On Numbers (1900-1909)

"Great numbers are not counted correctly to a unit, they are estimated; and we might perhaps point to this as a division between arithmetic and statistics, that whereas arithmetic attains exactness, statistics deals with estimates, sometimes very accurate, and very often sufficiently so for their purpose, but never mathematically exact." (Arthur L Bowley, "Elements of Statistics", 1901)

"[…] numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the difference of things." (Richard Dedekind, "Essays on the Theory of Numbers", 1901)

"Statistics may rightly be called the science of averages. […] Great numbers and the averages resulting from them, such as we always obtain in measuring social phenomena, have great inertia. […] It is this constancy of great numbers that makes statistical measurement possible. It is to great numbers that statistical measurement chiefly applies." (Sir Arthur L Bowley,"Elements of Statistics", 1901)

"It has been said that no science is established on a firm basis unless its generalisations can be expressed in terms of number, and it is the special province of mathematics to assist the investigator in finding numerical relations between phenomena. After experiment, then mathematics. While a science is in the experimental or observational stage, there is little scope for discerning numerical relations. It is only after the different workers have 'collected data' that the mathematician is able to deduce the required generalisation." (Joseph W Mellor, "Higher Mathematics for Students of Chemistry and Physics', 1902)

"All the modern higher mathematics is based on a calculus of operations, on laws of thought. All mathematics, from the first, was so in reality; but the evolvers of the modern higher calculus have known that it is so. Therefore elementary teachers who, at the present day, persist in thinking about algebra and arithmetic as dealing with laws of number, and about geometry as dealing with laws of surface and solid content, are doing the best that in them lies to put their pupils on the wrong track for reaching in the future any true understanding of the higher algebras. Algebras deal not with laws of number, but with such laws of the human thinking machinery as have been discovered in the course of investigations on numbers. Plane geometry deals with such laws of thought as were discovered by men intent on finding out how to measure surface; and solid geometry with such additional laws of thought as were discovered when men began to extend geometry into three dimensions." (Mary E Boole, "Lectures on the Logic of Arithmetic", 1903)

"Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians." (Bertrand Russell,"The Principles of Mathematics", 1903)

"[...] measurement demands some one-one relations between the numbers and magnitudes in question - a relation which may be direct or indirect, important or trivial, according to circumstances." (Bertrand Russel, "The Principles of Mathematics", 1903)

"It is the intuition of pure number, that of pure logical forms, which illumines and directs those we have called analysts. This it is which enables them not alone to demonstrate, but also to invent." (Henri Poincaré, "The Value of Science", 1905)

"Pythagoras says that number is the origin of all things, and certainly the law of number is the key that unlocks the secrets of the universe. But the law of number possesses an immanent order, which is at first sight mystifying, but on a more intimate acquaintance we easily understand it to be intrinsically necessary; and this law of number explains the wondrous consistency of the laws of nature." (Paul Carus, "Reflections on Magic Squares", Monist Vol. 16, 1906)

"[…] imaginary numbers made their own way into arithmetical calculation without the approval, and even against the desires of individual mathematicians, and obtained wider circulation only gradually and the extent to which they showed themselves useful." (Felix Klein,"Elementary Mathematics from an Advanced Standpoint", 1908)

"If one intends to base arithmetic on the theory of natural numbers as finite cardinals, one has to deal mainly with the definition of finite set; for the cardinal is, according to its nature, a property of a set, and any proposition about finite cardinals can always be expressed as a proposition about finite sets. In the following I will try to deduce the most important property of natural numbers, namely the principle of complete induction, from a definition of finite set which is as simple as possible, at the same time showing that the different definitions [of finite set] given so far are equivalent to the one given here." (Ernst Zermelo, "Ueber die Grundlagen der Arithmetik", Atti del IV Congresso Internazionale dei Matematici, 1908)

"If the activity of a science can be supplied by a machine, that science cannot amount to much, so it is said; and hence it deserves a subordinate place. The answer to such arguments, however, is that the mathematician, even when he is himself operating with numbers and formulas, is by no means an inferior counterpart of the errorless machine [...]" (Felix Klein, "Elementary Mathematics from an Advanced Standpoint", 1908)

