Richard von Mises - Collected Quotes

"A collective appropriate for the application of the theory of probability must fulfil two conditions. First, the relative frequencies of the attributes must possess limiting values. Second, these limiting values must remain the same in all partial sequences which may be selected from the original one in an arbitrary way. Of course, only such partial sequences can be taken into consideration as can be extended indefinitely, in the same way as the original sequence itself." (Richard von Mises, "Probability, Statistics and Truth", 1928)

"A great number of popular and more or less serious objections to the theory of probability disappear at once when we recognize that the exclusive purpose of this theory is to determine, from the given probabilities in a number of initial collectives, the probabilities in a new collective derived from the initial ones." (Richard von Mises, "Probability, Statistics and Truth", 1928)

"The rational concept of probability, which is the only basis of probability calculus, applies only to problems in which either the same event repeats itself again and again, or a great number of uniform elements are involved at the same time. Using the language of physics, we may say that in order to apply the theory of probability we must have a practically unlimited sequence of uniform observations." (Richard von Mises, "Probability, Statistics and Truth", 1928)

"The result of each calculation appertaining to the field of probability is always, as far as our theory goes, nothing else but a probability, or, using our general definition, the relative frequency of a certain event in a sufficiently long (theoretically, infinitely long) sequence of observations. The theory of probability can never lead to a definite statement concerning a single event. The only question that it can answer is: what is to be expected in the course of a very long sequence of observations? It is important to note that this statement remains valid also if the calculated probability has one of the two extreme values 1 or 0." (Richard von Mises, "Probability, Statistics and Truth", 1928)

"All followers of the axiomatic method and most mathematicians think that there is some such thing as an absolute ‘mathematical rigor’ which has to be satisfied by any deduction if it is to be valid. The history of mathematics shows that this is not the case, that, on the contrary, every generation is surpassed in rigor again and again by its successors." (Richard von Mises, "Positivism: A Study in Human Understanding", 1951)

"If the concept of probability and the formulae of the theory of probability are used without a clear understanding of the collectives involved, one may arrive at entirely misleading results." (Richard von Mises, "Probability, Statistics, and Truth"2nd Ed., 1957)

"Remember that algebra, with all its deep and intricate problems, is nothing but a development of the four fundamental operations of arithmetic. Everyone who understands the meaning of addition, subtraction, multiplication, and division holds the key to all algebraic problems." (Richard von Mises, "Probability, Statistics, and Truth"2nd Ed., 1957)

"Starting from statistical observations and applying to them a clear and precise concept of probability it is possible to arrive at conclusions which are just as reliable and ‘truth-full’ and quite as practically useful as those obtained in any other exact science." (Richard von Mises, "Probability, Statistics, and Truth"2nd Ed., 1957)

"The main interest of physical statistics lies in fact not so much in the distribution of the phenomena in space, but rather in their succession in time." (Richard von Mises, "Probability, Statistics, and Truth"2nd Ed., 1957)

"The problems of statistical physics are of the greatest in our time, since they lead to a revolutionary change in our whole conception of the universe." (Richard von Mises, "Probability, Statistics, and Truth"2nd Ed., 1957)

"The theory of probability can never lead to a definite statement concerning a single event." (Richard von Mises, "Probability, Statistics, and Truth" 2nd Ed., 1957)

"The probability concept used in probability theory has exactly the same structure as have the fundamental concepts in any field in which mathematical analysis is applied to describe and represent reality." (Richard von Mises, "Mathematical Theory of Probability and Statistics", 1964)

Carl G Hempel - Collected Quotes

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

"The most distinctive characteristic which differentiates mathematics from the various branches of empirical science, and which accounts for its fame as the queen of the sciences, is no doubt the peculiar certainty and necessity of its results." (Carl G Hempel, "Geometry and Empirical Science", 1945)

"Theories are usually introduced when previous study of a class of phenomena has revealed a system of uniformities. […] Theories then seek to explain those regularities and, generally, to afford a deeper and more accurate understanding of the phenomena in question. To this end, a theory construes those phenomena as manifestations of entities and processes that lie behind or beneath them, as it were." (Carl G Hempel, "Philosophy of Natural Science", 1966)

"A geometrical theory in physical interpretation can never be validated with mathematical certainty […] like any other theory of empirical science, it can acquire only a more or less high degree of confirmation." (Carl G Hempel)

"The propositions of mathematics are devoid of all factual content; they convey no information whatever on any empirical subject matter." (Carl G Hempel)

Lancelot L Whyte - Collected Quotes

“Science does not begin with facts; one of its tasks is to uncover the facts by removing misconceptions.” (Lancelot L Whyte, “Accent on Form”, 1954)

