexts that influence it so strongly. Facts are not pure and unsullied bits of information; culture also influences what we see and how we see it. Theories, moreover, are not inexorable inductions from facts. The most creative theories are often imaginative visions imposed upon facts; the source of imagination is also strongly cultural." (Stephen J Gould, "The Mismeasure of Man", 1980)

"In science 'fact' can only mean 'confirmed to such a degree that it would be perverse to withhold provisional assent'." (Stephen J Gould, 1983) 

"Perhaps randomness is not merely an adequate description for complex causes that we cannot specify. Perhaps the world really works this way, and many events are uncaused in any conventional sense of the word." (Stephen J Gould, "Hen's Teeth and Horse's Toes", 1983)

"Well evolution is a theory. It is also a fact. And facts and theories are different things, not rungs in a hierarchy of increasing certainty. Facts are the world’s data. Theories are structures of ideas that explain and interpret facts. Facts do not go away when scientists debate rival theories to explain them." (Stephen J Gould, "Hen’s Teeth and Horses Toes", 1983)

"The progress of science requires more than new data; it needs novel frameworks and contexts. And where do these fundamentally new views of the world arise? They are not simply discovered by pure observation; they require new modes of thought. And where can we find them, if old modes do not even include the right metaphors? The nature of true genius must lie in the elusive capacity to construct these new modes from apparent darkness. The basic chanciness and unpredictability of science must also reside in the inherent difficulty of such a task." (Stephen J Gould, "The Flamingo's Smile: Reflections in Natural History", 1985)

"We often think, naïvely, that missing data are the primary impediments to intellectual progress - just find the right facts and all problems will dissipate. But barriers are often deeper and more abstract in thought. We must have access to the right metaphor, not only to the requisite information. Revolutionary thinkers are not, primarily, gatherers of facts, but weavers of new intellectual structures." (Stephen J Gould, "The Flamingo's Smile: Reflections in Natural History", 1985)

"Numbers have undoubted powers to beguile and benumb, but critics must probe behind numbers to the character of arguments and the biases that motivate them." (Stephen J Gould, "An Urchin in the Storm: Essays About Books and Ideas", 1987)

"All great theories are expansive, and all notions so rich in scope and implication are underpinned by visions about the nature of things. You may call these visions ‘philosophy’, or ‘metaphor’, or ‘organizing principle’, but one thing they are surely not - they are not simple inductions from observed facts of the natural world." (Stephen J Gould, "Time’s Arrow, Time’s Cycle", 1987)

"Numbers have undoubted powers to beguile and benumb, but critics must probe behind numbers to the character of arguments and the biases that motivate them." (Stephen J Gould, "An Urchin in the Storm: Essays About Books and Ideas", 1987) 

"The concepts of science, in all their richness and ambiguity, can be presented without any compromise, without any simplification counting as distortion, in language accessible to all intelligent people." (Stephen J Gould, "Wonderful Life: The Burgess Shale and the Nature of History", 1990)

"Probability does pervade the universe, and in this sense, the old chestnut about baseball imitating life really has validity. The statistics of streaks and slumps, properly understood, do teach an important lesson about epistemology, and life in general. The history of a species, or any natural phenomenon, that requires unbroken continuity in a world of trouble, works like a batting streak. All are games of a gambler playing with a limited stake against a house with infinite resources. The gambler must eventually go bust. His aim can only be to stick around as long as possible, to have some fun while he's at it, and, if he happens to be a moral agent as well, to worry about staying the course with honor!" (Stephen J Gould, 1991)

"But our ways of learning about the world are strongly influenced by the social preconceptions and biased modes of thinking that each scientist must apply to any problem. The stereotype of a fully rational and objective ‘scientific method’, with individual scientists as logical (and interchangeable) robots, is self-serving mythology." (Stephen J Gould, "This View of Life: In the Mind of the Beholder", "Natural History", Vol. 103, No. 2, 1994)

"I do not know that my view is more correct; I do not even think that ‘right’ and ‘wrong’ are good categories for assessing complex mental models of external reality - for models in science are judged [as] useful or detrimental, not as true or false." (Stephen J Gould, "Dinosaur in a Haystack: Reflections in Natural History", 1995)

"Misunderstanding of probability may be the greatest of all impediments to scientific literacy." (Stephen J Gould, "Dinosaur in a  Haystack: Reflections in natural  history", 1995)

"Scientists reach their  conclusions  for the damnedest of reasons: intuition, guesses, redirections after wild-goose chases, all combing with a dollop of rigorous observation and logical  reasoning to be sure […] This  messy and personal side of science should not be  disparaged, or covered up, by  scientists for two  major reasons. First, scientists should proudly show this  human face to  display their kinship with all other  modes of creative human thought […] Second, while biases and references often impede understanding, these  mental idiosyncrasies  may  also serve as powerful, if  quirky and personal, guides to solutions." (Stephen J Gould, "Dinosaur in a  Haystack: Reflections in natural  history", 1995)

"Theories rarely arise as patient inferences forced by accumulated facts. Theories are mental constructs potentiated by complex external prods (including, in idealized cases, a commanding push from empirical reality)." (Stephen J Gould, "Leonardo's Mountain of Clams and the Diet of Worms", 1998) 

"The human mind delights in finding pattern - so much so that we often mistake coincidence or forced analogy for profound meaning. No other habit of thought lies so deeply within the soul of a small creature trying to make sense of a complex world not constructed for it." (Stephen J Gould, "The Flamingo's Smile: Reflections in Natural History", 2010)

Max Planck - Collected Quotes

“Nature prefers the more probable states to the less probable because in nature processes take place in the direction of greater probability. Heat goes from a body at higher temperature to a body at lower temperature because the state of equal temperature distribution is more probable than a state of unequal temperature distribution.” (Max Planck, “The Atomic Theory of Matter”, 1909)

"An indispensable hypothesis, even though still far from being a guarantee of success, is however the pursuit of a specific aim, whose lighted beacon, even by initial failures, is not betrayed." (Max Planck, [Nobel lecture] 1918) 

"If one wishes to obtain a definite answer from Nature one must attack the question from a more general and less selfish point of view." (Max Planck, “A Survey of Physics”, 1925)

"In all cases, the quantum hypothesis has given rise to the idea, that in Nature, changes occur which are not continuous, but of an explosive nature." (Max Planck, “A Survey of Physics”, 1925)

“Nothing is more interesting to the true theorist than a fact which directly contradicts a theory generally accepted up to that time, for this is his particular work.” (Max Planck, “A Survey of Physics”, 1925)

"The chief law of physics, the pinnacle of the whole system is, in my opinion, the principle of least action." (Max Planck, “A Survey of Physics”, 1925)

"The measure of the value of a new hypothesis in physics is not its obviousness but its utility."  (Max Planck, “A Survey of Physics”, 1925)

"Modern Physics impresses us particularly with the truth of the old doctrine which teaches that there are realities existing apart from our sense-perceptions, and that there are problems and conflicts where these realities are of greater value for us than the richest treasures of the world of experience." (Max Planck, "The Universe in the Light of Modern Physics", 1931)

"Physics would occupy an exceptional position among all the other sciences if it did not recognize the rule that the most far-reaching and valuable results of investigation can only be obtained by following a road leading to a goal which is theoretically unobtainable. This goal is the apprehension of true reality." (Max Planck, "The Universe in the Light of Modern Physics", 1931)

"We have no right to assume that any physical laws exist, or if they have existed up until now, that they will continue to exist in a similar manner in the future." (Max Planck, "The Universe in the Light of Modern Physics", 1931)