"Let us look for a moment at the general significance of the fact that calculating machines actually exist, which relieve mathematicians of the purely mechanical part of numerical computations, and which accomplish the work more quickly and with a greater degree of accuracy; for the machine is not subject to the slips of the human calculator. The existence of such a machine proves that computation is not concerned with the significance of numbers, but that it is concerned essentially only with the formal laws of operation; for it is only these that the machine can obey - having been thus constructed - an intuitive perception of the significance of numbers being out of the question." (Felix Klein, "Elementarmathematik vom hoheren Standpunkte aus", 1908)

"What, in fact, is mathematical discovery? It does not consist in making new combinations with mathematical entities that are already known. That can be done by anyone, and the combinations that could be so formed would be infinite in number, and the greater part of them would be absolutely devoid of interest. Discovery consists precisely in not constructing useless combinations, but in constructing those that are useful, which are an infinitely small minority. Discovery is discernment, selection." (Henri Poincaré, "Science and Method", 1908)

"Arithmetic does not present to us that feeling of continuity which is such a precious guide; each whole number is separate from the next of its kind and has in a sense individuality; each in a manner is an exception and that is why general theorems are rare in the theory of numbers; and that is why those theorems which may exist are more hidden and longer escape those who are searching for them." (Henri Poincaré,"Annual Report of the Board of Regents of the Smithsonian Institution", 1909)

On Numbers (1800-1829)

"It is very inaccurate to say that a negative number is less than 0, which is what many authors claim. A negative number is a positive number, but in another sense, and therefore relative." (Van Swinden, cca 1800)

"Number theory is revealed in its entire simplicity and natural beauty when the field of arithmetic is extended to the imaginary numbers" (Carl F Gauss, "Disquisitiones arithmeticae" ["Arithmetical Researches"], 1801)

"The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. […] The dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated." (Carl F Gauss, "Disquisitiones Arithmeticae" ["Arithmetical Researches"], 1801)

"The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers. […] Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated." (Carl F Gauss, "Disquisitiones Arithmeticae" ["Arithmetical Researches"], 1801)

"Regarding numbers and proportions, the best way to catch the imagination is to speak to the eyes." (William Playfair, "Elemens de statistique", 1802)

"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)

"It is the greatest that will ever be discovered, for, as they [perfect numbers] are merely curious without being useful, it is not likely that any person will attempt to find one beyond it." (Peter Barlow, "Theory of Numbers", 1811)"

"It is contrary to the usual order of things, that events so harmonious as those of the system of the world, should depend on such diversified agents as are supposed to exist in our artificial arrangements; and there is reason to anticipate a great reduction in the number of undecompounded bodies, and to expect that the analogies of nature will be found conformable to the refined operations of art. The more the phenomena of the universe are studied, the more distinct their connection appears, and the more simple their causes, the more magnificent their design, and the more wonderful the wisdom and power of their Author." (Sir Humphry Davy, "Elements of Chemical Philosophy", 1812)

"The most important questions of life are, for the most part, really only problems of probability. Strictly speaking one may even say that nearly all our knowledge is problematical; and in the small number of things which we are able to know with certainty, even in the mathematical sciences themselves, induction and analogy, the principal means for discovering truth, are based on probabilities, so that the entire system of human knowledge is connected with this theory." (Pierre-Simon Laplace, "Theorie Analytique des Probabilités", 1812)

"Probability has reference partly to our ignorance, partly to our knowledge [..] The theory of chance consists in reducing all the events of the same kind to a certain number of cases equally possible, that is to say, to such as we may be equally undecided about in regard to their existence, and in determining the number of cases favorable to the event whose probability is sought. The ratio of this number to that of all cases possible is the measure of this probability, which is thus simply a fraction whose number is the number of favorable cases and whose denominator is the number of all cases possible." (Pierre-Simon Laplace, Philosophical Essay on Probabilities", 1814)

"Here I am at the limit which God and nature has assigned to my individuality. I am compelled to depend upon word, language and image in the most precise sense, and am wholly unable to operate in any manner whatever with symbols and numbers which are easily intelligible to the most highly gifted minds." (Johann Wolfgang von Goethe, [Letter to Naumann] 1826)

On Numbers (1700-1724)

"To understand the theory of chance thoroughly, requires a great knowledge of numbers, and a pretty competent one of Algebra." (John Arbuthnot, "An Essay on the Usefulness of Mathematical Learning", 1701)