"Science starts with an assumption which is always present, though it may be unconscious, may be forgotten, and may sometimes even be denied." (Lancelot L Whyte, "Accent on Form: An Anticipation of the Science of Tomorrow", 1954)

“The true aim of science is to discover a simple theory which is necessary and sufficient to cover the facts, when they have been purified of traditional prejudices.” (Lancelot L Whyte, “Accent on Form”, 1954)

"Both science and art have to do with ordered complexity." (Lancelot L Whyte, "The Griffin", 1957)

"Every scientific generation, measured by its most vocal members, exaggerates the historical importance of its own members [...] there is a perpetual temptation to study the latest and to neglect the past." (Lancelot L Whyte, "Essay on Atomism to 1960", 1961)

"If the universe is a mingling of probability clouds spread through a cosmic eternity of space-time, how is there as much order, persistence, and coherent transformation as there is?" (Lancelot L Whyte, "Essay on Atomism to 1960", 1961)

"Systematic errors of theory can seldom be discovered by direct attack; it is easier to uncover them by studying how and why physical theory took the path it did. That is why a clue to the future can sometimes be found in the past, and this is my reason for studying the history of atomism." (Lancelot L Whyte, "Essay on Atomism to 1960", 1961)

"Every rule has its limits, and every concept its ambiguities. Most of all is this true in the science of life, where nothing quite corresponds to our ideas; similar ends are reached by varied means, and no causes are simple." (Lancelot L Whyte, "Internal Factors in Evolution", 1965)

Nicolas de Condorcet - Collected Quotes

"[…] we are far from having exhausted all the applications of analysis to geometry, and instead of believing that we have approached the end where these sciences must stop because they  have reached the limit of the forces of the human spirit, we ought to avow rather we are only at the first steps of an immense career. These new [practical] applications, independently of the utility which they may have in themselves, are necessary to the progress of analysis in general; they give birth to questions which one would not think to propose; they demand that one create new methods. Technical processes are the children of need; one can say the same for the methods of the most abstract sciences. But we owe the latter to the needs of a more noble kind, the need to discover the new truths or to know better the laws of nature." (Nicolas de Condorcet, 1781)

"[…] determine the probability of a future or unknown event not on the basis of the number of possible combinations resulting in this event or in its complementary event, but only on the basis of the knowledge of order of familiar previous events of this kind" (Nicolas de Condorcet, "Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix", 1785)

"We must therefore establish a form of decision-making in which voters need only ever pronounce on simple propositions, expressing their opinions only with a yes or a no. […] Clearly, if anyone’s vote was self-contradictory (intransitive), it would have to be discounted, and we should therefore establish a form of voting which makes such absurdities impossible." (Nicolas de Condorcet, "On the form of decisions made by plurality vote", 1788)

"It has never yet been supposed, that all the facts of nature, and all the means of acquiring precision in the computation and analysis of those facts, and all the connections of objects with each other, and all the possible combinations of ideas, can be exhausted by the human mind." (Nicolas de Condorcet, "Outlines Of An Historical View Of The Progress Of The Human Mind", 1795)

"[All phenomena] are equally susceptible of being calculated, and all that is necessary, to reduce the whole of nature to laws similar to those which Newton discovered with the aid of the calculus, is to have a sufficient number of observations and a mathematics that is complex enough." (Nicolas de Condorcet) 

"As the mind learns to understand more complicated combinations of ideas, simpler formulae soon reduce their complexity; so truths that were discovered only by great effort, that could at first only be understood by men capable of profound thought, are soon developed and proved by methods that are not beyond the reach of common intelligence. The strength and the limits of man." (Nicolas de Condorcet) 

"Mathematics is the science that yields the best opportunity to observe the working of the mind. Its study is the best training of our abilities as it develops both the power and the precision of our thinking. Mathematics is valuable on account of the number and variety of its applications. And it is equally valuable in another respect: By cultivating it, we acquire the habit of a method of reasoning which can be applied afterwards to the study of any subject and can guide us in life's great and little problems." (Nicolas de Condorcet)

"This adventure of the physical sciences […] could not be observed without enlightened men seeking to follow it up in the other sciences; at each step it held out to them the model to be followed." (Nicolas de Condorcet)

"Those sciences, created almost in our own days, the object of which is man himself, the direct goal of which is the happiness of man, will enjoy a progress no less sure than that of the physical sciences; and this sweet idea - that our nephews will surpass us in wisdom as in enlightenment - is no longer an illusion." (Nicolas de Condorcet)

Occam's Razor = The Law of Parsimony (1900-1949)

"Entia non sunt multiplicanda praeter necessitatem. That is to say; before you try a complicated hypothesis, you should make quite sure that no simplification of it will explain the facts equally well." (Charles S Peirce, "Pragmatism and Pragmaticism", [lecture] 1903)