"Axioms are instruments which are used in every department of science, and in every department there are purists who are inclined to oppose with all their might any expansion of the accepted axioms beyond the boundary of their logical application." (Max Planck,"Where is Science Going?", 1932) 

"Every measurement first acquires its meaning for physical science through the significance which a theory gives it." (Max Planck,"Where is Science Going?", 1932)

"It goes without saying that the laws of nature are in themselves independent of the properties of the instruments with which they are measured. Therefore in every observation of natural phenomena we must remember the principle that the reliability of the measuring apparatus must always play an important role." (Max Planck,"Where is Science Going?", 1932)

"It is not the possession of truth, but the success which attends the seeking after it, that enriches the seeker and brings happiness to him." (Max Planck, "Where is Science Going?", 1932)

"No doctrinal system in physical science, or indeed perhaps in any science, will alter its content of its own accord. Here we always need the pressure of outer circumstances. Indeed the more intelligible and comprehensive a theoretical system is the more obstinately it will resist all attempts at reconstruction or expansion."  (Max Planck, "Where is Science Going?", 1932)

"Scientific discovery and scientific knowledge have been achieved only by those who have gone in pursuit of them without any practical purpose whatsoever in view." (Max Planck, “Where is Science Going?”, 1932)

"There is scarcely a scientific axiom that is not nowadays denied by somebody. And at the same time almost any nonsensical theory that may be put forward in the name of science would be almost sure to find believers and disciples."  (Max Planck, “Where is Science Going?”, 1932) 

"We are in a position similar to that of a mountaineer who is wandering over uncharted spaces, and never knows whether behind the peak which he sees in front of him and which he tries to scale there may not be another peak still beyond and higher up." (Max Planck, “Where is Science Going?”, 1932)

"It is impossible to make a clear cut between science, religion, and art. The whole is never equal simply to the sum of its various parts." (Max Planck, "The Philosophy of Physics", 1936) 

"Physics is an exact Science and hence depends upon measurement, while all measurement itself requires sense-perception. Consequently all the ideas employed in Physics are derived from the world of sense-perception." (Max Planck, "The Universe in the Light of Modern Physics", 1937) 

"A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it." (Max Planck, "A Scientific Autobiography", 1949) 

"It is never possible to predict a physical occurrence with unlimited precision." (Max Planck, "A Scientific Autobiography", 1949)  

"Science is not contemplative repose amidst knowledge already gained, but is indefatigable work and an ever progressive development." (Max Planck, "A Scientific Autobiography", 1949) 

"Since the real world, in the absolute sense of the word, is independent of individual personalities, and in fact of all human intelligence, every discovery made by an individual acquires a completely universal significance. This gives the inquirer, wrestling with his problem in quiet seclusion, the assurance that every discovery will win the unhesitating recognition of all experts throughout the entire world, and in this feeling of the importance of his work lies his happiness. It compensates him fully for many a sacrifice which he must make in his daily life." (Max Planck, "The Meaning and Limits of Exact Science", 1949)

"The outside world is something independent from man, something absolute, and the quest for the laws which apply to this absolute appeared to me as the most sublime scientific pursuit in life." (Max Planck, "A Scientific Autobiography", 1949)  

"[...] when the pioneer in science sends for the groping feelers of his thoughts, he must have a vivid intuitive imagination, for new ideas are not generated by deduction, but by an artistically creative imagination. Nevertheless, the worth of a new idea is invariably determined, not by the degree of its intuitiveness - which, incidentally, is to a major extent a matter of experience and habit - but by the scope and accuracy of the individual laws to the discovery of which it eventually leads. (Max Planck, The Meaning and Limits of Exact Science, Science Vol. 110 (2857), 1949)

“Science does not mean an idle resting upon a body of certain knowledge; it means unresting endeavor and continually progressing development toward an end which the poetic intuition may apprehend, but which the intellect can never fully grasp.” (Max Planck, “The New Science”, 1959)

"Science does not mean an idle resting upon a body of certain knowledge; it means unresting endeavor and continually progressing development toward an end which the poetic intuition may apprehend, but which the intellect can never fully grasp." (Max Planck, “The New Science”, 1959)

“This other world is the so-called physical world image; it is merely an intellectual structure. To a certain extent it is arbitrary. It is a kind of model or idealization created in order to avoid the inaccuracy inherent in every measurement and to facilitate exact definition.” (Max Planck, “The Philosophy of Physics”, 1963)

"Experiments are the only means of knowledge at our disposal. The rest is poetry, imagination." (Max Planck)

Leonhard Euler - Collected Quotes

"[…] such numbers, which by their natures are impossible, are ordinarily called imaginary or fanciful numbers, because they exist only in the imagination." (Leonhard Euler, 1732)

"The branch of geometry that deals with magnitudes has been zealously studied throughout the past, but there is another branch that has been almost unknown up to now; Leibniz spoke of it first, calling it the ‘geometry of position’ (geometria situs). This branch of geometry deals with relations dependent on position; it does not take magnitudes into considerations, nor does it involve calculation with quantities. But as yet no satisfactory definition has been given of the problems that belong to this geometry of position or of the method to be used in solving them." (Leonhard Euler, 1735)

"In addition to that branch of geometry which is concerned with magnitudes, and which has always received the greatest attention, there is another branch, previously almost unknown, which Leibniz first mentioned, calling it the geometry of position. This branch is concerned only with the determination of position and its properties; it does not involve measurements, nor calculations made with them. It has not yet been satisfactorily determined what kind of problems are relevant to this geometry of position, or what methods should be used in solving them. Hence, when a problem was recently mentioned, which seemed geometrical but was so constructed that it did not require the measurement of distances, nor did calculation help at all, I had no doubt that it was concerned with the geometry of position, especially as its solution involved only position, and no calculation was of any use." (Leonhard Euler,"Solution of a problem relative to the geometry of position", 1735)

"For since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in the universe in which some rule of maximum or minimum does not appear." (Leonhard Euler, "De Curvis Elasticis", 1744)

"I have finally discovered the true solution: in the same way that to one sine there correspond an infinite number of different angles I have found that it is the same with logarithms, and each number has an infinity of different logarithms, all of them imaginary unless the number is real and positive; there is only one logarithm which is real, and we regard it as its unique logarithm." (Leonhard Euler, [letter to Cramer] 1746)

"A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities. […] Functions are divided into algebraic and transcendental. The former are those made up from only algebraic operations, the latter are those which involve transcendental operations."(Leonhard Euler, "Introduction to Analysis of the Infinite", 1748)

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

"Those quantities that depend on others in this way, namely, those that undergo a change when others change, are called functions of these quantities. This definition applies rather widely and includes all ways in which one quantity could be determined by another." (Leonhard Euler, "Foundations of differential calculus, with applications to finite analysis and series", 1755)

"It will seem not a little paradoxical to ascribe a great importance to observations even in that part of the mathematical sciences which is usually called Pure Mathematics, since the current opinion is that observations are restricted to physical objects that make impression on the senses. As we must refer the numbers to the pure intellect alone, we can hardly understand how observations and quasi-experiments can be of use in investigating the nature of the numbers. Yet, in fact, as I shall show here with very good reasons, the properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations. There are even many properties of the numbers with which we are well acquainted, but which we are not yet able to prove; only observations have led us to their knowledge. Hence we see that in the theory of numbers, which is still very imperfect, we can place our highest hopes in observations; they will lead us continually to new properties which we shall endeavor to prove afterwards. The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such a discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful. (Leonhard Euler, "Specimen de usu observationum in mathesi pura" , Novi Commentarii academiae scientiarum Petropolitanae 6, 1756/57)