"Imaginary numbers are a fine and wonderful refuge of the divine spirit almost an amphibian between being and non-being." (Gottfried Leibniz, 1702)

"Mathematicks therefore is a Science which teaches or contemplates whatever is capable of Measure or Number as such. When it relates to Number, it is called Arithmetick; but when to measure, as Length, Breadth, Depth, Degrees of Velocity in Motion, Intenseness or Remissness of Sounds, Augmentation or Diminution of Quality, etc. it is called Geometry." (Jacques Ozanam, "A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms", 1702)

"Arithmetic and geometry, according to Plato, are the two wings of the mathematician. The object indeed of all mathematical questions, is to determine the ratios of numbers, or of magnitudes ; and it may even be said, to continue the comparison of the ancient philosopher, that arithmetic is the mathematician's right wing; for it is an incontestable truth, that geometrical determinations would, for the most part, present nothing satisfactory to the mind, if the ratios thus determined could not be reduced to numerical ratios. This justifies the common practice, which we shall here follow, of beginning with arithmetic." (Jacques Ozanam et al, "Recreations in Mathematics and Natural Philosophy", 1703)

"Further, it cannot escape anyone that for judging in this way about any event at all, it is not enough to use one or two trials, but rather a great number of trials is required. And sometimes the stupidest man|by some instinct of nature per se and by no previous instruction" (this is truly amazing)| knows for sure that the more observations of this sort that are taken, the less the danger will be of straying from the mark." (Jacob Bernoulli, "Ars Conjectandi" ["The Art of Conjecturing"], 1713

"It seems that to make a correct conjecture about any event whatever, it is necessary to calculate exactly the number of possible cases and then to determine how much more likely it is that one case will occur than another." (Jacob Bernoulli, "Ars Conjectandi" ["The Art of Conjecturing"], 1713))

"The probability of an Event is greater, or less, according to the number of Chances by which it may Happen, compar’d with the number of all the Chances, by which it may either Happen or Fail. […] Therefore, if the Probability of Happening and Failing are added together, the Sum will always be equal to Unit." (Abraham De Moivre, "The Doctrine of Chances", 1718)

🪷On Mind: On Imagination (1700-1749)

"The mathematics are the friends to religion, inasmuch as they charm the passions, restrain the impetuosity of the imagination, and purge the mind from error and prejudice. Vice is error, confusion and false reasoning; and all truth is more or less opposite to it. Besides, mathematical truth may serve for a pleasant entertainment for those hours which young men are apt to throw away upon their vices; the delightfulness of them being such as to make solitude not only easy but desirable." (John Arbuthnot, "An Essay on the Usefulness of Mathematical Learning", 1701)

"The ideas imprinted on the senses by the Author of Nature are called real things: and those excited in the imagination being less regular, vivid and constant, are more properly termed ideas, or images of things, which they copy and represent. But then our sensations, be they never so vivid and distinct, are nevertheless ideas, that is, they exist in the mind, or are perceived by it, as truly as the ideas of its own framing. The ideas of sense are allowed to have more reality in them, that is, to be more strong, orderly, and coherent than the creatures of the mind; but this is no argument that they exist without the mind. They are also less dependent on the spirit, or thinking substance which perceives them, in that they are excited by the will of another and more powerful spirit: yet still they are ideas, and certainly no idea, whether faint or strong, can exist otherwise than in a mind perceiving it." (George Berkeley, "Principles of Human Knowledge", 1710)

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

"[...] things which do not now exist in the mind itself, can only be perceived, remembered, or imagined, by means of ideas or images of them in the mind, which are the immediate objects of perception, remembrance, and imagination. This doctrine appears evidently to be borrowed from the old system; which taught, that external things make impressions upon the mind, like the impressions of a seal upon wax; that it is by means of those impressions that we perceive, remember) or imagine them; and that those impressions must resemble the things from which they are taken. When we form our notions of the operations of the mind by analogy, this way of conceiving them seems to be very natural, and offers itself to our thoughts: for as every thing which is felt must make some impression upon the body, we are apt to think, that every thing which is understood must make some impression upon the mind." (Thomas Reid, "An Inquiry into the Human Mind", 1734)