"Nothing perhaps has so retarded the reception of the higher conclusions of Geology among men in general, as ... [the] instinctive parsimony of the human mind in matters where time is concerned. (Charles Lapworth, Proceedings of the Geological Society of London, 1903)

"The process of induction is the process of assuming the simplest law that can be made to harmonize with our experience. This process, however, has no logical foundation but only a psychological one. It is clear that there are no grounds for believing that the  simplest course of events will really happen." (Ludwig Wittgenstein, "Tractatus Logico-Philosophicus", 1922)

"Whenever possible, substitute constructions out of known entities for inferences to unknown entities." (Bertrand Russell, 1924)

"In scientific thought we adopt the simplest theory which will explain all the facts under consideration and enable us to predict new facts of the same kind. The catch in this criterion lies in the world 'simplest'. It is really an aesthetic canon such as we find implicit in our criticisms of poetry or painting. The layman finds such a law as dx/dt = K(d2x/dy2) much less simple than 'it oozes', of which it is the mathematical statement. The physicist reverses this judgment, and his statement is certainly the more fruitful of the two, so far as prediction is concerned. It is, however, a statement about something very unfamiliar to the plainman, namely, the rate of change of a rate of change." (John B S Haldane, "Possible Worlds", 1927)

"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 can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience." (Albert Einstein, "On the Method of Theoretical Physics", [The Herbert Spencer Lecture, delivered at Oxford] 1933)

"In the conception of a machine or the product of a machine there is a point where one may leave off for parsimonious reasons, without having reached aesthetic perfection; at this point perhaps every mechanical factor is accounted for, and the sense of incompleteness is due to the failure to recognize the claims of the human agent. Aesthetics carries with it the implications of alternatives between a number of mechanical solutions of equal validity; and unless this awareness is present at every stage of the process [...] it is not likely to come out with any success in the final stage of design." (Lewis Mumford," The Esthetic Assimilation of the Machine", Technics and Civilization, 1934)

"When two hypotheses are possible, we provisionally choose that which our minds adjudge to be simpler, on the supposition that this is the more likely to lead in the direction of truth. It includes as a special case the principle of Occam's razo - entia non multiplicana praeter necessitatem." (James Jeans,"Physics and Philosophy", 1942)

"When two hypotheses are possible, we provisionally choose that which our minds adjudge to the simpler on the supposition that this Is the more likely to lead in the direction of the truth." (James H Jeans, "Physics and Philosophy" 3rd Ed., 1943)

"A theory is the more impressive the greater the simplicity of its premises is, the more different kinds of things it relates, and the more extended its area of applicability." (Albert Einstein, "Autobiographical Notes", 1949)

Occam's Razor = The Law of Parsimony (-1499)

"We may assume the superiority ceteris paribus [other things being equal] of the demonstration which derives from fewer postulates or hypotheses." (Aristotle, “Posterior Analytics”, cca. 400 BC)

"We may assume the superiority ceteris paribus [all things being equal] of the demonstration which derives from fewer postulates or hypotheses - in short from fewer premisses; for [...] given that all these are equally well known, where they are fewer knowledge will be more speedily acquired, and that is a desideratum. The argument implied in our contention that demonstration from fewer assumptions is superior may be set out in universal form [...]" (Aristotle, “Posterior Analytics”, cca. 400 BC)

"O immortal gods! Men do not realize how great a revenue parsimony can be!"  (Marcus Tullius Cicero, "Paradoxa Stoicorum", [Paradox VI] 46 BC) 

 

"Always take the short cut; and that is the rational one. Therefore say and do everything according to soundest reason." (Marcus Aurelius, "Meditations". cca. 121–180 AD)

"We consider it a good principle to explain the phenomena by the simplest hypothesis possible." (Ptolemy)

"That is better and more valuable which requires fewer, other circumstances being equal. [...] For if one thing were demonstrated from many and another thing from fewer equally known premises, clearly that is better which is from fewer because it makes us know quickly, just as a universal demonstration is better than particular because it produces knowledge from fewer premises. Similarly in natural science, in moral science, and in metaphysics the best is that which needs no premises and the better that which needs the fewer, other circumstances being equal." (Robert Grosseteste,” Commentarius in Posteriorum Analyticorum Libros”, cca. 1217–1220)

"It is superfluous to suppose that what can be accounted for by a few principles has been produced by many." (Thomas Aquinas, “Summa Theologica”, cca. 1266-1273)

"All that is superfluous displeases God and nature. All that displeases God and nature is evil." (Dante Alighieri, "De Monarchia", cca. 1312-1313)

