Mental Models XXIX

"If the second principle [the context principle] is not observed, one is almost forced to take as the meanings of words mental pictures or acts of the individual mind, and so to offend against the first principle as well." (Gottlob Frege, "The Foundations of Arithmetic" , 1884)

"The unimaginability of the content of a word is no reason, then, to deny it any meaning or to exclude it from usage. That we are nevertheless inclined to do so is probably owing to the fact that we consider words individually and ask about their meaning [in isolation], for which we then adopt a mental picture. Thus a word for which we are lacking a corresponding inner picture will seem to have no content. However, we must always consider a complete sentence. Only in [the context of] the latter do the words really have a meaning. The inner pictures which somehow sway before us (in reading the sentence) need not correspond to the logical components of the judgment. It is enough if the sentence as a whole has a sense; by means of this its parts also receive their content." (Gottlob Frege, "The Foundations of Arithmetic" , 1884) 

"But surely it is self-evident that every theory is merely a framework or scheme of concepts together with their necessary relations to one another, and that the basic elements can be constructed as one pleases." (Gottlob Frege, "On the Foundations of Geometry and Formal Theories of Arithmetic" , cca. 1903-1909)

"The mind of man, learning consciously and unconsciously lessons of experience, gradually constructs a mental image of its surroundings - as the mariner draws a chart of strange coasts to guide him in future voyages, and to enable those that follow after him to sail the same seas with ease and safety." (William C Dampier, "The Recent Development of Physical Science" , 1904) 

"It seems that the human mind has first to construct forms independently, before we can find them in things. Kepler’s marvelous achievement is a particularly fine example of the truth that knowledge cannot spring from experience alone, but only from the comparison of the inventions of the intellect with observed fact." (Albert Einstein, 1930)

“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)

 "The evolutionary vision is agnostic in regard to systems in the universe of greater complexity than those of which human beings have clear knowledge. It recognizes aesthetic, moral, and religious ideas and experiences as a species, in this case of mental structures or of images, which clearly interacts with other species in the world's great' ecosystem." (Kenneth Boulding," Ecodynamics: A New Theory of Societal Evolution", 1978) 

"Perhaps we all lose our sense of reality to the precise degree to which we are engrossed in our own work, and perhaps that is why we see in the increasing complexity of our mental constructs a means for greater understanding, even while intuitively we know that we shall never be able to fathom the imponderables that govern our course through life." (Winfried G Sebald, "The Rings of Saturn", 1998)

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

"It's pretty much a fact that our entire universe is a mental construct. We don't actually deal with reality directly. We simply compose a picture of reality from what's going on in our retinas, in the timpani of our ears, and in our nerve endings." (Alan Moore, The Believer No. 99, [interview] 2013)

"Society exists only as a mental concept; in the real world there are only individuals." (Charley Reese) [attributed also to Oscar Wilde, improbable though]

John D Barrow - Collected Quotes

"A more interesting problem is the extent to which the brain is qualitatively adapted to understand the Universe. Why should its categories of thought and understanding be able to cope with the scope and nature of the real world? Why should be Theory of Everything be written in a 'language' that our minds can decode? Why has the process of natural selection so over-endowed us with mental faculties that we can understand the whole fabric of the Universe far beyond anything required for our past and present survival?"  (John D Barrow, "New Theories of Everything: The Quest for Ultimate Explanation", 1991)

"Each of the most basic physical laws that we know corresponds to some invariance, which in turn is equivalent to a collection of changes which form a symmetry group. [...] whilst leaving some underlying theme unchanged. [...] for example, the conservation of energy is equivalent to the invariance of the laws of motion with respect to translations backwards or forwards in time [...] the conservation of linear momentum is equivalent to the invariance of the laws of motion with respect to the position of your laboratory in space, and the conservation of angular momentum to an invariance with respect to directional orientation [...] discovery of conservation laws indicated that Nature possessed built-in sustaining principles which prevented the world from just ceasing to be. There were fewer roles for the Deity to play [...]"  (John D Barrow, "New Theories of Everything: The Quest for Ultimate Explanation", 1991)

"Everywhere […] in the Universe, we discern that closed physical systems evolve in the same sense from ordered states towards a state of complete disorder called thermal equilibrium. This cannot be a consequence of known laws of change, since […] these laws are time symmetric- they permit […] time-reverse. […] The initial conditions play a decisive role in endowing the world with its sense of temporal direction. […] some prescription for initial conditions is crucial if we are to understand […]"  (John D Barrow, "New Theories of Everything: The Quest for Ultimate Explanation", 1991)

"In practice, the intelligibility of the world amounts to the fact that we find it to be algorithmically compressible. We can replace sequences of facts and observational data by abbreviated statements which contain the same information content. These abbreviations we often call 'laws of Nature.' If the world were not algorithmically compressible, then there would exist no simple laws of nature. Instead of using the law of gravitation to compute the orbits of the planets at whatever time in history we want to know them, we would have to keep precise records of the positions of the planets at all past times; yet this would still not help us one iota in predicting where they would be at any time in the future. This world is potentially and actually intelligible because at some level it is extensively algorithmically compressible. At root, this is why mathematics can work as a description of the physical world. It is the most expedient language that we have found in which to express those algorithmic compressions." (John D Barrow, "New Theories of Everything", 1991)

"It is enigma enough that the world is described by mathematics; but by simple mathematics, of the sort that a few years energetic study now produces familiarity with, this is an enigma within an enigma."   (John D Barrow, "New Theories of Everything: The Quest for Ultimate Explanation", 1991)

"It is simplest to think of mathematics as the catalogue of all possible patterns. [...] When viewed in this way, it is inevitable that the world is described by mathematics. [...] In many ways the search for a Theory of Everything is a manifestation of a faith that this compression goes all the way down to the bedrock of reality [...]"  (John D Barrow, "New Theories of Everything: The Quest for Ultimate Explanation", 1991)

"On this view, we recognize science to be the search for algorithmic compressions. We list sequences of observed data. We try to formulate algorithms that compactly represent the information content of those sequences. Then we test the correctness of our hypothetical abbreviations by using them to predict the next terms in the string. These predictions can then be compared with the future direction of the data sequence. Without the development of algorithmic compressions of data all science would be replaced by mindless stamp collecting - the indiscriminate accumulation of every available fact. Science is predicated upon the belief that the Universe is algorithmically compressible and the modern search for a Theory of Everything is the ultimate expression of that belief, a belief that there is an abbreviated representation of the logic behind the Universe's properties that can be written down in finite form by human beings."  (John D Barrow, "New Theories of Everything: The Quest for Ultimate Explanation", 1991)

"Somehow the breathless world that we witness seems far removed from the timeless laws of Nature which govern the elementary particles and forces of Nature. The reason is clear. We do not observe the laws of Nature: we observe their outcomes. Since these laws find their most efficient representation as mathematical equations, we might say that we see only the solutions of those equations not the equations themselves. This is the secret which reconciles the complexity observed in Nature with the advertised simplicity of her laws."  (John D Barrow, "New Theories of Everything: The Quest for Ultimate Explanation", 1991)

"String theory promises to take a further step beyond that taken by Einstein's picture of force subsumed within curved space and time geometry. Indeed, string theory contains Einstein's theory of gravitation within itself. Loops of string behave like the exchange particles of the gravitational forces, or 'gravitons' as they are called in the point-particle picture of things. But it has been argued that it must be possible to extract even the geometry of space and time from the characteristics of the strings and their topological properties. At present, it is not known how to do this and we merely content ourselves with understanding how strings behave when they sit in a background universe of space and time."  (John D Barrow, "New Theories of Everything: The Quest for Ultimate Explanation", 1991)

"The inflationary period of expansion does not smooth out irregularity by entropy-producing processes like those explored by the cosmologies of the seventies. Rather it sweeps the irregularity out beyond the Horizon of our visible Universe, where we cannot see it . The entire universe of stars and galaxies on view to us. […] on this hypothesis, is but the reflection of a minute, perhaps infinitesimal, portion of the universe's initial conditions, whose ultimate extent and structure must remain forever unknowable to us. A theory of everything does not help here. The information contained in the observable part of the universe derives from the evolution of a tiny part of the initial conditions for the entire universe. The sum total of all the observations we could possibly make can only tell us about a minuscule portion of the whole."  (John D Barrow, "New Theories of Everything: The Quest for Ultimate Explanation", 1991)