"Nay farther, even with relation to that succession, we cou'd only admit of those perceptions, which are immediately present to our consciousness, nor cou'd those lively images, with which the memory presents us, be ever receiv'd as true pictures of past perceptions. The memory, senses, and understanding are, therefore, all of them founded on the imagination, or the vivacity of our ideas."(David Hume, "A Treatise of Human Nature A Treatise of Human Nature", 1739)

"Beware of determining and declaring your Opinion suddenly on any Object; for Imagination often gets the Start of judgment, and makes People believe they see Things, which better Observations will convince them could not possibly be seen: Therefore assert nothing till after repeated Experiments and Examinations in all Lights and in all Positions." (Henry Baker, "The Microscope Made Easy", 1742)

"For it ought to be considered that both –b   and –c  , as they stand alone, are, in some Sense, as much impossible Quantities as v(-b)  and v(-c) ; since the Sign –, according to the established Rules of Notation, shews the Quantity, to which it is prefixed, is to be subtracted, but to subtract something from nothing is impossible, and the Notion or Supposition of a Quantity actually less than Nothing, absurd and shocking to the Imagination." (Thomas Simpson, "A Treatise of Algebra", 1745)

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🆂David Stirzaker - Collected Quotes

"By its very nature a model cannot include all the details of the reality it seeks to represent, for then it would be just as hard to comprehend and describe as the reality we want to model. At best, our model should give a reasonable picture of some small part of reality. It has to be a simple (even crude) description; and we must always be ready to scrap or improve a model if it fails in this task of accurate depiction. That having been said, old models are often still useful." (David Stirzaker, "Probability and Random Variables: A Beginner's Guide", 1999) 

"Conversely, there are few features of life, the universe, or anything, in which chance is not in some way crucial. Nor is this merely some abstruse academic point; assessing risks and taking chances are inescapable facets of everyday existence. It is a trite maxim to say that life is a lottery; it would be more true to say that life offers a collection of lotteries that we can all, to some extent, choose to enter or avoid. And as the information at our disposal increases, it does not reduce the range of choices but in fact increases them." (David Stirzaker, "Probability and Random Variables: A Beginner's Guide", 1999)

"For several centuries that we know of, and probably for many centuries before that, flipping a coin (or rolling a die) has been the epitome of probability, the paradigm of randomness. You flip the coin (or roll the die), and nobody can accurately predict how it will fall. Nor can the most powerful computer predict correctly how it will fall, if it is flipped energetically enough. This is why cards, dice, and other gambling aids crop up so often in literature both directly and as metaphors. No doubt it is also the reason for the (perhaps excessive) popularity of gambling as entertainment. If anyone had any idea what numbers the lottery would show, or where the roulette ball will land, the whole industry would be a dead duck." (David Stirzaker, "Probability and Random Variables: A Beginner’s Guide", 1999)

"From the moment we first roll a die in a children’s board game, or pick a card (any card), we start to learn what probability is. But even as adults, it is not easy to tell what it is, in the general way." (David Stirzaker, "Probability and Random Variables: A Beginner’s Guide", 1999)

"It is difficult to make progress in any branch of mathematics without using the ideas and notation of sets and functions. Indeed it would be perverse to try to do so, since these ideas and notation are very helpful in guiding our intuition and solving problems." (David Stirzaker, "Probability and Random Variables: A Beginner’s Guide", 1999)

"Models form extraordinarily powerful and economical ways of thinking about the world. In fact they are often so good that the model is confused with reality. If you ever think about atoms, you probably imagine little billiard balls; more sophisticated readers may imagine little orbital systems of elementary particles. Of course atoms are not`really' like that; these visions are just convenient old models." (David Stirzaker, "Probability and Random Variables: A Beginner’s Guide", 1999)

"[...] the chance of a head (or a double six) is just a chance. The whole point of probability is to discuss uncertain eventualities before they occur. After this event, things are completely different. As the simplest illustration of this, note that even though we agree that if we flip a coin and roll two dice then the chance of a head is greater than the chance of a double six, nevertheless it may turn out that the coin shows a tail when the dice show a double six." (David Stirzaker, "Probability and Random Variables: A Beginner’s Guide", 1999)

"The problem is that to interpret probability as a relative frequency requires that we can repeat some game or activity as many times as we wish. Often this is clearly not the case." (David Stirzaker, "Probability and Random Variables: A Beginner’s Guide", 1999)

