On Measures (-1849)

"Numbers prime to one another are those which are measured by a unit alone as a common measure." (Euclid, "The Elements", Book VII)

"Measure, time and number are nothing but modes of thought or rather of imagination." (Baruch Spinoza, [Letter to Ludvicus Meyer] 1663)

"To measure motion, space is as necessary to be considered as time. [... They] are made use of to denote the position of finite: real beings, in respect one to another, in those infinite uniform oceans of duration and space." (John Locke, "An Essay Concerning Human Understanding", 1689)

"Most of our philosophical instruments are measures of effects. The progress made in natural philosophy increases every day by the number of these measures; by these it still continues to be improved." (George Adams, "Lectures on Natural and Experimental Philosophy" Vol. 1, 1794)

"Mathematical analysis is as extensive as nature itself; it defines all perceptible relations, measures times, spaces, forces, temperatures; this difficult science is formed slowly, but it preserves every principle which it has once acquired; it grows and strengthens itself incessantly in the midst of the many variations and errors of the human mind. It's chief attribute is clearness; it has no marks to express confused notations. It brings together phenomena the most diverse, and discovers the hidden analogies which unite them." (J B Joseph Fourier, "The Analytical Theory of Heat", 1822)

"The measure of the probability of an event is the ratio of the number of cases favourable to that event, to the total number of cases favourable or contrary, and all equally possible, or all of which have the same chance." (Siméon-Denis Poisson, "Recherches sur la Probabilités des Jugemens" ["An Investigation of the Laws of Thought"], 1837)

"Yet time and space are but inverse measures of the force of the soul. The spirit sports with time." (Ralph W Emerson, "Essays", 1841)

On Measures (1850-1899)

"Measure consists in the superposition of the magnitudes to be compared; it therefore requires a means of using one magnitude as the standard for another. In the absence of this, two magnitudes can only be compared when one is a part of the other; in which case we can only determine the more or less and not the how much." (Bernhard Riemann, "On the hypotheses which lie at the foundation of geometry", 1854)

"The purely formal sciences, logic and mathematics, deal with such relations which are independent of the definite content, or the substance of the objects, or at least can be. In particular, mathematics involves those relations of objects to each other that involve the concept of size, measure, number." (Hermann Hankel, "Theorie der Complexen Zahlensysteme", 1867)

"The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal." (William James, "Clifford's Lectures and Essays", 1879)

"Nothing that we can measure is inconceivably large or inconceivably small in physical science." (William T Kelvin, 1883)

"She [Nature] works with reference to no measure of time, no limit of space, and with an abundance of material not expressed by exhaustless." (John Burroughs, "Birds and Poets With Other Papers", 1884)

"I call a sign which stands for something merely because it resembles it, an icon. Icons are so completely substituted for their objects as hardly to be distinguished from them. Such are the diagrams of geometry. A diagram, indeed, so far as it has a general signification, is not a pure icon; but in the middle part of our reasonings we forget that abstractness in great measure, and the diagram is for us the very thing. So in contemplating a painting, there is a moment when we lose the consciousness that it is not the thing, the distinction of the real and the copy disappears, and it is for the moment a pure dream, - not any particular existence, and yet not general. At that moment we are contemplating an icon." (Charles S Peirce, "On The Algebra of Logic : A Contribution to the Philosophy of Notation" in The American Journal of Mathematics 7, 1885)

On Measure (1900-1949)

"Let us notice first of all, that every generalization implies in some measure the belief in the unity and simplicity of nature." (Jules H Poincaré, "Science and Hypothesis", 1905)

"So completely is nature mathematical that some of the more exact natural sciences, in particular astronomy and physics, are in their theoretic phases largely mathematical in character, while other sciences which have hitherto been compelled by the complexity of their phenomena and the inexactitude of their data to remain descriptive and empirical, are developing towards the mathematical ideal, proceeding upon the fundamental assumption that mathematical relations exist between the forces and the phenomena, and that nothing short, of the discovery and formulations of these relations would constitute definitive knowledge of the subject. Progress is measured by the closeness of the approximation to this ideal formulation." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"Just as data gathered by an incompetent observer are worthless - or by a biased observer, unless the bias can be measured and eliminated from the result - so also conclusions obtained from even the best data by one unacquainted with the principles of statistics must be of doubtful value." (William F White, "A Scrap-Book of Elementary Mathematics: Notes, Recreations, Essays", 1908)

"The second law of thermodynamics appears solely as a law of probability, entropy as a measure of the probability, and the increase of entropy is equivalent to a statement that more probable events follow less probable ones." (Max Planck, "A Survey of Physics", 1923)

"Our environment may and should mean something towards us which is not to be measured with the tools of the physicist or described by the metrical symbols of the mathematician." (Arthur S Eddington, "Science and the Unseen World", 1929)

"Search for measurable elements among your phenomena, and then search for relations between these measures of physical quantities." (Alfred N Whitehead, "Science and the Modern World", 1929)

"The discoveries in physical science, the triumphs in invention, attest the value of the process of trial and error. In large measure, these advances have been due to experimentation." (Louis Brandeis, "Judicial opinions", 1932)

"[…] reality is a system, completely ordered and fully intelligible, with which thought in its advance is more and more identifying itself. We may look at the growth of knowledge […] as an attempt by our mind to return to union with things as they are in their ordered wholeness. […] and if we take this view, our notion of truth is marked out for us. Truth is the approximation of thought to reality […] Its measure is the distance thought has travelled […] toward that intelligible system […] The degree of truth of a particular proposition is to be judged in the first instance by its coherence with experience as a whole, ultimately by its coherence with that further whole, all comprehensive and fully articulated, in which thought can come to rest." (Brand Blanshard, "The Nature of Thought" Vol. II, 1939)

"Hence the awkward expression ‘negative entropy’ can be replaced by a better one: entropy, taken with the negative sign, is itself a measure of order. Thus the device by which an organism maintains itself stationary at a fairly high level of orderliness ( = fairly low level of entropy) really consists in continually sucking orderliness from its environment." (Erwin Schrödinger, "What is Life?", 1944)

On Measures (1950-1974)

"Just as entropy is a measure of disorganization, the information carried by a set of messages is a measure of organization. In fact, it is possible to interpret the information carried by a message as essentially the negative of its entropy, and the negative logarithm of its probability. That is, the more probable the message, the less information it gives. Clichés, for example, are less illuminating than great poems." (Norbert Wiener, "The Human Use of Human Beings", 1950)

"The belief in science has replaced in large measure, the belief in God. Even where religion was regarded as compatible with science, it was modified by the mentality of the believer in scientific truth." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"Beauty had been born, not, as we so often conceive it nowadays, as an ideal of humanity, but as measure, as the reduction of the chaos of appearances to the precision of linear symbols. Symmetry, balance, harmonic division, mated and mensurated intervals - such were its abstract characteristics." (Herbert Read, "Icon and Idea: The Function of Art in the Development of Human Consciousness", 1955)

"Religion and science are the two conjugated faces of phases of one and the same act of complete knowledge - the only one which can embrace the past and future of evolution so as to contemplate, measure and fulfill them." (Pierre T de Chardin, "The Phenomenon of Man", 1955)

"Since we are assured that the all-wise Creator has observed the most exact proportions of number, weight and measure in the make of all things, the most likely way therefore to get any insight into the nature of those parts of the Creation which come within our observation must in all reason be to number, weigh and measure." (Stephen Hales, "Vegetable Staticks", 1961)

"Entropy is a measure of the heat energy in a substance that has been lost and is no longer available for work. It is a measure of the deterioration of a system." (William B Sill & Norman Hoss (Eds.), "Popular Science Encyclopedia of the Sciences", 1963)

"If our model is to be at all realistic, it will also need to be rather complex, It will in fact be too complex for easy handling by the traditional analytic measures, even after suitable simplifications." (Charles P Bonini, "Simulation of Information and Decision System in the Firm" , 1963)

"It is a commonplace of modern technology that there is a high measure of certainty that problems have solutions before there is knowledge of how they are to be solved." (John K Galbraith, "The New Industrial State", 1967)

"The aim of science is not so much to search for truth, or even truths, as to classify our knowledge and to establish relations between observable phenomena in order to be able to predict the future in a certain measure and to explain the sequence of phenomena in relation to ourselves." (Pierre L du Noüy, "Between Knowing and Believing", 1967)

"The 'flow of information' through human communication channels is enormous. So far no theory exists, to our knowledge, which attributes any sort of unambiguous measure to this 'flow'." (Anatol Rapoport, "Modern Systems Research for the Behavioral Scientist", 1969)

On Measures (1975-1999)

