On Induction (1975-1999)

"Mathematical induction […] is an entirely different procedure. Although it, too, leaps from the knowledge of particular cases to knowledge about an infinite sequence of cases, the leap is purely deductive. It is as certain as any proof in mathematics, and an indispensable tool in almost every branch of mathematics." (Martin Gardner, "Aha! Insight", 1978)

"The word ‘induction’ has two essentially different meanings. Scientific induction is a process by which scientists make observations of particular cases, such as noticing that some crows are black, then leap to the universal conclusion that all crows are black. The conclusion is never certain. There is always the possibility that at least one unobserved crow is not black." (Martin Gardner, "Aha! Insight", 1978)

"Science sometimes improves hypothesis and sometimes disproves them. But proof would be another matter and perhaps never occurs except in the realms of totally abstract tautology. We can sometimes say that if such and such abstract suppositions or postulates are given, then such and such abstract suppositions or postulates are given, then such and such must follow absolutely. But the truth about what can be perceived or arrived at by induction from perception is something else again." (Gregory Bateson, "Mind and Nature, a Necessary Unity", 1979)

"Science, since people must do it, is a socially embedded activity. It progresses by hunch, vision, and intuition. Much of its change through time does not record a closer approach to absolute truth, but the alteration of cultural contexts that influence it so strongly. Facts are not pure and unsullied bits of information; culture also influences what we see and how we see it. Theories, moreover, are not inexorable inductions from facts. The most creative theories are often imaginative visions imposed upon facts; the source of imagination is also strongly cultural." (Stephen J Gould, "The Mismeasure of Man", 1980)

"The purpose of scientific enquiry is not to compile an inventory of factual information, nor to build up a totalitarian world picture of Natural Laws in which every event that is not compulsory is forbidden. We should think of it rather as a logically articulated structure of justifiable beliefs about nature. It begins as a story about a Possible World - a story which we invent and criticize and modify as we go along, so that it winds by being, as nearly as we can make it, a story about real life." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"It is actually impossible in theory to determine exactly what the hidden mechanism is without opening the box, since there are always many different mechanisms with identical behavior. Quite apart from this, analysis is more difficult than invention in the sense in which, generally, induction takes more time to perform than deduction: in induction one has to search for the way, whereas in deduction one follows a straightforward path." (Valentino Braitenberg, "Vehicles: Experiments in Synthetic Psychology", 1984)

"All great theories are expansive, and all notions so rich in scope and implication are underpinned by visions about the nature of things. You may call these visions ‘philosophy’, or ‘metaphor’, or ‘organizing principle’, but one thing they are surely not - they are not simple inductions from observed facts of the natural world." (Stephen J Gould,"Time’s Arrow, Time’s Cycle", 1987)

"Nature is not ‘given’ to us - our minds are never virgin in front of reality. Whatever we say we see or observe is biased by what we already know, think, believe, or wish to see. Some of these thoughts, beliefs and knowledge can function as an obstacle to our understanding of the phenomena. […] mathematics is not a natural science. It is not about the phenomena of the real world, it is not about observation and induction. Mathematical induction is not a method for making generalizations." (Anna Sierpinska,"Understanding in Mathematics", 1994)

"Model building is the art of selecting those aspects of a process that are relevant to the question being asked. As with any art, this selection is guided by taste, elegance, and metaphor; it is a matter of induction, rather than deduction. High science depends on this art." (John H Holland," Hidden Order: How Adaptation Builds Complexity", 1995)

On Induction (1950-1974)

"To say that observations of the past are certain, whereas predictions are merely probable, is not the ultimate answer to the question of induction; it is only a sort of intermediate answer, which is incomplete unless a theory of probability is developed that explains what we should mean by ‘probable’ and on what ground we can assert probabilities." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference." (George Pólya, "Induction and Analogy in Mathematics", 1954)

"[…] the human reason discovers new relations between things not by deduction, but by that unpredictable blend of speculation and insight […] induction, which - like other forms of imagination - cannot be formalized." (Jacob Bronowski, "The Reach of Imagination", 1967)

"Innocent, unbiased observation is a myth." (Sir Peter B Medawar, "Induction and Intuition in Scientific Thought", 1969)