"The scope of Theories of Everything is infinite but bounded; they are necessary parts of a full understanding of things but they are far from sufficient to reveal everything about a Universe like ours. In the pages of this book, we have seen something of what a Theory of Everything might hope to teach us about the unity of the Universe and the way in which it may contain elements that transcend our present compartmentalized view of Nature's ingredients. But we have also learnt that there is more to Everything than meets the eye. Unlike many others that we can imagine, our world contains prospective elements. Theories of Everything can make no impression upon predicting these prospective attributes of reality; yet, strangely, many of these qualities will themselves be employed in the human selection and approval of an aesthetically acceptable Theory of Everything. There is no formula that can deliver all truth, all harmony, all simplicity. No Theory of Everything can ever provide total insight. For, to see through everything, would leave us seeing nothing at all."  (John D Barrow, "New Theories of Everything: The Quest for Ultimate Explanation", 1991)

"There is one qualitative aspect of reality that sticks out from all others in both profundity and mystery. It is the consistent success of mathematics as a description of the workings of reality and the ability of the human mind to discover and invent mathematical truths."  (John D Barrow, "New Theories of Everything: The Quest for Ultimate Explanation", 1991)

"Three laws governing black hole changes were thus found, but it was soon noticed that something unusual was going on. If one merely replaced the words 'surface area' by 'entropy' and 'gravitational field' by 'temperature', then the laws of black hole changes became merely statements of the laws of thermodynamics. The rule that the horizon surface areas can never decrease in physical processes becomes the second law of thermodynamics that the entropy can never decrease; the constancy of the gravitational field around the horizon is the so-called zeroth law of thermodynamics that the temperature must be the same everywhere in a state of thermal equilibrium. The rule linking allowed changes in the defining quantities of the black hole just becomes the first law of thermodynamics, which is more commonly known as the conservation of energy."  (John D Barrow, "New Theories of Everything: The Quest for Ultimate Explanation", 1991)

"That is, the physicist likes to learn from particular illustrations of a general abstract concept. The mathematician, on the other hand, often eschews the particular in pursuit of the most abstract and general formulation possible. Although the mathematician may think from, or through, particular concrete examples in coming to appreciate the likely truth of very general statements, he will hide all those intuitive steps when he comes to present the conclusions of his thinking to outsiders. It presents the results of research as a hierarchy of definitions, theorems and proofs after the manner of Euclid; this minimizes unnecessary words but very effectively disguises the natural train of thought that led to the original results."  (John D Barrow, "New Theories of Everything: The Quest for Ultimate Explanation", 1991)

"Topology is that branch of mathematics which is interested in the forms of things aside from their size and shape. Two things are said to be topologically equivalent if one can be deformed smoothly into the other without sticking, cutting, or puncturing it in any way. Thus an egg is equivalent to a sphere."  (John D Barrow, "New Theories of Everything: The Quest for Ultimate Explanation", 1991)

"A less inflexible picture of mathematics is one that focuses on the fact that it is an open-ended human activity. Inventionism is the belief that mathematics is nothing more than what mathematicians do. …We invent mathematics; we do not discover it. [...] The independent discovery of the same mathematical theorems by different mathematicians from totally different economic, cultural, and political backgrounds - often at widely separated times in history - argues against such a simple view. The inventionist could respond by pointing to the universality of human languages. [...]One might expect that those aspects of this universal grammar that share features of logic, and hence counting, would also make counting appear instinctive. In fact, although simple counting [...] is fairly universal in ancient and primitive cultures, virtually none of them went on to carry out mathematical operations more sophisticated than counting. This suggests that these higher mathematical operations are not genetically programmed into the human brain [...] They are more likely to be by-products of multi-purpose pattern-recognition capabilities." (John D Barrow, "The Artful Universe", 1995)

"Highly correlated brown and black noise patterns do not seem to have seem to have attractive counterparts in the visual arts. There, over-correlation is the order of the day, because it creates the same dramatic associations that we find in attractive natural scenery, or in the juxtaposition of symbols. Somehow, it is tediously predictable when cast in a one-dimensional medium, like sound." (John D Barrow, "The Artful Universe", 1995)

"The laws of Nature are based upon the existence of a pattern, linking one state of affairs to another; and where there is pattern, there is symmetry. Yet. [...] the symmetries that the laws enshrine are broken in [...] outcomes. Suppose that we balance a needle on its point and then release it. The law of gravity, which governs its subsequent motion, is perfectly democratic. It has no preference for any particular direction in the Universe: it is symmetrical in this respect. Yet, when the needle falls, it must fall in a particular direction. The directional symmetry of the underlying law is broken, therefore [...] By the same token, the fallen needle hides the symmetry of the law. [...] Such 'symmetry-breaking' governs much of what we see in the Universe... It allows a Universe governed by a small number of symmetrical laws to manifest an infinite diversity of complex, asymmetrical states. This is how the Universe can be at once, simple and complicated." (John D Barrow, "The Artful Universe", 1995)

"Where there is life there is a pattern, and where there is a pattern there is mathematics. Once that germ of rationality and order exists to turn a chaos into a cosmos, then so does mathematics. There could not be a non-mathematical Universe containing living observers."  (John D Barrow, "The Artful Universe", 1995)

"The physicist's concept of nothing - the vacuum [...] began as empty space - the void… turned into a stagnant ether through which all the motions of the Universe swam, vanished in Einstein's hands, then re-emerged in the twentieth-century quantum picture of how Nature works." (John D Barrow, "The Book of Nothing", 2000)

"Images and pictures [...] have played a key role in shaping our scientific picture of the world. [...] Carefully constructed families of pictures can act as a calculus all their own. Like any successful systems of symbols, with an appropriate grammar they enlarge the number of things that we can do without consciously thinking." (John D Barrow, "Cosmic Imagery: Key Images in the History of Science" 2008)

"Scientific pictures are often not just about science. They may [...] have an undeniable aesthetic quality. They may even have been primarily works of art that possess a scientific message." (John D Barrow, "Cosmic Imagery: Key Images in the History of Science" 2008)

"The advent of small, inexpensive computers with superb graphics has changed the way many sciences are practiced, and the way that all sciences present the results of experiments and calculations." (John D Barrow, "Cosmic Imagery: Key Images in the History of Science", 2008)

"If one looks at the special problems that were the mainsprings of progress along the oldest and most persistent lines of human inquiry, then one finds Nothing, suitably disguised as something, never far from the centre of things." ( John D Barrow, "The Book of Nothing", 2009)

"While we have no reason to expect that our position in the universe is special in every way, we would be equally misled were we to assume that it could not be special in any way." (John D Barrow, "The Book of Universes: Exploring the Limits of the Cosmos", 2011)

Arthur C. Clarke - Collected Quotes

"The urge to explore, to discover, to ‘follow knowledge like a sinking star’, is a primary human impulse which needs and can receive no further justification than its own existence." (Arthur C. Clarke, "The Challenge of the Spaceship", 1959)

"To obtain a mental picture of the distance to the nearest star, compared to the nearest planet, you must imagine a world in which the closest object to you is only five feet away - and there is nothing else to see until you have travelled a thousand miles." (Arthur C Clarke, "We'll Never Conquer Space", 1960)

"Any sufficiently advanced technology is indistinguishable from magic." (Arthur C Clarke, "Profiles of the Future", 1962)

"Anything that is theoretically possible will be achieved in practice, no matter what the technical difficulties are, if it is desired greatly enough." (Arthur C Clarke, "Profiles of the Future", 1962)

"It is really quite amazing by what margins competent but conservative scientists and engineers can miss the mark, when they start with the preconceived idea that what they are investigating is impossible." (Arthur C Clarke, "Profiles of the Future", 1962)

"No equation, however impressive and complex, can arrive at the truth if the initial assumptions are incorrect." (Arthur C Clarke, "Profiles of the Future", 1962)

"The only way of discovering the limits of the possible is to venture a little way past them into the impossible." (Arthur C Clarke, "Profiles of the Future", 1962)

"When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong." (Arthur C Clarke, "Profiles of the Future", 1962)

"Once you can reproduce a phenomenon, you are well on the way to understanding it." (Arthur C. Clarke, Voices from the Sky", 1965)

"Some facts are so incredible that they are believed at once, for no one could possibly have imagined them." (Arthur C. Clarke, "The Lost Worlds of 2001", 1972)

"There was no substitute for reality; one should be aware of imitations." (Arthur C Clarke, "The Fountains of Paradise", 1979)

"What is becoming more interesting than the myths themselves has been the study of how the myths were constructed from sparse or unpromising facts - indeed, sometimes from no facts - in a kind of mute conspiracy of longing, very rarely under anybody's conscious control." (Arthur C Clarke, "The Light of Other Days", 2000)

John M Keynes - Collected Quotes

"Transcending the flux of the sensuous universe, there exists a stable world of pure thought, a divinely ordered world of ideas, accessible to man, free from the mad dance of time, infinite and eternal." (John M Keynes, "The Human Worth of Rigorous Thinking", 1916)

"It has been pointed out already that no knowledge of probabilities, less in degree than certainty, helps us to know what conclusions are true, and that there is no direct relation between the truth of a proposition and its probability. Probability begins and ends with probability." (John M Keynes, "A Treatise on Probability", 1921)