"We must distinguish carefully the ratios that our ears really perceive from those that the sounds expressed as numbers include." (Leonhard Euler, "Conjecture into the reasons for some dissonances generally heard in music", 1760)

"In order to know the curvature of a curve, the determination of the radius of the osculating circle furnishes us the best measure, where for each point of the curve we find a circle whose curvature is precisely the same. However, when one looks for the curvature of a surface, the question is very equivocal and not at all susceptible to an absolute response, as in the case above. There are only spherical surfaces where one would be able to measure the curvature, assuming the curvature of the sphere is the curvature of its great circles, and whose radius could be considered the appropriate measure. But for other surfaces one doesn’t know even how to compare a surface with a sphere, as when one can always compare the curvature of a curve with that of a circle. The reason is evident, since at each point of a surface there are an infinite number of different curvatures. One has to only consider a cylinder, where along the directions parallel to the axis, there is no curvature, whereas in the directions perpendicular to the axis, which are circles, the curvatures are all the same, and all other oblique sections to the axis give a particular curvature. It’s the same for all other surfaces, where it can happen that in one direction the curvature is convex, and in another it is concave, as in those resembling a saddle." (Leonhard Euler, "Recherches sur la courbure des surfaces", 1767)

"Till now the mathematicians tried in vain to discover some order in the sequence of the prime numbers and we have every reason to believe that there is some mystery which the human mind shall never penetrate. To convince oneself, one has only to glance at the tables of the primes, which some people took the trouble to compute beyond a hundred thousand, and one perceives that there is no order and no rule. This is so much more surprising as the arithmetic gives us definite rules with the help of which we can continue the sequence of the primes as far as we please, without noticing, however, the least trace of order." (Leonhard Euler, "Letters of Euler on different subjects in physics and philosophy. Addressed to a German princess, 1768)

"All such expressions as √-1, √-2, etc., are consequently impossible or imaginary numbers, since they represent roots of negative quantities; and of such numbers we may truly assert that they are neither nothing, nor greater than nothing, nor less than nothing, which necessarily constitutes them imaginary or impossible." (Leonhard Euler, "Algebra" , 1770)

"First, everything will be said to be a magnitude, which is capable of increase or diminution, or to which something may be added or subtracted […] mathematics is nothing more than the science of magnitudes, which finds methods by which they can be measured." (Leonhard Euler, "Algebra" , 1770)

"I take the word 'mapping' in the widest possible sense; any point of the spherical surface is represented on the plane by any desired rule, so that every point of the sphere corresponds to a specified point in the plane, and inversely." (Leonhard Euler, "On the representation of Spherical Surfaces onto the Plane", 1777)

"The difference of two square numbers is always a product, and divisible both by the sum and by the difference of the roots of those two squares; consequently the difference of two squares can never be a prime number." (Leonhard Euler, "Elements of Algebra", 1810)

"Originally assuming the concept of the absolute integers, it extended its domain step by step; integers were supplemented by fractions, rational numbers by irrational numbers, positive numbers by negative numbers, and real numbers by imaginary numbers. This advance, however, occurred initially with a fearfully hesitant step. The first algebraists preferred to call negative roots of equations false roots, and it is precisely these where the problem to which they refer was always termed in such a way as to ensure that the nature of the quantity sought did not admit any opposite." (Carl F Gauss, "Theoria residuorum biquadraticum. Commentatio secunda. [Selbstanzeige]", Göttingische gelehrte Anzeigen 23 (4), 1831)

"Every syllogism, then, consists of three propositions; the two first of which are called the premises and the third the conclusion. Now, the advantage of the all these [valid] forms to direct our reasoning is this, that if the premises are both true, the conclusion infallibly is so.

This is likewise the only method of discovering unknown truths. Every truth must always be the conclusion of a syllogism, whose premises are indubitably true." (Leonhard Euler, "Letters of Euler on Different Subjects in Natural Philosophy Addressed to a German Princess" Vol. 1, 1833)

"The properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations. There are even many properties of the numbers with which we are well acquainted, but which we are not yet able to prove; only observations have led us to their knowledge. Hence we see that in the theory of numbers, which is still very imperfect, we can place our highest hopes in observations." (Leonhard Euler, cca. 1856/57)

"There must be a double method for solving mechanical problems: one is the direct method founded on the laws of equilibrium or of motion; but the other one is by knowing which formula must provide a maximum or a minimum. The former way proceeds by efficient causes: both ways lead to the same solution, and it is such a harmony which convinces us of the truth of the solution, even if each method has to be separately founded on indubitable principles. But is often very difficult to discover the formula which must be a maximum or minimum, and by which the quantity of action is represented." (Leonhard Euler, "Specimen de usu observationum in mathesi pura", Novi Commentarii academiae scientiarum Petropolitanae 6, 1756/57)

"[…] nothing takes place in the world whose meaning is not that of some maximum or minimum." (Leonhard Euler)

"[…] we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such a discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful." (Leonhard Euler)

"Although to penetrate into the intimate mysteries of nature and hence to learn the true causes of phenomena is not allowed to us, nevertheless it can happen that a certain fictive hypothesis may suffice for explaining many phenomena." (Leonhard Euler)

"Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate." (Leonhard Euler)

"Since the divine plan is the most perfect thing there is, there can be no doubt that all actions in the world can be determined by the calculus of minima and maxima from the corresponding causes." (Leonhard Euler)

"Some facts can be seen more clearly by example than by proof." (Leonard Euler)

Morris Kline - Collected Quotes

"The history of mathematics shows that the introduction of better and better symbolism and operations has made a commonplace of processes that would have been impossible with the unimproved techniques." (Morris Kline, "Mathematics in Western culture", 1953)

"Mathematicians create by acts of insight and intuition. Logic then sanctions the conquests of intuition. It is the hygiene that mathematics practice to keep its ideas healthy and strong. Moreover, the whole structure rests fundamentally on uncertain ground, the intuitions of man." (Morris Kline, "Mathematics in Western Culture", 1953)

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

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

"The creative act owes little to logic or reason. In their accounts of the circumstances under which big ideas occurred to them, mathematicians have often mentioned that the inspiration had no relation to the work they happened to be doing. Sometimes it came while they were traveling, shaving or thinking about other matters. The creative process cannot be summoned at will or even cajoled by sacrificial offering. Indeed, it seems to occur most readily when the mind is relaxed and the imagination roaming freely." (Morris Kline, Scientific American, 1955) 

"For many parts of nature can neither be invented with sufficient subtlety, nor demonstrated with sufficient perspicuity, nor accommodated unto use with sufficient dexterity without the aid and intervention of mathematics."(Morris Kline, "Mathematics and the Physical World", 1959)

"Mathematics is a model of exact reasoning, an absorbing challenge to the mind, an esthetic experience for creators and some students, a nightmarish experience to other students, and an outlet for the egotistic display of mental power." (Morris Kline, "Mathematics and the Physical World", 1959)

"But both managed to understand mathematics and to make a 'fair' number of contributions to the subject. Rigorous proof is not nearly so important as proving the worth of what we are teaching; and most teachers, instead of being concerned about their failure to be sufficiently rigorous, should really be concerned about their failure to provide a truly intuitive approach. The general principle, then, is that the rigor should be suited to the mathematical age of the student and not to the age of mathematics." (Morris Kline, "Mathematics: A Cultural Approach", 1962)

