On Method VI (Scientific Method)

"[...] scientific method is simply the attempt to acquire knowledge of general laws directly or indirectly by experience, by the use of our five senses. The only limitations that can be assigned to the applicability of this process are those due to the character of experience. Anything that is logically related to experience by discoverable laws and is capable of description in general terms can be dealt with by the scientific method." (Arthur D Ritchie, "Scientific Method: An Inquiry Into the Character and Validity of Natural Laws", 1923)

"Science attempts to establish an understanding of all types of phenomena. Many different explanations can sometimes be given that agree qualitatively with experiments or observations. However, when theory and experiment quantitatively agree, then we can usually be more confident in the validity of the theory. In this manner mathematics becomes an integral part of the scientific method." (Richard Haberman, "Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow", 1998)

"Scientific method is not much different from our day-to-day ways of learning about the world. Without really thinking about the steps or the standards, common sense invokes the same process of evidence and reasoning as scientists more explicitly follow." (Peter Kosso, "A Summary of Scientific Method", 2011)

"Scientific method is the gateway into scientific discoveries that in turn prompt technological advances and cultural influences." (Hugh G Gauch Jr., "Scientific Method in Brief", 2012)

"The traditional scientific method is hypothesis driven. The researcher formulates a theory of how the world works, and then seeks to support or reject this hypothesis based on data." (Steven S Skiena, "The Data Science Design Manual", 2017)

"Its primary function was to make previously invisible phenomena subject to direct inspection in a graphic display […] The graphic method had another function, that of communication to the scientific community and educated readers. These displays made complex phenomena palpable and concrete." (Michael Friendly & Howard Wainer, "A History of Data Visualization and Graphic Communication", 2021)

"The general principles of starting with a well-defined question, engaging in careful observation, and then formulating hypotheses and assessing the strength of evidence for and against them became known as the scientific method." (Michael Friendly & Howard Wainer, "A History of Data Visualization and Graphic Communication", 2021)

"We are accustomed to intellectual diffusion taking place from the natural and physical sciences into the social sciences; certainly that is the direction taken for both calculus and the scientific method. But statistical graphics in particular, and statistics in general, took the reverse route." (Michael Friendly & Howard Wainer, "A History of Data Visualization and Graphic Communication", 2021)

On Least Squares Method

"From the foregoing we see that the two justifications each leave something to be desired. The first depends entirely on the hypothetical form of the probability of the error; as soon as that form is rejected, the values of the unknowns produced by the method of least squares are no more the most probable values than is the arithmetic mean in the simplest case mentioned above. The second justification leaves us entirely in the dark about what to do when the number of observations is not large. In this case the method of least squares no longer has the status of a law ordained by the probability calculus but has only the simplicity of the operations it entails to recommend it." (Carl Friedrich Gauss, "Anzeige: Theoria combinationis observationum erroribus minimis obnoxiae: Pars prior", Göttingische gelehrte Anzeigen, 1821)

"[…] in the Law of Errors we are concerned only with the objective quantities about which mathematical reasoning is ordinarily exercised; whereas in the Method of Least Squares, as in the moral sciences, we are concerned with a psychical quantity - the greatest possible quantity of advantage." (Francis Y Edgeworth, "The method of least squares", 1883) 

"The method of least squares is used in the analysis of data from planned experiments and also in the analysis of data from unplanned happenings. The word 'regression' is most often used to describe analysis of unplanned data. It is the tacit assumption that the requirements for the validity of least squares analysis are satisfied for unplanned data that produces a great deal of trouble." (George E P Box, "Use and Abuse of Regression", 1966)

"At the heart of probabilistic statistical analysis is the assumption that a set of data arises as a sample from a distribution in some class of probability distributions. The reasons for making distributional assumptions about data are several. First, if we can describe a set of data as a sample from a certain theoretical distribution, say a normal distribution (also called a Gaussian distribution), then we can achieve a valuable compactness of description for the data. For example, in the normal case, the data can be succinctly described by giving the mean and standard deviation and stating that the empirical (sample) distribution of the data is well approximated by the normal distribution. A second reason for distributional assumptions is that they can lead to useful statistical procedures. For example, the assumption that data are generated by normal probability distributions leads to the analysis of variance and least squares. Similarly, much of the theory and technology of reliability assumes samples from the exponential, Weibull, or gamma distribution. A third reason is that the assumptions allow us to characterize the sampling distribution of statistics computed during the analysis and thereby make inferences and probabilistic statements about unknown aspects of the underlying distribution. For example, assuming the data are a sample from a normal distribution allows us to use the t-distribution to form confidence intervals for the mean of the theoretical distribution. A fourth reason for distributional assumptions is that understanding the distribution of a set of data can sometimes shed light on the physical mechanisms involved in generating the data." (John M Chambers et al, "Graphical Methods for Data Analysis", 1983)

"Least squares' means just what it says: you minimise the (suitably weighted) squared difference between a set of measurements and their predicted values. This is done by varying the parameters you want to estimate: the predicted values are adjusted so as to be close to the measurements; squaring the differences means that greater importance is placed on removing the large deviations." (Roger J Barlow, "Statistics: A guide to the use of statistical methods in the physical sciences", 1989)

"Principal components and principal factor analysis lack a well-developed theoretical framework like that of least squares regression. They consequently provide no systematic way to test hypotheses about the number of factors to retain, the size of factor loadings, or the correlations between factors, for example. Such tests are possible using a different approach, based on maximum-likelihood estimation." (Lawrence C Hamilton, "Regression with Graphics: A second course in applied statistics", 1991)

"Fuzzy models should make good predictions even when they are asked to predict on regions that were not excited during the construction of the model. The generalization capabilities can be controlled by an appropriate initialization of the consequences (prior knowledge) and the use of the recursive least squares to improve the prior choices. The prior knowledge can be obtained from the data." (Jairo Espinosa et al, "Fuzzy Logic, Identification and Predictive Control", 2005)

"Often when people relate essentially the same variable in two different groups, or at two different times, they see this same phenomenon - the tendency of the response variable to be closer to the mean than the predicted value. Unfortunately, people try to interpret this by thinking that the performance of those far from the mean is deteriorating, but it’s just a mathematical fact about the correlation. So, today we try to be less judgmental about this phenomenon and we call it regression to the mean. We managed to get rid of the term 'mediocrity', but the name regression stuck as a name for the whole least squares fitting procedure - and that’s where we get the term regression line." (Richard D De Veaux et al, "Stats: Data and Models", 2016)

On the Monte Carlo Method

"The Monte Carlo method is a numerical method of solving mathematical problems by the simulation of random variables. [...] One advantageous feature of the Monte Carlo method is the simple structure of the computation algorithm. As a rule, a program is written to carry out one random trial [...]. This trial is repeated N times, each trial being independent of the rest, and then the results of all trials are averaged. Therefore, the Monte Carlo method is sometimes called the method of statistical trials." (Ilya M Sobol, "A Primer for the Monte Carlo Method", 1994)

"To understand what kinds of problems are solvable by the Monte Carlo method, it is important to note that the method enables simulation of any process whose development is influenced by random factors. Second, for many mathematical problems involving no chance, the method enables us to artificially construct a probabilistic model (or several such models), making possible the solution of the problems." (Ilya M Sobol, "A Primer for the Monte Carlo Method", 1994)

"Indeed, the frequency of crashes in the Monte Carlo simulations was much smaller than the frequency of crashes in the real data: if one of the most frequently used benchmarks of the industry is incapable of reproducing the observed frequency of crashes, this indeed means that there is something to explain that may require new concepts and methods." (Didier Sornette, "Why Stock Markets Crash: Critical Events in Complex Systems", 2003)

"Monte Carlo simulations handle uncertainty by using a computer’s random number generator to determine outcomes. Done over and over again, the simulations show the distribution of the possible outcomes. [...] The beauty of these Monte Carlo simulations is that they allow users to see the probabilistic consequences of their decisions, so that they can make informed choices. [...] Monte Carlo simulations are one of the most valuable applications of data science because they can be used to analyze virtually any uncertain situation where we are able to specify the nature of the uncertainty [...]" (Gary Smith & Jay Cordes, "The 9 Pitfalls of Data Science", 2019)

"The Monte Carlo tree search method is naturally suited to non-deterministic settings such as card games or backgammon. Minimax trees are not well suited to non-deterministic settings because of the inability to predict the opponent’s moves while building the tree. On the other hand, Monte Carlo tree search is naturally suited to handling such settings, since the desirability of moves is always evaluated in an expected sense. The randomness in the game can be naturally combined with the randomness in move sampling in order to learn the expected outcomes from each choice of move." (Charu C Aggarwal, "Artificial Intelligence: A Textbook", 2021)