 "The amount of information conveyed by the message increases as the amount of uncertainty as to what message actually will be produced becomes greater. A message which is one out of ten possible messages conveys a smaller amount of information than a message which is one out of a million possible messages. The entropy of communication theory is a measure of this uncertainty and the uncertainty, or entropy, is taken as the measure of the amount of information conveyed by a message from a source. The more we know about what message the source will produce, the less uncertainty, the less the entropy, and the less the information." (John R Pierce, "An Introduction to Information Theory: Symbols, Signals and Noise", 1979)

"The term closed loop-learning process refers to the idea that one learns by determining what s desired and comparing what is actually taking place as measured at the process and feedback for comparison. The difference between what is desired and what is taking place provides an error indication which is used to develop a signal to the process being controlled." (Harold Chestnut, 1984)

"Just like a computer, we must remember things in the order in which entropy increases. This makes the second law of thermodynamics almost trivial. Disorder increases with time because we measure time in the direction in which disorder increases."  (Stephen Hawking, "A Brief History of Time", 1988)

"Engineers, always looking for optimal values for the measures of magnitudes which interest them, think of mathematicians as custodians of a fund of formulae, to be supplied to them on demand." (Jean Dieudonné, "Mathematics - The Music of Reason", 1992)

"It has long been appreciated by science that large numbers behave differently than small numbers. Mobs breed a requisite measure of complexity for emergent entities. The total number of possible interactions between two or more members accumulates exponentially as the number of members increases. At a high level of connectivity, and a high number of members, the dynamics of mobs takes hold. " (Kevin Kelly, "Out of Control: The New Biology of Machines, Social Systems and the Economic World", 1995)

"Quantify. If whatever it is you’re explaining has some measure, some numerical quantity attached to it, you’ll be much better able to discriminate among competing hypotheses. What is vague and qualitative is open to many explanations." (Carl Sagan, "The Demon-Haunted World: Science as a Candle in the Dark", 1995)

"Yet everything has a beginning, everything comes to an end, and if the universe actually began in some dense explosion, thus creating time and space, so time and space are themselves destined to disappear, the measure vanishing with the measured, until with another ripple running through the primordial quantum field, something new arises from nothingness once again." (David Berlinski, "A Tour of the Calculus", 1995)

"Probabilities aren't just numbers, and they aren't just frequencies-on-average. They are also rather like a substance that flows, dividing according to the likelihood of various outcomes, subdividing when several trials are performed in succession, and adding together when several outcomes are combined to give an event. This is a metaphor, but an accurate and powerful one. It is, in a sense, the metaphor that mathematicians formalise when they offer a definition of probability. In this sense, probability behaves like volume, mass, or area. The technical term is 'measure'. The technical definition of probability is 'a measure such that various nice things happen'. Probability is a quantity that flows through the conceptual maze of possible events, and it behaves just like water flowing through pipes." (Ian Stewart, "The Magical Maze: Seeing the World Through Mathematical Eyes", 1997)

On Measures (2000-2009)

"One measure of the depth of a physical theory is the extent to which it poses serious challenges to aspects of our worldview that had previously seemed immutable." (Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest, for the Ultimate Theory", 2000)

"[…] interval mathematics and fuzzy logic together can provide a promising alternative to mathematical modeling for many physical systems that are too vague or too complicated to be described by simple and crisp mathematical formulas or equations. When interval mathematics and fuzzy logic are employed, the interval of confidence and the fuzzy membership functions are used as approximation measures, leading to the so-called fuzzy systems modeling." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)

"One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?" (Stephen W Hawking, "The Universe in a Nutshell", 2001)

"The total disorder in the universe, as measured by the quantity that physicists call entropy, increases steadily steadily as we go from past to future. On the other hand, the total order in the universe, as measured by the complexity and permanence of organized structures, also increases steadily as we go from past to future." (Freeman J Dyson, [Page-Barbour lecture], 2004)

"All models (whether mental or those turned into computer maps/models) are developed using a particular lens of what we value - what we think is important to understand, or what performance we wish to develop or improve. Although organizations can build forum models focusing on the performance measure du jour, they would be well advised to use a systemic or integral framework for what to include." (Peggy Holman et al, "The Change Handbook", 2007)

"The total disorder in the universe, as measured by the quantity that physicists call entropy, increases steadily as we go from past to future. On the other hand, the total order in the universe, as measured by the complexity and permanence of organized structures, also increases steadily as we go from past to future." (Freeman J Dyson, "A Many-Colored Glass: Reflections on the Place of Life in the Universe", 2007)

"This new model of development would be based clearly on the goal of sustainable human well-being. It would use measures of progress that clearly acknowledge this goal. It would acknowledge the importance of ecological sustainability, social fairness, and real economic efficiency. Ecological sustainability implies recognizing that natural and social capital are not infinitely substitutable for built and human capital, and that real biophysical limits exist to the expansion of the market economy." (Robert Costanza, "Toward a New Sustainable Economy", 2008)

On Measure (2010-2019)

"[...] according to the quantum theory, randomness is a basic trait of reality, whereas in classical physics it is a derivative property, though an equally objective one. Note, however, that this conclusion follows only under the realist interpretation of probability as the measure of possibility. If, by contrast, one adopts the subjectivist or Bayesian conception of probability as the measure of subjective uncertainty, then randomness is only in the eye of the beholder." (Mario Bunge, "Matter and Mind: A Philosophical Inquiry", 2010)

"Second Law of thermodynamics is not an equality, but an inequality, asserting merely that a certain quantity referred to as the entropy of an isolated system - which is a measure of the system’s disorder, or ‘randomness’ - is greater (or at least not smaller) at later times than it was at earlier times." (Roger Penrose, "Cycles of Time: An Extraordinary New View of the Universe", 2010)

"In an information economy, entrepreneurs master the science of information in order to overcome the laws of the purely physical sciences. They can succeed because of the surprising power of the laws of information, which are conducive to human creativity. The central concept of information theory is a measure of freedom of choice. The principle of matter, on the other hand, is not liberty but limitation - it has weight and occupies space." (George Gilder, "Knowledge and Power: The Information Theory of Capitalism and How it is Revolutionizing our World", 2013)

"In negative feedback regulation the organism has set points to which different parameters (temperature, volume, pressure, etc.) have to be adapted to maintain the normal state and stability of the body. The momentary value refers to the values at the time the parameters have been measured. When a parameter changes it has to be turned back to its set point. Oscillations are characteristic to negative feedback regulation […]" (Gaspar Banfalvi, "Homeostasis - Tumor – Metastasis", 2014)

"Fuzzy thinking can never be proven wrong. And only when we are proven wrong so clearly that we can no longer deny it to ourselves will we adjust our mental models of the world - producing a clearer picture of reality. Forecast, measure, revise: it is the surest path to seeing better." (Philip E Tetlock, "Superforecasting: The Art and Science of Prediction", 2015)

"The proper measure of a philosophical system or a scientific theory is not the degree to which it anticipated modern thought, but its degree of success in treating the philosophical and scientific problems of its own day." (Steven Weinberg, "To Explain the World: The Discovery of Modern Science", 2015)

"Information theory leads to the quantification of the information content of the source, as denoted by entropy, the characterization of the information-bearing capacity of the communication channel, as related to its noise characteristics, and consequently the establishment of the relationship between the information content of the source and the capacity of the channel. In short, information theory provides a quantitative measure of the information contained in message signals and help determine the capacity of a communication system to transfer this information from source to sink over a noisy channel in a reliable fashion." (Ali Grami, "Information Theory", 2016)

"Statistics is the science of collecting, organizing, summarizing, and analyzing information to draw conclusions or answer questions. In addition, statistics is about providing a measure of confidence in any conclusions." (Michael Sullivan, "Statistics: Informed Decisions Using Data", 5th Ed., 2017)

On Sampling (-1949)

"By a small sample we may judge of the whole piece." (Miguel de Cervantes, "Don Quixote de la Mancha", 1605–1615)

"To a very striking degree our culture has become a Statistical culture. Even a person who may never have heard of an index number is affected [...] by [...] of those index numbers which describe the cost of living. It is impossible to understand Psychology, Sociology, Economics, Finance or a Physical Science without some general idea of the meaning of an average, of variation, of concomitance, of sampling, of how to interpret charts and tables." (Carrol D Wright, 1887)

"If the number of experiments be very large, we may have precise information as to the value of the mean, but if our sample be small, we have two sources of uncertainty: (I) owing to the 'error of random sampling' the mean of our series of experiments deviates more or less widely from the mean of the population, and (2) the sample is not sufficiently large to determine what is the law of distribution of individuals." (William S Gosset, "The Probable Error of a Mean", Biometrika, 1908)