"Induction is aimed at revealing regularities and relationships that are hidden behind the outer aspects of the phenomena under study. Its most common tools are generalization, specialization, and analogy. Generalization arises from an attempt to grasp the significance of observed facts and is then verified by further particular cases." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"Induction is the process of eliciting general laws via observation and the correlation of particular instances. All sciences, including mathematics, make use of the induction method. Now, mathematical induction is applied only by mathematicians in the proof of theorems of a particular kind." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

On Induction (1875-1899)

"Whatever lies beyond the limits of experience, and claims another origin than that of induction and deduction from established data, is illegitimate." (George H Lewes, "The Foundations of a Creed", 1875)

"In the Theory of Numbers it happens rather frequently that, by some unexpected luck, the most elegant new truths spring up by induction." (Carl Friedrich Gauss, Werke, 1876)

"The general mental qualification necessary for scientific advancement is that which is usually denominated 'common sense', though added to this, imagination, induction, and trained logic, either of common language or of mathematics, are important adjuncts." (Joseph Henry, [Address] 1877)

"Mathematics is not the discoverer of laws, for it is not induction; neither is it the framer of theories, for it is not hypothesis; but it is the judge over both, and it is the arbiter to which each must refer its claims; and neither law can rule nor theory explain without the sanction of mathematics." (Benjamin Peirce, "Linear Associative Algebra", American Journal of Mathematics Vol. 4, 1881)

"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)

"There is as great a distinction between mathematics and the mathematical sciences as there is between induction and the inductive sciences. Practically, few cases of induction do not involve, to a greater or less extent, deductions; so few mathematical processes do not involve some strictly logical procedure." (Charles C Everett, "The Science of Thought", 1890)

"In every science, after having analysed the ideas, expressing the more complicated by means of the more simple, one finds a certain number that cannot be reduced among them, and that one can define no further. These are the primitive ideas of the science; it is necessary to acquire them through experience, or through induction; it is impossible to explain them by deduction." (Giuseppe Peano, "Notations de Logique Mathématique", 1894)

"Observe, finally, that this induction is possible only if the same operation can be repeated indefinitely. That is why the theory of chess can never become a science: the different moves of the game do not resemble one another." (Henri Poincaré, "On the Nature of Mathematical Reasoning", 1894)

"If men of science owe anything to us, we may learn much from them that is essential. For they can show how to test proof, how to secure fulness and soundness in induction, how to restrain and to employ with safety hypothesis and analogy." (Lord John Acton, [Lecture] "The Study of History", 1895)

"As for me (and probably I am not alone in this opinion), I believe that a single universally valid principle summarizing an abundance of established experimental facts according to the rules of induction, is more reliable than a theory which by its nature can never be directly verified; so I prefer to give up the theory rather than the principle, if the two are incompatible." (Ernst Zermelo, "Über mechanische Erklärungen irreversibler Vorgänge. Eine Antwort auf Hrn. Boltzmann’s ‘Entgegnung’" Annalen der Physik und Chemie 59, 1896)

"The ordinary logic has a great deal to say about genera and species, or in our nineteenth century dialect, about classes. Now a class is a set of objects compromising all that stand to one another in a special relation of similarity. But where ordinary logic talks of classes the logic of relatives talks of systems. A system is a set of objects compromising all that stands to one another in a group of connected relations. Induction according to ordinary logic rises from the contemplation of a sample of a class to that of a whole class; but according to the logic of relatives it rises from the contemplation of a fragment of a system to the envisagement of the complete system." (Charles S Peirce, "Cambridge Lectures on Reasoning and the Logic of Things: Detached Ideas on Vitally Important Topics", 1898)