"The nice thing with Monte Carlo is that you play a game of let’s pretend, like this: first of all there are ten scenarios with different probabilities, so let’s first pick a probability. The dice in this case is a random number generator in the computer. You roll the dice and pick a scenario to work with. Then you roll the dice for a certain speed, and you roll the dice again to see what direction it took. The last thing is that it collided with the bottom at an unknown time so you roll dice for the unknown time. So now you have speed, direction, starting point, time. Given them all, I know precisely where it [could have] hit the bottom. You have the computer put a point there. Rolling dice, I come up with different factors for each scenario. If I had enough patience, I could do it with pencil and paper. We calculated ten thousand points. So you have ten thousand points on the bottom of the ocean that represent equally likely positions of the sub. Then you draw a grid, count the points in each cell of the grid, saying that 10% of the points fall in this cell, 1% in that cell, and those percentages are what you use for probabilities for the prior for the individual distributions." (Henry R Richardson) [in (Sharon B McGrayne, "The Theory That Would Not Die", 2011)]

Out of Context: On Scientific method (Definitions)

"[...] scientific method is simply the attempt to acquire knowledge of general laws directly or indirectly by experience, by the use of our five senses. The only limitations that can be assigned to the applicability of this process are those due to the character of experience. Anything that is logically related to experience by discoverable laws and is capable of description in general terms can be dealt with by the scientific method." (Arthur D Ritchie, "Scientific Method: An Inquiry Into the Character and Validity of Natural Laws", 1923)

"Scientific method is what working scientists do, not what other people or even they themselves may say about it." (Percy W Bridgman, "Reflections of a Physicist", 1950)

"Scientific method is the way to truth, but it affords, even in principle, no unique definition of truth. Any so-called pragmatic definition of truth is doomed to failure equally." (Willard v O Quine, "Word and Object", 1960)

"The scientific method is a potentiation of common sense, exercised with a specially firm determination not to persist in error if any exertion of hand or mind can deliver us from it." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"Scientific method is concerned with efficient ways of generating knowledge." (George E P Box, "Total Quality: Its Origins and its Future", 1995)

"Scientific method is not much different from our day-to-day ways of learning about the world. Without really thinking about the steps or the standards, common sense invokes the same process of evidence and reasoning as scientists more explicitly follow." (Peter Kosso, "A Summary of Scientific Method, "2011")

"Scientific method is the gateway into scientific discoveries that in turn prompt technological advances and cultural influences." (Hugh G Gauch Jr., "Scientific Method in Brief", 2012)

"The scientific method is the foundation of modern research. It’s how we prove a theory. It’s how we demonstrate cause and effect. It’s how we discover, innovate, and invent." (Kristin H Jarman, "The Art of Data Analysis: How to answer almost any question using basic statistics", 2013)

The traditional scientific method is hypothesis driven. The researcher formulates a theory of how the world works, and then seeks to support or reject this hypothesis based on data." (Steven S Skiena, "The Data Science Design Manual", 2017)

Out of Context: On Science (Definitions)

"Science is, I believe, nothing but trained and organized common sense [...]" (Thomas H Huxley, "On the Educational Value of the Natural History Sciences", 1854)

"Science is organized knowledge; and before knowledge can be organized, some of it must first be possessed." (Herbert Spencer, "The Art of Education", The North British Review, 1854)

"In its broadest sense science is organised knowledge, and its methods consist of the observation and classification of the phenomena of which we become conscious through our senses, and the investigation of the causes of which these are the effects." (Richard Strachey, Nature Vol. 12 [E], 1875)

"Science is the observation of things possible, whether present or past; prescience is the knowledge of things which may come to pass, though but slowly." (Leonardo da Vinci, "The Notebooks of Leonardo da Vinci", 1883)

"Science is not the monopoly of the naturalist or the scholar, nor is it anything mysterious or esoteric. Science is the search for truth, and truth is the adequacy of a description of facts." (Paul Carus, "Philosophy as a Science", 1909)

"Science is the only truth and it is the great lie. It knows nothing, and people think it knows everything. It is misrepresented. People think that science is electricity, automobilism, and dirigible balloons. It is something very different. It is life devouring itself. It is the sensibility transformed into intelligence. It is the need to know stifling the need to live. It is the genius of knowledge vivisecting the vital genius." (Rémy de Gourmont, "Art and Science", cca. 1905-1909)

"Science is frankly empirical in method and aim; it seeks to discover the laws of concrete being and becoming, and to formulate these in the simplest terms, which are either immediate data of experience or verifiably derived therefrom." (J Arthur Thomson, "The System of Animate Nature" Vol. 1, 1920)

"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)

"Science is a magnificent force, but it is not a teacher of morals." (William J Bryan, "Undelivered Trial Summation Scopes Trial", 1925)

"Science is either an important statement of systematic theory correlating observations of a common world or is the daydream of a solitary intelligence with a taste for the daydream of publication." (Alfred N Whitehead, "Process and Reality: An Essay in Cosmology", 1929)

"Science is feasible when the variables are few and can be enumerated; when their combinations are distinct and clear." (Paul Valéry, "Moralités" ["Morality"], 1932)

"Science is the attempt to discover, by means of observation, and reasoning based upon it, first, particular facts about the world, and then laws connecting facts with one another and (in fortunate cases) making it possible to predict future occurrences." (Bertrand Russell, "Religion and Science, Grounds of Conflict", 1935)

"Science, in the broadest sense, is the entire body of the most accurately tested, critically established, systematized knowledge available about that part of the universe which has come under human observation." (Austin L Porterfield, "Creative Factors in Scientific Research", 1941)

"Physics is not about the real world, it is about ‘abstractions’ from the real world, and this is what makes it so scientific." (Anthony Standen, "Science is a Sacred Cow", 1950)

"Science, then, is the attentive consideration of common experience; it is common knowledge extended and refined." (George Santayana, "The Life of Reason, or the Phases of Human Progress", 1954)

"Science is the creation of concepts and their exploration in the facts." (Jacob Bronowski, "Science and Human Values", 1956)

"Science is the reduction of the bewildering diversity of unique events to manageable uniformity within one of a number of symbol systems [...]" (Aldous L Huxley, "Essay", Daedalus, 1962)

"Science is what scientists do. Science is knowledge, a body of information about the external world. Science is the ability to predict. Science is power, it is engineering. Science explains, or gives causes and reasons." (John Bremer "What Is Science?" [in "Notes on the Nature of Science"], 1962)

"Science is the combined effort to find out what sort of behavior ensues when various conditions are fulfilled." (H Rom Harré, "Philosophical Issues and Conceptual Change", Theory Into Practice Vol. 10 (2), 1971)

"Science is a product of the human mind, but this product is as objective as a cathedral."  (Karl R Popper, "The Logic and Evolution of Scientific Theory", [in "All Life is Problem Solving", 1999] 1972)

"Science is systematic organisation of knowledge about the universe on the basis of explanatory hypotheses which are genuinely testable." (Francisco J Ayala, "Studies in the Philosophy of Biology: Reduction and Related Problems", 1974)

"Within a Metaphysics of Quality, science is a set of static intellectual patterns describing this reality, but the patterns are not the reality they describe." (Robert M Pirsig, "Zen and the Art of Motorcycle Maintenance", 1974)

"Science is a quintessentially human activity, not a mechanized, robot-like accumulation of objective information, leading by laws of logic to inescapable interpretation." (Stephen J Gould, "Ever Since Darwin", 1977)

"Engineering or Technology is the making of things that did not previously exist, whereas science is the discovering of things that have long existed." (David Billington, "The Tower and the Bridge: The New Art of Structural Engineering", 1983)

"Science is human experience systematically extended (by intent, methodology and instrumentation) for the purpose of learning more about the natural world and for the critical empirical testing and possible falsification of all ideas about the natural world." (Robert E Kofahl, Correctly Redefining Distorted Science: A Most Essential Task", Creation Research Society Quarterly Vol. 23, 1986)

"Science is a mechanism. It's a way of trying to improve your knowledge of nature. It's a system for testing your thoughts against the universe and seeing whether they match." (Isaac Asimov, [Interview by Bill Moyers] 1988)

"Science is more than a mere attempt to describe nature as accurately as possible. Frequently the real message is well hidden, and a law that gives a poor approximation to nature has more significance than one which works fairly well but is poisoned at the root." (Robert H March, "Physics for Poets", 1996)

"Science is like photographing a series of close-ups with your back to the sun. No matter which way you move, your shadow always falls across the scene you photograph. No matter what you do, you can never efface yourself from the photographed scene." (F David Peat, "From Certainty to Uncertainty", 2002)

"Science is that story our society tells itself about the cosmos." (F David Peat, "From Certainty to Uncertainty", 2002)

"Science is the art of the appropriate approximation." (Byron K Jennings, "On the Nature of Science", Physics in Canada Vol. 63 (1), 2007)

"Science is not only the enterprise of harnessing nature to serve the practical needs of humankind. It is also part of man’s unending search for knowledge about the universe and his place within it." (Henry P Stapp, "Mindful Universe: Quantum Mechanics and the Participating Observer", 2007)

"Science is about discovery, but it is also about communication. An idea can hardly be said to exist if you do not awaken that same idea in someone else." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"Science isn’t about being right. It is about convincing others of the correctness of an idea through a methodology all will accept using data everyone can trust. New ideas take time to be accepted because they compete with others that have already passed the test." (Tom Koch, "Commentary: Nobody loves a critic: Edmund A Parkes and John Snow’s cholera", International Journal of Epidemiology Vol. 42 (6), 2013)