"The postulate of randomness thus resolves itself into the question, 'of what population is this a random sample?' which must frequently be asked by every practical statistician." (Ronald Fisher, "On the Mathematical Foundation of Theoretical Statistics", Philosophical Transactions of the Royal Society of London Vol. A222, 1922)

"The principle underlying sampling is that a set of objects taken at random from a larger group tends to reproduce the characteristics of that larger group: this is called the Law of Statistical Regularity. There are exceptions to this rule, and a certain amount of judgment must be exercised, especially when there are a few abnormally large items in the larger group. With erratic data, the accuracy of sampling can often be tested by comparing several samples. On the whole, the larger the sample the more closely will it tend to resemble the population from which it is taken; too small a sample would not give reliable results." (Lewis R Connor, "Statistics in Theory and Practice", 1932)

"If the chance of error alone were the sole basis for evaluating methods of inference, we would never reach a decision, but would merely keep increasing the sample size indefinitely." (C West Churchman, "Theory of Experimental Inference", 1948)

"If significance tests are required for still larger samples, graphical accuracy is insufficient, and arithmetical methods are advised. A word to the wise is in order here, however. Almost never does it make sense to use exact binomial significance tests on such data - for the inevitable small deviations from the mathematical model of independence and constant split have piled up to such an extent that the binomial variability is deeply buried and unnoticeable. Graphical treatment of such large samples may still be worthwhile because it brings the results more vividly to the eye." (Frederick Mosteller & John W Tukey, "The Uses and Usefulness of Binomial Probability Paper?", Journal of the American Statistical Association 44, 1949) 

♾️On Complex Numbers X (Quantum Mechanics I)

"Meantime, there is no doubt a certain crudeness in the use of a complex wave function. If it were unavoidable in principle, and not merely a facilitation of the calculation, this would mean that there are in principle two wave functions, which must be used together in order to obtain information on the state of the system. [...] Our inability to give more accurate information about this is intimately connected with the fact that, in the pair of equations [considered], we have before us only the substitute - extraordinarily convenient for the calculation, to be sure - for a real wave equation of probably the fourth order, which, however, I have not succeeded in forming for the non-conservative case."(Edwin Schrödinger, "Quantisation as a Problem of Proper Values" , Annalen der Physik Vol. 81 (4), 1926)

"Our bra and ket vectors are complex quantities, since they can be multiplied by complex numbers and are then of the same nature as before, but they are complex quantities of a special kind which cannot be split up into real and pure imaginary parts. The usual method of getting the real part of a complex quantity, by taking half the sum of the quantity itself and its conjugate, cannot be applied since a bra and a ket vector are of different natures and cannot be added." (Paul Dirac, "The Principles of Quantum Mechanics", 1930)

"In his desire to consider at any cost the propagation phenomenon of the waves ψ as something real in the classical sense of the word, the author had refused to acknowledge that the whole development of the theory increasingly tended to highlight the essential complex nature of the wave function." (Edwin Schrödinger. "Mémoires sur la mécanique ondulatoire", 1933) [author‘s comment in the French translation] 

"One might think one could measure a complex dynamical variable by measuring separately its real and pure imaginary parts. But this would involve two measurements or two observations, which would be alright in classical mechanics, but would not do in quantum mechanics, where two observations in general interfere with one another - it is not in general permissible to consider that two observations can be made exactly simultaneously, and if they are made in quick succession the first will usually disturb the state of the system and introduce an indeterminacy that will affect the second." (Ernst C K Stückelberg, "Quantum Theory in Real Hilbert Space", 1960) 

"It has been generally believed that only the complex numbers could legitimately be used as the ground field in discussing quantum-mechanical operators. Over the complex field, Frobenius' theorem is of course not valid; the only division algebra over the complex field is formed by the complex numbers themselves. However, Frobenius' theorem is relevant precisely because the appropriate ground field for much of quantum mechanics is real rather than complex." (Freeman Dyson, "The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics" , Journal of Mathematical Physics Vol. 3, 1962)

"Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the ‘three-fold way’ […] This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics." (John C Baez, "Division Algebras and Quantum Theory", 2011)

"It is particularly helpful to use complex numbers to model periodic phenomena, especially to operate with phase differences. Mathematically, one can treat a physical quantity as being complex, but address physical meaning only to its real part. Another possibility is to treat the real and imaginary parts of a complex number as two related (real) physical quantities. In both cases, the structure of complex numbers is useful to make calculations more easily, but no physical meaning is actually attached to complex variables." (Ricardo Karam, "Why are complex numbers needed in quantum mechanics? Some answers for the introductory level", American Journal of Physics Vol. 88 (1), 2020)

"What is essentially different in quantum mechanics is that it deals with complex quantities (e.g. wave functions and quantum state vectors) of a special kind, which cannot be split up into pure real and imaginary parts that can be treated independently. Furthermore, physical meaning is not attached directly to the complex quantities themselves, but to some other operation that produces real numbers (e.g. the square modulus of the wave function or of the inner product between state vectors)." (Ricardo Karam, "Why are complex numbers needed in quantum mechanics? Some answers for the introductory level", American Journal of Physics Vol. 88 (1), 2020) 

From Parts to Wholes (1000-1699)

"Now the kind of philosophy under which we proceed in the whole and in the part is moral philosophy or ethics; because the whole was undertaken not for speculation but for practice."  (Dante Alighieri, "Epistolae" ["Letters"], cca. 14th century)

"Those who devised the eccentrics seen thereby in large measure to have solved the problem of apparent motions with approximate calculations. But meanwhile they introduced a good many ideas which apparently contradict the first principles of uniform motion. Nor could they elicit or deduce from the eccentrics the principal consideration, that is, the structure of the universe and the true symmetry of its parts."  (Nicolaus Copernicus, "De revolutionibus orbium coelestium", 1543)

"Given that annihilation of nature in its entirety is impossible, and that death and dissolution are not appropriate to the whole mass of this entire globe or star, from time to time, according to an established order, it is renewed, altered, changed, and transformed in all its parts." (Giordano Bruno, "The Ash Wednesday Supper", 1584)

"I tell you that if natural bodies have it from Nature to be moved by any movement, this can only be circular motion, nor is it possible that Nature has given to any of its integral bodies a propensity to be moved by straight motion. I have many confirmations of this proposition, but for the present one alone suffices, which is this. I suppose the parts of the universe to be in the best arrangement, so that none is out of its place, which is to say that Nature and God have perfectly arranged their structure. This being so, it is impossible for those parts to have it from Nature to be moved in straight, or in other than circular motion, because what moves straight changes place, and if it changes place naturally, then it was at first in a place preternatural to it, which goes against the supposition. Therefore, if the parts of the world are well ordered, straight motion is superfluous and not natural, and they can only have it when some body is forcibly removed from its natural place, to which it would then return by a straight line, for thus it appears that a part of the earth does [move] when separated from its whole. I said 'it appears to us', because I am not against thinking that not even for such an effect does Nature make use of straight line motion." (Galileo Galilei, [Letter to Francesco Ingoli] 1624)

"For any number there exists a corresponding even number which is its double. Hence the number of all numbers is not greater than the number of even numbers, that is, the whole is not greater than the part." (Gottfried W Leibniz, "De Arte Combinatoria", 1666)

"Consider however (imitating Mathematicians) certainty or truth to be like the whole; and probabilities [to be like] parts, such that probabilities would be to truths what an acute angle [is] to a right [angle]." (Gottfried W Leibniz, [letter to Vincent Placcius] 1687)

"To decrease geometrically is this, that in equal times, first the whole quantity then each of its successive remainders is diminished, always by a like proportional part.“ (John Napier, "The Construction of the Wonderful Canon of Logarithms", 1889)

From Parts to Wholes (BC)

"Further, the state is by nature clearly prior to the family and to the individual, since the whole is of necessity prior to the part; for example, if the whole body be destroyed, there will be no foot or hand, except in an equivocal sense, as we might speak of a stone hand; for when destroyed the hand will be no better than that. But things are defined by their working and power; and we ought not to say that they are the same when they no longer have their proper quality, but only that they have the same name." (Aristotle, Politics, 4th century BC)

"The infinite […] happens to subsist in a way contrary to what is asserted by others: for the infinite is not that beyond which there is nothing, but it is that of which there is always something beyond. […] But that pertaining to which there is nothing beyond is perfect and whole. […] that of which nothing is absent pertaining to the parts […] the whole is that pertaining to which there is nothing beyond. But that pertaining to which something external is absent, that is not all […] But nothing is perfect which has not an end; and the end is a bound. On this account […] Parmenides spoke better than Melissus: for the latter says that the infinite is a whole; but the former, that the whole is finite, and equally balanced from the middle: for to conjoin the infinite with the universe and the whole, is not to connect line with line." (Aristotle, Physics, cca. 4th century BC)