"Mathematics then is a formidable and bold bridge between ourselves and the external world. Though it is a purely human creation, the access it has given us to some domains of nature enable us to progress far beyond all expectations. Indeed it is paradoxical that abstractions so remote from reality should achieve so much. Artificial the mathematical account may be, a fairy tale perhaps, but one with a moral." (Morris Kline,"Mathematics: A Cultural Approach", 1962)

"Mathematics is a body of knowledge, but it contains no truths." (Morris Kline, "Mathematics in Western Culture", 1964)

"The tantalizing and compelling pursuit of mathematical problems offers mental absorption, peace of mind amid endless challenges, repose in activity, battle without conflict, ‘refuge from the goading urgency of contingent happenings’, and the sort of beauty changeless mountains present to senses tried by the present-day kaleidoscope of events." (Morris Kline, "Mathematics in Western Culture", 1964)
  
"The history of arithmetic and algebra illustrates one of the striking and curious features of the history of mathematics. Ideas that seem remarkably simple once explained were thousands of years in the making." (Morris Kline, "Mathematics for the Nonmathematician", 1967)

"[…] although mathematical concepts and operations are formulated to represent aspects of the physical world, mathematics is not to be identified with the physical world. However, it tells us a good deal about that world if we are careful to apply it and interpret it properly." (Morris Kline, "Mathematics for the Nonmathematician", 1967)

"[…] mathematics is not portraying laws inherent in the design of the universe but is merely providing man-made schemes or models which we can use to deduce conclusions about our world only to the extent that the model is a good idealization." (Morris Kline, "Mathematics for the Nonmathematician", 1967)

"[...] the Greeks, Descartes, Newton, Euler, and many others believed mathematics to be the accurate description of real phenomena and that they regarded their work as the uncovering of the mathematical design of the universe. Mathematicians did deal with abstractions, but these were no more than the ideal forms of physical objects or happenings." (Morris Kline, "Mathematical Thought from Ancient to Modern Times", 1972)

"[...] the quaternions [...] were not quite what the physicists wanted. They sought a concept that was not divorced from but more closely associated with Cartesian coordinates than quaternions were. (Morris Kline, "Mathematical Thought from Ancient to Modern Times", 1972)

"The introduction and gradual acceptance of concepts that have no immediate counterparts in the real world certainly forced the recognition that mathematics is a human, somewhat arbitrary creation, rather than an idealization of the realities in nature, derived solely from nature. But accompanying this recognition and indeed propelling its acceptance was a more profound discovery - mathematics is not a body of truths about nature." (Morris Kline, "Mathematical Thought from Ancient to Modern Times" Vol. III, 1972)

"While vector analysis was being created and afterward there was much controversy between the proponents of quaternions and the proponents of vectors as to which was more useful. The quaternionists were fanatical about the value of quaternions but the proponents of vector analysis were equally partisan. On one side were aligned the leading supporters of quaternions such as Tait and, on the other, Gibbs and Heaviside. Apropos of the controversy, Heaviside remarked sarcastically that for the treatment of quaternions, quaternions are the best instrument. On the other hand Tait described Heaviside’s vector algebra as 'a sort of hermaphrodite monster, compounded of the notations of Grassmann and Hamilton'. [...] The issue was finally decided in favor of vectors. Engineers welcomed Gibbs’s and Heaviside’s vector analysis, though the mathematicians did not. By the beginning of the present century the physicists too were quite convinced that vector analysis was what they wanted. Textbooks on the subject soon appeared in all countries and are now standard. The mathematicians finally followed suit and introduced vector methods in analytic and differential geometry." (Morris Kline, "Mathematical Thought from Ancient to Modern Times", 1972)

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

"We are now compelled to accept the fact that there is no such thing as an absolute proof or a universally acceptable proof. We know that, if we question the statements we accept on an intuitive basis, we shall be able to prove them only if we accept others on an intuitive basis." (Morris Kline, "Mathematics: The loss of certainty", 1980)

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

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

"Contrary to the impression students acquire in school, mathematics is not just a series of techniques. Mathematics tells us what we have never known or even suspected about notable phenomena and in some instances even contradicts perception. It is the essence of our knowledge of the physical world. It not only transcends perception but outclasses it." (Morris Kline, "Mathematics and the Search for Knowledge", 1985)

"A proof tells us where to concentrate our doubts. […] An elegantly executed proof is a poem in all but the form in which it is written." (Morris Kline)

"Mathematics is a spirit of rationality. It is this spirit that challenges, simulates, invigorates and drives human minds to exercise themselves to the fullest. It is this spirit that seeks to influence decisively the physical, normal and social life of man, that seeks to answer the problems posed by our very existence, that strives to understand and control nature and that exerts itself to explore and establish the deepest and utmost implications of knowledge already obtained." (Morris Kline)

"Statistics: The mathematical theory of ignorance." (Morris Kline)

Clifford A Pickover - Collected Quotes

"I do not know if God is a mathematician, but mathematics is the loom upon which God weaves the fabric of the universe. [...] The fact that reality can be described or approximated by simple mathematical expressions suggests to me that nature has mathematics at its core." (Clifford A Pickover, "The Loom of God: Mathematical Tapestries at the Edge of Time", 1997) 

“In many ways, the mathematical quest to understand infinity parallels mystical attempts to understand God. Both religions and mathematics attempt to express the relationships between humans, the universe, and infinity. Both have arcane symbols and rituals, and impenetrable language. Both exercise the deep recesses of our mind and stimulate our imagination. Mathematicians, like priests, seek ‘ideal’, immutable, nonmaterial truths and then often try to apply theses truth in the real world.” (Clifford A Pickover, "The Loom of God: Mathematical Tapestries at the Edge of Time", 1997)

“Is God a mathematician? Certainly, the world, the universe, and nature can be reliably understood using mathematics. Nature is mathematics.” (Clifford A Pickover, “The Loom of God”, 1997)

”Throughout both ancient and modern history the feverish hunt for perfect numbers became a religion.” (Clifford A Pickover, "Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning", 2001)

"Intellectual brilliance is no guarantee against being dead wrong." (Clifford A Pickover, "The Mathematics of Oz", 2002)

"In our modern era, God and mathematics are usually placed in totally separate arenas of human thought. But [...] this has not always been the case, and even today many mathematicians find the exploration of mathematics akin to a spiritual journey. The line between religion and mathematics becomes indistinct. In the past, the intertwining of religion and mathematics has produced useful results and spurred new areas of scientific thought. […] In many ways, the mathematical quest to understand infinity parallels mystical attempts to understand God. Both religion and mathematics attempt to express relationships between humans, the universe, and infinity. Both have arcane symbols and rituals, and impenetrable language. Both exercise the deep recesses of our minds and stimulate our imagination. Mathematicians, like priests, seek ‘ideal’, immutable truths and then often try to apply these truths to the real world. Some atheists claim another similarity: mathematics and religion are the most powerful evidence of the inventive genius of the human race.
Of course, there are also many differences between mathematics and religion." (Clifford A Pickover, "The Loom of God: Tapestries of Mathematics and Mysticism", 2009)

“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)

"For me mathematics cultivates a perpetual state of wonder about the nature of mind, the limits of thoughts, and our place in this vast cosmos." (Clifford A Pickover)

Ian Stewart - Collected Quotes

"An intuitive proof allows you to understand why the theorem must be true; the logic merely provides firm grounds to show that it is true." (Ian Stewart, "Concepts of Modern Mathematics", 1975)

"Many pages have been expended on polemics in favor of rigor over intuition, or of intuition over rigor. Both extremes miss the point: the power of mathematics lies precisely in the combination of intuition and rigor." (Ian Stewart, "Concepts of Modern Mathematics", 1975)