"Science, at its core, is simply a method of practical logic that tests hypotheses against experience." (John M Greer, "After Progress: Reason and Religion at the End of the Industrial Age", 2015)

"Science is an endeavor to understand and describe the real world out there to (at best) alleviate and enrich human existence." (Achim Zielesny, "From Curve Fitting to Machine Learning" 2^nd Ed., 2016)

"Science is a game - but a game with reality, a game with sharpened knives." (Erwin Schrödinger)

"Science is not a technique or a body of knowledge, though it uses both. It is rather an attitude of inquiry, or observation and reasoning, with respect to the world. It can be developed, not by memorizing facts or juggling formulas to get an answer, but only by actual practice of scientific observation and reasoning." (Karl T Compton)

"Science is the knowledge of the many, orderly and methodically-arranged, so as to become comprehended by one." (Sir John Herschel)

On Knowledge (1875-1899)

"[…] it must be noticed that these diagrams do not naturally harmonize with the propositions of ordinary life or ordinary logic. […] The great bulk of the propositions which we commonly meet with are founded, and rightly founded, on an imperfect knowledge of the actual mutual relations of the implied classes to one another. […] one very marked characteristic about these circular diagrams is that they forbid the natural expression of such uncertainty, and are therefore only directly applicable to a very small number of such propositions as we commonly meet with." (John Venn, "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings", 1880)

"Science is the observation of things possible, whether present or past; prescience is the knowledge of things which may come to pass, though but slowly." (Leonardo da Vinci, "The Notebooks of Leonardo da Vinci", 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)

"I am convinced that it is impossible to expound the methods of induction in a sound manner, without resting them on the theory of probability. Perfect knowledge alone can give certainty, and in nature perfect knowledge would be infinite knowledge, which is clearly beyond our capacities. We have, therefore, to content ourselves with partial knowledge, - knowledge mingled with ignorance, producing doubt." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1887)

"Just as, in the map of a half-explored country, we see detached bits of rivers, isolated mountains, and undefined plains, not connected into any complete plan, so a new branch of knowledge consists of groups of facts, each group standing apart, so as not to allow us to reason from one to another." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1887)

"We say it is ‘explanation’ but it is only in ‘description’ that we are in advance of the older stages of knowledge and science. We describe better we explain just as little as our predecessors." (Friedrich W Nietzsche, "The Joyful Wisdom", 1887)

"The smallest group of facts, if properly classified and logically dealt with, will form a stone which has its proper place in the great building of knowledge, wholly independent of the individual workman who has shaped it." (Karl Pearson, "The Grammar of Science", 1892)

"The true aim of the teacher must be to impart an appreciation of method and not a knowledge of facts." (Karl Pearson, "The Grammar of Science", 1892)

"There is no short cut to truth, no way to gain a knowledge of the universe except through the gateway of scientific method." (Karl Pearson, "The Grammar of Science", 1892)

"Facts are not much use, considered as facts. They bewilder by their number and their apparent incoherency. Let them be digested into theory, however, and brought into mutual harmony, and it is another matter. Theory is of the essence of facts. Without theory scientific knowledge would be only worthy of the mad house." (Oliver Heaviside, "Electromagnetic Theory", 1893)

"Mature knowledge regards logical clearness as of prime importance: only logically clear images does it test as to correctness; only correct images does it compare as to appropriateness. By pressure of circumstances the process is often reversed. Images are found to be suitable for a certain purpose; are next tested as to their correctness ; and only in the last place purged of implied contradictions." (Heinrich Hertz, "The Principles of Mechanics Presented in a New Form", 1894)

"Without a theory all our knowledge of nature would be reduced to a mere inventory of the results of observation. Every scientific theory must be regarded as an effort of the human mind to grasp the truth, and as long as it is consistent with the facts, it forms a chain by which they are linked together and woven into harmony." (Thomas Preston, "The Theory of Heat", 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)

"In order to comprehend and fully control arithmetical concepts and methods of proof, a high degree of abstraction is necessary, and this condition has at times been charged against arithmetic as a fault. I am of the opinion that all other fields of knowledge require at least an equally high degree of abstraction as mathematics, - provided, that in these fields the foundations are also everywhere examined with the rigour and completeness which is actually necessary." (David Hilbert, "Die Theorie der algebraischen Zahlkorper", 1897)

On Differential Equations IV

"Problems relative to the uniform propagation, or to the varied movements of heat in the interior of solids, are reduced […] to problems of pure analysis, and the progress of this part of physics will depend in consequence upon the advance which may be made in the art of analysis. The differential equations […] contain the chief results of the theory; they express, in the most general and concise manner, the necessary relations of numerical analysis to a very extensive class of phenomena; and they connect forever with mathematical science one of the most important branches of natural  philosophy." (Jean-Baptiste-Joseph Fourier, "The Analytical Theory of Heat", 1822)

"It is well known that the central problem of the whole of modern mathematics is the study of the transcendental functions defined by differential equations." (Felix Klein, "Lectures on Mathematics", 1911)

"Men have fallen in love with statues and pictures. I find it easier to imagine a man falling in love with a differential equation, and I am inclined to think that some mathematicians have done so. Even in a nonmathematician like myself, some differential equations evoke fairly violent physical sensations to those described by Sappho and Catallus when viewing their mistresses. Personally, I obtain an even greater 'kick' from finite difference equations, which are perhaps more like those which an up-to-date materialist would use to describe human behavior." (John B S Haldane, "The Inequality of Man and Other Essays", 1932)

"The method of successive approximations is often applied to proving existence of solutions to various classes of functional equations; moreover, the proof of convergence of these approximations leans on the fact that the equation under study may be majorised by another equation of a simple kind. Similar proofs may be encountered in the theory of infinitely many simultaneous linear equations and in the theory of integral and differential equations. Consideration of semiordered spaces and operations between them enables us to easily develop a complete theory of such functional equations in abstract form." (Leonid V Kantorovich, "On one class of functional equations", 1936)

"The emphasis on mathematical methods seems to be shifted more towards combinatorics and set theory - and away from the algorithm of differential equations which dominates mathematical physics." (John von Neumann & Oskar Morgenstern, "Theory of Games and Economic Behavior", 1944)

"The study of changes in the qualitative structure of the flow of a differential equation as parameters are varied is called bifurcation theory. At a given parameter value, a differential equation is said to have stable orbit structure if the qualitative structure of the flow does not change for sufficiently small variations of the parameter. A parameter value for which the flow does not have stable orbit structure is called a bifurcation value, and the equation is said to be at a bifurcation point." (Jack K Hale & Hüseyin Kocak, "Dynamics and Bifurcations", 1991)

"Dynamical systems that vary in discrete steps […] are technically known as mappings. The mathematical tool for handling a mapping is the difference equation. A system of difference equations amounts to a set of formulas that together express the values of all of the variables at the next step in terms of the values at the current step. […] For mappings, the difference equations directly express future states in terms of present ones, and obtaining chronological sequences of points poses no problems. For flows, the differential equations must first be solved. General solutions of equations whose particular solutions are chaotic cannot ordinarily be found, and approximations to the latter are usually determined by numerical methods." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Faced with the overwhelming complexity of the real world, time pressure, and limited cognitive capabilities, we are forced to fall back on rote procedures, habits, rules of thumb, and simple mental models to make decisions. Though we sometimes strive to make the best decisions we can, bounded rationality means we often systematically fall short, limiting our ability to learn from experience." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"Following the traditional classification in the field of control systems, a system that describes the input-output behavior in a way similar to a mathematical mapping without involving a differential operator or equation is called a static system. In contrast, a system described by a differential operator or equation is called a dynamic system." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)

"The standard view among most theoretical physicists, engineers and economists is that mathematical models are syntactic (linguistic) items, identified with particular systems of equations or relational statements. From this perspective, the process of solving a designated system of (algebraic, difference, differential, stochastic, etc.) equations of the target system, and interpreting the particular solutions directly in the context of predictions and explanations are primary, while the mathematical structures of associated state and orbit spaces, and quantity algebras – although conceptually important, are secondary." (Zoltan Domotor, "Mathematical Models in Philosophy of Science" [Mathematics of Complexity and Dynamical Systems, 2012])

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Frederick Mosteller - Collected Quotes

 "As usual we may make the errors of I) rejecting the null hypothesis when it is true, II) accepting the null hypothesis when it is false. But there is a third kind of error which is of interest because the present test of significance is tied up closely with the idea of making a correct decision about which distribution function has slipped furthest to the right. We may make the error of III) correctly rejecting the null hypothesis for the wrong reason." (Frederick Mosteller, "A k-Sample Slippage Test for an Extreme Population", The Annals of Mathematical Statistics 19, 1948)

"Errors of the third kind happen in conventional tests of differences of means, but they are usually not considered, although their existence is probably recognized. It seems to the author that there may be several reasons for this among which are 1) a preoccupation on the part of mathematical statisticians with the formal questions of acceptance and rejection of null hypotheses without adequate consideration of the implications of the error of the third kind for the practical experimenter, 2) the rarity with which an error of the third kind arises in the usual tests of significance." (Frederick Mosteller, "A k-Sample Slippage Test for an Extreme Population", The Annals of Mathematical Statistics 19, 1948)