“Numquam ponenda est pluralitas sine necessitate.”
“Plurality is never to be posited without necessity.” (William of Occam, “Quaestiones et decisiones in quattuor libros Sententiarum Petri Lombardi”, 1495)

Mental Models XXVIII (Limitations V)

"Thought often leads us far beyond the imaginable without thereby depriving us of the basis for our conclusions. Even if, as it appears, thought without mental pictures is impossible for us men, still their connection with the object of thought can be wholly superficial, arbitrary, and conventional." (Gottlob Frege, "The Foundations of Arithmetic" , 1884)

“The classical tradition has been to consider the world to be an association of observable objects (particles, fluids, fields, etc.) moving according to definite laws of force, so that one could form a mental picture in space and time of the whole scheme. This led to a physics whose aim was to make assumptions about the mechanism and forces connecting these observable objects in the simplest possible way. It has become increasingly evident in recent times, however, that nature works on a different plan. Her fundamental laws do not govern the world as it appears in our mental picture in any very direct way, but instead they control a substratum of which we cannot form a mental picture without introducing irrelevancies.” (Paul A M Dirac, “The Principles of Quantum Mechanics”, 1930)

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

"[…] many philosophers have found it difficult to accept the hypothesis that an object is just about what it appears to be, and so is like the mental picture it produces in our minds. For an object and a mental picture are of entirely different natures - a brick and the mental picture of a brick can at best no more resemble one another than an orchestra and a symphony. In any case, there is no compelling reason why phenomena - the mental visions that a mind constructs out of electric currents in a brain - should resemble the objects that produced these currents in the first instance." (James H Jeans," Physics and Philosophy" 3rd Ed., 1943)

"What a man sees depends both upon what he looks at and also upon what his previous visual-conceptual experience has taught him to see." (Thomas Kuhn, "The Structure of Scientific Revolutions", 1962)

"The major problem of the mental model approach lies in the fact that the external world is to be represented in a highly specific way. Representing indeterminacy in terms of mental models thus poses difficulties, casting some doubt on the contention that mental models can do without variables." (Gert Rickheit & Lorenz Sichelschmidt, "Mental Models: Some Answers, Some Questions, Some Suggestions", 1999)

"To think that the world can ever change without changes in our mental models is folly." (Joseph Jaworski, "Synchronicity: The Inner Path of Leadership", 2011)

“When we face a challenge, most of us rely on our tried-and-true skills and their underlying mental models to get us through. But these patterns themselves could be the source of the problem. Often they enabled some level of success or survival in a former environment. But they may not be successful in our present circumstance. Now we need a new way. We need to build alternative mental models that lead to more adaptive behaviors and results. The first step in this change is to understand our current models and related behaviors. Because mental models are so central to our way of life, changing even one of them can be difficult. Changing involves abandoning long-held ties to a comfortable pattern of thinking and behavior and then replacing it with a substitute mental model and new behavior. Changing a mental model involves changing a part of ourselves. As you might guess, this process can feel like the psychological equivalent of surgery. It can be uncomfortable, even painful.” (Marshall Goldsmith & Daniel White, "Coaching Leaders: Guiding People Who Guide Others", 2013)

“The social world that humans have made for themselves is so complex that the mind simplifies the world by using heuristics, customs, and habits, and by making models or assumptions about how things generally work (the ‘causal structure of the world’). And because people rely upon (and are invested in) these mental models, they usually prefer that they remain uncontested.” (Dr James Brennan, “Psychological  Adjustment to Illness and Injury”, West of England Medical Journal Vol. 117 (2), 2018)

James H Jeans - Collected Quotes

"The concepts which now prove to be fundamental to our understanding of nature- a space which is finite; a space which is empty, so that one point [of our 'material' world] differs from another solely in the properties of space itself; four-dimensional, seven- and more dimensional spaces; a space which for ever expands; a sequence of events which follows the laws of probability instead of the law of causation - or alternatively, a sequence of events which can only be fully and consistently described by going outside of space and time - all these concepts seem to my mind to be structures of pure thought, incapable of realisation in any sense which would properly be described as material." (James Jeans, "The Mysterious Universe", 1930)

"The final truth about phenomena resides in the mathematical description of it; so long as there is no imperfection in this, our knowledge is complete. We go beyond the mathematical formula at our own risk; we may find a [nonmathematical] model or picture that helps us to understand it, but we have no right to expect this, and our failure to find such a model or picture need not indicate that either our reasoning or our knowledge is at fault." (James Jeans, "The Mysterious Universe", 1930)

"Today there is a wide measure of agreement, which on the physical side of science approaches almost to unanimity, that the stream of knowledge is heading towards a non-mechanical reality; the universe begins to look more like a great thought than like a great machine. Mind no longer appears as an accidental intruder into the realm of matter; we are beginning to suspect that we ought rather to hail it as a creator and governor of the realm of matter [...]" (James Jeans, "The Mysterious Universe", 1930)