"It is difficult to find an intelligible account of the meaning of ‘probability’, or of how we are ever to determine the probability of any particular proposition; and yet treatises on the subject profess to arrive at complicated results of the greatest precision and the most profound practical importance." (John M Keynes, "A Treatise on Probability", 1921)

"Part of our knowledge we obtain direct; and part by argument. The Theory of Probability is concerned with that part which we obtain by argument, and it treats of the different degrees in which the results so obtained are conclusive or inconclusive." (John M Keynes, "A Treatise on Probability", 1921)

"Probability is, so far as measurement is concerned, closely analogous to similarity." (John M Keynes, "A Treatise on Probability", 1921)

"We know that the probability of a well-established induction is great, but, when we are asked to name its degree, we cannot. Common sense tells us that some inductive arguments are stronger than others, and that some are very strong. But how much stronger or how strong we cannot express." (John M Keynes, "A Treatise on Probability", 1921)

"We must have a logical intuition of the probable relations between propositions. Once the existence of this relation between evidence and conclusion, the latter becomes the subject of the degree of belief." (John M Keynes, "Treatise on Probability", 1921)

"Economics is a science of thinking in terms of models joined to the art of choosing models which are relevant to the contemporary world. It is compelled to be this, because, unlike the typical natural science, the material to which it is applied is, in too many respects, not homogeneous through time. The object of a model is to segregate the semi-permanent or relatively constant factors from those which are transitory or fluctuating so as to develop a logical way of thinking about the latter, and of understanding the time sequences to which they give rise in particular cases." (John M Keynes, [letter to Roy Harrod] 1938)

"Good economists are scarce because the gift for using "vigilant observation" to choose good models, although it does not require a highly specialised intellectual technique, appears to be a very rare one."  (John M Keynes, [letter to Roy Harrod] 1938)

"Education is the inculcation of the incomprehensible into the indifferent by the incompetent." (John M Keynes)

"The difficulty lies, not in the new ideas, but in escaping the old ones, which ramify, for those brought up as most of us have been, into every corner of our minds." (John M Keynes)

Northrop Frye - Collected Quotes

"Imagination creates reality, and as desire is a part of imagination, the world we desire is more real than the world we passively accept." (Northrop Frye, "Fearful Symmetry", 1947)

"Just as a new scientific discovery manifests sometimes that was already latent in the order of nature, and at the same time is logically related to the total structure of the existing science, so the new poem manifests something that was already latent in the order of words." (Northrop Frye, "Anatomy of Criticism: Four Essays", 1957)

"Physics is an organized body of knowledge about nature, and a student of it says that he is learning physics, not nature. Art, like nature, has to be distinguished from the systematic study of it, which is criticism." (Northrop Frye, "Anatomy of Criticism: Four Essays", 1957)

"Science begins with the world we have to live in, accepting its data and trying to explain its laws. From there, it moves toward the imagination: it becomes a mental construct, a model of a possible way of interpreting experience. The further it goes in this direction, the more it tends to speak the language of mathematics, which is really one of the languages of the imagination, along with literature and music." (Northrop Frye, "The Educated Imagination", 1963)

"Art is not simply an identity of illusion and reality, but a counter-illusion: its world is a material world, but the material of an intelligible spiritual world."  (Northrop Frye)

"In imaginative thought there is no real knowledge of anything but similarities (ultimately identities): knowledge of differences is merely a transition to a new knowledge of similarities."  (Northrop Frye)

"We are always in the place of beginning; there is no advance in infinity."  (Northrop Frye)

Gregory Chaitin - Collected Quotes

"The message is that mathematics is quasi-empirical, that mathematics is not the same as physics, not an empirical science, but I think it's more akin to an empirical science than mathematicians would like to admit." (Gregory Chaitin, [interview] 2000)

"And although mathematical ideas and thought are constantly evolving, you will also see that the most basic fundamental problems never go away. Many of these problems go back to the ancient Greeks, and maybe even to ancient Sumer, although we may never know for sure. The fundamental philosophical questions like the continuous versus the discrete or the limits of knowledge are never definitively solved. Each generation formulates its own answer, strong personalities briefly impose their views, but the feeling of satisfaction is always temporary, and then the process continues, it continues forever." (Gregory Chaitin, "Meta Math: The Quest for Omega", 2005)

"And the map of our mathematical knowledge resembles a highway running through the desert or a dangerous jungle; if you stray off the road, you’ll be hopelessly lost and die! In other words, the current map of mathematics reflects what our tools are currently able to handle, not what is really out there. Mathematicians don’t like to talk about what they don’t know, they like to talk about the questions that current technique, current mathematical technology, is capable of handling."  (Gregory Chaitin, "Meta Math: The Quest for Omega", 2005)

"Mathematical facts are not isolated, they are woven into a vast spider's web of interconnections." (Gregory Chaitin, "Meta Math: The Quest for Omega", 2005)

"Mathematical truth is not totally objective. If a mathematical statement is false, there will be no proofs, but if it is true, there will be an endless variety of proofs, not just one! Proofs are not impersonal, they express the personality of their creator/discoverer just as much as literary efforts do. If something important is true, there will be many reasons that it is true, many proofs of that fact. [...] each proof will emphasize different aspects of the problem, each proof will lead in a different direction. Each one will have different corollaries, different generalizations. [...] the world of mathematical truth has infinite complexity […]" (Gregory Chaitin, "Meta Math: The Quest for Omega", 2005) 

"[...] the view that math provides absolute certainty and is static and perfect while physics is tentative and constantly evolving is a false dichotomy. Math is actually not that different from physics. Both are attempts of the human mind to organize, to make sense, of human experience; in the case of physics, experience in the laboratory, in the physical world, and in the case of math, experience in the computer, in the mental mindscape of pure mathematics.
And mathematics is far from static and perfect; it is constantly evolving, constantly changing, constantly morphing itself into new forms. New concepts are constantly transforming math and creating new fields, new viewpoints, new emphasis, and new questions to answer. And mathematicians do in fact utilize unproved new principles suggested by computational experience, just as a physicist would." (Gregory Chaitin, "Meta Math: The Quest for Omega", 2005)

"To survive, mathematical ideas must be beautiful, they must be seductive, and they must be illuminating, they must help us to understand, they must inspire us. […] Part of that beauty, an essential part, is the clarity and sharpness that the mathematical way of thinking about things promotes and achieves. Yes, there are also mystic and poetic ways of relating to the world, and to create a new math theory, or to discover new mathematics, you have to feel comfortable with vague, unformed, embryonic ideas, even as you try to sharpen them."  (Gregory Chaitin, "Meta Math: The Quest for Omega", 2005)

"Mathematics is not placid, static and eternal. […] Most mathematicians are happy to make use of those axioms in their proofs, although others do not, exploring instead so-called intuitionist logic or constructivist mathematics. Mathematics is not a single monolithic structure of absolute truth!" (Gregory J Chaitin, "A century of controversy over the foundations of mathematics", 2000)

"In a way, math isn't the art of answering mathematical questions, it is the art of asking the right questions, the questions that give you insight, the ones that lead you in interesting directions, the ones that connect with lots of other interesting questions - the ones with beautiful answers." (Gregory Chaitin)

Linus Carl Pauling - Collected Quotes

"The scientist, if he is to be more than a plodding gatherer of bits of information, needs to exercise an active imagination." (Linus Pauling, "Tomorrow", 1943)

"Science cannot be stopped. Men will gather knowledge no matter what the consequences. Science will go on whether we are pessimistic or optimistic, as I am. More interesting discoveries than we can imagine will be made, and I am awaiting them, full of curiosity and enthusiasm." (Linus Pauling, "Chemical Achievement and Hope for the Future", 1947)

"It is structure that we look for whenever we try to understand anything. All science is built upon this search; […] We like to understand, and to explain, observed facts in terms of structure."  (Linus Pauling, "Main Currents in Modern Thought", 1950)

"Science is the search for truth. It is not a game in which one tries to beat his opponent, to do harm to others. We need to have the spirit of science in international affairs, to make the conduct of international affairs the effort to find the right solution, the just solution of international problems, not the effort by each nation to get the better of other nations, to do harm to them when it is possible." (Linus Pauling, "No More War!", 1958)

"One never knows how hard a problem is until it has been solved. You don’t necessarily know that you will succeed if you work harder or longer." (Linus Pauling, [interview] 1986)

"A scientist can be productive in various ways. One is having the ability to plan and carry out experiments, but the other is having the ability to formulate new ideas, which can be about what experiments can be carried out […] by making [the] proper calculations. Individual scientists who are successful in their work are successful for different reasons." (Linus Pauling)

"Facts are the air of scientists. Without them you can never fly." (Linus Pauling)

"I have always wanted to know as much as possible about the world." (Linus Pauling)