"The proof that the state is a creation of nature and prior to the individual is that the individual, when isolated, is not self-sufficing; and therefore he is like a part in relation to the whole." (Aristotle, Politics, 4th century BC)

"The totality is not, as it were, a mere heap, but the whole is something besides the parts." (Aristotle, "Metaphysics", cca. 335-323 BC)
 
"The whole is more than the sum of its parts." (Aristotle, "Metaphysics", cca. 335-323 BC)
 
"And the whole [is] greater than the part." (Euclid, "Elements", cca. 300 BC)

"We can get some idea of a whole from a part, but never knowledge or exact opinion. Special histories therefore contribute very little to the knowledge of the whole and conviction of its truth. It is only indeed by study of the interconnexion of all the particulars, their resemblances and differences, that we are enabled at least to make a general survey, and thus derive both benefit and pleasure from history." (Polybius, "The Histories", cca. 150 BC)
 
"I say, then, that the universe and all its parts both received their first order from divine providence, and are at all times administered by it." (Marcus T Cicero, "De Natura Deorum" ["On the Nature of the Gods"], 45 BC)
 
"Order gives due measure to the members of a work considered separately, and symmetrical agreement to the proportions of the whole. It is an adjustment according to quantity. By this I mean the selection of modules from the members of the work itself and, starting from these individual parts of members, constructing the whole work to correspond." (Marcus Vitruvius Pollio, "De architectura" ["On Architecture"], cca. 15 BC)

From Parts to Wholes (1930-1939)

"Everything abstract is ultimately part of the concrete. Everything inanimate finally serves the living. That is why every activity dealing in abstraction stands in ultimate service to a living whole." (Edith Stein, "The Ethos of Woman's Professions", 1930)

"To apply the category of cause and effect means to find out which parts of nature stand in this relation. Similarly, to apply the gestalt category means to find out which parts of nature belong as parts to functional wholes, to discover their position in these wholes, their degree of relative independence, and the articulation of larger wholes into sub-wholes." (Kurt Koffka, 1931)

"Even these humble objects reveal that our reality is not a mere collocation of elemental facts, but consists of units in which no part exists by itself, where each part points beyond itself and implies a larger whole. Facts and significance cease to be two concepts belonging to different realms, since a fact is always a fact in an intrinsically coherent whole. We could solve no problem of organization by solving it for each point separately, one after the other; the solution had to come for the whole. Thus we see how the problem of significance is closely bound up with the problem of the relation between the whole and its parts. It has been said: The whole is more than the sum of its parts. It is more correct to say that the whole is something else than the sum of its parts, because summing is a meaningless procedure, whereas the whole-part relationship is meaningful." (Kurt Koffka, "Principles of Gestalt Psychology", 1935)

"Maximal knowledge of a total system does not necessarily include total knowledge of all its parts, not even when these are fully separated from each other and at the moment are not influencing each other at all. Thus it may be that some part of what one knows may pertain to relations […] between the two subsystems (we shall limit ourselves to two), as follows: if a particular measurement on the first system yields this result, then for a particular measurement on the second the valid expectation statistics are such and such; but if the measurement in question on the first system should have that result, then some other expectation holds for that one the second. […] In this way, any measurement process at all or, what amounts to the same, any variable at all of the second system can be tied to the not-yet-known value of any variable at all of the first, and of course vice versa also." (Erwin Schrödinger, "The Present Situation in Quantum Mechanics", 1935)

"The existence of finality within the organism is undeniable. Each part seems to know the present and future needs of the whole, and acts accordingly. The significance of time and space is not the same for our tissues as for our mind. The body perceives the remote as well as the near, the future as well as the present." (Alexis Carrel, 1935)

"[Gestalt:] a system whose parts are dynamically connected in such a way that a change of one part results in a change of all other parts." (Kurt Lewin, "Principles of topological psychology", 1936)

"The realist method starts with the whole in order to distinguish the parts." (Étienne Gilson, "Methodical Realism", 1936)

"An organization is a subordinate system of a specific larger system, the cooperative system, whose components are physical, biological, and personal systems. The relations with other organizations… are outside this specific cooperative system. Other organizations are a part of the social environment of the organization. It is for this reason I have used the phrase 'complex of organizations' rather than 'system'. Usually the most significant relationships of a unit organization are those with the specific cooperative system of which it is a part. It is this system which primarily and on the whole in most instances determines the chief conditions of the organization's existence." (Chester Barnard, "The Functions of the Executive The Functions of the Executive", 1938)

"I see the tasks of social sciences to discover what kinds of order actually do exist in the whole range of the behavior of human beings; what kind of functional relationships between different parts of culture exist in space and over time, and what functionally more useful kinds of order can be created." (Robert S Lynd, "Knowledge of What?", 1939)

"The failure of the social sciences to think through and to integrate their several responsibilities for the common problem of relating the analysis of parts to the analysis of the whole constitutes one of the major lags crippling their utility as human tools of knowledge." (Robert S Lynd, "Knowledge of What?", 1939)

"The whole, though larger than any of its parts, does not necessarily obscure their separate identities." (William O Douglas, "Judicial Opinions", 1939)

On Spacetime (1800-1849)

"Is it not truly wonderful, that, in the constitution of the universe, time and space should every where be so happily combined, that notwithstanding the infinity of wheels and springs which mutually depend on each other, and which are all necessary to the play of the machine, the visible order of nature should, nevertheless, every where preserve the same air of simplicity aud uniformity." (Johann H Lambert, "The System of the World", 1800)

"Genius and science have burst the limits of space, and few observations, explained by just reasoning, have unveiled the mechanism of the universe. Would it not also be glorious for man to burst the limits of time, and, by a few observations, to ascertain the history of this world, and the series of events which preceded the birth of the human race?" (Georges Cuvier, "Essays on the Theory of the Earth", 1822)

"History in general is therefore the development of Spirit in Time, as Nature is the development of the Idea is Space." (Georg W F Hegel, "Lectures on the Philosophy of History", 1837)

"Yet time and space are but inverse measures of the force of the soul. The spirit sports with time." (Ralph W Emerson, "Essays", 1841)

"Great distance in either time or space has wonderful power to lull and render quiescent the human mind." (Abraham Lincoln, [An Address Delivered by Abraham Lincoln], 1842)

"Poetry is the universal art of the spirit which has become free in itself and which is not tied down for its realization to external sensuous material; instead, it launches out exclusively in the inner space and the inner time of ideas and feelings." (G W Friedrich Hegel, "Introduction to Aesthetics", 1842)

"Language has time as its element; all other media have space as their element." (Søren Kierkegaard, "Either/Or: A Fragment of Life", 1843)

On Spacetime (-1799)

"In time you will know all with certainty; Time is the only test of honest men, one day is space enough to know a rogue." (Sophocles, "Oedious the King", 429 BC)

"The objection we are dealing with argues from the standpoint of an agent that presupposes time and acts in time, but did not institute time. Hence the question about 'why God's eternal will produces an effect now and and not earlier' presupposes that time exists; for 'now' and 'earlier' are segments of time. With regard to the universal production of things, among which time is also to be counted, we should not ask, 'Why now and not earlier?' Rather we should ask: 'Why did God wish this much time to intervene?' And this depends on the divine will, which is perfectly free to assign this or any other quantity to time. The same may be noted with respect to the dimensional quantity of the world. No one asks why God located the material world in such and such a place rather than higher up or lower down or in some other position; for there is no place outside the world. The fact that God portioned out so much quantity to the world that no part of it would be beyond the place occupied in some other locality, depends on the divine will. However, although there was no time prior to the world and no place outside the world, we speak as if there were. Thus we say that before the world existed there was nothing except God, and that there is no body lying outside the world. But in thus speaking of 'before' and 'outside,' we have in mind nothing but time and place as they exist in our imagination." (Thomas Aquinas, "Compendium Theologiae" ["Compendium of Theology"], cca. 1265 [unfinished])

"There is a very different relationship between [...] space and duration. For we do not ascribe various durations to the different parts of space, but say that all endure together. The moment of duration is the same at Rome and at London, on the Earth and on the stars, and throughout the heavens. And just as we understand any moment of duration to be diffused throughout all spaces, according to its kind, without any thought of its parts, so it is no more contradictory that Mind also, according to its kind, can be diffused through space without any thought of its parts." (Isaac Newton, De Gravitatione et Aequipondio Fluridorum, cca. 1664-1668) 