"In brief, a mathematical formula can never tell us what a thing is, but only how it behaves; it can only specify an object through its properties. And these are unlikely to coincide in toto with the properties of any single macroscopic object of our everyday life.” (James H Jeans, "The Mysterious Universe", 1930)

"The essential fact is simply that all the pictures which science now draws of nature, and which alone seem capable of according with observational fact, are mathematical pictures." (James H Jeans, "The Mysterious Universe", 1930)

"The making of models or pictures to explain mathematical formulae and the phenomena they describe is not a step towards, but a step away from reality; it is like making graven images of a spirit." (James H Jeans, "The Mysterious Universe", 1930)

"[…] our knowledge of the external world must always consist of numbers, and our picture of the universe - the synthesis of our knowledge - must necessarily be mathematical in form. All the concrete details of the picture, the apples, the pears and bananas, the ether and atoms and electrons, are mere clothing that we ourselves drape over our mathematical symbols - they do not belong to Nature, but to the parables by which we try to make Nature comprehensible." (James H Jeans, "The New World-Picture of Modern Physics", Supplement to Nature Vol. 134 (3384), 1934)

“A science which confines itself to correlating the phenomena can never learn anything about the reality underlying the phenomena, while a science which goes further than this, and introduces hypotheses about reality can never acquire certain knowledge of a positive kind about reality […]” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

“Exhaustive studies by many investigators have shown that the fundamental laws of nature do not control the phenomena directly. We must picture them as operating in a substratum of which we can form no mental picture unless we are willing to introduce a number of irrelevant and therefore unjustifiable suppositions.” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

“In time they [physicists] hoped to devise a model which would reproduce all the phenomena of physics, and so make it possible to predict them all. […] To-day we not only have no perfect model, but we know that it is of no use to search for one - it could have no intelligible meaning for us. For we have found out that nature does not function in a way that can be made comprehensible to the human mind through models or pictures. […] Although we can never devise a pictorial representation which shall be both true to nature and intelligible to our minds, we may still be able to make partial aspects of the truth comprehensible through pictorial representations or parables. As the whole truth does not admit of intelligible representation, every such pictorial representation or parable must fail somewhere. The physicist of the last generation was continually making pictorial representations and parables, and also making the mistake of treating the half-truths of pictorial representations and parables as literal truths.” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

“It was supposed that a model which reproduced all the phenomena of a science, and so made it possible to predict them all, must in some way correspond to the reality underlying the phenomena. But obviously this cannot be so. After one perfect model had been found, a second of equal perfection might appear, and as both models could not correspond to reality, we should have at least one perfect model which did not correspond to reality. Thus we could never be sure that any model corresponded to reality. In brief, we can never have certain knowledge as to the nature of reality.” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

“[…] many philosophers have found it difficult to accept the hypothesis that an object is just about what it appears to be, and so is like the mental picture it produces in our minds. For an object and a mental picture are of entirely different natures - a brick and the mental picture of a brick can at best no more resemble one another than an orchestra and a symphony. In any case, there is no compelling reason why phenomena - the mental visions that a mind constructs out of electric currents in a brain - should resemble the objects that produced these currents in the first instance.” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

“Nothing is left in the world but happenings for which no explanation or interpretation is offered or even attempted, and science has now for its single aim the discovery of the laws to which these happenings conform - the pattern of events.” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

“Physicists who are trying to understand nature may work in many different fields and by many different methods; one may dig, one may sow, one may reap. But the final harvest will always be a sheaf of mathematical formulae. These will never describe nature itself, hut only our observations on nature. Our studies can never put us into contact with reality; we can never penetrate beyond the impressions that reality implants in our minds.” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

"[…] physics tries to discover the pattern of events which controls the phenomena we observe. But we can never know what this pattern means or how it originates; and even if some superior intelligence were to tell us, we should find the explanation unintelligible. Our studies can never put us into contact with reality, and its true meaning and nature must be for ever hidden from us." (James H Jeans, "Physics and Philosophy" 3rd Ed., 1943)

“Science usually advances by a succession of small steps, through a fog in which even the most keen-sighted explorer can seldom see more than a few paces ahead. Occasionally the fog lifts, an eminence is gained, and a wider stretch of territory can be surveyed - sometimes with startling results. A whole science may then seem to undergo a kaleidoscopic ‘rearrangement’, fragments of knowledge being found to fit together in a hitherto unsuspected manner. Sometimes the shock of readjustment may spread to other sciences; sometimes it may divert the whole current of human thought.” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