"If you want to have good ideas you must have many ideas. Most of them will be wrong, and what you have to learn is which ones to throw away." (Linus Pauling)

"Science is the search for truth, that is the effort to understand the world: it involves the rejection of bias, of dogma, of revelation, but not the rejection of morality." (Linus Pauling)

"There is no area of the world that should not be investigated by scientists. There will always remain some questions that have not been answered. In general, these are the questions that have not yet been posed." (Linus Pauling)

William S Jevons - Collected Quotes

"Most statistical arguments depend upon a few figures picked out at random." (William S Jevons, [letter to Richard Hutton] 1863)

"They [diagrams] are designed not so much to allow of reference to particular numbers, which can be better had from printed tables of figures, as to exhibit to the eye the general results of large masses of figures which it is hopeless to attack in any other way than by graphical representation." (William S Jevons, [letter to Richard Hutton] 1863)

"Logic is not only an exact science, but is the most simple and elementary of all sciences; it ought therefore undoubtedly to find some place in every course of education." (William S Jevons, "Elementary Lessons on Logic", 1870)

"Logic should no longer be considered an elegant and learned accomplishment; it should take its place as an indispensable study for every well-informed person." (William S Jevons, "Elementary Lessons on Logic", 1870)

"A correct theory is the first step towards improvement, by showing what we need and what we might accomplish." (William S Jevons, "The Theory of Political Economy", 1871)

"As a science progresses, its power of foresight rapidly increases, until the mathematician in his library acquires the power of anticipating nature, and predicting what will happen in circumstances which the eye of man has never examined." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"By induction we gain no certain knowledge; but by observation, and the inverse use of deductive reasoning, we estimate the probability that an event which has occurred was preceded by conditions of specified character, or that such conditions will be followed by the event. [...] I have no objection to use the words cause and causation, provided they are never allowed to lead us to imagine that our knowledge of nature can attain to certainty. [...] We can never recur too often to the truth that our knowledge of the laws and future events of the external world are only probable." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"Deduction is certain and infallible, in the sense that each step in deductive reasoning will lead us to some result, as certain as the law itself. But it does not follow that deduction will lead the reasoner to every result of a law or combination of laws." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"Every strange phenomenon may be a secret spring which, if rightly touched, will open the door to new chambers in the palace of nature." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"Experience gives us the materials of knowledge: induction digests those materials, and yields us general knowledge." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"I am convinced that it is impossible to expound the methods of induction in a sound manner, without resting them on the theory of probability. Perfect knowledge alone can give certainty, and in nature perfect knowledge would be infinite knowledge, which is clearly beyond our capacities. We have, therefore, to content ourselves with partial knowledge, - knowledge mingled with ignorance, producing doubt." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"In abstract mathematical theorems the approximation to absolute truth is perfect, because we can treat of infinitesimals. In physical science, on the contrary, we treat of the least quantities which are perceptible." (William S Jevons, „The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"It is the prerogative of Intellect to discover what is uniform and unchanging in the phenomena around us." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"It must be the ground of all reasoning and inference that what is true of one thing will be true of its equivalent, and that under carefully ascertained conditions Nature repeats herself." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"It [probability] is the very guide of life, and hardly can we take a step or make a decision of any kind without correctly or incorrectly making an estimation of probabilities." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"It would be an error to suppose that the great discoverer seizes at once upon the truth, or has any unerring method of divining it. In all probability the errors of the great mind exceed in number those of the less vigorous one. Fertility of imagination and abundance of guesses at truth are among the first requisites of discovery; but the erroneous guesses must be many times as numerous as those that prove well founded. The weakest analogies, the most whimsical notions, the most apparently absurd theories, may pass through the teeming brain, and no record remain of more than the hundredth part. […] The truest theories involve suppositions which are inconceivable, and no limit can really be placed to the freedom of hypotheses." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"Just as, in the map of a half-explored country, we see detached bits of rivers, isolated mountains, and undefined plains, not connected into any complete plan, so a new branch of knowledge consists of groups of facts, each group standing apart, so as not to allow us to reason from one to another." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"Nature is a spectacle continually exhibited to our senses, in which phenomena are mingled in combinations of endless variety and novelty." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"Numerical facts, like other facts, are but the raw materials of knowledge, upon which our reasoning faculties must be exerted in order to draw forth the principles of nature. [...] Numerical precision is the soul of science [...]" (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"Our ultimate object in induction must be to obtain the complete relation between the conditions and the effect, but this relation will generally be so complex that we can only attack it in detail." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"Perfect readiness to reject a theory inconsistent with fact is a primary requisite of the philosophic mind. But it, would be a mistake to suppose that this candour has anything akin to fickleness; on the contrary, readiness to reject a false theory may be combined with a peculiar pertinacity and courage in maintaining an hypothesis as long as its falsity is not actually apparent." (William S Jevons, "The Principles of Science", 1874)

"Quantities which are called errors in one case, may really be most important and interesting phenomena in another investigation. When we speak of eliminating error we really mean disentangling the complicated phenomena of nature." (William S Jevons, "The Principles of Science", 1874)

"Science arises from the discovery of Identity amid Diversity." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"Simplicity is naturally agreeable to a mind of limited powers, but to an infinite mind all things are simple." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"Summing up, then, it would seem as if the mind of the great discoverer must combine contradictory attributes. He must be fertile in theories and hypotheses, and yet full of facts and precise results of experience. He must entertain the feeblest analogies, and the merest guesses at truth, and yet he must hold them as worthless till they are verified in experiment. When there are any grounds of probability he must hold tenaciously to an old opinion, and yet he must be prepared at any moment to relinquish it when a clearly contradictory fact is encountered." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"The man of one idea has but a single chance of truth." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"The truth or untruth of a natural law, when carefully investigated, resolves itself into a high or low degree of probability, and this is the case whether or not we are capable of producing precise numerical data." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"The whole value of science consists in the power which it confers upon us of applying to one object the knowledge acquired from like objects; and it is only so far, therefore, as we can discover and register resemblances that we can turn our observations to account." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

Wolfgang Pauli - Collected Quotes

"[…] inner images are rather psychic manifestations of the archetypes which, however, would also have to put forth, create, condition anything lawlike in the behavior of the corporeal world. The laws of this world would then be the physical manifestations of the archetypes. […] Each law of nature should then have an inner correspondence and vice versa, even though this is not always directly visible today." (Wolfgang Pauli, [letter to Markus Fierz] 1948)

"The layman always means, when he says 'reality' that he is speaking of something self-evidently known; whereas to me it seems the most important and exceedingly difficult task of our time is to work on the construction of a new idea of reality." (Wolfgang Pauli, [letter to Markus Fierz] 1948)

"When one analyzes the pre-conscious step to concepts, one always finds ideas which consist of 'symbolic images.' The first step to thinking is a painted vision of these inner pictures whose origin cannot be reduced only and firstly to the sensual perception but which are produced by an 'instinct to imagining' and which are re-produced by different individuals independently, i. e. collectively [...] But the archaic image is also the necessary predisposition and the source of a scientific attitude. To a total recognition belong also those images out of which have grown the rational concepts." (Wolfgang Pauli, [letter to Markus Fierz] 1948)

"The process of understanding in nature, together with the joy that man feels in understanding, i.e., in becoming acquainted with new knowledge, seems therefore to rest upon a correspondence, a coming into congruence of preexistent internal images of the human psyche with external objects and their behavior. […] the place of clear concepts is taken by images of strongly emotional content, which are not thought but are seen pictorially, as it were, before the minds eye." (Wolfgang Pauli, "Der Einfluss archetypischer Vorstellungen auf die Bildung  naturwissenschaftlicher Theorien bei Kepler", 1952)

"To us […] the only acceptable point of view appears to be the one that recognizes both sides of reality - the quantitative and the qualitative, the physical and the psychical - as compatible with each other, and can embrace them simultaneously […] It would be most satisfactory of all if physis and psyche (i.e., matter and mind) could be seen as complementary aspects of the same reality." (Wolfgang Pauli', "The Influence of Archetypal Ideas on the Scientific Theories of Kepler", [Lecture at the Psychological Club of Zurich], 1948)

"The best that most of us can hope to achieve in physics is simply to misunderstand at a deeper level." (Wolfgang Pauli, [letter to Jagdish Mehra], 1958)

Benjamin Disraeli - Collected Quotes

"It is remarkable that when great discoveries are effected, their simplicity always seems to detract from their originality: on these occasions we are reminded of the egg of Columbus!" (Benjamin Disraeli, "Curiosities of Literature" Vol. 3, 1824)

"Knowledge must be gained by ourselves. Mankind may supply us with facts; but the results, even if they agree with previous ones, must be the work of our own minds." (Benjamin Disraeli, "The Young Duke", 1831)