"For many purposes graphical accuracy is sufficient. The speed of graphical processes, and more especially the advantages of visual presentation in pointing out facts or clues which might otherwise be overlooked, make graphical analysis very valuable." (Frederick Mosteller & John W Tukey, "The Uses and Usefulness of Binomial Probability Paper?", Journal of the American Statistical Association 44, 1949)

"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)

"Scientific and technological advances have made the world we live in complex and hard to understand. […] Science itself shows the same growing complexity. We often hear that 'one man can no longer cover a broad enough field' and that 'there is too much narrow specialization'. And yet these complexities must be met - and resolved. At all levels, decisions must be made which involve consideration of more than a single field." (Frederick Mosteller et al, "The Education of a Scientific Generalist", Science 109,1949)

"Mathematical models for empirical phenomena aid the development of a science when a sufficient body of quantitative information has been accumulated. This accumulation can be used to point the direction in which models should be constructed and to test the adequacy of such models in their interim states. Models, in turn, frequently are useful in organizing and interpreting experimental data and in suggesting new directions for experimental research." (Robert R. Bush & Frederick Mosteller, "A Mathematical Model for Simple Learning", Psychological Review 58, 1951)

"Almost any sort of inquiry that is general and not particular involves both sampling and measurement […]. Further, both the measurement and the sampling will be imperfect in almost every case. We can define away either imperfection in certain cases. But the resulting appearance of perfection is usually only an illusion." (Frederick Mosteller et al, "Principles of Sampling", Journal of the American Statistical Association Vol. 49 (265), 1954)

"Because representativeness is inherent in the sampling plan and not in the particular sample at hand, we can never make adequate use of sample results without some measure of how well the results of this particular sample are likely to agree with the results of other samples which the same sampling plan might have provided. The ability to assess stability fairly is as important as the ability to represent the population fairly. Modern sampling plans concentrate on both." (Frederick Mosteller et al, "Principles of Sampling", Journal of the American Statistical Association Vol. 49 (265), 1954)

"By sampling we can learn only about collective properties of populations, not about properties of individuals. We can study the average height, the percentage who wear hats, or the variability in weight of college juniors [...]. The population we study may be small or large, but there must be a population - and what we are studying must be a population characteristic. By sampling, we cannot study individuals as particular entities with unique idiosyncrasies; we can study regularities (including typical variabilities as well as typical levels) in a population as exemplified by the individuals in the sample." (Frederick Mosteller et al, "Principles of Sampling", Journal of the American Statistical Association Vol. 49 (265), 1954)

"In many cases general probability samples can be thought of in terms of (1) a subdivision of the population into strata, (2) a self-weighting probability sample in each stratum, and (3) combination of the stratum sample means weighted by the size of the stratum." (Frederick Mosteller et al, "Principles of Sampling", Journal of the American Statistical Association Vol. 49 (265), 1954)

"That which can be and should be representative is the sampling plan, which includes the manner in which the sample was drawn (essentially a specification of what other samples might have been drawn and what the relative chances of selection were for any two possible samples) and how it is to be analyzed. [...] It is clear that many [...] groups fail to be represented in any particular sample, yet this is not a criticism of that sample. Representation is not, and should not be, by groups. It is, and should be, by individuals as members of the sampled population. Representation is not, and should not be, in any particular sample. It is, and should be, in the sampling plan." (Frederick Mosteller et al, "Principles of Sampling", Journal of the American Statistical Association Vol. 49 (265), 1954)

"The main purpose of a significance test is to inhibit the natural enthusiasm of the investigator." (Frederick Mosteller, "Selected Quantitative Techniques", 1954)

"We must emphasize that such terms as 'select at random', 'choose at random', and the like, always mean that some mechanical device, such as coins, cards, dice, or tables of random numbers, is used." (Frederick Mosteller et al, "Principles of Sampling", Journal of the American Statistical Association Vol. 49 (265), 1954)

"We have made the sampling plan representative, not by giving each individual an equal chance to enter the sample and then weighting them equally, but by a more noticeable process of compensation, where those individuals very likely to enter the sample are weighted less, while those unlikely to enter are weighted more when they do appear. The net result is to give each individual an equal chance of affecting the (weighted) sample mean." (Frederick Mosteller et al, "Principles of Sampling", Journal of the American Statistical Association Vol. 49 (265), 1954)

"We realize that if someone just 'grabs a handful', the individuals in the handful almost always resemble one another (on the average) more than do the members of a simple random sample. Even if the 'grabs' [sampling] are randomly spread around so that every individual has an equal chance of entering the sample, there are difficulties. Since the individuals of grab samples resemble one another more than do individuals of random samples, it follows (by a simple mathematical argument) that the means of grab samples resemble one another less than the means of random samples of the same size. From a grab sample, therefore, we tend to underestimate the variability in the population, although we should have to overestimate it in order to obtain valid estimates of variability of grab sample means by substituting such an estimate into the formula for the variability of means of simple random samples. Thus using simple random sample formulas for grab sample means introduces a double bias, both parts of which lead to an unwarranted appearance of higher stability." (Frederick Mosteller et al, "Principles of Sampling", Journal of the American Statistical Association Vol. 49 (265), 1954)

"Weighing a sample appropriately is no more fudging the data than is correcting a gas volume for barometric pressure." (Frederick Mosteller et al, "Principles of Sampling", Journal of the American Statistical Association Vol. 49 (265), 1954)

"A primary goal of any learning model is to predict correctly the learning curve - proportions of correct responses versus trials. Almost any sensible model with two or three free parameters, however, can closely fit the curve, and so other criteria must be invoked when one is comparing several models." (Robert R Bush & Frederick Mosteller, "A Comparison of Eight Models?", Studies in Mathematical Learning Theory, 1959)

"A satisfactory prediction of the sequential properties of learning data from a single experiment is by no means a final test of a model. Numerous other criteria - and some more demanding - can be specified. For example, a model with specific numerical parameter values should be invariant to changes in independent variables that explicitly enter in the model." (Robert R Bush & Frederick Mosteller,"A Comparison of Eight Models?", Studies in Mathematical Learning Theory, 1959)

"In the testing of a scientific model or theory, one rarely has a general measure of goodness-of-fit, a universal yardstick by which one accepts or rejects the model. Indeed, science does not and should not work this way; a theory is kept until a better one is found. One way that science does work is by comparing two or more theories to determine their relative merits in handling relevant data."(Robert R Bush & Frederick Mosteller, "A Comparison of Eight Models?", Studies in Mathematical Learning Theory, 1959)

"In a problem, the great thing is the challenge. A problem can be challenging for many reasons: because the subject matter is intriguing, because the answer defies unsophisticated intuition, because it illustrates an important principle, because of its vast generality, because of its difficulty, because of a clever solution, or even because of the simplicity or beauty of the answer." (Frederick Mosteller, "Fifty Challenging Problems in Probability with Solutions", 1965)

"Using data from the population as it stands is a dangerous substitute for testing." (Frederick Mosteller & Gale Mosteller, "New Statistical Methods in Public Policy. Part I: Experimentation", Journal of Contemporary Business 8, 1979)

"Although we often hear that data speak for themselves, their voices can be soft and sly." (Frederick Mosteller, "Beginning Statistics with Data Analysis", 1983)

"The law of truly large numbers states: With a large enough sample, any outrageous thing is likely to happen." (Frederick Mosteller, "Methods for Studying Coincidences", Journal of the American Statistical Association Vol. 84, 1989)

"It is easy to lie with statistics, but easier to lie without them [...]" (Frederick Mosteller)

On Generalization (1920-1929)

"If we are not content with the dull accumulation of experimental facts, if we make any deductions or generalizations, if we seek for any theory to guide us, some degree of speculation cannot be avoided. Some will prefer to take the interpretation which seems to be most immediately indicated and at once adopted as an hypothesis; others will rather seek to explore and classify the widest possibilities which are not definitely inconsistent with the facts. Either choice has its dangers: the first may be too narrow a view and lead progress into a cul-de-sac; the second may be so broad that it is useless as a guide and diverge indefinitely from experimental knowledge." (Sir Arthur S Eddington, "The Internal Constitution of the Stars Observatory", Vol. 43, 1920)

"It is well to be explicit when a positive generalization is made from negative experimental evidence." (Arthur Eddington, "Space, Time and Gravitation: An Outline of the General Relativity", 1920)

"Generalization is the golden thread which binds many facts into one simple description." (Joseph W Mellor, "A Comprehensive Treatise on Inorganic and Theoretical Chemistry", 1922)

"[…] a history of mathematics is largely a history of discoveries which no longer exist as separate items, but are merged into some more modern generalization, these discoveries have not been forgotten or made valueless. They are not dead, but transmuted." (John W N Sullivan, "The History of Mathematics in Europe", 1925)

"Number knows no limitations, either from the side of the infinitely great or from the side of the infinitely small, and the facility it offers for generalization is too great for us not to be tempted by it." (Émile Borel, "Space and Time", 1926)