Mayme I Logsdon - Collected Quotes

"An incontestable claim of mathematics to importance in our civilization is that it is indispensable in a scientific explanation of what we observe in nature, i.e., the phenomena of nature. Of the several fields of elementary mathematics, the calculus may be called the motion-picture machine of mathematics which catches natural phenomena in the act of changing, or, as Newton called it, in a state of flux. Other fields of mathematics are to be likened to the camera which shows a still picture (of nature) as it appears at a given instant without regard to the possible appearance the following instant." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"In mathematics, as in the world about us, when one quantity depends on a second quantity, or when the value of one symbol depends on the value of another symbol, the first is said to be a function of the second. If the second quantity, or the second symbol, is thought of as taking on a number of arbitrary values (e.g., the angle A when it increases or decreases), it is called an independent variable and the function which depends on it is called a dependent variable. It may happen that a function depends on more than one independent variable." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"Mathematical theories have been of great service in many experimental sciences in correlating the results of observations and in predicting new data afterward verified by observation. This has happened particularly in geometry, physics, and astronomy. But the relationship between a mathematical theory and the data which it is designed to relate is often misunderstood. When such a theory has been successful as a correlating agent, the conviction is likely to become established that the theory has a unique relationship to nature as interpreted for us by the observations. Furthermore, it is sometimes inferred that nature behaves in precisely the way which the mathematics indicates. As a matter of fact, nature never does behave in this way, and there are always more mathematical theories than one whose results depart from a given set of data by less than the errors of observation." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"Neither the principle of cause and effect nor the principle of uncertainty can be precisely characteristic of the behavior of nature. They are merely most interesting theorems in two different theories by means of which we endeavor to correlate and interpret observed data. The ultimate choice between the two theories must be determined by convenience or by their relative accuracies of fit with observation, and not because of any supposedly precise correspondence with nature on the part of either one of them." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"[…] the purposes of an applied mathematical science are twofold: first to correlate and systematize data which may otherwise appear heterogeneous and unrelated in character, and second to predict by logical processes new results which might be difficult or impossible to discover by experimental methods alone." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"The purpose of a coordinate system is twofold. It enables one person to describe the position of points or objects in such a manner that others listening or reading may know exactly what points or objects are meant. And it is a device for linking algebra and geometry so that an algebraic equation corresponds to a geometric locus and, conversely, a geometric locus corresponds to one or more algebraic equations." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"The underlying notion of the integral calculus is also that of finding a limiting value, but this time it is the limiting value of a sum of terms when the number of terms increases without bound at the same time that the numerical value of each term approaches Zero. The area bounded by one or more curves is found as the limiting value of a sum of small rectangles; the length of an arc of a curve is found as the limiting value of a sum of lengths of straight lines (chords of the arc); the volume of a solid bounded by one or more curved surfaces is found as the limiting value of a sum of volumes of small solids bounded by planes; etc." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"The words 'maximum' and 'minimum' are used here in a technical sense. Maximum value of the function, for example, does not mean (as one might well suppose) the greatest value which the function attains for any value of x but, merely, the greatest value which it attains when, having been increasing, it ceases increasing and begins to decrease. In other words, the ordinate of a maximum point on a curve is greater than the ordinates of other nearby points. In a similar manner the ordinate of a minimum point is less than the ordinates of other nearby points." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"There is probably no one word which is more closely associated in everyone's mind with the mathematician than the word equation. The reason for this is easy to find. In the language of mathematics the word 'equation' plays the same role as that played by the word 'sentence' in a spoken and written language. Now the sentence is the unit for the expression of thought; the equation is the unit for the expression of a mathematical idea." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"To square a circle means to find a square whose area is equal to the area of a given circle. In its first form this problem asked for a rectangle whose dimensions have the same ratio as that of the circumference of a circle to its radius. The proof of the impossibility of solving this by use of ruler and compasses alone followed immediately from the proof, in very recent times, that π cannot be the root of a polynomial equation with rational coefficients." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"When an induction, based on observations, is made, it is not intended that it shall be accepted as a universal truth, but it is advanced as a hypothesis for further study. Additional observations are then made and the results compared with the results expected from the hypothesis. If there is more deviation between the experimental results and the computed results than can be expected from the inaccuracies of observation and measurement, the scientist discards the' hypothesis and tries to formulate another." (Mayme I Logsdon, "A Mathematician Explains", 1935)

On Experiments (2011-2019)

"To get a true understanding of the work of mathematicians, and the need for proof, it is important for you to experiment with your own intuitions, to see where they lead, and then to experience the same failures and sense of accomplishment that mathematicians experienced when they obtained the correct results. Through this, it should become clear that, when doing any level of mathematics, the roads to correct solutions are rarely straight, can be quite different, and take patience and persistence to explore." (Alan Sultan & Alice F Artzt,"The Mathematics that every Secondary School Math Teacher Needs to Know", 2011)