"All our dignity consists of thought. It is from there that we must be lifted up and not from space and time, which we could never fill. So let us work on thinking well. That is the principle of morality." (Blaise Pascal, "Pensées", 1670)

"Not at all as far as its absolute, intrinsic nature is concerned. [,,,] Whether things run or stand still, whether we sleep or wake, time flows in its even tenor (aequo tenore tempus labitur). Even if all the stars would have remained at the places where they had been created, nothing would have been lost to time (nihil inde quicquam tempori decessisset). The temporal relations of earlier, afterwards, and simultaneity, even in that tranquil state, would have had their proper existence (prius, posterius, simul etiam in illo transquillo statu fuisset in se)."  (Isaac Barrow, "Lectiones Geometricae", 1672)

"Just as space existed before the world was created and even now there exists an infinite space beyond the world (with which God coexists) [...] so time exists before the world and simultaneously with the world (prius mundo et simul cum mundo)." (Isaac Barrow, "Lectiones Geometricae", 1672)

"Time absolutely is quantity, admitting in some manner the chief affections of quantity, equality, inequality, and proportion; nor do I believe there is anyone but allows that those things existed equal times, which rose and perished simultaneously." (Isaac Barrow, "Lectiones Geometricae", 1672)

"Although time, space, place, and motion are very familiar to everyone, it must be noted that these quantities are popularly conceived solely with reference to the objects of sense perception." (Isaac Newton, "The Principia: Mathematical Principles of Natural Philosophy", 1687)

"To measure motion, space is as necessary to be considered as time....[They] are made use of to denote the position of finite: real beings, in respect one to another, in those infinite uniform oceans of duration and space." (John Locke, "An Essay Concerning Human Understanding", 1689)

"If a plurality of states of things is assumed to exist which involves no opposition to each other, they are said to exist simultaneously. Thus we deny that what occurred last year and this year are simultaneous, for they involve incompatible states of the same thing.  If one of two states which are not simultaneous involves a reason for the other, the former is held to be prior, the latter posterior. My earlier state involves a reason for the existence of my later state. And since my prior state, by reason of the connection between all things, involves the prior state of other things as well, it also involves a reason for the later state of these other things and is thus prior to them. Therefore whatever exists is either simultaneous with other existences or prior or posterior." (Gottfried W Leibniz, "Initium rerum Mathematicarum metaphysica", 1715)

"Time is the order of existence of those things which are not simultaneous. Thus time is the universal order of changes when we do not take into consideration the particular kinds of change. Duration is magnitude of time. If the magnitude of time is diminished uniformly and continuously, time disappears into moment, whose magnitude is zero. Space is the order of coexisting things, or the order of existence for things which are simultaneous." (Gottfried W Leibniz, "Initium rerum Mathematicarum metaphysica", 1715)

"Time, matter, space - all, it may be, are no more than a point." (Denis Diderot, "Lettre sur les aveugles", 1749)

"Any point has a real mode of existence, through which it is where it is; & another, due to which it exists at the time when it does exist. These real modes of existence are to me real time & space ; the possibility of these modes, hazily apprehended by us, is, to my mind, empty space & again empty time, so to speak ; in other words, space & imaginary time." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

"Further we believe that GOD Himself is present everywhere throughout the whole of the undoubtedly divisible space that all bodies occupy; & yet He is onefold in the highest degree & admits not of any composite nature whatever. Moreover, the same idea seems to depend on an analogy between space & time. For, just as rest is a conjunction with a continuous series of all the instants In the interval of time during which the rest endures; so also this virtual extension is a conjunction of one instant of time with a continuous series of all the points of space throughout which this one-fold entity extends virtually. Hence, just as rest is believed to exist in Nature, so also are we bound to admit virtual extension; & if this is admitted, then it will be possible for the primary elements of matter to be simple, & yet not absolutely non-extended." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

"Nature is the system of laws established by the Creator for the existence of things and for the succession of creatures. Nature is not a thing, because this thing would be everything. Nature is not a creature, because this creature would be God. But one can consider it as an immense vital power, which encompasses all, which animates all, and which, subordinated to the power of the first Being, has begun to act only by his order, and still acts only by his concourse or consent. […] Time, space and matter are its means, the universe its object, motion and life its goal." (Georges-Louis L de Buffon, "Histoire Naturelle, Générale et Particulière, Avec la Description du Cabinet du Roi", 1764)

On Spacetime (1975-1999)

"Our look is as bound by time-space as our brain. We never look, we never see beyond this limitation; we do not know how to look through and beyond these fragmentary frontiers. But the eyes have to see beyond them, penetrating deeply and widely, without choosing, without shelter; they have to wander beyond man-made frontiers of ideas and values and to feel beyond love. Then there is a benediction which no god can give." (Jiddu Krishnamurti," Krishnamurti's Notebook", 1976)

"According to the special theory there is a finite limit to the speed of causal chains, whereas classical causality allowed arbitrarily fast signals. Foundational studies […] soon revealed that this departure from classical causality in the special theory is intimately related to its most dramatic consequences: the relativity of simultaneity, time dilation, and length contraction. By now it had become clear that these kinematical effects are best seen as consequences of Minkowski space-time, which in turn incorporates a nonclassical theory of causal structure. However, it has not widely been recognized that the converse of this proposition is also true: the causal structure of Minkowski space-time contains within itself the entire geometry (topological and metrical structure) of Minkowski space-time." (John A. Winnie," The Causal Theory of Space-Time", 1977) 

"The structure of space-time, taken as a whole, is the subject matter of the science called cosmology. Since you are asking about all space and all time in cosmology, you are interested in the entire universe, everywhere and everywhen, viewed as a static geometrical object." (Rudolf B Rucker," Geometry, Relativity and the Fourth Dimension", 1977)

"Instead of thinking of space and time as a stage, on which the drama of matter unfolds, we have to imagine some ultra- modern theater, in which the stage itself becomes one of the actors." (John Stachel, "Einstein's Odyssey: His Journey from Special to General Relativity", 1979)

"It is unscientific to say that within the many billions of galactic systems, ours is the only planet that supports life in advanced form. Nature shuns one of a kind as much as it abhors a vacuum. Given infinite time and space, anything that occurs at one place or time in the universe will occur elsewhere or ‘elsewhen’." (Norman Cousins, "Rendezvous with Infinity", Cosmic Search Magazine Vol. 1 (1), 1979)

"No matter how finely you subdivide time and space, each tiny division contains infinity." (Frank Herbert, "Dune Genesis", 1980)

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

"The quantum is that embarrassing little piece of thread that always hangs from the sweater of space-time. Pull it and the whole thing unravels." (Fred A Wolf, "Star Wave: Mind Consciousness of Quantum Physics", 1984)

"The nothingness ‘before’ the creation of the universe is the most complete void that we can imagine - no space, time, or matter existed. It is a world without place, without duration or eternity, without number - it is what mathematicians call ‘the empty set’. Yet this unthinkable void converts itself into the plenum of existence - a necessary consequence of physical laws. Where are these laws written into that void? What ‘tells’ the void that is pregnant with a possible universe? It would seem that, even the void is subject to law, a logic that exists prior to space and time." (Heinz R Pagels, "Perfect Symmetry: The Search for the Beginning of Time", 1985)

"The theory of relativity does, however, force us to change fundamentally our ideas of space and time. We must accept that time is not completely separate from and independent of space, but is combined with it to form an object called space-time." (Stephen Hawking, "A Brief History of Time: From the Big Bang to Black Holes", 1988)

"Theoretical physicists are accustomed to living in a world which is removed from tangible objects by two levels of abstraction. From tangible atoms we move by one level of abstraction to invisible fields and particles. A second level of abstraction takes us from fields and particles to the symmetry-groups by which fields and particles are related. The superstring theory takes us beyond symmetry-groups to two further levels of abstraction. The third level of abstraction is the interpretation of symmetry-groups in terms of states in ten-dimensional space-time. The fourth level is the world of the superstrings by whose dynamical behavior the states are defined." (Freeman J Dyson, "Infinite in All Directions", 1988)

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

"Time and space are finite in extent, but they don't have any boundary or edge. They would be like the surface of the earth, but with two more dimensions." (Stephen Hawking, "Black Holes and Baby Universes and Other Essays", 1993)

"Minkowski, building on Einstein's work, had now discovered that the Universe is made of a four-dimensional ‘spacetime’ fabric that is absolute, not relative." (Kip S Thorne, "Black Holes and Time Warps: Einstein's Outrageous Legacy" , 1994)

"Arm chair reflections on the concept of causation [are] not going to yield new insights. The grandfather paradox is simply a way of pointing to the fact that if the usual laws of physics are supposed to hold true in a chronology violating spacetime, then consistency constraints emerge. [To understand these constraints] involves solving problems in physics, not armchair philosophical reflections." (John Earman, “Recent Work on Time Travel", 1995) 