"The whole point of probability is to discuss uncertain eventualities before they occur. After this event, things are completely different." (David Stirzaker, "Probability and Random Variables: A Beginner’s Guide", 1999)

"[...] unlike the apparatus for choosing numbers, gamblers choose numbers for various reasons. Very few choose at random; they use birthdays, ages, patterns, and so on. However, you might suppose that for any gambler chosen at random, that choice of numbers would be evenly distributed over the possibilities." (David Stirzaker, "Probability and Random Variables: A Beginner’s Guide", 1999)

"Use of the term 'model' makes it easier to keep in mind this distinction between theory and reality. By its very nature a model cannot include all the details of the reality it seeks to represent, for then it would be just as hard to comprehend and describe as the reality we want to model. At best, our model should give a reasonable picture of some small part of reality. It has to be a simple (even crude) description; and we must always be ready to scrap or improve a model if it fails in this task of accurate depiction. That having been said, old models are often still useful. The theory of relativity supersedes the Newtonian model, but all engineers use Newtonian mechanics when building bridges or motor cars, or probing the solar system." (David Stirzaker, "Probability and Random Variables: A Beginner’s Guide", 1999)

"We cannot really have a perfectly shuffled pack of perfect cards; this ‘collection of equally likely hands’ is actually a fiction. We create the idea, and then use the rules of arithmetic to calculate the required chances. This is characteristic of all mathematics, which concerns itself only with rules defining the behaviour of entities which are themselves undefined (such as ‘numbers’ or ‘points’)." (David Stirzaker, "Probability and Random Variables: A Beginner’s Guide", 1999)

🅰Hannah Arendt - Collected Quotes

"The new always happens against the overwhelming odds of statistical laws and their probability, which for all practical, everyday purposes amounts to certainty; the new therefore always appears in the guise of a miracle." (Hannah Arendt, "The Human Condition", 1958)

"Storytelling reveals meaning without committing the error of defining it." (Hannah Arendt, "Men in Dark Times", 1968)

"Predictions of the future are never anything but projections of present automatic processes and procedures, that is, of occurrences that are likely to come to pass if men do not act and if nothing unexpected happens; every action, for better or worse, and every accident necessarily destroys the whole pattern in whose frame the prediction moves and where it finds its evidence." (Hannah Arendt, "On Violence", 1970)

"It interrupts any doing, any ordinary activities, no matter what they happen to be. All thinking demands a stop-and-think." (Hannah Arendt, "The Life of the Mind", 1977)

"It was mathematics, the non-empirical science par excellence, wherein the mind appears to play only with itself, that turned out to be the science of sciences, delivering the key to those laws of nature and the universe that are concealed by appearances." (Hannah Arendt, "The Life of the Mind", 1977)

"Metaphysical fallacies contain the only clues we have to what thinking means to those who engage in it." (Hannah Arendt, "The Life of the Mind", 1977)

"The need of reason is not inspired by the quest for truth but by the quest for meaning. And truth and meaning are not the same. The basic fallacy, taking precedence over all specific metaphysical fallacies, is to interpret meaning on the model of truth." (Hannah Arendt, "The Life of the Mind", 1977)

"To expect truth to come from thinking signifies that we mistake the need to think with the urge to know." (Hannah Arendt, "The Life of the Mind", 1977)

"What science and the quest for knowledge are after is irrefutable truth; that is, propositions that human beings are not free to reject - that are compelling. They are of two kinds, as we have known since Leibnitz: truths of reasoning and truths of fact." (Hannah Arendt, The New Yorker, 1977)

"Analogies, metaphors, and emblems are the threads by which the mind holds on to the world even when, absentmindedly, it has lost direct contact with it, and they guarantee the unity of human experience. Moreover, in the thinking process itself they serve as models to give us our bearings lest we stagger blindly among experiences that our bodily senses with their relative certainty of knowledge cannot guide us through." (Hannah Arendt, "The Life of the Mind", 1981)

🅼Robert Musil - Collected Quotes

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

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

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

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

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

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

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

On Principles (1975-1999)