"Replication is at the heart of science. If you read a study claiming that toast usually lands buttered side down, you’ll probably want to find at least one replication before you begin to take the result seriously. That’s good scientific practice. A replication experiment inevitably differs a little from the initial experiment - the toast was a little different in shape and dropped in a slightly different way - and so again finding a similar result gives some reassurance that the initial result was not caused by some quirk of the initial experiment." (Geoff Cumming, "Understanding the New Statistics", 2012)

"The ultimate arbiter of truth is experiment, not the comfort one derives from one's a priori beliefs, nor the beauty or elegance one ascribes to one's theoretical models." (Lawrence M Krauss, "A Universe from Nothing: Why There Is Something Rather than Nothing", 2012)

"There are limits on the data we can gather and the kinds of experiments we can perform." (Charles Wheelan, "Naked Statistics: Stripping the Dread from the Data", 2012)

"One good experiment is worth a thousand models […]; but one good model can make a thousand experiments unnecessary." (David Lloyd & Evgenii I Volkov,"The Ultradian Clock: Timekeeping for Intracelular Dynamics" [in"Complexity, Chaos, and Biological Evolution", Ed. by Erik Mosekilde & Lis Mosekilde, 2013)

"Self-selection bias occurs when people choose to be in the data - for example, when people choose to go to college, marry, or have children. […] Self-selection bias is pervasive in 'observational data', where we collect data by observing what people do. Because these people chose to do what they are doing, their choices may reflect who they are. This self-selection bias could be avoided with a controlled experiment in which people are randomly assigned to groups and told what to do. (Gary Smith, "Standard Deviations", 2014)

"We encounter regression in many contexts - pretty much whenever we see an imperfect measure of what we are trying to measure. Standardized tests are obviously an imperfect measure of ability. [...] Each experimental score is an imperfect measure of 'ability', the benefits from the layout. To the extent there is randomness in this experiment - and there surely is - the prospective benefits from the layout that has the highest score are probably closer to the mean than was the score." (Gary Smith, "Standard Deviations", 2014)

"Mathematics is both abstract and concrete, revealing much of the mental experiment, working with unobserved abstractions and objects, and the current scientific progress depended on the ability to operate precisely with abstractions and force of reasoning; […]" (Octavian Stanasila, Metabolism of Mathematics and Computer Science No. 8, 2015)

"The correlational technique known as multiple regression is used frequently in medical and social science research. This technique essentially correlates many independent (or predictor) variables simultaneously with a given dependent variable (outcome or output). It asks, 'Net of the effects of all the other variables, what is the effect of variable A on the dependent variable?' Despite its popularity, the technique is inherently weak and often yields misleading results. The problem is due to self-selection. If we don’t assign cases to a particular treatment, the cases may differ in any number of ways that could be causing them to differ along some dimension related to the dependent variable. We can know that the answer given by a multiple regression analysis is wrong because randomized control experiments, frequently referred to as the gold standard of research techniques, may give answers that are quite different from those obtained by multiple regression analysis." (Richard E Nisbett, "Mindware: Tools for Smart Thinking", 2015)

"The work around the complex systems map supported a concentration on causal mechanisms. This enabled poor system responses to be diagnosed as the unanticipated effects of previous policies as well as identification of the drivers of the sector. Understanding the feedback mechanisms in play then allowed experimentation with possible future policies and the creation of a coherent and mutually supporting package of recommendations for change." (David C Lane et al, "Blending systems thinking approaches for organisational analysis: reviewing child protection", 2015)

"[…] people attempt to use highly flexible mathematical structures with large numbers of parameters that can be adjusted to fit the data, the result often being models that fit the data well but lack structural representation of the phenomena and thus are not predictive outside the range of the data. The situation is exacerbated by uncertainty regarding model parameters on account of insufficient data relative to model complexity, which in fact means uncertainty regarding the models themselves. More importantly from the standpoint of epistemology, the amount of available data is often miniscule in comparison to the amount needed for validation. The desire for knowledge has far outstripped experimental/observational capability. We are starved for data." (Edward R Dougherty, "The Evolution of Scientific Knowledge: From certainty to uncertainty", 2016)

"Limitations on experimentation can result in limitations on the complexity or details of a theory. To be validated, a theory cannot exceed the experimentalist’s ability to conceive and perform appropriate experiments. With the uncertainty theory, modern physics appears to have brought us beyond the situation where limitations on observation result only from insufficient experimental apparatus to a point where limitations are unsurpassable in principle." (Edward R Dougherty, "The Evolution of Scientific Knowledge: From certainty to uncertainty", 2016)