"Yet everything has a beginning, everything comes to an end, and if the universe actually began in some dense explosion, thus creating time and space, so time and space are themselves destined to disappear, the measure vanishing with the measured, until with another ripple running through the primordial quantum field, something new arises from nothingness once again." (David Berlinski, "A Tour of the Calculus", 1995)

"And of course the space the wave function live in, and (therefore) the space we live in, the space in which any realistic understanding of quantum mechanics is necessarily going to depict the history of the world as playing itself out […] is configuration-space. And whatever impression we have to the contrary (whatever impression we have, say, of living in a three-dimensional space, or in a four dimensional spacetime) is somehow flatly illusory." (David Albert, "Elementary Quantum Metaphysics", 1996)

"A line normally represents space, but if we reinterpret that line as representing the passage of time, then presumably we are looking at 'shapes that repeat indefinitely along a temporal line at regular intervals'. Well, nearly. 'Shape' is the wrong word for something that exists in time. What word should we use? Let's reason it out by analogy - one of the sharpest weapons in the mathematical armoury. A shape is a (more or less) complicated collection of points in space; the temporal analogue is a complicated collection of points in time. A point in time is usually called an 'event'. So instead of 'shape' we have a 'series of events'. That is, the temporal pattern corresponding to the simplest frieze is a series of events that repeats indefinitely at regular intervals of time." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"A single snapshot of the pattern. A fixed spiral is elegant in form, but it does not possess symmetry in the mathematical sense. However, a rotating spiral does possess a dynamic symmetry - as does any continuously rotating object. It is a symmetry that takes place in both space and time. Start with the spiral, let time pass. It rotates to a new position. Now rotate it backwards through the appropriate angle, and it looks exactly as it did to begin with. This is a space-time symmetry: translate time and rotate space. Indeed, the rotating spiral has lots of space-time symmetries, because every period of time corresponds to some spatial rotation." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"Dynamics involves a crucial element: time. Symmetry is about shapes, which are things in space. How can symmetries apply to time? [...] We can even put both together, and look at patterns in space and time combined. Turning space into time is a typical mathematician's trick, a kind of 'technology transfer'. However, the 'technology' involved is conceptual technology - ideas - rather than gadgetry. No mathematician would ever waste a good idea by restricting it to just one incarnation. Many physically different systems possess an underlying mathematical identity: they obey rules which, in the abstract, are exactly the same - and must therefore lead to corresponding conclusions." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"From a physical point of view, time is very different from space - and this difference is often built into mathematical systems that model the physical world. Sometimes, however, we can exploit the very general nature of mathematics, and from this point of view a spatial distance and an interval of time are simply the values of certain numerical quantities, or variables. The difference between space and time then becomes a matter of interpretation - the same underlying mathematics can have several different meanings." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"Where did those symmetries come from? From the even more extensive set of symmetries of the (idealised) uniform state in the infinite dish. The instability of that state caused certain symmetries to be eliminated, but others persist. For target patterns, some rotations and reflections persist. For spirals, what persists is the space-time symmetries 'let time pass and then rotate back'. In a very curious sense, the patterns that we see in the spirals are evidence of other patterns that might have been - the unstable uniform state with its enormous amount of (totally boring) symmetry. They are 'caused' by something that doesn't actually happen." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"In an infinite universe, every point in space-time is the center." (David Zindell, "War in Heaven", 1998) 

"Spacetime tells matter how to move; matter tells spacetime how to curve." (John A Wheeler, "Geons, Black Holes and Quantum Foam: A Life in Physics" , 1998) 

On Spacetime (1900-1924)

"The most ordinary things are to philosophy a source of insoluble puzzles. In order to explain our perceptions it constructs the concept of matter and then finds matter quite useless either for itself having or for causing perceptions in a mind. With infinite ingenuity it constructs a concept of space or time and then finds it absolutely impossible that there be objects in this space or that processes occur during this time [...] The source of this kind of logic lies in excessive confidence in the so-called laws of thought." (Ludwig E Boltzmann, "On Statistical Mechanics", 1904)

"Time and Space [...] It is not nature which imposes them upon us, it is we who impose them upon nature because we find them convenient." (Henri Poincaré, "The Value of Science", 1905)

"The most violent revolutions in an individual's beliefs leave most of his old order standing. Time and space, cause and effect, nature and history, and one's own biography remain untouched. New truth is always a go-between, a smoother-over of transitions. It marries old opinion to new fact so as ever to show a minimum of jolt, a maximum of continuity." (William James, "What Pragmatism Means", 1907)

"The objects of our perception invariably include places and times in combination. Nobody has ever noticed a place except at a time, or a time except at a place. But I still respect the dogma that both space and time have independent significance. A point of space at a point of time, that is a system of values x, y, z, t, I will call a world-point." (Hermann Minkowski, "Space and Time", [Address to the 80th Assembly of German Natural Scientists and Physicians] 1908)

"The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality." (Hermann Minkowski, "Space and Time", [Address to the 80th Assembly of German Natural Scientists and Physicians] 1908)

"The belief that mathematics, because it is abstract, because it is static and cold and gray, is detached from life, is a mistaken belief. Mathematics, even in its purest and most abstract estate, is not detached from life. It is just the ideal handling of the problems of life, as sculpture may idealize a human figure or as poetry or painting may idealize a figure or a scene. Mathematics is precisely the ideal handling of the problems of life, and the central ideas of the science, the great concepts about which its stately doctrines have been built up, are precisely the chief ideas with which life must always deal and which, as it tumbles and rolls about them through time and space, give it its interests and problems, and its order and rationality." (Cassius J Keyser,"The Humanization of the Teaching of Mathematics", 1912)

"The true scientific mind is not to be tied down by its own conditions of time and space. It builds itself an observatory erected upon the border line of present, which separates the infinite past from the infinite future. From this sure post it makes its sallies even to the beginning and to the end of all things." (Arthur C Doyle, "The Poison Belt", 1913)

"Science is simply setting out on a fishing expedition to see whether it cannot find some procedure which it can call measurement of space and some procedure which it can call the measurement of time, and something which it can call a system of forces, and something which it can call masses." (Alfred N Whitehead, "The Concept of Nature", 1920)

"The discovery of Minkowski […] is to be found […] in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space. In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude, √-1*ct, proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same role as the three space-coordinates. Formally, these four co-ordinates correspond exactly to the three space co-ordinates in Euclidean geometry." (Albert Einstein,"Relativity: The Special and General Theory", 1920)

"And now, in our time, there has been unloosed a cataclysm which has swept away space, time, and matter hitherto regarded as the firmest pillars of natural science, but only to make place for a view of things of wider scope, and entailing a deeper vision." (Hermann Weyl, "Space, Time, Matter", 1922)

"Science derives its conclusions by the laws of logic from our sense perceptions, Thus it does not deal with the real world, of which we know nothing, but with the world as it appears to our senses. […] All our sense perceptions are limited by and attached to the conceptions of time and space. […] Modern physics has come to the same conclusion in the relativity theory, that absolute space and absolute time have no existence, but, time and space exist only as far as things or events fill them, that is, are forms of sense perception." (Charles P. Steinmetz, "Religion and Modern Science", The Christian Register 1922)

"The scene of action of reality is not a three-dimensional Euclidean space but rather a four-dimensional world, in which space and time are linked together indissolubly. However deep the chasm may be that separates the intuitive nature of space from that of time in our experience, nothing of this qualitative difference enters into the objective world which physics endeavors to crystallize out of direct experience. It is a four-dimensional continuum, which is neither 'time' nor 'space'. Only the consciousness that passes on in one portion of this world experiences the detached piece which comes to meet it and passes behind it as history, that is, as a process that is going forward in time and takes place in space." (Hermann Weyl, "Space, Time, Matter", 1922) 

"In the grandeur of its sweep in space and time, and the beauty and simplicity of the relations which it discloses between the greatest and the smallest things of which we know, it reveals as perhaps nothing else does, the majesty of the order about us which we call nature, and, as I believe, of that Power behind the order, of which it is but a passing shadow." (Henry N Russell, "Annual Report of the Board of Regents of the Smithsonian Institution", 1923)

On Abstraction (1960-1969)

"It is of our very nature to see the universe as a place that we can talk about. In particular, you will remember, the brain tends to compute by organizing all of its input into certain general patterns. It is natural for us, therefore, to try to make these grand abstractions, to seek for one formula, one model, one God, around which we can organize all our communication and the whole business of living." (John Z Young, "Doubt and Certainty in Science: A Biologist’s Reflections on the Brain", 1960)