“The pictures we draw of nature show similar limitations; these are the price we pay for limiting our pictures of nature to the kinds that can be understood by our minds. As we cannot draw one perfect picture, we make two imperfect pictures and turn to one or the other according as we want one property or another to be accurately delineated. Our observations tell us which is the right picture to use for each particular purpose […] . Yet some properties of nature are so far-reaching and general that neither picture can depict them properly of itself. In such cases we must appeal to both pictures, and these sometimes give us different and inconsistent information. Where, then, shall we find the truth?” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

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

“When two hypotheses are possible, we provisionally choose that which our minds adjudge to the simpler on the supposition that this Is the more likely to lead in the direction of the truth.” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

 “Whenever a man increases the content of his mind he gains new knowledge, and this occurs each time a new relation is established between the worlds on the two sides of the sense-organs - the world of ideas in an individual mind, and the world of objects existing outside individual minds which is common to us all.” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

 “Yet a review of receipt physics has shown that all attempts at mechanical models or pictures have failed and must fail. For a mechanical model or picture must represent things as happening in space and time, while it has recently become clear that the ultimate processes of nature neither occur in, nor admit of representation in, space and time. Thus an understanding of the ultimate processes of nature is for ever beyond our reach: we shall never be able - even in imagination - to open the case of our watch and see how the wheels go round. The true object of scientific study can never be the realities of nature, but only our own observations on nature.” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

"All the pictures which science draws of Nature, and which alone seem capable of according with observational facts, are mathematical pictures." (Sir James Jeans)

Yuri I Manin - Collected Quotes

"A proof only becomes a proof after the social act of ‘accepting it as a proof’." (Yuri I Manin, "A Course in Mathematical Logic", 1977)

"Most likely, logic is capable of justifying mathematics to no greater extent than biology is capable of justifying life." (Yuri Manin, "A Course in Mathematical Logic", 1977)

"A real change of theory is not a change of equations - it is a change of mathematical structure, and only fragments of competing theories, often not very important ones conceptually, admit comparison with each other within a limited range of phenomena." (Yuri I Manin, "Mathematics and Physics", 1981)

"After all of this it is a miracle that our models describe anything at all successfully. In fact, they describe many things well: we observe what they have predicted, and we understand what we observe. However, this last act of observation and understanding always eludes physical description." (Yuri I Manin, "Mathematics and Physics", 1981)

"In the limit of idealization, all of mathematics can be regarded as the collection of grammatically correct potential texts in a formal language." (Yuri I Manin, "Mathematics and Physics", 1981)

"'Infinity' is not a phenomenon - it is only a word  which enables us somehow to learn truths about finite things." (Yuri I Manin, "Mathematics and Physics", 1981)

"Mathematics associates new mental images with […] physical abstractions; these images are almost tangible to the trained mind but are far removed from those that are given directly by life and physical experience." (Yuri I Manin, "Mathematics and Physics", 1981)

"Moreover, life - perhaps the most interesting physical phenomenon - is embroidered on a delicate quilt made up of an interplay of instabilities, where a few quanta of action can have great informational value, and neglect of the small terms in equations means death." (Yuri I Manin, "Mathematics and Physics", 1981)

"The ‘eyes of the mind’ must be able to see in the phase space of mechanics, in the space of elementary events of probability theory, in the curved four-dimensional space-time of general relativity, in the complex infinite dimensional projective space of quantum theory. To comprehend what is visible to the ‘actual eyes’, we must understand that it is only the projection of an infinite dimensional world on the retina." (Yuri I Manin, "Mathematics and Physics", 1981)

"The computational formalism of mathematics is a thought process that is externalised to such a degree that for a time it becomes alien and is turned into a technological process. A mathematical concept is formed when this thought process, temporarily removed from its human vessel, is transplanted back into a human mold. To think […] means to calculate with critical awareness." (Yuri I. Manin, "Mathematics and Physics", 1981)

"The ‘mad idea’ which will lie at the basis of a future fundamental physical theory will come from a realization that physical meaning has some mathematical form not previously associated with reality. From this point of view the problem of the ‘mad idea’ is the problem of choosing, not of generating, the right idea." (Yuri I. Manin, "Mathematics and Physics", 1981) 

"The principal aim of physical theories is understanding. A theory's ability to find a number is merely a useful criterion for a correct understanding." (Yuri I Manin, "Mathematics and Physics", 1981)

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

"Symmetries of a geometric object are traditionally described by its automorphism group, which often is an object of the same geometric class (a topological space, an algebraic variety, etc.). Of course, such symmetries are only a particular type of morphisms, so that Klein’s Erlanger program is, in principle, subsumed by the general categorical approach." (Yuri I Manin, "Topics in Noncommutative Geometry", 1991)