"A great many problems are easier to solve rigorously if you know in advance what the answer is." (Ian Stewart, "From Here to Infinity", 1987)

"Computer scientists working on algorithms for factorization would be well advised to brush up on their number theory." (Ian Stewart, "Geometry Finds Factor Fast", Nature Vol. 325, 1987) 

"Mathematics is good if it enriches the subject, if it opens up new vistas, if it solves old problems, if it fills gaps, fitting snugly and satisfyingly into what is already known, or if it forges new links between previously unconnected parts of the subject It is bad if it is trivial, overelaborate, or lacks any definable mathematical purpose or direction It is pure if its methods are pure - that is, if it doesn't cheat and tackle one problem while pretending to tackle another, and if there are no gaping holes in its logic It is applied if it leads to useful insights outside mathematics. By these criteria, today's mathematics contains as high a proportion of good work as at any other period, and as any other area, and much of it manages to be both pure and applied at the same time." (Ian Stewart, "The Problems of Mathematics", 1987)

"Symmetries abound in nature, in technology, and - especially - in the simplified mathematical models we study so assiduously. Symmetries complicate things and simplify them. They complicate them by introducing exceptional types of behavior, increasing the number of variables involved, and making vanish things that usually do not vanish. They simplify them by introducing exceptional types of behavior, increasing the number of variables involved, and making vanish things that usually do not vanish. They violate all the hypotheses of our favorite theorems, yet lead to natural generalizations of those theorems. It is now standard to study the 'generic' behavior of dynamical systems. Symmetry is not generic. The answer is to work within the world of symmetric systems and to examine a suitably restricted idea of genericity." (Ian Stewart, "Bifurcation with symmetry", 1988)

"[...] an apparently random universe could be obeying every whim of a deterministic deity who chooses how the dice roll; a universe that has obeyed perfect mathematical laws for the last ten billion years could suddenly start to play truly random dice. So the distinction is about how we model the system, and what point of view seems most useful, rather than about any inherent feature of the system itself." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"Chaos teaches us that anybody, God or cat, can play dice deterministically, while the naïve onlooker imagines that something random is going on." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"If sinks, sources, saddles, and limit cycles are coins landing heads or tails, then the exceptions are a coin landing on edge. Yes, it might happen, in theory; but no, it doesn't, in practice." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"In contrast, the system may be a pack of cards, and the dynamic may be to shuffle the pack and then take the top card. Imagine that the current top card is the ace of spades, and that after shuffling the pack the top card becomes the seven of diamonds. Does that imply that whenever the top card is the ace of spades then the next top card will always be the seven of diamonds? Of course not. So this system is random." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"In modelling terms, the difference between randomness and determinacy is clear enough. The randomness in the pack of cards arises from our failure to prescribe unique rules for getting from the current state to the next one. There are lots of different ways to shuffle a pack. The determinism of the cannonball is a combination of two things: fully prescribed rules of behaviour, and fully defined initial conditions. Notice that in both systems we are thinking on a very short timescale: it is the next state that matters - or, if time is flowing continuously, it is the state a tiny instant into the future. We don't need to consider long-term behaviour to distinguish randomness from determinacy." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"Indeed a deterministic die behaves very much as if it has six attractors, the steady states corresponding to its six faces, all of whose basins are intertwined. For technical reasons that can't quite be true, but it is true that deterministic systems with intertwined basins are wonderful substitutes for dice; in fact they're super-dice, behaving even more ‘randomly’ - apparently - than ordinary dice. Super-dice are so chaotic that they are uncomputable. Even if you know the equations for the system perfectly, then given an initial state, you cannot calculate which attractor it will end up on. The tiniest error of approximation - and there will always be such an error - will change the answer completely." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"Mathematicians are beginning to view order and chaos as two distinct manifestations of an underlying determinism. And neither state exists in isolation. The typical system can exist in a variety of states, some ordered, some chaotic. Instead of two opposed polarities, there is a continuous spectrum. As harmony and discord combine in musical beauty, so order and chaos combine in mathematical [and physical] beauty." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"It's a bit like having a theory about coins that move in space, but only being able to measure their state by interrupting them with a table. We hypothesize that the coin may be able to revolve in space, a state that is neither ‘heads’ nor ‘tails’ but a kind of mixture. Our experimental proof is that when you stick a table in, you get heads half the time and tails the other half - randomly. This is by no means a perfect analogy with standard quantum theory - a revolving coin is not exactly in a superposition of heads and tails - but it captures some of the flavour." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"Mathematicians are beginning to view order and chaos as two distinct manifestations of an underlying determinism. And neither state exists in isolation. The typical system can exist in a variety of states, some ordered, some chaotic. Instead of two opposed polarities, there is a continuous spectrum. As harmony and discord combine in musical beauty, so order and chaos combine in mathematical [and physical] beauty." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"Mathematics is a remarkable sprawling riot of imagination, ranging from pure intellectual curiosity to nuts-and-bolts utility; and it is all one thing." (Ian Stewart, "Game, Set, and Math: Enigmas and Conundrums", 1989)

"Perhaps God can play dice, and create a universe of complete law and order, in the same breath." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"The chance events due to deterministic chaos, on the other hand, occur even within a closed system determined by immutable laws. Our most cherished examples of chance - dice, roulette, coin-tossing – seem closer to chaos than to the whims of outside events. So, in this revised sense, dice are a good metaphor for chance after all. It's just that we've refined our concept of randomness. Indeed, the deterministic but possibly chaotic stripes of phase space may be the true source of probability." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"The randomness of the card-shuffle is of course caused by our lack of knowledge of the precise procedure used to shuffle the cards. But that is outside the chosen system, so in our practical sense it is not admissible. If we were to change the system to include information about the shuffling rule – for example, that it is given by some particular computer code for pseudo-random numbers, starting with a given ‘seed value’ - then the system would look deterministic. Two computers of the same make running the same ‘random shuffle’ program would actually produce the identical sequence of top cards." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"The flapping of a single butterfly’s wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done." (Ian Stewart, "Does God Play Dice?", 1989)

"[…] a symmetry isn't a thing; it's a transformation. Not any old transformation, though: a symmetry of an object is a transformation that leaves it apparently unchanged." (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"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)

"Nature behaves in ways that look mathematical, but nature is not the same as mathematics. Every mathematical model makes simplifying assumptions; its conclusions are only as valid as those assumptions." (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"Nature is never perfectly symmetric. Nature's circles always have tiny dents and bumps. There are always tiny fluctuations, such as the thermal vibration of molecules. These tiny imperfections load Nature's dice in favour of one or other of the set of possible effects that the mathematics of perfect symmetry considers to be equally possible." (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"Scientists use mathematics to build mental universes. They write down mathematical descriptions - models - that capture essential fragments of how they think the world behaves. Then they analyse their consequences. This is called 'theory'. They test their theories against observations: this is called 'experiment'. Depending on the result, they may modify the mathematical model and repeat the cycle until theory and experiment agree. Not that it's really that simple; but that's the general gist of it, the essence of the scientific method." (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"To a mathematician, an object possesses symmetry if it retains its form after some transformation. A circle, for example, looks the same after any rotation; so a mathematician says that a circle is symmetric, even though a circle is not really a pattern in the conventional sense - something made up from separate, identical bits. Indeed the mathematician generalizes, saying that any object that retains its form when rotated - such as a cylinder, a cone, or a pot thrown on a potter's wheel - has circular symmetry." (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 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)