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

"Relativity is inherently convergent, though convergent toward a plurality of centers of abstract truths. Degrees of accuracy are only degrees of refinement and magnitude in no way affects the fundamental reliability, which refers, as directional or angular sense, toward centralized truths. Truth is a relationship." (R Buckminster Fuller, "The Designers and the Politicians", 1962)

"Scientists, it should already be clear, never learn concepts, laws, and theories in the abstract and by themselves. Instead, these intellectual tools are from the start encountered in a historically and pedagogically prior unit that displays them with and through their applications." (Thomas Kuhn, "The Structure of Scientific Revolutions", 1962)

"With even a superficial knowledge of mathematics, it is easy to recognize certain characteristic features: its abstractions, its precision, its logical rigor, the indisputable character of its conclusions, and finally, the exceptionally broad range for its applications." (Aleksandr D Aleksandrov, 1963)

"A quantity like time, or any other physical measurement, does not exist in a completely abstract way. We find no sense in talking about something unless we specify how we measure it. It is the definition by the method of measuring a quantity that is the one sure way of avoiding talking nonsense..." (Hermann Bondi. "Relativity and Common Sense", 1964)

"If you have a large number of unrelated ideas, you have to get quite a distance away from them to get a view of all of them, and this is the role of abstraction. If you look at each too closely you see too many details. If you get far away things may appear simpler because you can only see the large, broad outlines; you do not get lost in petty details." (John G Kemeny, "Random Essays on Mathematics, Education, and Computers", 1964)

"The interplay between generality and individuality, deduction and construction, logic and imagination - this is the profound essence of live mathematics. Anyone or another of these aspects of mathematics can be found at the center of a given achievement. In a far reaching development all of them will be involved. Generally speaking, such a development will start from the 'concrete', then discard ballast by abstraction and rise to the lofty layers of thin air where navigation and observation are easy: after this flight comes the crucial test for learning and reaching specific goals in the newly surveyed low plains of individual 'reality'. In brief, the flight into abstract generality must start from and return again to the concrete and specific." (Richard Courant, "Mathematics in the Modern World", Scientific American Vol. 211 (3), 1964) 

"A more problematic example is the parallel between the increasingly abstract and insubstantial picture of the physical universe which modern physics has given us and the popularity of abstract and non-representational forms of art and poetry. In each case the representation of reality is increasingly removed from the picture which is immediately presented to us by our senses." (Harvey Brooks, "Scientific Concepts and Cultural Change", 1965)

"Learning is any change in a system that produces a more or less permanent change in its capacity for adapting to its environment. Understanding systems, especially systems capable of understanding problems in new task domains, are learning systems." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"The more we are willing to abstract from the detail of a set of phenomena, the easier it becomes to simulate the phenomena. Moreover we do not have to know, or guess at, all the internal structure of the system but only that part of it that is crucial to the abstraction." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"We realize, however, that all scientific laws merely represent abstractions and idealizations expressing certain aspects of reality. Every science means a schematized picture of reality, in the sense that a certain conceptual construct is unequivocally related to certain features of order in reality […]" (Ludwig von Bertalanffy, "General System Theory", 1968)

"Pure mathematics are concerned only with abstract propositions, and have nothing to do with the realities of nature. There is no such thing in actual existence as a mathematical point, line or surface. There is no such thing as a circle or square. But that is of no consequence. We can define them in words, and reason about them. We can draw a diagram, and suppose that line to be straight which is not really straight, and that figure to be a circle which is not strictly a circle. It is conceived therefore by the generality of observers, that mathematics is the science of certainty." (William Godwin, "Thoughts on Man", 1969)

On Discovery (1850-1899)

"To get to know, to discover, to publish - this is the destiny of a scientist." (François Arago, "De L’Utilité des Pensions", 1855)

"We learn wisdom from failure much more than from success. We often discover what will do, by finding out what will not do; and probably he who never made a mistake never made a discovery." (Samuel Smiles, “Facilities and Difficulties”, 1859)

"Nothing can be more puerile than the complaints sometimes made by certain cultivators of a science, that it is very difficult to make discoveries now that the soil has been exhausted, whereas they were so easily made when the ground was first broken. It is an error begotten by ignorance out of indolence. The first discovery did not drop upon the expectant idler who, with placid equanimity waited for the goods the gods might send, but was heavily obtained by patient, systematic, and intelligent labour; and, beyond all question, the same labour of the same mind which made the first discoveries in the new science, would now succeed in making many more, trampled though the field may be by the restless feet of those unmethodical inquirers who, running to and fro, anxiously exclaim, 'Who will show us any good thing?'" (George Gore, "Psychological Inquiries", Journal of Mental Science, 1862)

"The process of discovery is very simple. An unwearied and systematic application of known laws to nature, causes the unknown to reveal themselves. Almost any mode of observation will be successful at last, for what is most wanted is method." (Henry Thoreau, "A Week on the Concord and Merrimack Rivers", 1862)

"It has often been said that, to make discoveries, one must be ignorant. This opinion, mistaken in itself, nevertheless conceals a truth. It means that it is better to know nothing than to keep in mind fixed ideas based on theories whose confirmation we constantly seek, neglecting meanwhile everything that fails to agree with them." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"The discoverer and the poet are inventors; and they are so because their mental vision detects the unapparent, unsuspected facts, almost as vividly as ocular vision rests on the apparent and familiar." (George H Lewes, "Principles of Success in Literature", 1865)

"Every process has laws, known or unknown, according to which it must take place. A consciousness of them is so far from being necessary to the process, that we cannot discover what they are, except by analyzing the results it has left us." (Lord William T Kelvin , "An Outline of the Necessary Laws of Thought", 1866)

"It is notorious that the same discovery is frequently made simultaneously and quite independently, by different persons. […] It would seem, that discoveries are usually made when the time is ripe for them - that is to say, when the ideas from which they naturally flow are fermenting in the minds of many men." (Sir Francis Galton, "Hereditary Genius", 1869)

"Accurate and minute measurement seems to the nonscientific imagination a less lofty and dignified work than looking for something new. But nearly all the grandest discoveries of science have been but the rewards of accurate measurement and patient long contained labor in the minute sifting of numerical results." (William T Kelvin, "Report of the British Association For the Advancement of Science" Vol. 41, 1871)

"Modern discoveries have not been made by large collections of facts, with subsequent discussion, separation, and resulting deduction of a truth thus rendered perceptible. A few facts have suggested an hypothesis, which means a supposition, proper to explain them. The necessary results of this supposition are worked out, and then, and not till then, other facts are examined to see if their ulterior results are found in Nature." (Augustus de Morgan, "A Budget of Paradoxes", 1872)

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

"Great inventions are never, and great discoveries are seldom, the work of any one mind. Every great invention is really an aggregation of minor inventions, or the final step of a progression. It is not usually a creation, but a growth, as truly so as is the growth of the trees in the forest." (Robert H Thurston, "The Growth of the Steam Engine", Popular Science, 1877) 

"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." (W Stanley Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1877)

"A discoverer is a tester of scientific ideas; he must not only be able to imagine likely hypotheses, and to select suitable ones for investigation, but, as hypotheses may be true or untrue, he must also be competent to invent appropriate experiments for testing them, and to devise the requisite apparatus and arrangements." (George Gore, "The Art of Scientific Discovery", 1878)

"The philosopher believes that the value of his philosophy lies in the whole, in the building: posterity discovers it in the bricks with which he built and which are then often used again for better building: in the fact, that is to say, that building can be destroyed and nonetheless possess value as material." (Friedrich Nietzsche, "Human, all-too-human", 1878)

"Historical investigation not only promotes the understanding of that which now is, but also brings new possibilities before us, by showing that which exists to be in great measure conventional and accidental. From the higher point of view at which different paths of thought converge we may look about us with freer vision and discover routes before unknown." (Ernst Mach, "The Science of Mechanics", 1883)

"In that pure enjoyment experienced on approaching to the ideal, in that eagerness to draw aside the veil from the hidden truth, and even in that discord which exists between the various workers, we ought to see the surest pledges of further scientific success. Science thus advances, discovering new truths, and at the same time obtaining practical results." (Dmitry I Mendeleev, "The Principles of Chemistry" Vol. 1, 1891)

"All great scientists have, in a certain sense, been great artists; the man with no imagination may collect facts, but he cannot make great discoveries." (Karl Pearson, "The Grammar of Science", 1892)

"There is no subject more captivating, more worthy of study, than nature. To understand this great mechanism, to discover the forces which are active, and the laws which govern them, is the highest aim of the intellect of man." (Nikola Tesla, "The Inventions, Researches and Writings of Nikola Tesla|, 1894)