"Scientific knowledge is worldly knowledge in the sense that it points into the future by making predictions about events that have yet to take place. Scientific knowledge is contingent, always awaiting the possibility of its invalidation. Its truth or falsity lies in the verity of its predictions and, since these predictions depend upon the outcomes of experiments, ultimately the validity of scientific knowledge is relative to the methodology of verification." (Edward R Dougherty, "The Evolution of Scientific Knowledge: From certainty to uncertainty", 2016)

"The foundations of a discipline are inseparable from the rules of its game, without which there is no discipline, just idle talk. The foundations of science reside in its epistemology, meaning that they lie in the mathematical formulation of knowledge, structured experimentation, and statistical characterization of validity. Rules impose limitations. These may be unpleasant, but they arise from the need to link ideas in the mind to natural phenomena. The mature scientist must overcome the desire for intuitive understanding and certainty, and must live with stringent limitations and radical uncertainty." (Edward R Dougherty, "The Evolution of Scientific Knowledge: From certainty to uncertainty", 2016)

"A good estimator has to be more than just consistent. It also should be one whose variance is less than that of any other estimator. This property is called minimum variance. This means that if we run the experiment several times, the 'answers' we get will be closer to one another than 'answers' based on some other estimator." (David S Salsburg, "Errors, Blunders, and Lies: How to Tell the Difference", 2017)

"The crucial concept that brings all of this together is one that is perhaps as rich and suggestive as that of a paradigm: the concept of a model. Some models are concrete, others are abstract. Certain models are fairly rigid; others are left somewhat unspecified. Some models are fully integrated into larger theories; others, or so the story goes, have a life of their own. Models of experiment, models of data, models in simulations, archeological modeling, diagrammatic reasoning, abductive inferences; it is difficult to imagine an area of scientific investigation, or established strategies of research, in which models are not present in some form or another. However, models are ultimately understood, there is no doubt that they play key roles in multiple areas of the sciences, engineering, and mathematics, just as models are central to our understanding of the practices of these fields, their history and the plethora of philosophical, conceptual, logical, and cognitive issues they raise. "(Otávio Bueno, [in" Springer Handbook of Model-Based Science", Ed. by Lorenzo Magnani & Tommaso Bertolotti, 2017])

"How do you know when a correlation indicates causation? One way is to conduct a controlled experiment. Another is to apply logic. But be careful - it’s easy to get bogged down in semantics." (Daniel J Levitin, "Weaponized Lies", 2017)

"One kind of probability - classic probability - is based on the idea of symmetry and equal likelihood […] In the classic case, we know the parameters of the system and thus can calculate the probabilities for the events each system will generate. […] A second kind of probability arises because in daily life we often want to know something about the likelihood of other events occurring […]. In this second case, we need to estimate the parameters of the system because we don’t know what those parameters are. […] A third kind of probability differs from these first two because it’s not obtained from an experiment or a replicable event - rather, it expresses an opinion or degree of belief about how likely a particular event is to occur. This is called subjective probability […]." (Daniel J Levitin, "Weaponized Lies", 2017)

"The crucial concept that brings all of this together is one that is perhaps as rich and suggestive as that of a paradigm: the concept of a model. Some models are concrete, others are abstract. Certain models are fairly rigid; others are left somewhat unspecified. Some models are fully integrated into larger theories; others, or so the story goes, have a life of their own. Models of experiment, models of data, models in simulations, archeological modeling, diagrammatic reasoning, abductive inferences; it is difficult to imagine an area of scientific investigation, or established strategies of research, in which models are not present in some form or another. However, models are ultimately understood, there is no doubt that they play key roles in multiple areas of the sciences, engineering, and mathematics, just as models are central to our understanding of the practices of these fields, their history and the plethora of philosophical, conceptual, logical, and cognitive issues they raise." (Otávio Bueno, [in" Springer Handbook of Model-Based Science", Ed. by Lorenzo Magnani & Tommaso Bertolotti, 2017])

"The mental model is the arena where imagination takes place. It enables us to experiment with different scenarios by making local alterations to the model. […] To speak of causality, we must have a mental model of the real world. […] Our shared mental models bind us together into communities." (Judea Pearl & Dana Mackenzie, "The Book of Why: The new science of cause and effect", 2018)