"[…] the statistical prediction of the future from the past cannot be generally valid, because whatever is future to any given past, is in tum past for some future. That is, whoever continually revises his judgment of the probability of a statistical generalization by its successively observed verifications and failures, cannot fail to make more successful predictions than if he should disregard the past in his anticipation of the future. This might be called the ‘Principle of statistical accumulation’." (Clarence I Lewis, "Mind and the World-Order: Outline of a Theory of Knowledge", 1929)

"The true method of discovery is like the flight of an aeroplane. It starts from the ground of particular observation; it makes a flight in the thin air of imaginative generalization; and it again lands for renewed observation rendered acute by rational interpretation." (Alfred N Whitehead, "Process and Reality", 1929)

"Without doubt, if we are to go back to that ultimate, integral experience, unwarped by the sophistications of theory, that experience whose elucidation is the final aim of philosophy, the flux of things is one ultimate generalization around which we must weave our philosophical system." (Alfred N Whitehead, "Process and Reality: An Essay in Cosmology", 1929)

On Axioms (1900-1909)

"If geometry is to serve as a model for the treatment of physical axioms, we shall try first by a small number of axioms to include as large a class as possible of physical phenomena, and then by adjoining new axioms to arrive gradually at the more special theories. […] The mathematician will have also to take account not only of those theories coming near to reality, but also, as in geometry, of all logically possible theories. We must be always alert to obtain a complete survey of all conclusions derivable from the system of axioms assumed." (David Hilbert, 1900)

"When we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. The axioms so set up are at the same time the definitions of those elementary ideas; and no statement within the realm of the science... is held to be correct unless it can be derived from axioms by means of a finite number of logical steps. Upon closer consideration the question arises: Whether, in any way, certain statements of single axioms depend upon one another, and whether the axioms may not therefore contain certain parts in common, which must be isolated if one wishes to arrive at a system of axioms that shall be altogether independent of one another." (David Hilbert, "Mathematische Probleme", Gŏttinger Nachrichten, 1900)

"No theorem can be new unless a new axiom intervenes in its demonstration; reasoning can only give us immediately evident truths borrowed from direct intuition; it would only be an intermediary parasite." (Henri Poincaré, "Science and Hypothesis", 1901)

"Syllogistic reasoning remains incapable of adding anything to the data that are given it; the data are reduced to axioms, and that is all we should find in the conclusions." (Henri Poincaré, "Science and Hypothesis", 1901)

"Like almost every subject of human interest, this one [mathematics] is just as easy or as difficult as we choose to make it. A lifetime may be spent by a philosopher in discussing the truth of the simplest axiom. The simplest fact as to our existence may fill us with such wonder that our minds will remain overwhelmed with wonder all the time." (John Perry, "Teaching of Mathematics", 1902)

"No theorem can be new unless a new axiom intervenes in its demonstration; reasoning can only give us immediately evident truths borrowed from direct intuition; it would only be an intermediary parasite." (Henri Poincaré, "Science and Hypothesis", 1902)

"The requisites for the axioms are various. They should be simple, in the sense that each axiom should enumerate one and only one statement. The total number of axioms should be few. A set of axioms must be consistent, that is to say, it must not be possible to deduce the contradictory of any axiom from the other axioms. According to the logical 'Law of Contradiction,' a set of entities cannot satisfy inconsistent axioms. Thus the existence theorem for a set of axioms proves their consistency. Seemingly this is the only possible method of proof of consistency." (Alfred N Whitehead, "The axioms of projective geometry, 1906) 

"Every definition implies an axiom, since it asserts the existence of the object defined. The definition then will not be justified, from the purely logical point of view, until we have proved that it involves no contradiction either in its terms or with the truths previously admitted." (Henri Poincaré," Science and Method", 1908)

"It has been argued that mathematics is not or, at least, not exclusively an end in itself; after all it should also be applied to reality. But how can this be done if mathematics consisted of definitions and analytic theorems deduced from them and we did not know whether these are valid in reality or not. One can argue here that of course one first has to convince oneself whether the axioms of a theory are valid in the area of reality to which the theory should be applied. In any case, such a statement requires a procedure which is outside logic." (Ernst Zermelo, "Mathematische Logik - Vorlesungen gehalten von Prof. Dr. E. Zermelo zu Göttingen im S. S", 1908)

"It is by logic that we prove, but by intuition that we discover. [...] Every definition implies an axiom, since it asserts the existence of the object defined. The definition then will not be justified, from the purely logical point of view, until we have proved that it involves no contradiction either in its terms or with the truths previously admitted." (Henri Poincaré, "Science and Method", 1908)

"I do in no wise share this view [that the axioms are arbitrary propositions which we assume wholly at will, and that in like manner the fundamental conceptions are in the end only arbitrary symbols with which we operate] but consider it the death of all science: in my judgment the axioms of geometry are not arbitrary, but reasonable propositions which generally have the origin in space intuition and whose separate content and sequence is controlled by reasons of expediency." (Felix Klein, "Elementarmathematik vom hoheren Standpunkte aus", 1909)

Harold Jeffreys - Collected Quotes

"A prior probability used to express ignorance is merely the formal statement of that ignorance." (Harold Jeffreys, "Theory of Probability", 1939)

"An hypothesis that may be true may be rejected because it has not predicted observable results that have not occurred." (Harold Jeffreys, "Theory of Probability", 1939)

"An observation, strictly, is only a sensation. Nobody means that we should reject everything but sensations. But as soon as we go beyond sensations we are making inferences." (Harold Jeffreys, "Theory of Probability", 1939)

"The theory of probability makes it possible to respect the great men on whose shoulders we stand." (Harold Jeffreys, "Theory of Probability", 1939)

"If we have no information relevant to the actual value of the parameter, the probability must be chosen so as to express the fact that we have none." (Harold Jeffreys, "Theory of Probability", 1939)

"It is essential to the possibility of induction that we shall be prepared for occasional wrong decisions." (Harold Jeffreys, "Theory of Probability", 1939)

"The best way of testing differences from a systematic rule is always to arrange our work so as to ask and answer one question at a time." (Harold Jeffreys, "Theory of Probability", 1939)

"The difference made by any ordinary change of the prior probability is comparable with the effect of one extra observation." (Harold Jeffreys, "Theory of Probability", 1939)

"The posterior probabilities of the hypotheses are proportional to the products of the prior probabilities and the likelihoods." (Harold Jeffreys, "Theory of Probability", 1939)

"The whole of the information contained in the observations that is relevant to the posterior probabilities of different hypotheses is summed up in the values that they give to the likelihood." (Harold Jeffreys, "Theory of Probability", 1939)

"What the use of P [the significance level] implies, therefore, is that a hypothesis that may be true may be rejected because it has not predicted observable results that have not occurred." (Harold Jeffreys, "Theory of Probability", 1939)

“Nature does not consist entirely, or even largely, of problems designed by a Grand Examiner to come out neatly in finite terms, and whatever subject we tackle the first need is to overcome timidity about approximating.” (Harold Jeffreys & Bertha S Jeffreys, “Methods of Mathematical Physics”, 1946)

On Automata I

"Ninety-nine [students] out of a hundred are automata, careful to walk in prescribed paths, careful to follow the prescribed custom. This is not an accident but the result of substantial education, which, scientifically defined, is the subsumption of the individual." (William T Harris, "The Philosophy of Education", 1889)

"We are automata entirely controlled by the forces of the medium being tossed about like corks on the surface of the water, but mistaking the resultant of the impulses from the outside for free will. The movements and other actions we perform are always life preservative and tho seemingly quite independent from one another, we are connected by invisible links." (Nikola Tesla, "My Inventions", 1919)

"Besides electrical engineering theory of the transmission of messages, there is a larger field [cybernetics] which includes not only the study of language but the study of messages as a means of controlling machinery and society, the development of computing machines and other such automata, certain reflections upon psychology and the nervous system, and a tentative new theory of scientific method." (Norbert Wiener, "Cybernetics", 1948)

"Automata have begun to invade certain parts of mathematics too, particularly but not exclusively mathematical physics or applied mathematics. The natural systems (e.g., central nervous system) are of enormous complexity and it is clearly necessary first to subdivide what they represent into several parts that to a certain extent are independent, elementary units. The problem then consists of understanding how these elements are organized as a whole. It is the latter problem which is likely to attract those who have the background and tastes of the mathematician or a logician. With this attitude, he will be inclined to forget the origins and then, after the process of axiomatization is complete, concentrate on the mathematical aspects." (John Von Neumann, "The General and Logical Theory of Automata", 1951)

"A world of automata – of creatures that worked like machines – would hardly be worth creating." (Clive S Lewis, Mere Christianity, 1952)

"Cellular automata are discrete dynamical systems with simple construction but complex self-organizing behaviour. Evidence is presented that all one-dimensional cellular automata fall into four distinct universality classes. Characterizations of the structures generated in these classes are discussed. Three classes exhibit behaviour analogous to limit points, limit cycles and chaotic attractors. The fourth class is probably capable of universal computation, so that properties of its infinite time behaviour are undecidable." (Stephen Wolfram, "Nonlinear Phenomena, Universality and complexity in cellular automata", Physica 10D, 1984)