"Nature is more powerful than education; time will develop everything." (Benjamin Disraeli, "Contarini Fleming", 1832)

"What we anticipate seldom occurs; what we least expected generally happens." (Benjamin Disraeli, "Henrietta Temple", 1837)

"Extreme views are never just; something always turns up which disturbs the calculations formed upon their data." (Benjamin Disraeli, 1847)

"As a general rule the most successful man in life is the man who has the best information." (Benjamin Disraeli, "Endymion", 1880)

"Imagination is too often accompanied by somewhat irregular logic." (Benjamin Disraeli, "Wit and Wisdom, Imagination", 1881)

"No one for a moment can pretend that printing is so great a discovery as writing, or algebra as a language." (Benjamin Disraeli, "Lothair", 1870)

"The world is devoted to physical science, because it believes theses discoveries will increase its capacity of luxury and self-indulgence. But the pursuit of science only leads to the insoluble. (Benjamin Disraeli, "Lothair", 1870)

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

"What art was to the ancient world, Science is to the modern; the distinctive faculty. In the minds of men, the useful has succeeded to the beautiful." (Benjamin Disraeli)

Jean-Baptiste Lamarck - Collected Quotes

"Time is insignificant and never a difficulty for Nature. It is always at her disposal and represents an unlimited power with which she accomplishes her greatest and smallest tasks." (Jean-Baptiste Lamarck, "Hydrogeology", 1802)

"It is not enough to discover and prove a useful truth previously unknown, but that it is necessary also to be able to propagate it and get it recognized." (Jean-Baptiste Lamarck, "Zoological Philosophy: An Exposition with Regard to the Natural, History of Animals" Vol. 2, 1809)

"Imagination is one of the finest faculties of man: it ennobles and elevates his thoughts and relieves him from the domination of minute details; and when it reaches a very high development, it makes him superior to the great majority of other people." (Jean-Baptiste Lamarck, "Zoological Philosophy: An Exposition with Regard to the Natural, History of Animals" Vol. 2, 1809)

"Man is condemned to exhaust all possible errors when he examines any set of facts before he recognizes the truth. (Jean-Baptiste Lamarck, "Zoological Philosophy: An Exposition with Regard to the Natural, History of Animals", 1809) 

"Reason is not a faculty; still less is it a torch or entity of any kind; but it is a special condition of the individual's intellectual faculties; a condition that is altered by experience, gradually improves and controls the judgments, according as the individual exercises his intellect." (Jean-Baptiste Lamarck, "Zoological Philosophy: An Exposition with Regard to the Natural, History of Animals" Vol. 2, 1809)

"All knowledge that is not the real product of observation, or of consequences deduced from observation, is entirely groundless and illusory." (Jean-Baptiste Lamarck)

"The most important discoveries of the laws, methods and progress of Nature have nearly always sprung from the examination of the smaller objects which she contains." (Jean-Baptiste Lamarck)

Gottlob Frege - Collected Quotes

"When we consider that the whole of geometry rests ultimately on axioms which derive their validity from the nature of our intuitive faculty, we seem well justified in questioning the sense of imaginary forms, since we attribute to them properties which not infrequently contradict all our intuitions." (Gottlob Frege, "On a Geometrical Representation of Imaginary forms in the Plane", 1873)

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

"When we consider complex numbers and their geometrical representation, we leave the field of the original concept of quantity, as contained especially in the quantities of Euclidean geometry: its lines, surfaces and volumes. According to the old conception, length appears as something material which fills the straight line between its end points and at the same time prevents another thing from penetrating into its space by its rigidity. In adding quantities, we are therefore forced to place one quantity against another. Something similar holds for surfaces and solid contents. The introduction of negative quantities made a dent in this conception, and imaginary quantities made it completely impossible. Now all that matters is the point of origin and the end point; whether there is a continuous line between them, and if so which, appears to make no difference whatsoever; the idea of filling space has been completely lost. All that has remained is certain general properties of addition, which now emerge as the essential characteristic marks of quantity. The concept has thus gradually freed itself from intuition and made itself independent. This is quite unobjectionable, especially since its earlier intuitive character was at bottom mere appearance. Bounded straight lines and planes enclosed by curves can certainly be intuited, but what is quantitative about them, what is common to lengths and surfaces, escapes our intuition." (Gottlob Frege, "Methods of Calculation based on an Extension of the Concept of Quantity", 1874)

"If the second principle [the context principle] is not observed, one is almost forced to take as the meanings of words mental pictures or acts of the individual mind, and so to offend against the first principle as well." (Gottlob Frege, "The Foundations of Arithmetic" , 1884)

"The aim of proof is, in fact, not merely to place the truth of a proposition beyond all doubt, but also to afford us insight into the dependence of one truth upon another. After we have convinced ourselves that a boulder is immovable, by trying unsuccessfully to move it, there remains the further question, what is it that supports it so securely." (Gottlob Frege, "The Foundations of Arithmetic", 1884)

"The basis of arithmetic lies deeper, it seems, than that of any of the empirical sciences, and even than that of geometry. The truths of arithmetic governs all that is numerable. This is the widest domain of all; for to it belongs not only the existent, not only the intuitable, but everything thinkable. Should not the laws of number, then, be connected very intimately with the laws of thought?"  (Gottlob Frege, "The Foundations of Arithmetic", 1884)

“The unimaginability of the content of a word is no reason, then, to deny it any meaning or to exclude it from usage. That we are nevertheless inclined to do so is probably owing to the fact that we consider words individually and ask about their meaning [in isolation], for which we then adopt a mental picture. Thus a word for which we are lacking a corresponding inner picture will seem to have no content. However, we must always consider a complete sentence. Only in [the context of] the latter do the words really have a meaning. The inner pictures which somehow sway before us (in reading the sentence) need not correspond to the logical components of the judgment. It is enough if the sentence as a whole has a sense; by means of this its parts also receive their content.” (Gottlob Frege, “The Foundations of Arithmetic”, 1884) 

“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)

"I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction." (Gottlob Frege, "The Foundations of Arithmetic", 1884)

"To apply arithmetic in the physical sciences is to bring logic to bear on observed facts, calculation becomes deduction. The laws of number, therefore, are not really applicable to external things, they are not laws of nature. They are, however, applicable to judgements holding good of things in the external world they are laws of the laws of nature. They assert not connections between phenomena, but connections between judgements, and among judgements are included the laws of nature." (Gottlob Frege, "The Foundations of Arithmetic", 1884) 

"If I compare arithmetic with a tree that unfolds upward into a multitude of techniques and theorems while its root drives into the depths, then it seems to me that the impetus of the root." (Gottlob Frege, "Grundgesetze der Arithmetik" ["Basic Laws of Arithmetic"], 1893)

"There is more danger of numerical sequences continued indefinitely than of trees growing up to heaven. Each will some time reach its greatest length."  (Gottlob Frege, "Grundgesetze der Arithmetik" ["Basic Laws of Arithmetic"], 1893)

"Whereas in meaningful arithmetic equations and inequations are sentences expressing thoughts, in formal arithmetic they are comparable with the positions of chess pieces, transformed in accordance with certain rules without considerations for any sense. For if they were viewed as having sense, the rules could not be arbitrarily stipulated; they would have to be so chosen that from formulas expressing true propositions could be derived only formulas likewise expressing true propositions. Then the standpoint of formal arithmetic would have to be abandoned, which insists that the rules for the manipulation of signs are quite arbitrarily stipulated. Only subsequently may one ask whether the signs can be given a sense compatible with the rules previously laid down. Such matters, however, lie entirely outside formal arithmetic and only arise when applications are to be made. Then, however, they must be considered; for an arithmetic with no thought as its content will also be without possibility of application. Why can no application be made of a configuration of chess pieces? Obviously, because it expresses no thought. If it did so and every chess move conforming to the rules corresponded to a transition from one thought to another, applications of chess would also be conceivable. Why can arithmetical equations be applied? Only because they express thoughts. How could we possibly apply an equation which expressed nothing and was nothing more than a group of figures, to be transformed into another group of figures in accordance with certain rules? Now, it is applicability alone which elevates arithmetic from a game to the rank of a science. So applicability necessarily belongs to it. Is it good, then, to exclude from arithmetic what it needs in order to be a science?"  (Gottlob Frege, "Grundgesetze der Arithmetik" ["Basic Laws of Arithmetic"], 1893)

"But surely it is self-evident that every theory is merely a framework or scheme of concepts together with their necessary relations to one another, and that the basic elements can be constructed as one pleases." (Gottlob Frege, "On the Foundations of Geometry and Formal Theories of Arithmetic" , cca. 1903-1909)

"No sharp boundary can be drawn between logic and arithmetic. If this formal theory is correct, then logic cannot be as barren as it may appear upon superficial examination - an appearance for which logicians themselves must be assigned part of the blame." (Gottlob Frege, "On the Foundations of Geometry and Formal Theories of Arithmetic", cca. 1903-1909)