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

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

"A good theorem will almost always have a wide-ranging influence on later mathematics, simply by virtue of the fact that it is true. Since it is true, it must be true for some reason; and if that reason lies deep, then the uncovering of it will usually require a deeper understanding of neighboring facts and principles." (Ian Richards, "Number theory", 1978)

"Real progress in understanding nature is rarely incremental. All important advances are sudden intuitions, new principles, new ways of seeing." (Marilyn Ferguson, "The Aquarian Conspiracy: Personal and Social Transformation in the 1980s", 1980)

"For the great majority of mathematicians, mathematics is […] a whole world of invention and discovery - an art. The construction of a new theorem, the intuition of some new principle, or the creation of a new branch of mathematics is the triumph of the creative imagination of the mathematician, which can be compared to that of a poet, the painter and the sculptor." (George F J Temple, "100 Years of Mathematics: a Personal Viewpoint", 1981)

"Both religion and science must preserve their autonomy and their distinctiveness. Religion is not founded on science nor is science an extension of religion. Each should possess its own principles, its pattern of procedures, its diversities of interpretation and its own conclusions." (Pope John Paul II, [Letter to Father George V Coyne], 1988)

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

"Principles are the territory. Values are maps. When we value correct principles, we have truth - a knowledge of things as they are." (Stephen R Covey, "The 7 Habits of Highly Effective People", 1989)

"The ‘objective reality’, or the territory itself, is composed of ‘lighthouse’ principles that govern human growth and happiness - natural laws that are woven into the fabric of every civilized society throughout history and comprise the roots of every family and institution that has endured and prospered. The degree to which our mental maps accurately describe the territory does not alter its existence." (Stephen Covey, "The 7 Habits of Highly Effective People", 1989)

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

"The word theory, as used in the natural sciences, doesn’t mean an idea tentatively held for purposes of argument - that we call a hypothesis. Rather, a theory is a set of logically consistent abstract principles that explain a body of concrete facts. It is the logical connections among the principles and the facts that characterize a theory as truth. No one element of a theory [...] can be changed without creating a logical contradiction that invalidates the entire system. Thus, although it may not be possible to substantiate directly a particular principle in the theory, the principle is validated by the consistency of the entire logical structure." (Alan Cromer, "Uncommon Sense: The Heretical Nature of Science", 1993)

"Within image theory, it is suggested that important components of decision-making processes are the different 'images' that a person may use to evaluate choice options. Images may represent a person's principles, goals, or plans. Decision options may then match or not match these images and be adopted, rejected, considered further, depending on circumstances." (Deborah J Terry & Michael A Hogg, "Attitudes, Behavior, and Social Context: The Role of Norms and Group Membership", 1999)

On Principles (1950-1974)

 "It is more important to have a clear understanding of general principles, without, however, thinking of them as fixed laws, than to load the mind with a mass of detailed technical information which can readily be found in reference books or card indexes." (William I B Beveridge, "The Art of Scientific Investigation", 1950)

"[…] the chief reason in favor of any theory on the principles of mathematics must always be inductive, i.e., it must lie in the fact that the theory in question enables us to deduce ordinary mathematics. In mathematics, the greatest degree of self-evidence is usually not to be found quite at the beginning, but at some later point; hence the early deductions, until they reach this point, give reasons rather from them, than for believing the premises because true consequences follow from them, than for believing the consequences because they follow from the premises." (Alfred N Whitehead, "Principia Mathematica", 1950)

"The principle of complementarity states that no single model is possible which could provide a precise and rational analysis of the connections between these phenomena [before and after measurement]. In such a case, we are not supposed, for example, to attempt to describe in detail how future phenomena arise out of past phenomena. Instead, we should simply accept without further analysis the fact that future phenomena do in fact somehow manage to be produced, in a way that is, however, necessarily beyond the possibility of a detailed description. The only aim of a mathematical theory is then to predict the statistical relations, if any, connecting the phenomena." (David Bohm, "A Suggested Interpretation of the Quantum Theory in Terms of ‘Hidden’ Variables", 1952)

"We have thus assigned to pure reason and experience their places in a theoretical system of physics. The structure of the system is the work of reason: the empirical contents and their mutual relations must find their representation in the conclusions of the theory. In the possibility of such a representation lie the sole value and justification of the whole system, and especially of the concepts and fundamental principles which underlie it. Apart from that, these latter are free inventions of human intellect, which cannot be justified either by the nature of that intellect or in any other fashion a priori." (Albert Einstein, "Ideas and Opinions", 1954)