"A number is a process that has long ago been thingified so thoroughly that everybody thinks of it as a thing. It is just as feasible-though less familiar to most of us-to think of an operation or a function as a thing. For example, we might talk of "square root" as if it were a thing- and I mean here not the square root of any particular number, but the function itself. In this image, the square-root function is a kind of sausage machine: you stuff a number in at one end and its square root pops out at the other." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"At every turn, new vistas arise-an  unexpected river that must be crossed using stepping stones, a vast, tranquil lake, an impassable crevasse. The user of mathematics walks only the well-trod parts of this mathematical territory. The creator of mathematics explores its unknown mysteries, maps them, and builds roads through them to make them more easily accessible to everybody else." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 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)

"If you start with a number and form its square root, you get another number. The term for such an 'object' is function. You can think of a function as a mathematical rule that starts with a mathematical object-usually a number-and associates to it another object in a specific manner. Functions are often defined using algebraic formulas, which are just shorthand ways to explain what the rule is, but they can be defined by any convenient method. Another term with the same meaning as 'function' is transformation: the rule transforms the first object into the second. […] Operations and functions are very similar concepts. Indeed, on a suitable level of generality there is not much to distinguish them. Both of them are processes rather than things." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Mathematical 'things' have no existence in the real world: they are abstractions. But mathematical processes are also abstractions, so processes are no less 'things' than the "things" to which they are applied. The thingification of processes is commonplace." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Mathematics is not just a collection of isolated facts: it is more like a landscape; it has an inherent geography that its users and creators employ to navigate through what would otherwise be an impenetrable jungle." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"No, nature is, in its own subtle way, simple. However, those simplicities do not present themselves to us directly. Instead, nature leaves clues for the mathematical detectives to puzzle over. It's a fascinating game, even to a spectator. And it's an absolutely irresistible one if you are a mathematical Sherlock Holmes." (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)

"Proofs knit the fabric of mathematics together, and if a single thread is weak, the entire fabric may unravel." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"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: The unreal reality of mathematics", 1995)

"The image of mathematics raised by this description of its basic objects is something like a tree, rooted in numbers and branching into ever more esoteric data structures as you proceed from trunk to bough, bough to limb, limb to twig…" (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"The ingredient that knits this landscape together is proof. Proof determines the route from one fact to another. To professional mathematicians, no statement is considered valid unless it is proved beyond any possibility of logical error. But there are limits to what can be proved, and how it can be proved. A great deal of work in philosophy and the foundations of mathematics has established that you can't prove everything, because you have to start somewhere; and even when you've decided where to start, some statements may be neither provable nor disprovable." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"[…] the meaning of the word 'solve' has undergone a series of major changes. First that word meant 'find a formula'. Then its meaning changed to 'find approximate numbers'. Finally, it has in effect become 'tell me what the solutions look like'. In place of quantitative answers, we seek qualitative ones." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"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 mathematician should never leave home without them […]" (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"The story of calculus brings out two of the main things that mathematics is for: providing tools that let scientists calculate what nature is doing, and providing new questions for mathematicians to sort out to their own satisfaction. These are the external and internal aspects of mathematics, often referred to as applied and pure mathematics." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"There is much beauty in nature's clues, and we can all recognize it without any mathematical training. There is beauty, too, in the mathematical stories that start from the clues and deduce the underlying rules and regularities, but it is a different kind of beauty, applying to ideas rather than things." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 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)

"However, random walk theory also tells us that the chance that the balance never returns to zero - that is, that H stays in the lead for ever - is 0. This is the sense in which the 'law of averages' is true. If you wait long enough, then almost surely the numbers of heads and tails will even out. But this fact carries no implications about improving your chances of winning, if you're betting on whether H or T turns up. The probabilities are unchanged, and you don't know how long the 'long run' is going to be. Usually it is very long indeed." (Ian Stewart, The Magical Maze: Seeing the world through mathematical eyes", 1997)

"In fact, mathematics is the closest that we humans get to true magic. How else to describe the patterns in our heads that - by some mysterious agency - capture patterns of the universe around us?" (Ian Stewart, "The Magical Maze: Seeing the World Through Mathematical Eyes", 1997)

"The basis of many misconceptions about probability is a belief in something usually referred to as 'the law of averages', which alleges that any unevenness in random events gets ironed out in the long run. For example, if a tossed coin keeps coming up heads, then it is widely believed that at some stage there will be a predominance of tails to balance things out." (Ian Stewart, The Magical Maze: Seeing the world through mathematical eyes", 1997)

"The 'law of averages' asserts itself not by removing imbalances, but by swamping them. Random walk theory tells us that if you wait long enough - on average, infinitely long - then eventually the numbers will balance out. If you stop at that very instant, then you may imagine that your intuition about a 'law of averages' is justified. But you're cheating: you stopped when you got the answer you wanted. Random walk theory also tells us that if you carry on for long enough, you will reach a situation where the number of H's is a billion more than the number of T's." (Ian Stewart, The Magical Maze: Seeing the world through mathematical eyes", 1997) 

"A mathematical circle, then, is something more than a shared delusion. It is a concept endowed with extremely specific features; it 'exists' in the sense that human minds can deduce other properties from those features, with the crucial caveat that if two minds investigate the same question, they cannot, by correct reasoning, come up with contradictory answers." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"A proof provides a cast-iron guarantee that some idea is correct. No amount of experimental evidence can substitute for that." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"As caricatured by applied mathematicians, pure math is abstract ivory-tower intellectual nonsense with no practical implications. Applied math, respond the pure-math diehards, is intellectually sloppy, lacks rigor, and substitutes number crunching for understanding. Like all good caricatures, both statements contain grains of truth, but you should not take them literally."  (Ian Stewart, "Letters to a Young Mathematician", 2006)

"By 'network' I mean a set of dynamical systems that are 'coupled together', with some influencing the behavior of others. The systems themselves are the nodes of the network- think of them as blobs - and two nodes are joined by an arrow if one of them (at the tail end) influences the other (at the head end). For example, each node might be a nerve cell in some organism, and the arrows might be connections along which signals pass from one cell to another." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Definitions pin things down, they limit the prospects for creativity and diversity. A definition, implicitly, attempts to reduce all possible variations of a concept to a single pithy phrase." (Ian Stewart, "Letters to a Young Mathematician", 2006)