"The conception of logical laws must be the decisive factor in the treatment of logic, and that conception depends upon what we understand by the word ‘true’. It is generally admitted at the very beginning that logical laws must be rules of conduct to guide thought to truth […]" (Gottlob Frege," Grundgesetze", The Monist, 1915) 

"It really is worth the trouble to invent a new symbol if we can thus remove not a few logical difficulties and ensure the rigour of the proofs. But many mathematicians seem to have so little feeling for logical purity and accuracy that they will use a word to mean three or four different things, sooner than make the frightful decision to invent a new word." (Gottlob Frege)

Michael F Atiyah - Collected Quotes

"A theorem is never arrived at in the way that logical thought would lead you to believe or that posterity thinks. It is usually much more accidental, some chance discovery in answer to some kind of question. Eventually you can rationalize it and say that this is how it fits. Discoveries never happen as neatly as that. You can rewrite history and make it look much more logical, but actually it happens quite differently." (Michael F Atiyah, 2004)

"Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine." (Michael F Atiyah, 2004)

"Mathematics is always a continuum, linked to its history, the past - nothing comes out of zero." (Michael F Atiyah, [interview] 2004)

"At every major step physics has required, and frequently stimulated, the introduction of new mathematical tools and concepts. Our present understanding of the laws of physics, with their extreme precision and universality, is only possible in mathematical terms." (Michael F Atiyah, 2005)

"We all know what we like in music, painting or poetry, but it is much harder to explain why we like it. The same is true in mathematics, which is, in part, an art form. We can identify a long list of desirable qualities: beauty, elegance, importance, originality, usefulness, depth, breadth, brevity, simplicity, clarity. However, a single work can hardly embody them all; in fact, some are mutually incompatible. Just as different qualities are appropriate in sonatas, quartets or symphonies, so mathematical compositions of varying types require different treatment." (Michael F Atiyah, "Mathematics: Art and Science" Bulletin of the AMS 43, 2006)

"In mathematics, beauty is a very important ingredient. Beauty exists in mathematics as in architecture and other things. It is a difficult thing to define but it is something you recognise when you see it. It certainly has to have elegance, simplicity, structure and form. All sorts of things make up real beauty. There are many different kinds of beauty and the same is true of mathematical theorems. Beauty is an important criterion in mathematics because basically there is a lot of choice in what you can do in mathematics and science. It determines what you regard as important and what is not." (Michael F Atiyah, 2009)

"In the broad light of day mathematicians check their equations and their proofs, leaving no stone unturned in their search for rigour. But, at night, under the full moon, they dream, they float among the stars and wonder at the miracle of the heavens. They are inspired. Without dreams there is no art, no mathematics, no life." (Michael F Atiyah, "The Art of Mathematics", 2010)

"[…] a lot of mathematics is predictable. Somebody shows you how to solve one problem, and you do the same thing again. Every time you take a step forward you’re following in the steps of the person who came before. Every now and again, somebody comes along with a totally new idea and shakes everybody up." (Michael F Atiyah, [interview] 2013)

"People think mathematics begins when you write down a theorem followed by a proof. That’s not the beginning, that’s the end. For me the creative place in mathematics comes before you start to put things down on paper, before you try to write a formula. You picture various things, you turn them over in your mind. You’re trying to create, just as a musician is trying to create music, or a poet. There are no rules laid down. You have to do it your own way. But at the end, just as a composer has to put it down on paper, you have to write things down. But the most important stage is understanding. A proof by itself doesn’t give you understanding." (Michael F Atiyah, [interview] 2013)

"[…] there’s atomic physics - electrons and protons and neutrons, all the stuff of which atoms are made. At these very, very, very small scales, the laws of physics are much the same, but there is also a force you ignore, which is the gravitational force. Gravity is present everywhere because it comes from the entire mass of the universe. It doesn’t cancel itself out, it doesn’t have positive or negative value, it all adds up." (Michael F Atiyah, [interview] 2013)

"Any good theorem should have several proofs, the more the better. For two reasons: usually, different proofs have different strengths and weaknesses, and they generalise in different directions: they are not just repetitions of each other." (Michael F Atiyah)

"If you attack a mathematical problem directly, very often you come to a dead end, nothing you do seems to work and you feel that if only you could peer round the corner there might be an easy solution. There is nothing like having somebody else beside you, because he can usually peer round the corner." (Michael F Atiyah)

"The aim of mathematics is to explain as much as possible in simple terms." (Michael F Atiyah)

John Dewey - Collected Quotes

"It [science] involves an intelligent and persistent endeavor to revise current beliefs so as to weed out what is erroneous, to add to their accuracy, and, above all, to give them such shape that the dependencies of the various facts upon one another may be as obvious as possible." (John Dewey, "Democracy and Education", 1916)

"The way to enable a student to apprehend the instrumental value of arithmetic is not to lecture him on the benefit it will be to him in some remote and uncertain future, but to let him discover that success in something he is interested in doing depends on ability to use numbers." (John Dewey, "Democracy and Education", 1916)

"Thinking is the accurate and deliberate instituting of connections between what is done and its consequences." (John Dewey, "Democracy and Education", 1916)

"Reason is experimental intelligence, conceived after the pattern of science, and used in the creation of social arts; it has something to do. It liberates man from the bondage of the past, due to ignorance and accident hardened into custom. It projects a better future and assists man in its realization. And its operation is always subject to test in experience. […] The principles which man projects as guides […] are not dogmas. They are hypotheses to be worked out in practice, and to be rejected, corrected and expanded as they fail or succeed in giving our present experience the guidance it requires. We may call them programmes of action, but since they are to be used in making our future acts less blind, more directed, they are flexible. Intelligence is not something possessed once for all. It is in constant process of forming, and its retention requires constant alertness in observing consequences, an open-minded will to learn and courage in re-adjustment." (John Dewey, "Reconstruction in Philosophy", 1920)

"Scientific principles and laws do not lie on the surface of nature. They are hidden, and must be wrested from nature by an active and elaborate technique of inquiry." (John Dewey, "Reconstruction in Philosophy", 1920) 

"The first distinguishing characteristic of thinking is facing the facts - inquiry, minute and extensive scrutinizing, observation." (John Dewey, "Reconstruction in Philosophy", 1920)

"Factual science may collect statistics, and make charts. But its predictions are, as has been well said, but past history reversed." (John Dewey, "Art as Experience", 1934)

"The intellect is at home in that which is fixed only because it is done and over with, for intellect is itself just as much a deposit of past life as is the matter to which it is congenial. Intuition alone articulates in the forward thrust of life and alone lays hold of reality." (John Dewey, "Time and Individuality", 1940)

"A problem well-defined is a problem half solved." (John Dewey)

"Education is not preparation for life; education is life itself." (John Dewey)

"The truth is that which works." (John Dewey)

Félix E Borel - Collected Quotes

"All of mathematics can be deduced from the sole notion of an integer; here we have a fact universally acknowledged today." (Émile Borel, "Contribution a l'analyse arithmetique du continu", 1903)

"Can there be laws of chance? The answer, it would seem should be negative, since chance is in fact defined as the characteristic of the phenomena which follow no law, phenomena whose causes are too complex to permit prediction." (Félix E Borel, "Probabilities and Life", 1943)

"Events with a sufficiently small probability never occur, or at least we must act, in all circumstances, as if they were impossible." (Félix E Borel, "Probabilities and Life", 1943)

"Probabilities must be regarded as analogous to the measurement of physical magnitudes; that is to say, they can never be known exactly, but only within certain approximation." (Félix E Borel, "Probabilities and Life", 1943)

"The sweeping development of mathematics during the last two centuries is due in large part to the introduction of complex numbers; paradoxically, this is based on the seemingly absurd notion that there are numbers whose squares are negative." (Félix E Borel, 1952)

"Number knows no limitations, either from the side of the infinitely great or from the side of the infinitely small, and the facility it offers for generalization is too great for us not to be tempted by it." (Félix E Borel, "Space and Time", 1960)

"Numbers are the landmarks which enable us to speak in a language common to all men, of successive moments of duration." (Félix E Borel, "Space and Time", 1960)

"The search for truth is the most noble aim of science, and there are no degrees between truth and error; the insignificance and smallness of a phenomenon will perhaps diminish its practical interest, but not its scientific value." (Félix E Borel, "Space and Time", 1960)

"Incomplete knowledge must be considered as perfectly normal in probability theory; we might even say that, if we knew all the circumstances of a phenomenon, there would be no place for probability, and we would know the outcome with certainty." (Félix E Borel, "Probability and Certainty", 1963)