"Cellular automata are mathematical models for complex natural systems containing large numbers of simple identical components with local interactions. They consist of a lattice of sites, each with a finite set of possible values. The value of the sites evolve synchronously in discrete time steps according to identical rules. The value of a particular site is determined by the previous values of a neighbourhood of sites around it." (Stephen Wolfram, "Nonlinear Phenomena, Universality and complexity in cellular automata", Physica 10D, 1984)

"Cellular automata may be considered as discrete dynamical systems. In almost all cases, cellular automaton evolution is irreversible. Trajectories in the configuration space for cellular automata therefore merge with time, and after many time steps, trajectories starting from almost all initial states become concentrated onto 'attractors'. These attractors typically contain only a very small fraction of possible states. Evolution to attractors from arbitrary initial states allows for 'self-organizing' behaviour, in which structure may evolve at large times from structureless initial states. The nature of the attractors determines the form and extent of such structures." (Stephen Wolfram, "Nonlinear Phenomena, Universality and complexity in cellular automata", Physica 10D, 1984)

"Finite Nature is a hypothesis that ultimately every quantity of physics, including space and time, will turn out to be discrete and finite; that the amount of information in any small volume of space-time will be finite and equal to one of a small number of possibilities. [...] We take the position that Finite Nature implies that the basic substrate of physics operates in a manner similar to the workings of certain specialized computers called cellular automata." (Edward Fredkin, "A New Cosmogony", PhysComp ’92: Proceedings of the Workshop on Physics and Computation, 1993)

Complexity vs Mathematics II

"[Mathematics] guides our minds in an orderly way, and furnishes us simple and rational principles by means of which ambiguities are clarified, disorder is converted into order, and complexities are analyzed into their component parts." (Johann B Mencken, "The Charlatanry of the Learned", 1715)

"These sciences, Geometry, Theoretical Arithmetic and Algebra, have no principles besides definitions and axioms, and no process of proof but deduction; this process, however, assuming a most remarkable character; and exhibiting a combination of simplicity and complexity, of rigour and generality, quite unparalleled in other subjects." (William Whewell, "The Philosophy of the Inductive Sciences", 1840)

"The value of mathematical instruction as a preparation for those more difficult investigations, consists in the applicability not of its doctrines but of its methods. Mathematics will ever remain the past perfect type of the deductive method in general; and the applications of mathematics to the simpler branches of physics furnish the only school in which philosophers can effectually learn the most difficult and important of their art, the employment of the laws of simpler phenomena for explaining and predicting those of the more complex." (John S Mill, "System of Logic", 1843)

"It is certainly true that all physical phenomena are subject to strictly mathematical conditions, and mathematical processes are unassailable in themselves. The trouble arises from the data employed. Most phenomena are so highly complex that one can never be quite sure that he is dealing with all the factors until the experiment proves it. So that experiment is rather the criterion of mathematical conclusions and must lead the way." (Amos E Dolbear, "Matter, Ether, Motion", 1894)

"Mathematics, the science of the ideal, becomes the means of investigating, understanding and making known the world of the real. The complex is expressed in terms of the simple. From one point of view mathematics may be defined as the science of successive substitutions of simpler concepts for more complex [...]" (William F White, "A Scrap-book of Elementary Mathematics", 1908)

"A great department of thought must have its own inner life, however transcendent may be the importance of its relations to the outside. No department of science, least of all one requiring so high a degree of mental concentration as Mathematics, can be developed entirely, or even mainly, with a view to applications outside its own range. The increased complexity and specialisation of all branches of knowledge makes it true in the present, however it may have been in former times, that important advances in such a department as Mathematics can be expected only from men who are interested in the subject for its own sake, and who, whilst keeping an open mind for suggestions from outside, allow their thought to range freely in those lines of advance which are indicated by the present state of their subject, untrammelled by any preoccupation as to  applications to other departments of science." (Ernst W Hobson, Nature Vol. 84, [address] 1910)

"Elegance may produce the feeling of the unforeseen by the unexpected meeting of objects we are not accustomed to bring together; there again it is fruitful, since it thus unveils for us kinships before unrecognized. It is fruitful even when it results only from the contrast between the simplicity of the means and the complexity of the problem set; it makes us then think of the reason for this contrast and very often makes us see that chance is not the reason; that it is to be found in some unexpected law. In a word, the feeling of  mathematical elegance is only the satisfaction due to any adaptation of the solution to the needs of our mind, and it is because of this very adaptation that this solution can be for us an instrument. Consequently this esthetic satisfaction is bound up with the economy of thought." (Jules Henri Poincaré, "The Future of Mathematics", Monist Vol. 20, 1910)

"The mathematical laws presuppose a very complex elaboration. They are not known exclusively either a priori or a posteriori, but are a creation of the mind; and this creation is not an arbitrary one, but, owing to the mind’s resources, takes place with reference to experience and in view of it. Sometimes the mind starts with intuitions which it freely creates; sometimes, by a process of elimination, it gathers up the axioms it regards as most suitable for producing a harmonious development, one that is both simple and fertile. The mathematics is a voluntary and intelligent adaptation of thought to things, it represents the forms that will allow of qualitative diversity being surmounted, the moulds into which reality must enter in order to become as intelligible as possible." (Émile Boutroux, "Natural Law in Science and Philosophy", 1914)

"No equation, however impressive and complex, can arrive at the truth if the initial assumptions are incorrect." (Arthur C Clarke, "Profiles of the Future: An Inquiry into the Limits of the Possible", 1973)

"Economists are all too often preoccupied with petty mathematical problems of interest only to themselves. This obsession with mathematics is an easy way of acquiring the appearance of scientificity without having to answer the far more complex questions posed by the world we live in." (Thomas Piketty, Capital in the Twenty-First Century, 2013)

John W N Sullivan - Collected Quotes

"Knowledge for the sake of knowledge, as the history of science proves, is an aim with an irresistible fascination for mankind, and which needs no defense. The mere fact that science does, to a great extent, gratify our intellectual curiosity, is a sufficient reason for its existence." (John W N Sullivan, "The Limitations of Science", 1915)

"[…] science deals with but a partial aspect of reality, and there is no faintest reason for supposing that everything science ignores is less real than what it accepts. [...] Why is it that science forms a closed system? Why is it that the elements of reality it ignores never come in to disturb it? The reason is that all the terms of physics are defined in terms of one another. The abstractions with which physics begins are all it ever has to do with." (John W N Sullivan, "The Limitations of Science", 1915)

"Science deals with but a partial aspect of reality, and […] there is no faintest reason for supposing that everything science ignores is less real than what it accepts. […] Why is it that science forms a closed system? Why is is that the elements of reality it ignores never come in to disturb it? The reason is that all the terms of physics are defined in terms of one another. The abstractions with which physics begins are all it ever has to do with […]" (John W N Sullivan, "The Limitations of Science", 1915)

"Science, like everything else that man has created, exists, of course, to gratify certain human needs and desires. The fact that it has been steadily pursued for so many centuries, that it has attracted an ever-wider extent of attention, and that it is now the dominant intellectual interest of mankind, shows that it appeals to a very powerful and persistent group of appetites." (John W N Sullivan, "The Limitations of Science", 1915)

"Since the primary object of the scientific theory is to express the harmonies which are found to exist in nature, we see at once that these theories must have an aesthetic value. The measure of the success of a scientific theory is, in fact, a measure of its aesthetic value, since it is a measure of the extent to which it has introduced harmony in what was before chaos." (John W N Sullivan, "The Justification of the Scientific Method", The Athenaeum (4644), 1919)

"The measure in which science falls short of art is the measure in which it is incomplete as science." (John W N Sullivan, "The Justification of the Scientific Method", The Athenaeum, 1919)

"[…] a history of mathematics is largely a history of discoveries which no longer exist as separate items, but are merged into some more modern generalization, these discoveries have not been forgotten or made valueless. They are not dead, but transmuted." (John W N Sullivan, "The History of Mathematics in Europe", 1925)

"The electron is not, for example, an enduring something that can be tracked through time. Its mathematical description does not involve that degree of definiteness. Any picture we form of the atom errs, as it were, by excess of solidity. The mathematical symbols refer to entities more indefinite than our pictorial imagination, limited as it is by experience of 'gross matter', can construct." (John W N Sullivan, "The Bases of Modern Science", 1929)

"The present tendency of physics is toward describing the universe in terms of mathematical relations between unimaginable entities." (John W N Sullivan, "The Bases of Modern Science", 1929)

"Science, indeed, tells us a very great deal less about the universe than we have been accustomed to suppose, and there is no reason to believe that all we can ever know must be couched in terms of its thin and largely arbitrary abstractions."  (John W N Sullivan, "Art and Reality", 1964)

"The mathematician is entirely free, within the limits of his imagination, to construct what worlds he pleases. What he is to imagine is a matter for his own caprice; he is not thereby discovering the fundamental principles of the universe nor becoming acquainted with the ideas of God." (John W N Sullivan)

On Engineering VI

"Can one think that because we are engineers, beauty does not preoccupy us or that we do not try to build beautiful, as well as solid and long lasting structures? Aren’t the genuine functions of strength always in keeping with unwritten conditions of harmony? [...] Besides, there is an attraction, a special charm in the colossal to which ordinary theories of art do not apply." (Gustave Eiffel, [interview in 'Le Temps'] 1887)