 "Even when it comes to something as simple as counting, we mathematicians see the world differently from other folk." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Like many mathematicians, I get my inspiration from nature. Nature may not look very mathematical; you don't see sums written on the trees. But math is not about sums, not really. It's about patterns and why they occur. Nature's patterns are both beautiful and inexhaustible." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Math is actually very important, but because it genuinely is difficult, nearly all of the teaching slots are occupied with making sure that students learn how to solve certain types of problem and get the answers right. There isn't time to tell them about the history of the subject, about its connections with our culture and society, about the huge quantity of new mathematics that is created every year, or about the unsolved questions, big and little, that litter the mathematical landscape." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Mathematical facts fit together and lead, via logic, to new facts. A deduction is only as strong as its weakest link. To be safe, all weak links must be removed." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Mathematicians do not spend most of their time doing numerical calculations, even though calculations are sometimes essential to making progress. They do not Letters to a Young Mathematician occupy themselves with grinding out symbolic formulas, but formulas can nonetheless be indispensable." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Mathematicians need proofs to keep them honest. All technical areas of human activity need reality checks. It is not enough to believe that something works, that it is a good way to proceed, or even that it is true. We need to know why it's true. Otherwise, we don't know anything at all." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Nature is always deeper, richer, and more interesting than you thought, and mathematics gives you a very powerful way to appreciate this." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Now let me explain a wonderful thing: the more mathematics you learn, the more opportunities you will find for asking new questions. As our knowledge of mathematics grows, so do the opportunities for fresh discoveries. This may sound unlikely, but it is a natural consequence of how new mathematical ideas build on older ones." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Once you understand a problem, many aspects of it suddenly become much simpler. As mathematicians the world over say, everything is either impossible or trivial." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Physicists use mathematics to study what they amusingly call the real world. It is real, in a sense, but much of physics addresses rather artificial aspects of reality, such as a lone electron or a solar system with only one planet. Physicists are often scathing about proofs, partly out of fear, but also because experiment gives them a very effective way to check their assumptions and calculations." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Sometimes in math the best way to make progress is to introduce simplifications [...]. The simplifications are not assertions about the outside world: they are ways to restrict the domain of discourse, to keep it manageable." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"The further we push out the boundaries of mathematics, the bigger the boundary itself becomes. There is no danger that we will ever run out of new problems to solve." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"The mathematician's circle, with its infinitely thin circumference and a radius that remains constant to infinitely many decimal places, cannot take physical form. If you draw it in sand, as Archimedes did, its boundary is too thick and its radius too variable." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Then there's the inner beauty of mathematics, which should not be underrated. Math done 'for its own sake' can be exquisitely beautiful and elegant. Not the 'sums' we all do at school; as individuals those are mostly ugly and formless, although the general principles that govern them have their own kind of beauty. It's the ideas, the generalities, the sudden flashes of insight, the realization that trying to trisect an angle with straightedge and compass is like trying to prove that 3 is an even number, that it makes perfect sense that you can't construct a regular seven-sided polygon but you can construct one with seventeen sides, that there is no way to untie an overhand knot, and why some infinities are bigger than others whereas some that ought to be bigger are actually equal [...]" (Ian Stewart, "Letters to a Young Mathematician", 2006)

"What is mathematics? It is the shared social construct created by people who are aware of certain opportunities, and we call those people mathematicians. The logic is still slightly circular, but mathematicians can always recognize a fellow spirit. Find out what that fellow spirit does; it will be one more aspect of our shared social construct." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"When you are doing math, it feels as though what you are working on is real. You can almost pick things up and turn them around, squash them and stroke them and pull them to pieces. On the other hand, you often make progress by forgetting what it all means and focusing solely on how the symbols dance." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Equations are the mathematician's way of working out the value of some unknown quantity from circumstantial evidence. ‘Here are some known facts about an unknown number: deduce the number.’ An equation, then, is a kind of puzzle, centered upon a number. We are not told what this number is, but we are told something useful about it. Our task is to solve the puzzle by finding the unknown number." (Ian Stewart, "Why Beauty Is Truth", 2007)

"The complex numbers extend the real numbers by throwing in a new kind of number, the square root of minus one. But the price we pay for being able to take square roots of negative numbers is the loss of order. The complex numbers are a complete system but are spread out across a plane rather than aligned in a single orderly sequence." (Ian Stewart, "Why Beauty Is Truth", 2007)

"When you get to know them, equations are actually rather friendly. They are clear, concise, sometimes even beautiful. The secret truth about equations is that they are a simple, clear language for describing certain ‘recipes’ for calculating things." (Ian Stewart, "Why Beauty Is Truth", 2007)

"A complex number is just a pair of real numbers, manipulated according to a short list of simple rules. Since a pair of real numbers is surely just as ‘real’ as a single real number, real and complex numbers are equally closely related to reality, and ‘imaginary’ is misleading." (Ian Stewart, "Why Beauty Is Truth", 2007)

"Equations have hidden powers. They reveal the innermost secrets of nature. […] The power of equations lies in the philosophically difficult correspondence between mathematics, a collective creation of human minds, and an external physical reality. Equations model deep patterns in the outside world. By learning to value equations, and to read the stories they tell, we can uncover vital features of the world around us." (Ian Stewart, "In Pursuit of the Unknown", 2012)

 

"There are two kinds of equations in mathematics, which on the surface look very similar. One kind presents relations between various mathematical quantities: the task is to prove the equation is true. The other kind provides information about an unknown quantity, and the mathematician’s task is to solve it - to make the unknown known." (Ian Stewart, "In Pursuit of the Unknown", 2012)

"A proof is a story told to and dissected by people who have spent much of their life learning how to read such stories and find mistakes or inconsistencies: people whose main aim is to prove the storyteller wrong, and who possess the uncanny knack of spotting weaknesses and hammering away at them until they collapse in a cloud of dust." (Ian Stewart, "Visions of Infinity", 2013)

"A symmetry of some mathematical structure is a transformation of that structure, of a specified kind, that leaves specified properties of the structure unchanged." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"Although the four colour problem is ostensibly about maps, it has no useful applications to cartography. Practical criteria for colouring maps mainly reflect political differences, and if that means that adjacent regions must have the same colour, so be it. The problem’s interest lay entirely within pure mathematics, in a new area that had only just started to develop: topology. This is ‘rubber-sheet geometry’ in which shapes can be deformed in any continuous manner. But even there, the four colour problem didn’t belong to the mainstream. It seemed to be no more than a minor curiosity." (Ian Stewart, "Visions of Infinity", 2013)

"An algebraic variety, be it real, complex, projective, or not, is a topological space. Therefore it has a shape. To find out useful things about the shape, we think like topologists and calculate the homology and cohomology groups. But the natural ingredients in algebraic geometry aren’t geometric objects like triangulations and cycles. They are the things we can most easily describe by algebraic equations." (Ian Stewart, "Visions of Infinity", 2013)

"Commutativity is a very pleasant mathematical property. Its absence is a bit of a nuisance, and this is just one of the reasons why quantising a field turns out to be tricky. Nonetheless, it can sometimes be done." (Ian Stewart, "Visions of Infinity", 2013)

"Having agreed on the axioms, a proof of some statement is a series of steps, each of which is a logical consequence of either the axioms, or previously proved statements, or both. In effect, the mathematician is exploring a logical maze, whose junctions are statements and whose passages are valid deductions. A proof is a path through the maze, starting from the axioms. What it proves is the statement at which it terminates." (Ian Stewart, "Visions of Infinity", 2013)

"Hermite’s transcendence proof is magical, a rabbit extracted with a flourish from the top hat of analysis. The rabbit is a complicated formula related to a hypothetical algebraic equation that e is assumed to satisfy. Using algebra, Hermite proves that this formula is equal to some nonzero integer. Using analysis, he proves that it must lie between - 1/2 and 1/2. Since the only integer in this range is zero, these results are contradictory. Therefore the assumption that e satisfies an algebraic equation must be false, so e is transcendental." (Ian Stewart, "Visions of Infinity", 2013)

"Homology and cohomology don’t tell us everything we would like to know about the shape of a topological space - distinct spaces can have the same homology and cohomology – but they do provide a lot of useful information, and a systematic framework in which to calculate it and use it." (Ian Stewart, "Visions of Infinity", 2013)

"Homotopy is a geometrical construction that can be carried out entirely inside the space, and it provides information on the topological type of that space. It does that using an abstract algebraic structure known as a group. A group is a collection of mathematical objects, any two of which can be combined to give another object in the group. This law of combination - often called multiplication or addition, even when it’s not the usual arithmetical operation with that name - is required to satisfy a few simple and natural conditions." (Ian Stewart, "Visions of Infinity", 2013)