"The problem of error has preoccupied philosophers since the earliest antiquity. According to the subtle remark made by a famous Greek philosopher, the man who makes a mistake is twice ignorant, for he does not know the correct answer, and he does not know that he does not know it." (Félix Borel, "Probability and Certainty", 1963)

Johannes Kepler - Collected Quotes

"It is very difficult to write mathematics books today. If one does not take pains with the fine points of theorems, explanations, proofs and corollaries, then it won’t be a mathematics book; but if one does these things, then the reading of it will be extremely boring." (Johannes Kepler, "Astronomia Nova", 1609)

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

"A mind accustomed to mathematical deduction, when confronted with the faulty foundations resists a long, long time, like an obstinate mule, until compelled by beating and curses to put its foot into that dirty puddle." (Johannes Kepler)

"As in every discipline, so in astronomy, too, the conclusions that we teach the reader are seriously intended, and our discussion is no mere game." (Johannes Kepler)

"It is a right, yes a duty, to search in cautious manner for the numbers, sizes, and weights, the norms for everything [God] has created. For He himself has let man take part in the knowledge of these things […] For these secrets are not of the kind whose research should be forbidden; rather they are set before our eyes like a mirror so that by examining them we observe to some extent the goodness and wisdom of the Creator." (Johannes Kepler)

"[...] it is the most widely accepted axiom in the natural science that Nature makes use of the fewest possible means" (Johannes Kepler)

"The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics." (Johannes Kepler)

"What else can the human mind hold besides numbers and magnitudes? These alone we apprehend correctly, and if piety permits to say so, our comprehension is in this case of the same kind as God’s, at least insofar as we are able to understand it in this mortal life." (Johannes Kepler)

"We find, therefore, under this orderly arrangement, a wonderful symmetry in the universe, and a definite relation of harmony in the motion and magnitude of the orbs, of a kind that is not possible to obtain in any other way." (Johannes Kepler)

Paul C W Davies - Collected Quotes

"A model of the universe does not require faith, but a telescope. If it is wrong, it is wrong." (Paul C W Davies, "Space and Time in the Modern Universe", 1977)

"Mathematics and beauty are the foundation stones of the universe. No one who has studied the forces of nature can doubt that the world about us is a manifestation of something very, very clever indeed." (Paul C W Davies, "The Forces of Nature", 1979)

"The basis of this theory is that in nature there is an inherent uncertainty or unpredictability that manifests itself only on an atomic scale. For example, the position of a subatomic particle such as an electron may not be a well-defined concept at all; it should be envisaged as jiggling around in a random sort of a way. Energy, too, becomes a slightly nebulous concept, subject to capricious and unpredictable changes." (Paul C W Davies, "The Edge of Infinity: Where the Universe Came from and How It Will End", 1981)

"Whether mathematical simplicity is God’s affair or our, the fact remains that this feature more than any other remains the mainspring of progress in the physical sciences." (Paul C W Davies, "The Edge of Infinity", 1981)

"The equations of physics have in them incredible simplicity, elegance and beauty. That in itself is sufficient to prove to me that there must be a God who is responsible for these laws and responsible for the universe" (Paul C W Davies, 1984)

"Until now, physical theories have been regarded as merely models with approximately describe the reality of nature. As the models improve, so the fit between theory and reality gets closer. Some physicists are now claiming that supergravity is the reality, that the model and the real world are in mathematically perfect accord." (Paul C W Davies, "Superforce", 1984)

"That the universe has organized its own self-awareness - is for me powerful evidence that there is 'something going on' behind it all. The impression of design is overwhelming. Science may explain all the processes whereby the universe evolves its own destiny, but that still leaves room for there to be a meaning behind existence." (Paul C W Davies, "The Cosmic Blueprint: New Discoveries In Nature's Creative Ability To Order Universe", 1987)

"When it comes to very highly organized systems, such as a living cell, the task of modeling by approximation to simple, continuous and smoothly varying quantities is hopeless. It is for this reason that attempts by sociologists and economists to imitate physicists and describe their subject matter by simple mathematical equations is rarely convincing." (Paul C W Davies, "The Cosmic Blueprint: New Discoveries in Nature’s Creative Ability to Order the Universe", 1987)

"The belief that the underlying order of the world can be expressed in mathematical form lies at the very heart of science. So deep does this belief run that a branch of science is considered not to be properly understood until it can be cast in mathematics." (Paul C W Davies, "The Mind of God: The Scientific Basis for a Rational World", 1992)

"The scientific quest is a journey into the unknown." (Paul C W Davies, "The Mind of God: The Scientific Basis for a Rational World", 1992)

"It suggests to me that consciousness and our ability to do mathematics are no mere accident, no trivial detail, no insignificant by-product of evolution that is piggy-backing on some other mundane property. It points to what I like to call the cosmic connection, the existence of a really deep relationship between minds that can do mathematics and the underlying laws of nature that produce them. We have a closed system of consistency here: the laws of physics produce complex systems, and these complex systems lead to consciousness, which then produces mathematics, which can encode [...] the very laws of physics that gave rise to it." (Paul Davies, "Are We Alone?: Philosophical Implications of the Discovery of Extraterrestrial Life", 1995)

"Underpinning everything [...] are the laws of physics. These remarkably ingenious laws are able to permit matter to self-organize to the point where consciousness emerges in the cosmos - mind from matter - and the most striking product of the human mind is mathematics. This is the baffling thing. Mathematics is [...] produced by the human mind. Yet if we ask where mathematics works best, it is in areas like particle physics and astrophysics, areas of fundamental science that are very, very far removed from everyday affairs. [...] at the opposite end of spectrum of complexity from the human brain. [...] a product of the most complex system we know in nature, the human brain, finds a consonance with the underlying, simplest and most fundamental level, the basic building blocks that make up the world." (Paul C W Davies, "Are We Alone?: Philosophical Implications of the Discovery of Extraterrestrial Life", 1995)

"Maybe it is not complexity per se that is significant, but organized complexity." (Paul Davies, "The Origin of Life", 2003)

"The laws of nature are rigged not only in favor of complexity, or just in favor of life, but also in favor of mind. To put it dramatically, it implies that mind is written into the laws of nature in a fundamental way." (Paul C W  Davies, "The Fifth Miracle: The Search for the Origin and Meaning of Life", 1999)

"Physicists have been drawn to elegant mathematical relationships that bind the subject together with economy and style, melding disparate qualities in subtle and harmonious ways. But this is to import a new factor into the argument - questions of aesthetics and taste. We are then on shaky ground indeed. It may be that M theory looks beautiful to its creators, but ugly to N theorists, who think that their theory is the most elegant. But then the O theorists disagree with both groups [...]" (Paul C W Davies, "Cosmic Jackpot: Why Our Universe Is Just Right for Life", 2007)

"Disorder is a collective property of large assemblages; it makes no sense to say a single molecule is disordered or random. Thermodynamic quantities like entropy and heat energy are defined by reference to enormous numbers of particles - for example, molecules of gas careering about - and averaging across them without considering the details of individual particles. (Such averaging is sometimes called a ‘coarse-grained view’.) Thus, the temperature of a gas is related to the average energy of motion of the gas molecules. The point is that whenever one takes an average some information is thrown away, that is, we accept some ignorance. The average height of a Londoner tells us nothing about the height of a specific person. Likewise, the temperature of a gas tells us nothing about the speed of a specific molecule. In a nutshell: information is about what you know, and entropy is about what you don’t know." (Paul Davies, "The Demon in the Machine: How Hidden Webs of Information Are Solving the Mystery of Life", 2019)

"Gödel shattered the ancient dream that cast-iron logical reasoning would always produce irrefutable truth. His result is arguably the highest product of the human intellect. All other discoveries about the world of physical things or the world of reason tell us something we didn’t know before. Gödel’s theorem tells us that the world of mathematics embeds inexhaustible novelty; even an unbounded intellect, a god, can never know everything. It is the ultimate statement of open-endedness." (Paul Davies, "The Demon in the Machine: How Hidden Webs of Information Are Solving the Mystery of Life", 2019)

"Information makes a difference in the world. We might say it has ‘causal power’. The challenge to science is to figure out how to couple abstract information to the concrete world of physical objects." (Paul Davies, "The Demon in the Machine: How Hidden Webs of Information Are Solving the Mystery of Life", 2019)

"It would be wrong of me to give the impression that information flow in biology is restricted to gene regulatory networks. Unfortunately, the additional complexity of some other networks makes them even harder to model computationally, especially as the simple version of 0s and 1s (off and on) mostly won’t do. On top of that, the number of components skyrockets when it comes to more finely tuned functions like metabolism. The general point remains: biology will ‘stand out’ from random complexity in the manner of its information patterning and processing, and though complex, the software account of life will still be vastly simpler than the underlying molecular systems that support it, as it is for electronic circuits." (Paul Davies, "The Demon in the Machine: How Hidden Webs of Information Are Solving the Mystery of Life", 2019)