"The characteristic feature of our age results from the wedding of science and engineering. It is the working together of disciplined curiosity and purposeful ingenuity to create new materials, new forces, and new opportunities which powerfully affect our manner of living and ways of thinking." (Karl T Compton, "A Scientist Speaks: Excerpts from Addresses by Karl Taylor Compton - During the Years 1930-1949", 1955)

"[An engineer's] invention causes things to come into existence from ideas, makes world conform to thought; whereas science, by deriving ideas from observation, makes thought conform to existence." (Carl Mitcham, "Types of Technology", Research in Philosophy & Technology Vol. 1, 1978)

"Engineers use knowledge primarily to design, produce, and operate artifacts. [...] Scientists, by contrast, use knowledge primarily to generate more knowledge." (Walter Vincenti, What Engineers Know and How They Know It, 1990)

"Engineering is quite different from science. Scientists try to understand nature. Engineers try to make things that do not exist in nature. Engineers stress invention. To embody an invention the engineer must put his idea in concrete terms, and design something that people can use. That something can be a device, a gadget, a material, a method, a computing program, an innovative experiment, a new solution to a problem, or an improvement on what is existing. Since a design has to be concrete, it must have its geometry, dimensions, and characteristic numbers. Almost all engineers working on new designs find that they do not have all the needed information. Most often, they are limited by insufficient scientific knowledge. Thus they study mathematics, physics, chemistry, biology and mechanics. Often they have to add to the sciences relevant to their profession. Thus engineering sciences are born." (Yuan-Cheng Fung & Pin Tong, "Classical and Computational Solid Mechanics", 2001)

"Engineering isn't about perfect solutions; it's about doing the best you can with limited resources." (Randy Pausch, "The Last Lecture", 2008)

"The central activity of engineering, as distinguished from science, is the design of new devices, processes and systems." (Myron Tribus, "Rational Descriptions, Decisions and Designs", 2016)

"Engineering is a living branch of human activity and its frontiers are by no means exhausted." (Igor I Sikorsky)

"Engineering is not merely knowing and being knowledgeable, like a walking encyclopedia; engineering is not merely analysis; engineering is not merely the possession of the capacity to get elegant solutions to non-existent engineering problems; engineering is practicing the art of the organizing forces of technological change. [...] Engineers operate at the interface between science and society." (Gordon S Brown)

"The scientist describes what is; the engineer creates what never was." (Theodore von Kármán)

On Engineering IV

"The engineer must be able not only to design, but to execute. A draftsman may be able to design, but unless he is able to execute his designs to successful operation he cannot be classed as an engineer. The production engineer must be able to execute his work as he has planned it. This requires two qualifications in addition to technical engineering ability: He must know men, and he must have creative ability in applying good statistical, accounting, and 'system' methods to any particular production work he may undertake." (Hugo Diemer, "Industrial Engineering", 1905)

"An engineering science aims to organize the design principles used in engineering practice into a discipline and thus to exhibit the similarities between different areas of engineering practice and to emphasize the power of fundamental concepts. In short, an engineering science is predominated by theoretical analysis and very often uses the tool of advanced mathematics." (Qian Xuesen, "Engineering cybernetics", 1954)

"Engineering is a profession, an art of action and synthesis and not simply a body of knowledge. Its highest calling is to invent and innovate." (Hardy Cross, "Education for Innovation", 1968)

"A good scientist is a person with original ideas. A good engineer is a person who makes a design that works with as few original ideas as possible. There are no prima donnas in engineering." (Freeman J Dyson, "Disturbing the Universe", 1979)

"Engineering, like poetry, is an attempt to approach perfection. And engineers, like poets, are seldom completely satisfied with their creations. They notice, even if no one else does, the word that is not quite 'le mot juste' or the hairline crack that blemishes the structure." (Henry Petroski, "To Engineer Is Human: The Role of Failure in Successful Design", 1985)

"All of engineering involves some creativity to cover the parts not known, and almost all of science includes some practical engineering to translate the abstractions into practice." (Richard Hamming, "The Art of Doing Science and Engineering: Learning to Learn", 1991)

"Engineering is the application of scientific principles toward practical ends. If the engineering isn't practical, it's bad engineering." (Steve McConnell, "After the Gold Rush: Creating a True Profession of Software Engineering", 1999)

"All feats of engineering, whether stone dwellings or space stations, require a particular sequence of events. First, the engineer must understand the needs and wants of the society or subgroup of society that is to be served. Second, the engineer must formulate concepts of potential designs that might serve the designated needs and wants. Third, the engineer must analyze the concepts to determine their functionality. Fourth, the engineer must optimize selected candidate designs and choose a single preferred design. And fifth, the engineer must design a production system to realize the selected design." (George A Hazelrigg, "Laws and Models: An Introduction", 2000)

"Indeed, the most important part of engineering work - and also of other scientific work - is the determination of the method of attacking the problem, whatever it may be." (Charles P Steinmetz)

"Designers need to be part engineer. Good design only exists in concert with engineering. That is because form has to follow function, so you focus on function and then give the object a shape to make it appealing." (Ferdinand A Porsche)

On Engineering V (Systems Engineering I)

"The analysis of engineering systems and the understanding of economic structure have advanced since then, and the time is now more ripe to bring these topics into a potentially fruitful marriage." (Arnold Tustin, "The Mechanism of Economic Systems", 1953)

"In a society which is producing more people, more materials, more things, and more information than ever before, systems engineering is indispensable in meeting the challenge of complexity." (Harold Chestnut, "Systems Engineering Tools," 1965)

"Systems engineering is most effectively conceived of as a process that starts with the detection of a problem and continues through problem definition, planning and designing of a system, manufacturing or other implementing section, its use, and finally on to its obsolescence. Further, Systems engineering is not a matter of tools alone; It is a careful coordination of process, tools and people." (Arthur D. Hall, "Systems Engineering from an Engineering Viewpoint" In: Systems Science and Cybernetics. Vol.1 Issue.1, 1965) 

"The Systems engineering method recognizes each system is an integrated whole even though composed of diverse, specialized structures and sub-functions. It further recognizes that any system has a number of objectives and that the balance between them may differ widely from system to system. The methods seek to optimize the overall system functions according to the weighted objectives and to achieve maximum compatibility of its parts." (Harold Chestnut, "Systems Engineering Tools," 1965)

"In the minds of many writers systems engineering is synonomous with component selection and interface design; that is, the systems engineer does not design hardware but decides what types of existing hardware shall be coupled and how they shall be coupled. Complete agreement that this function is the essence of systems engineering will not be found here, for, besides the very important function of systems engineering in systems analysis, there is the role played by systems engineering in providing boundary conditions for hardware design." (A Wayne Wymore, "A Mathematical Theory of Systems Engineering", 1967)

"The purpose and real value of systems engineering is [...] to keep going around the loop; find inadequacies and make improvements." (Robert E Machol, "Mathematicians are useful", 1971)

"System engineering is the art and science of creating effective systems, using whole system, whole life principles." (Derek Hitchins, 1995)

"With the subsequent strong support from cybernetics, the concepts of systems thinking and systems theory became integral parts of the established scientific language, and led to numerous new methodologies and applications - systems engineering, systems analysis, systems dynamics, and so on." (Fritjof Capra, "The Web of Life", 1996)

"If all the theories pertinent to systems engineering could be discussed within a common framework by means of a standard set of nomenclature and definitions, many separate courses might not be required." (A Wayne Wymore)

"The central activity of engineering, as distinguished from science, is the design of new devices, processes and systems." (Myron Tribus, "Rational Descriptions, Decisions and Designs", 2016)

On Engineering I

"In fact 'engineering' now often signifies a new system of thought, a fresh method of attack upon the world’s problems the antithesis of traditionalism, with its precedents and dogmas. (Alfred D Flinn, "Leadership in Economic Progress", Civil Engineering Vol. 2 (4), 1932)

"There may be said to be two kinds of engineering, that which is essentially creative, and that which is practiced in pursuit of known methods." (William L Emmet, "The Autobiography of an Engineer", 1940)

"Science acquires knowledge but has no interest in its practical applications. The applications are the work of engineers." (Edwin P Hubble, "The Nature of Science and Other Lectures", 1954)

"Doing engineering is practicing the art of the organized forcing of technological change." (George Spencer-Brown, Electronics, Vol. 32 (47),  1959)

"Science aims at the discovery, verification, and organization of fact and information [...] engineering is fundamentally committed to the translation of scientific facts and information to concrete machines, structures, materials, processes, and the like that can be used by men." (Eric A Walker, "Engineers and/or Scientists", Journal of Engineering Education Vol. 51, 1961)

"What, then, is science according to common opinion? Science is what scientists do. Science is knowledge, a body of information about the external world. Science is the ability to predict. Science is power, it is engineering. Science explains, or gives causes and reasons." (John Bremer "What Is Science?" [in "Notes on the Nature of Science"], 1962)