"In order to understand how mathematics is applied to understanding of the real world it is convenient to subdivide it into the following three modes of functioning: model, theory, metaphor. A mathematical model describes a certain range of phenomena qualitatively or quantitatively. […] A (mathematical) metaphor, when it aspires to be a cognitive tool, postulates that some complex range of phenomena might be compared to a mathematical construction." (Yuri I Manin," Mathematics as Metaphor: Selected Essays of Yuri I. Manin" , 2007)

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

William Byers - Collected Quotes

"Logic moves in one direction, the direction of clarity, coherence, and structure. Ambiguity moves in the other direction, that of fluidity, openness, and release. Mathematics moves back and forth between these two poles. Mathematics is not a fixed, static entity that can be structured definitively. It is dynamic, alive: its dynamism a function of the relationship between the two poles that have been described above. It is the interactions between these different aspects that give mathematics its power." (William Byers, "How Mathematicians Think", 2007)

"Mathematics, far from being stymied by this situation, finds enormous value in it. The fecundity of ‘randomness’ is astounding; it is an inexhaustible source of scientific riches. Could ‘randomness’ be such a rich notion because of the inner contradiction that it contains, not despite it? The depth we sense in ‘randomness’ comes from something that lies behind any specific mathematical definition." (William Byers, "How Mathematicians Think", 2007)

"Mathematics is about truth: discovering the truth, knowing the truth, and communicating the truth to others. It would be a great mistake to discuss mathematics without talking about its relation to the truth, for truth is the essence of mathematics. In its search for the purity of truth, mathematics has developed its own language and methodologies - its own way of paring down reality to an inner essence and capturing that essence in subtle patterns of thought. Mathematics is a way of using the mind with the goal of knowing the truth, that is, of obtaining certainty." (William Byers, "How Mathematicians Think", 2007)

"Mathematics provides a good part of the cultural context for the worlds of science and technology. Much of that context lies not only in the explicit mathematics that is used, but also in the assumptions and worldview that mathematics brings along with it." (William Byers, "How Mathematicians Think", 2007)

"The concept of zero is so familiar that it takes a great deal of effort to recapture how mysterious, subtle, and contradictory the idea really is." (William Byers, "How Mathematicians Think", 2007)

"The immediate evidence from the natural world may seem to be chaotic and without any inner regularity, but mathematics reveals that under the surface the world of nature has an unexpected simplicity - an extraordinary beauty and order." (William Byers, "How Mathematicians Think", 2007)

"The infinite more than anything else is what characterizes mathematics and defines its essence. […] To grapple with infinity is one of the bravest and extraordinary endeavors that human beings have ever undertaken." (William Byers, "How Mathematicians Think", 2007)