"It is they who hold the secret of the mysterious property of the mind by which error ministers to truth, and truth slowly but irrevocably prevails. Theirs is the logic of discovery, the demonstration of the advance of knowledge and the development of ideas, which as the earthly wants and passions of men remain almost unchanged, are the charter of progress, and the vital spark in history." (Lord John Acton, "The Study of History", [lecture delivered at Cambridge] 1895)

"The folly of mistaking a paradox for a discovery, a metaphor for a proof, a torrent of verbiage for a spring of capital truths, and oneself for an oracle, is inborn in us." (Paul Valéry, "Introduction to the Method of Leonardo da Vinci", 1895)

♾️On Complex Numbers IIX

"There seems to me to be something analogous to polarized intensity in the pure imaginary part; and to unpolarized energy (indifferent to direction) in the real part of a quaternion: and thus we have some slight glimpse of a future Calculus of Polarities. This is certainly very vague […]" (Sir William R Hamilton, "On Quaternions; or on a new System of Imaginaries in Algebra", 1844) 

"Those who can, in common algebra, find a square root of -1, will be at no loss to find a fourth dimension in space in which ABC may become ABCD: or, if they cannot find it, they have but to imagine it, and call it an impossible dimension, subject to all the laws of the three we find possible. And just as √-1 in common algebra, gives all its significant combinations true, so would it be with any number of dimensions of space which the speculator might choose to call into impossible existence." (Augustus De Morgan, "Trigonometry and Double Algebra", 1849)

“The conception of the inconceivable [imaginary], this measurement of what not only does not, but cannot exist, is one of the finest achievements of the human intellect. No one can deny that such imaginings are indeed imaginary. But they lead to results grander than any which flow from the imagination of the poet. The imaginary calculus is one of the master keys to physical science. These realms of the inconceivable afford in many places our only mode of passage to the domains of positive knowledge. Light itself lay in darkness until this imaginary calculus threw light upon light. And in all modern researches into electricity, magnetism, and heat, and other subtile physical inquiries, these are the most powerful instruments.” (Thomas Hill, “The Imagination in Mathematics”, North American Review Vol. 85, 1857)

“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)

"The discovery of Minkowski […] is to be found […] in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space. In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude, √-1*ct, proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same role as the three space-coordinates. Formally, these four co-ordinates correspond exactly to the three space co-ordinates in Euclidean geometry." (Albert Einstein,"Relativity: The Special and General Theory", 1920) 

"As an operation, multiplication by i x i has the same effect as multiplication by -1; multiplication by i has the same effect as a rotation by a right angle, and these interpretations […] are consistent. […] Although the interpretation by means of rotations proves nothing, it may suggest that there is no occasion for anyone to muddle himself into a state of mystic wonderment over nothing about the grossly misnamed ‘imaginaries’." (Eric T Bell, "Gauss, the Prince of Mathematicians", 1956)

"How are we to explain the contrast between the matter-of-fact way in which √-1 and other imaginary numbers are accepted today and the great difficulty they posed for learned mathematicians when they first appeared on the scene? One possibility is that mathematical intuitions have evolved over the centuries and people are generally more willing to see mathematics as a matter of manipulating symbols according to rules and are less insistent on interpreting all symbols as representative of one or another aspect of physical reality. Another, less self-congratulatory possibility is that most of us are content to follow the computational rules we are taught and do not give a lot of thought to rationales." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2009)

 "All these questions he [the master?] does not pose. So we have to ask them: is the ‘Ring I’ a trap to catch the master or is the ‘Ring I’ a vessel of understanding? In quantum theory it specifies a formula which includes the irrational in a symbol of totality, in a holistic ‘cosmogramm’. But the formula has a catch. If one squares i = √-1, although a negative, one obtains a rationally understandable negative number -1. So one can make the irrational disappear through a slight of hand. This formula does not correspond to reality because the irrational that we call the collective unconscious or the objective psyche can never be rational. It remains always creatively spontaneous, not predictable, not manipulatable. Each holistic formula is in that sense also a trap, because it brings about the illusion that one has understood the whole." (Marie Louise von Franz, "Reflexionen zum ‘Ring I’")

William T Thomson [Lord Kelvin] - Collected Quotes

"Every process has laws, known or unknown, according to which it must take place. A consciousness of them is so far from being necessary to the process, that we cannot discover what they are, except by analyzing the results it has left us." (William T Kelvin , "An Outline of the Necessary Laws of Thought", 1866)

"A full mathematical investigation of the mutual action between two vortex rings of any 

given magnitudes and velocities passing one another in any two lines, so directed that 

they never come nearer one another than a large multiple of the diameter of either, is 

a perfect mathematical problem; and the novelty of the circumstances contemplated 

presents difficulties of an exciting character. Its solution will become the foundation of 

the proposed new kinetic theory of gases." (William T Kelvin, 1867)

"Nothing can be more fatal to progress than a too confident reliance on mathematical symbols; for the student is only too apt to take the easier course, and consider the formula not the fact as the physical reality." (William T Kelvin & Peter G Tait, "Treatise on Natural Philosophy", 1867)

"Accurate and minute measurement seems to the nonscientific imagination a less lofty and dignified work than looking for something new. But nearly all the grandest discoveries of science have been but the rewards of accurate measurement and patient long contained labor in the minute sifting of numerical results." (William T Kelvin, "Report of the British Association For the Advancement of Science" Vol. 41, 1871)

"Simplification of modes of proof is not merely an indication of advance in our knowledge of a subject, but is also the surest guarantee of readiness for farther progress." (William T Kelvin, "Elements of Natural Philosophy", 1873)

"All that is shown to the eye; and one of the most beautiful results of mathematics is the means of showing to the eye the law of variation, however complicated, of one independent variable." (William T Kelvin, [Presidential Address] 1883)

"[…] when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely in your thoughts advanced to the state of science." (William T Kelvin, "Electrical Units of Measurement", 1883)

"There cannot be a greater mistake than that of looking superciliously upon the practical applications of science. The life and soul of science is its practical application; and just as the great advances in mathematics have been made through the desire of discovering the solution of problems which were of a highly practical kind in mathematical science, so in physical science many of the greatest advances that have been made from the beginning of the world to the present time have been made in earnest desire to turn the knowledge of the properties of matter to some purpose useful to mankind." (William T Kelvin, "Electrical Units of Measurement", 1883)

"I have no satisfaction in formulas unless I feel their arithmetical magnitude." (William T Kelvin, [lecture] 1884)

"The scientific man sees and feels beauty as much as any mere observer - as much as any artist or painter. But he also sees something underlying that beauty; he wishes to learn something of the actions and forces producing those beautiful results." (William T Kelvin, "The Bangor Laboratories", 1885) 

"I do think [...] that you would find it would lose nothing by omitting the word 'vector' throughout. It adds nothing to the clearness or simplicity of the geometry, whether of two dimensions or three dimensions. Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell." (William T Kelvin, [Letter to Robert B Hayward] 1892)

"Quaternions came from Hamilton [...] and have been an unmixed evil to those who have touched them in any way. Vector is a useless survival [...] and has never been of the slightest use to any creature." (William T Kelvin, 1892)

"Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell." (William T Kelvin, 1892) 

"Symmetrical equations are good in their place, but "vector" is a useless survival, or offshoot, from quaternions, and has never been of the slightest use to any creature. Hertz wisely shunted it, but unwisely he adopted temporarily Heaviside’s nihilism. He even tended to nihilism in dynamics, as I warned you soon after his death. He would have grown out of all this, I believe, if he had lived. He certainly was the opposite pole of nature to a nihilist in his experimental work, and in his Doctorate Thesis on the impact of elastic bodies." (William T Kelvin, [footnote in Letter to George F FitzGerald] 1896) 

"I never satisfy myself until I can make a mechanical model of a thing. If I can make a mechanical model, I understand it." (William T Kelvin, 1904) 

"A scientific theory is a tool and not a creed." (William T Kelvin)

"Do not imagine that mathematics is hard and crabbed, and repulsive to common sense. It is merely the etherealization of common sense." (William T Kelvin)

"Mathematics is the only good metaphysics." (William T Kelvin)

"Paradoxes have no place in science. Their removal is the substitution of true for false statements and thoughts." (William T Kelvin, "On Sun's Heat", [Lecture])

"Science is bound, by the everlasting vow of honour, to face fearlessly every problem which can be fairly presented to it." (William T Kelvin)

"When you are face to face with a difficulty, you are up against a discovery." (William T Kelvin)

"When you call a thing mysterious, all that means is that you don't understand it." (William T Kelvin)