"Engineering is the art of skillful approximation; the practice of gamesmanship in the highest form. In the end it is a method broad enough to tame the unknown, a means of combing disciplined judgment with intuition, courage with responsibility, and scientific competence within the practical aspects of time, of cost, and of talent. This is the exciting view of modern-day engineering that a vigorous profession can insist be the theme for education and training of its youth. It is an outlook that generates its strength and its grandeur not in the discovery of facts but in their application; not in receiving, but in giving. It is an outlook that requires many tools of science and the ability to manipulate them intelligently In the end, it is a welding of theory and practice to build an early, strong, and useful result. Except as a valuable discipline of the mind, a formal education in technology is sterile until it is applied." (Ronald B Smith, "Professional Responsibility of Engineering", Mechanical Engineering Vol. 86 (1), 1964)

"Engineering is a method and a philosophy for coping with that which is uncertain at the earliest possible moment and to the ultimate service to mankind. It is not a science struggling for a place in the sun. Engineering is extrapolation from existing knowledge rather than interpolation between known points. Because engineering is science in action - the practice of decision making at the earliest moment - it has been defined as the art of skillful approximation. No situation in engineering is simple enough to be solved precisely, and none worth evaluating is solved exactly. Never are there sufficient facts, sufficient time, or sufficient money for an exact solution, for if by chance there were, the answer would be of academic and not economic interest to society. These are the circumstances that make engineering so vital and so creative." (Ronald B Smith, "Engineering Is…", Mechanical Engineering Vol. 86 (5), 1964)

"Engineering is the conscious application of science to the problems of economic production." (Halbert P Gillette)

"Engineering is the professional and systematic application of science to the efficient utilization of natural resources to produce wealth." (T J Hoover & J C L Fish)

On Tektology

"Tectology, or the theory of structure in organisms, is the comprehensive science of individuality among living natural bodies, which usually represent an aggregate of individuals of various orders.  The task of organic tectology is therefore to identify and explain organic individuality, i.e. to identify the precise natural laws according to which organic matter individualises itself, and according to which most organisms construct a unified form-complex composed of individuals of various orders." (Ernst  Häckel, "Generelle Morphologie der Organismen" ["General Morphology of Organisms"], 1866)

"It should therewith be remembered that as mathematics studies neutral complexes, mathematical thinking is an organizational process and hence its methods, as well as the methods of all other sciences and those of any practice, fall within the province of a general tektology. Tektology is a unique science which must not only work out its own methods by itself but must study them as well; therefore it is the completion of the cycle of sciences." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"Mathematics abstracts from all the particular properties of the elements hidden behind its schemata. This is achieved by mathematics with the help of indifferent symbols, like numbers or letters. Tektology must do likewise. Its generalizations should abstract from the concreteness of elements whose organizational relationships they express, and conceal this concreteness behind indifferent symbols." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"Tektology must discover what modes of organization are observed in nature and human activities; then generalize and systemize these modes; further it should explain them, that is, elaborate abstract schemes of their tendencies and regularities; finally, based on these schemes it must determine the directions of organizational modes development and elucidate their role in the economy of world processes. This general plan is similar to the plan of any other science but the object studied differs essentially. Tektology deals with the organizational experience not of some particular branch but with that of all of them in the aggregate; to put it in other words, tektology embraces the material of all the other sciences, as well as of all the vital practices from which those sciences arose, but considers this material only in respect of methods, i.e. everywhere it takes an interest in the mode of the organization of this material."  (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"The methods of tektology, as is seen, combine the abstract symbolism of mathematics and the experimental character Of the natural sciences. Furthermore, the very formulation of its problems, the very treatment of organizedness by tektology, as has been elucidated, should stick to the social historical viewpoint. And whatever the subject matter, or the content, of tektology , it embraces the whole world of experience. So tektology is really a universal science by its methods and its content."  (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"[…] there is a special relationship, a profound affinity between mathematics and tektology. Mathematical laws do not refer to a particular area of natural phenomena, as the laws of the other, special, sciences do, but to each and all phenomena, considered merely in their quantitative aspect; mathematics is in its own way universal, like tektology."  (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"Two divisions are distinguished in all natural sciences - 'statics' which deals with forms in equilibrium, and 'dynamics' which deals with the same forms, as well as their motion, in the process of change. […] Statics always evolves earlier than dynamics, the former being then reconstructed under the influence of the latter. The relationship between mathematics and tektology is seen to be similar: one represents the standpoint of organizational statics and the other - that of organizational dynamics. The latter standpoint is the more general, for equilibrium is only a particular case of motion, and in essence, is just an ideal case resulting from changes which are completely equal but quite opposite in direction." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"We shall call this universal organizational science the 'Tektology'. The literal translation of this word from the Greek is 'the theory of construction'. 'Construction' is the most generaI and suitable synonym for the modern concept of 'organization'. [...] The aim of tektology is to systematize organizational experience; this science is clearly empirical and should draw its conclusions by way of induction." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"For tektology the unity of experience is not 'discovered', but actively created by organizational means: ‘philosophers wanted to explain the world, but the main point is it change it’ said the greater precursor of organizational science, Karl Marx. The explanation of organizational forms and methods by tektology is directed not to a contemplation of their unity, but to a practical mastery over them." (Alexander Bogdanov, "Tektology: The Universal Organizational Science", 1922)

"Tektology must clarify the modes of organization that are perceived to exist in nature and human activity; then it must generalize and systematize these modes; further it must explain them, that is, propose abstract schemes of their tendencies and laws; finally, based on these schemes, determine the direction of organizational methods and their role in the universal process. This general plan is similar to the plan of any natural science; but the objective of tektology is basically different. Tektology deals with organizational experiences not of this or that specialized field, but of all these fields together. In other words, tektology embraces the subject matter of all the other sciences and of all the human experience giving rise to these sciences, but only from the aspect of method, that is, it is interested only in the modes of organization of this subject matter." (Alexander Bogdanov, "Tektology: The Universal Organizational Science", 1922)

"The strength of an organization lies in precise coordination of its parts, in strict correspondence of various mutually connected functions. This coordination is maintained through constant growth in tektological variety, but not without bounds […] there comes a moment when the parts of the whole become too differentiated in their organization and their resistance to the surrounding environment weakens. This leads sooner or later to disorganization." (Alexander Bogdanov, "Tektology: The Universal Organizational Science", 1922)

"Tektology was the first attempt in the history of science to arrive at a systematic formulation of the principles of organization operating in living and nonliving systems." (Fritjof Capra, "The Web of Life", 1996)

"Tektology is concerned only with activities, but activities are characterized by the fact that they produce changes. From this point of view it is out of the question to think about a simple and pure 'preservation' of forms, one that would constitute a real absence of changes. Preservation is always only a result of immediately equilibrating each of the appearing changes by another opposing change; it Is a dynamic equilibrium of changes."(Alexander Bogdanov) 

On Machines IV (Science vs. Machine I)

"As soon as an Analytical Engine exists, it will necessarily guide the future course of the science. Whenever any result is sought by its aid, the question will then arise - by what course of calculation can these results be arrived at by the machine in the shortest time?"(Charles Babbage, "Passages from the Life of a Philosopher", 1864)

"In Science, all tends to stir, to change, to form fresh surfaces. All denies, destroys, creates, replaces all. What was ground yesterday is put into the hopper again today. The colossal machine, Science, never rests. It is never satisfied; it is insatiable for improvement, of which the absolute knows nothing." (Victor Hugo, "William Shakespeare", 1864)

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

"Science is a magnificent force, but it is not a teacher of morals. It can perfect machinery, but it adds no moral restraints to protect society from the misuse of the machine. It can also build gigantic intellectual ships, but it constructs no moral rudders for the control of storm tossed human vessel. It not only fails to supply the spiritual element needed but some of its unproven hypotheses rob the ship of its compass and thus endangers its cargo." (William J Bryan, "Undelivered Trial Summation Scopes Trial", 1925)

"Science is a magnificent force, but it is not a teacher of morals. It can perfect machinery, but it adds no moral restraints to protect society from the misuse of the machine. It can also build gigantic intellectual ships, but it constructs no moral rudders for the control of storm tossed human vessel. It not only fails to supply the spiritual element needed but some of its unproven hypotheses rob the ship of its compass and thus endangers its cargo." (William J Bryan, 1925)

"The common view of science is that it is a sort of machine for increasing the race’s store of dependable facts. It is that only in part; in even larger part it is a machine for upsetting undependable facts." (Will Durant, 1931)

"The overwhelming presence of machines and instrumentation must be one of the most salient features of the modern scientific laboratory […] The development of science depends at least as much on new machinery as it does on new ideas." (Ronald Giere, "Explaining Science: A Cognitive Approach", 1988)

"Physics is the basic science of matter and energy, and engineering is physics applied to structures and machines. They and chemistry are the sciences that biologists need to explain the structure and mechanism of living things." (R McNeill Alexander, "Dynamics of Dinosaurs and Other Extinct Giants", 1989)

"The fallibility of methods means that there is no cookbook approach to doing science, no formula that can be applied or machine that can be built to generate scientific knowledge […] The skillful application of methods to a challenging problem is one of the great pleasures of science." (Committee on the Conduct of Science [National Academy of Sciences], "On Being a Scientist", 1989)