"A conceptual system is an integrated system of concepts that supports a coherent vision of some aspect of the world. A conceptual system is personal; it is a 'way of seeing', that is, a 'way of knowing'. [...] You cannot do mathematics or science without a conceptual system but such systems are not objective and permanent. They are subject to change and development. Therefore we cannot claim that the reality that we experience and work with in science is independent of the mind of the scientist." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"A conceptual system is inevitably associated with a particular way of thinking. Mathematics and science involve different modes of thinking of which deep thinking is the most difficult, radical, and important." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"Abstraction is indeed an essential element of concept formation and mathematics is the discipline that has investigated the process of abstraction in greater depth than anywhere else. [...] Mathematics is simultaneously the most abstract and the most concrete of disciplines." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"All cultures organize themselves around a story, which tells them how the world came into being - a creation myth." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"All scientific theories are incomplete and approximate. They may function well at the centre of the domain that they describe but tend to break down at the boundaries. Paradigms are usually deep insights into some aspect of reality but they never capture reality definitively." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"An act of creativity is the result of an insight that arises discontinuously. Of course the insight must be preceded by something that is deeply problematic; it is so deeply problematic that a resolution may well seem impossible. This is why the resolution does not arise through systematic means but only occurs when all systematic approaches have been exhausted to no effect, that is, if you want to be creative you must sometimes be prepared to fly blind. This is not easy to do. Creativity involves living for protracted periods with the kind of tension that arises in situations of cognitive dissonance." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"Any simulation, no matter how brilliant in conception, is qualitatively different from what it simulates. Human intelligence and creativity are primary phenomena that are real and immediate, whereas simulations are not real in the same way - they are secondary phenomena. Simulations arise from applying deep thinking to a real situation that exists outside of and prior to the model." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"Change is real; the unchanging is an illusion. Classical science focused on systems that were in equilibrium whereas modern science also looks at states that may be far from equilibrium. Equilibrium situations can give you the feeling that things are unchanging but this is always only a temporary condition. Equilibriums beak down and when they do, when the system is far from equilibrium, then the dynamism of the system is most visible. In such situations one is studying change directly." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"Creativity does not merely involve the production of novelty. It does not arise from applying the same procedure to a series of different situations. Creativity involves coming to see some situation or phenomenon in a substantially different way - it is not the phenomenon that has changed but rather the manner in which one views the phenomenon." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"Facts and concepts only acquire real meaning and significance when viewed through the lens of a conceptual system. [...] Facts do not exist independently of knowledge and understanding for without some conceptual basis one would not know what data to even consider. The very act of choosing implies some knowledge. One could say that data, knowledge, and understanding are different ways of describing the same situation depending on the type of human involvement implied - 'data' means a de-emphasis on the human dimension whereas 'understanding' highlights it." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"For every conceptual system there comes a time when the system is overwhelmed by a critical mass of new phenomena and problematic elements that do not fit within the old paradigm." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"For knowledge to be useful it must be understood. Whereas knowledge is primarily social, understanding is individual - it is tied to a particular individual." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"Functions are ambiguous creatures - they come with multiple representations. [...] These representations break down into two categories: those that see the function as a static object - a graph or list - versus those that see it as a process - the calculator button, the input-output machine." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"Learning is a dynamic event and so the belief that learning is primarily about the acquisition of facts is fundamentally flawed - the acquisition and manipulation of data is at best a prerequisite to learning. Real learning involves acquiring knowledge and understanding." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"Machines live in a universe of data; but the existence of knowledge necessitates a human presence." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"Mathematics courses are hierarchical but every new course begins with the assumption that the student is at the level of conceptual development that would be implied by an optimal understanding of the previous course. Unfortunately many mathematical ideas are so subtle and logically complex that it may take students many years to develop an adequate conceptual understanding. As a result, in practice there is a lot of 'faking it' going on and not merely on the part of the students." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"Paradigm change necessarily involves a discontinuous jump. Reality is singular and each paradigm evokes its own reality. This is the reason that scientific paradigms are not changed without a great deal of conflict; the reason why deep thinking is so difficult and involves overcoming so much resistance both in the individual and in the larger culture. In fact it has been said that a scientist never really gives up the paradigm within which she has been trained." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"Reality is necessarily viewed through a conceptual system and is inseparable from the system through which it is viewed. But reality is by definition singular - there is only one reality; there cannot be two or three. Something is either real or it is not. The notion that reality is relative or that there can be two competing and inconsistent realities is disorienting and produces untenable cognitive dissonance." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"Systems always contain problematic elements in a sense that is usually not clear until the implications of the new system are sufficiently explored. Often problems can be stated in the language of the initial system but can only be resolved by creating a new system. [...] Problems that can be stated in the language of one system often cannot be solved within that system because the solution depends upon the development of a new system." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"Teaching mathematics does not involve writing down a set of axioms and deducing their logical consequences. It involves introducing students to a new system of concepts. These concepts are often extremely subtle and deep." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"Technological change is discontinuous and difficult. It is a radical change in that it forces people to deal with the world in a different way, that is, it changes the world of experience."

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

"The introduction of a new concept in mathematics conventionally begins with a definition, which is followed by examples, and then the logical consequences of the definition—properties of the defined object and connections with other mathematical objects or processes. In this sequence the mathematical object [...] is carefully and precisely defined because we regard the concept as identical with its definition." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"The moment of insight is the moment in which one grasps the concept or makes the creative leap from one conceptual system to another." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015) 

"The problem of teaching is the problem of introducing concepts and conceptual systems. In this crucial task the procedures of formal mathematical argument are of little value. The way we reason in formal mathematics is itself a conceptual system - deductive logic - but it is a huge mistake to identify this with mathematics. [...] Mathematics lives in its concepts and conceptual systems, which need to be explicitly addressed in the teaching of mathematics." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"The problem with artificial intelligence and information technology is that they promise a methodology that would lead to a way of solving all problems - a self-generating technology that would apply to all situations without the need for new human insights and leaps of creativity." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"The true foundations of mathematics do not lie in axioms, definitions, and logical inference, which are the foundational elements of formal mathematics. The true foundations of mathematics lie in the minds of mathematicians as they interact with and try to make sense of their world - in their ideas, their intuitions, and their aesthetic sensibility." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"The words 'imaginary' and 'complex' again demonstrate how difficult it is to make a major change in conceptual systems - a difficulty that we already encountered with negative numbers, fractions, zero, and irrational numbers. The word 'imaginary' tells us that these numbers are unreal from the perspective of someone grounded in the real number system." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"What is this reality? It is our internal concepts, our mental models, which are real objects with real properties, (italics in original) and which are congruent to each other, which fit together and match." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)