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

"In a loose analogy, every finite symmetry group can be broken up, in a well-defined manner, into ‘indivisible’ symmetry groups - atoms of symmetry, so to speak. These basic building blocks for finite groups are known as simple groups - not because anything about them is easy, but in the sense of ‘not made up from several parts’. Just as atoms can be combined to build molecules, so these simple groups can be combined to build all finite groups." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"In homotopy, we ask whether a given loop can be shrunk continuously to a point. In homology, we ask a different question: does the loop form the boundary of a topological disc? That is, can you fit one or more triangular patches together so that the result is a region without any holes, and the boundary of this region is the loop concerned?" (Ian Stewart, "Visions of Infinity", 2013)

"It is often difficult to reconstruct how the mathematicians of the past arrived at new discoveries, because they had a habit of presenting only the final outcome of their deliberations, not the many false steps they took along the way. This problem is often compounded, because the natural thought patterns in past ages were different from today’s." (Ian Stewart, "Visions of Infinity", 2013) 

"Many of the great mathematical problems stem from deep and difficult questions in well-established areas of the subject. They are the big challenges that emerge when a major area has been thoroughly explored. They tend to be quite technical, and everyone in the area knows they’re hard to answer, because many experts have tried and failed. The area concerned will already possess many powerful techniques, massive mathematical machines whose handles can be cranked if you’ve done your homework - but if the problem is still open, then all of the plausible ways to use those! techniques have already been tried, and they didn’t work. So either there is a less plausible way to use the tried-and-tested techniques of the area, or you need new techniques." (Ian Stewart, "Visions of Infinity", 2013) 

"Mathematical intuition is the mind’s ability to sense form and structure, to detect patterns that we cannot consciously perceive. Intuition lacks the crystal clarity of conscious logic, but it makes up for that by drawing attention to things we would never have consciously considered." (Ian Stewart, "Visions of Infinity", 2013)

"Mathematical symmetry is an idealized model. However, slightly imperfect symmetry requires explanation; it’s not enough just to say ‘it’s asymmetric’."(Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

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

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

"Rotations and translations are global symmetries: they apply uniformly across the whole of space and time. A rotation about some axis rotates every point in space through the same angle. Gauge symmetries are different: they are local symmetries, which can vary from point to point in space. In the case of electromagnetism, these local symmetries are changes of phase. A local oscillation of the electromagnetic field has both an amplitude (how big it is) and a phase (the time at which it reaches its peak)." (Ian Stewart, "Visions of Infinity", 2013)

"Solving great problems requires a deep understanding of mathematics, plus persistence and ingenuity. It can involve years of concentrated effort, most of it apparently fruitless." (Ian Stewart, "Visions of Infinity", 2013)

"Squaring the circle sounds like a problem in geometry. That’s because it is a problem in geometry. But its solution turned out to lie not in geometry at all, but in algebra. Making unexpected connections between apparently unrelated areas of mathematics often lies at the heart of solving a great problem." (Ian Stewart, "Visions of Infinity", 2013)

"The classical theories of space, time, and matter were brought to their peak in James Clerk Maxwell’s equations for electromagnetism. This elegant system of equations unified two of nature’s forces, previously thought to be distinct. In place of electricity and magnetism, there was a single electromagnetic field. A field pervades the whole of space, as if the universe were filled with some kind of invisible fluid. At each point of space we can measure the strength and direction of the field, as if that fluid were flowing in mathematical patterns. For some purposes the electromagnetic field can be split into two components, the electric field and the magnetic field. But a moving magnetic field creates an electric one, and conversely, so when it comes to dynamics, both fields must be combined into a single more complex one." (Ian Stewart, "Visions of Infinity", 2013)

"The four colour problem is [...] equivalent to a different question: Given a network in the plane whose lines do not cross, is it possible to 4-colour the dots, so that two dots joined by a line always have different colours? The same reformulation applies with any number of colours." (Ian Stewart, "Visions of Infinity", 2013)

"The group of transformations of space-time that fixes the origin and leaves the interval invariant is called the Lorentz group after the physicist Hendrik Lorentz. The Lorentz group specifies how relative motion works in relativity, and is responsible for the theory’s counterintuitive features in which objects shrink, time slows down, and mass increases, as a body nears the speed of light." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"The most wonderful feature of elliptic functions outdoes trigonometric functions in a 

spectacular way. Not only are elliptic functions periodic: they are doubly periodic. A line is one-dimensional, so patterns can repeat in only one direction, along the line. The complex plane is twodimensional, so patterns can repeat like wallpaper: down the strip of paper, and also sideways along the wall into adjacent strips of paper. Associated with each elliptic function are two independent complex numbers, its periods, and adding either of them to the variable does not change the value of the function." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"[…] the symmetry group of the infinite logarithmic spiral is an infinite group, with one element for each real number . Two such transformations compose by adding the corresponding angles, so this group is isomorphic to the real numbers under addition." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"There are highly symmetric tiling patterns in hyperbolic geometry. For each of them, we can construct complex functions that repeat the same values on every tile. These are known as modular functions, and they are natural generalisations of elliptic functions. Hyperbolic geometry is a very rich subject, and the range of tiling patterns is much more extensive than it is for the Euclidean plane. So complex analysts started thinking seriously about non-Euclidean geometry. A profound link between analysis and number theory then appeared. Modular functions do for elliptic curves what trigonometric functions do for the circle." (Ian Stewart, "Visions of Infinity", 2013)

"We can find the minimax strategy by exploiting the game’s symmetry. Roughly speaking, the minimax strategy must have the same kind of symmetry." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"What makes a great mathematical problem great? Intellectual depth, combined with simplicity and elegance. Plus: it has to be hard. Anyone can climb a hillock; Everest is another matter entirely. A great problem is usually simple to state, although the terms required may be elementary or highly technical." (Ian Stewart, "Visions of Infinity", 2013)

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

"When we think of mathematics, what springs to mind is endless pages of dense symbols and formulas. However, those two million pages generally contain more words than symbols. The words are there to explain the background to the problem, the flow of the argument, the meaning of the calculations, and how it all fits into the evergrowing edifice of  mathematics." (Ian Stewart, "Visions of Infinity", 2013)

"A mathematical concept, then, is an organised pattern of ideas that are somehow interrelated, drawing on the experience of concepts already established. Psychologists call such an organised pattern of ideas a ‘schema’." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

"Although it is certainly possible to build up the whole of mathematics by axiomatic methods starting from the empty set, using no outside information whatsoever, it is also totally unintelligible to anyone who does not already understand the mathematics being built up." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

"Complex numbers do not fit readily into many people’s schema for ‘number’, and students often reject the concept when it is first presented. Modern mathematicians look at the situation with the aid of an enlarged schema in which the facts make sense." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

"It is not that the human mind cannot think logically. It is a question of different kinds of understanding. One kind of understanding is the logical, step-by-step way of understanding a formal mathematical proof. Each individual step can be checked but this may give no idea how they fit together, of the broad sweep of the proof, of the reasons that lead to it being thought of in the first place. Another kind of understanding arises by developing a global viewpoint, from which we can comprehend the entire argument at a glance. This involves fitting the ideas concerned into the overall pattern of mathematics, and linking them to similar ideas from other areas." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

"The essential quality of mathematics that binds it together in a coherent way is the use of mathematical proof to deduce new results from known ones, building up a strong and consistent theory." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

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

"When we extend the system of natural numbers and counting to embrace infinite cardinals, the larger system need not have all of the properties of the smaller one. However, familiarity with the smaller system leads us to expect certain properties, and we can become confused when the pieces don’t seem to fit. Insecurity arose when the square of a complex number violated the real number principle that all squares are positive. This was resolved when we realised that the complex numbers cannot be ordered in the same way as their subset of reals." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)