"Living organisms are not just bags of information: they are computers. It follows that a full understanding of life will come only from unravelling its computational mechanisms. And that requires an excursion into the esoteric but fascinating foundations of logic, mathematics and computing." (Paul Davies, "The Demon in the Machine: How Hidden Webs of Information Are Solving the Mystery of Life", 2019)

"[...] living organisms manifest deep new physical principles, and that we are on the threshold of uncovering and harnessing those principles. What is different this time, and why it has taken so many decades to discover the real secret of life, is that the new physics is not simply a matter of an additional type of force - a 'life force' - but something altogether more subtle, something that interweaves matter and information, wholes and parts, simplicity and complexity." (Paul Davies, "The Demon in the Machine: How Hidden Webs of Information Are Solving the Mystery of Life", 2019)

"Network theory confirms the view that information can take on 'a life of its own'. In the yeast network my colleagues found that 40 per cent of node pairs that are correlated via information transfer are not in fact physically connected; there is no direct chemical interaction. Conversely, about 35 per cent of node pairs transfer no information between them even though they are causally connected via a 'chemical wire' (edge). Patterns of information traversing the system may appear to be flowing down the 'wires' (along the edges of the graph) even when they are not. For some reason, 'correlation without causation' seems to be amplified in the biological case relative to random networks." (Paul Davies, "The Demon in the Machine: How Hidden Webs of Information Are Solving the Mystery of Life", 2019)

"One of the hallmarks of life is its limitless exuberance: its open-ended variety and complexity. If life represents something truly fundamental and extraordinary, then this quality of unconstrained possibility is surely key." (Paul Davies, "The Demon in the Machine: How Hidden Webs of Information Are Solving the Mystery of Life", 2019)

"Patterns of information flow can literally take on a life of their own, surging through cells, swirling around brains and networking across ecosystems and societies, displaying their own systematic dynamics. It is from this rich and complex ferment of information that the concept of agency emerges, with its links to consciousness, free will and other vexing puzzles. It is here, in the way living systems arrange information into organized patterns, that the distinctive order of life emerges from the chaos of the molecular realm." (Paul Davies, "The Demon in the Machine: How Hidden Webs of Information Are Solving the Mystery of Life", 2019)

"The concept of integrated information is clearest when applied to networks. Imagine a black box with input and output terminals. Inside are some electronics, such as a network with logic elements (AND, OR, and so on) wired together. Viewed from the outside, it will usually not be possible to deduce the circuit layout simply by examining the cause–effect relationship between inputs and outputs, because functionally equivalent black boxes can be built from very different circuits. But if the box is opened, it’s a different story. Suppose you use a pair of cutters to sever some wires in the network. Now rerun the system with all manner of inputs. If a few snips dramatically alter the outputs, the circuit can be described as highly integrated, whereas in a circuit with low integration the effect of some snips may make no difference at all." (Paul Davies, "The Demon in the Machine: How Hidden Webs of Information Are Solving the Mystery of Life", 2019)

"The flip side of reductionism is emergence - the recognition that new qualities and principles may emerge at higher levels of complexity that can themselves be relatively simple and grasped without knowing much about the levels below. Emergence has acquired something of a mystical air but in truth it has always played a role in science." (Paul Davies, "The Demon in the Machine: How Hidden Webs of Information Are Solving the Mystery of Life", 2019)

"[...] the Game of Life, in which a few simple rules executed repeatedly can generate a surprising degree of complexity. Recall that the game treats squares, or pixels, as simply on or off (filled or blank) and the update rules are given in terms of the state of the nearest neighbours. The theory of networks is closely analogous. An electrical network, for example, consists of a collection of switches with wires connecting them. Switches can be on or off, and simple rules determine whether a given switch is flipped, according to the signals coming down the wires from the neighbouring switches. The whole network, which is easy to model on a computer, can be put in a specific starting state and then updated step by step, just like a cellular automaton. The ensuing patterns of activity depend both on the wiring diagram (the topology of the network) and the starting state. The theory of networks can be developed quite generally as a mathematical exercise: the switches are called ‘nodes’ and the wires are called ‘edges’. From very simple network rules, rich and complex activity can follow." (Paul Davies, "The Demon in the Machine: How Hidden Webs of Information Are Solving the Mystery of Life", 2019)

"The story of life is really two narratives tightly interwoven. One concerns complex chemistry, a rich and elaborate network of reactions. The other is about information, not merely passively stored in genes but coursing through organisms and permeating biological matter to bestow a unique form of order. Life is thus an amalgam of two restlessly shifting patterns, chemical and informational. These patterns are not independent but are coupled together to form a system of cooperation and coordination that shuffles bits of information in a finely choreographed ballet." (Paul Davies, "The Demon in the Machine: How Hidden Webs of Information Are Solving the Mystery of Life", 2019)

"Whatever the minimal complexity for life may be, there is no doubt that even the simplest known life form is already stupendously complex. Indeed, life’s complexity is so daunting that it is tempting to give up trying to understand it in physical terms." (Paul Davies, "The Demon in the Machine: How Hidden Webs of Information Are Solving the Mystery of Life", 2019)

Arthur L Bowley - Collected Quotes

"A knowledge of statistics is like a knowledge of foreign languages or of algebra; it may prove of use at any time under any circumstances." (Arthur L Bowley, "Elements of Statistics", 1901)

"A statistical estimate may be good or bad, accurate or the reverse; but in almost all cases it is likely to be more accurate than a casual observer’s impression, and the nature of things can only be disproved by statistical methods." (Arthur L Bowley, "Elements of Statistics", 1901)

"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)

"Some of the common ways of producing a false statistical argument are to quote figures without their context, omitting the cautions as to their incompleteness, or to apply them to a group of phenomena quite different to that to which they in reality relate; to take these estimates referring to only part of a group as complete; to enumerate the events favorable to an argument, omitting the other side; and to argue hastily from effect to cause, this last error being the one most often fathered on to statistics. For all these elementary mistakes in logic, statistics is held responsible." (Arthur L Bowley, "Elements of Statistics", 1901)

"[…] statistics is the science of the measurement of the social organism, regarded as a whole, in all its manifestations." (Arthur L Bowley, "Elements of Statistics", 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." (Arthur L Bowley, "Elements of Statistics", 1901)

"Statistics may, for instance, be called the science of counting. Counting appears at first sight to be a very simple operation, which any one can perform or which can be done automatically; but, as a matter of fact, when we come to large numbers, e.g., the population of the United Kingdom, counting is by no means easy, or within the power of an individual; limits of time and place alone prevent it being so carried out, and in no way can absolute accuracy be obtained when the numbers surpass certain limits." (Arthur L Bowley, "Elements of Statistics", 1901)

"By [diagrams] it is possible to present at a glance all the facts which could be obtained from figures as to the increase,  fluctuations, and relative importance of prices, quantities, and values of different classes of goods and trade with various countries; while the sharp irregularities of the curves give emphasis to the disturbing causes which produce any striking change." (Arthur L Bowley, "A Short Account of England's Foreign Trade in the Nineteenth Century, its Economic and Social Results", 1905)

"Of itself an arithmetic average is more likely to conceal than to disclose important facts; it is the nature of an abbreviation, and is often an excuse for laziness." (Arthur L Bowley, "The Nature and Purpose of the Measurement of Social Phenomena", 1915)

"[...] the problems of the errors that arise in the process of sampling have been chiefly discussed from the point of view of the universe, not of the sample; that is, the question has been how far will a sample represent a given universe? The practical question is, however, the converse: what can we infer about a universe from a given sample? This involves the difficult and elusive theory of inverse probability, for it may be put in the form, which of the various universes from which the sample may a priori have been drawn may be expected to have yielded that sample?" (Arthur L Bowley, "Elements of Statistics. 5th Ed., 1926)

"Statistics are numerical statements of facts in any department of inquiry, placed in relation to each other; statistical methods are devices for abbreviating and classifying the statements and making clear the relations." (Arthur L Bowley, "An Elementary Manual of Statistics", 1934)

"Averages are statistical constants which enable us to comprehend in a single effort the

significance of the whole." (Arthur L Bowley)

"Dispersion is the measure of the variation of the items." (Arthur L Bowley)

"Great numbers and averages resulting from them, such as we always obtain in measuring social phenomena, have a great inertia." (Arthur L Bowley)

"[tabulation is] the intermediate process between the accumulation of data in whatever form they are obtained, and the final reasoned account of the result shown by the statistics." (Arthur L Bowley)

"The discussion of proper weights to be used has occupied a space in statistical literature out of all proportions to its significance, for it may be said at once that no great importance need be attached to the special choice of weights ; one of the most convenient facts of statistical theory is that, given certain conditions, the same result is obtained with sufficient closeness whatever logical system of weights is applied." (Arthur L Bowley)