"For a physicist mathematics is not just a tool by means of which phenomena can be calculated, it is the main source of concepts and principles by means of which new theories can be created." (Freeman Dyson, "Mathematics in the Physical Sciences", Scientific American, 1964)

"The method of guessing the equation seems to be a pretty effective way of guessing new laws. This shows again that mathematics is a deep way of expressing nature, and any attempt to express nature in philosophical principles, or in seat-of-the-pants mechanical feelings, is not an efficient way." (Richard Feynman, "The Character of Physical Law", 1965)

"Traditional organizational theories have tended to view the human organization as a closed system. This tendency has led to a disregard of differing organizational environments and the nature of organizational dependency on environment. It has led also to an over-concentration on principles of internal organizational functioning, with consequent failure to develop and understand the processes of feedback which are essential to survival." (Daniel Katz, "The Social Psychology of Organizations", 1966)

"The parallelism of general conceptions or even special laws in different fields therefore is a consequence of the fact that these are concerned with 'systems' and that certain general principles apply to systems irrespective of their nature. Hence principles such as those of wholeness and sum, mechanization, hierarchic order, approached to steady states, equifinality, etc., may appear in quite different disciplines. The isomorphism found in different realms is based of the existence of general system principles, of a more or less well-developed ‘general system theory’." (Ludwig von Bertalanffy, "General System Theory", 1968)

"Thus, there exist models, principles, and laws that apply to generalized systems or their subclasses, irrespective of their particular kind, the nature of their component elements, and the relations or 'forces' between them. It seems legitimate to ask for a theory, not of systems of a more or less special kind, but of universal principles applying to systems in general. In this way we postulate a new discipline called General System Theory. Its subject matter is the formulation and derivation of those principles which are valid for ‘systems’ in general." (Ludwig von Bertalanffy, „General System Theory: Foundations, Development, Applications", 1968)

"Modern scientific principle has been drawn from the investigation of natural laws, technology has developed from the experience of doing, and the two have been combined by means of mathematical system to form what we call engineering." (George S Emmerson, "Engineering Education: A Social History", 1973)

On Principles (1900-1924)

"The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself." (Bertrand Russell, "Principles of Mathematics", 1903)

"A physical theory is not an explanation. It is a system of mathematical propositions, deduced from a small number of principles, which aim to represent as simply, as completely, and as exactly as possible a set of experimental laws. […] Thus a true theory is not a theory which gives an explanation of physical appearances in conformity with reality; it is a theory which represents in a satisfactory manner a group of experimental laws. A false theory is not an attempt at an explanation based on assumptions contrary to reality; it is a ·group of propositions which do not agree with the experimental laws. Agreement with experiment is the sole criterion of truth for a physical theory." (Pierre-Maurice-Marie Duhem, "La théorie physique. Son objet, sa structure", 1906)

"Today it is no longer questioned that the principles of the analysts are the more far-reaching. Indeed, the synthesists lack two things in order to engage in a general theory of algebraic configurations: these are on the one hand a definition of imaginary elements, on the other an interpretation of general algebraic concepts. Both of these have subsequently been developed in synthetic form, but to do this the essential principle of synthetic geometry had to be set aside. This principle which manifests itself so brilliantly in the theory of linear forms and the forms of the second degree, is the possibility of immediate proof by means of visualized constructions." (Felix Klein, "Riemannsche Flächen", 1906)

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

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

"Every one knows there are mathematical axioms. Mathematicians have, from the days of Euclid, very wisely laid down the axioms or first principles on which they reason. And the effect which this appears to have had upon the stability and happy progress of this science, gives no small encouragement to attempt to lay the foundation of other sciences in a similar manner, as far as we are able." (William K Clifford et al, "Scottish Philosophy of Common Sense", 1915)

"The mathematician is free in his imagination, to construct what worlds he pleases. What he is to imagine is a matter for his own caprice; he is not thereby discovering the fundamental principles of the universe nor becoming acquainted with the ideas of God. If he can find, in experience, sets of entities which obey the same logical scheme as his mathematical entities, then he has applied his mathematics to the external world; he has created a branch of science." (John W N Sullivan, "Aspects of Science", 1923)