On Experiments (2000-2009)

"It is sometimes said that mathematics is not an experimental subject. This is not true! Mathematicians often use the evidence of lots of examples to help form a conjecture, and this is an experimental approach. Having formed a conjecture about what might be true, the next task is to try to prove it." (George M Phillips,"Mathematics Is Not a Spectator Sport", 2000)

"What does a rigorous proof consist of? The word ‘proof’ has a different meaning in different intellectual pursuits. A ‘proof’ in biology might consist of experimental data confirming a certain hypothesis; a ‘proof’ in sociology or psychology might consist of the results of a survey. What is common to all forms of proof is that they are arguments that convince experienced practitioners of the given field. So too for mathematical proofs. Such proofs are, ultimately, convincing arguments that show that the desired conclusions follow logically from the given hypotheses." (Ethan Bloch, "Proofs and Fundamentals", 2000)

"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)

"Heisenberg’s principle must be considered a special case of the complementarity principle […]. This states that an experiment on one aspect of a system (of atomic dimensions) destroys the possibility of learning about a complementarity aspect of the same system. Together these principles have shocking consequences for the comprehension of entropy and determinism." (Lars Skyttner, "General Systems Theory: Ideas and Applications", 2001)

"The primes have tantalized mathematicians since the Greeks, because they appear to be somewhat randomly distributed but not completely so. […] Although the prime numbers are rigidly determined, they somehow feel like experimental data." (Timothy Gowers,"Mathematics: A Very Short Introduction", 2002)

"A theory makes certain predictions and allows calculations to be made that can be tested directly through experiments and observations. But a theory such as superstrings talks about quantum objects that exist in a multidimensional space and at incredibly short distances. Other grand unified theories would require energies close to those experienced during the creation of the universe to test their predictions." (F David Peat, "From Certainty to Uncertainty", 2002)

"It's a bit like having a theory about coins that move in space, but only being able to measure their state by interrupting them with a table. We hypothesize that the coin may be able to revolve in space, a state that is neither ‘heads’ nor ‘tails’ but a kind of mixture. Our experimental proof is that when you stick a table in, you get heads half the time and tails the other half - randomly. This is by no means a perfect analogy with standard quantum theory - a revolving coin is not exactly in a superposition of heads and tails - but it captures some of the flavour." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)

"Science proceeds by abstracting what is essential from the accidental details of matter and process. […] Science begins with our relationship to nature. The facts it discovers about the universe are answers to human questions and involve human-designed experiments." (F David Peat, "From Certainty to Uncertainty", 2002)

"The primes have tantalized mathematicians since the Greeks, because they appear to be somewhat randomly distributed but not completely so. […] Although the prime numbers are rigidly determined, they somehow feel like experimental data." (Timothy Gowers, "Mathematics: A Very Short Introduction", 2002)

"In string theory one studies strings moving in a fixed classical spacetime. […] what we call a background-dependent approach. […] One of the fundamental discoveries of Einstein is that there is no fixed background. The very geometry of space and time is a dynamical system that evolves in time. The experimental observations that energy leaks from binary pulsars in the form of gravitational waves - at the rate predicted by general relativity to the […] accuracy of eleven decimal place - tell us that there is no more a fixed background of spacetime geometry than there are fixed crystal spheres holding the planets up." (Lee Smolin, "Loop Quantum Gravity", The New Humanists: Science at the Edge, 2003)

"Science does not speak of the world in the language of words alone, and in many cases it simply cannot do so. The natural language of science is a synergistic integration of words, diagrams, pictures, graphs, maps, equations, tables, charts, and other forms of visual and mathematical expression. [… Science thus consists of] the languages of visual representation, the languages of mathematical symbolism, and the languages of experimental operations." (Jay Lemke, "Teaching all the languages of science: Words, symbols, images and actions", 2003)

"A sudden change in the evolutive dynamics of a system (a ‘surprise’) can emerge, apparently violating a symmetrical law that was formulated by making a reduction on some (or many) finite sequences of numerical data. This is the crucial point. As we have said on a number of occasions, complexity emerges as a breakdown of symmetry (a system that, by evolving with continuity, suddenly passes from one attractor to another) in laws which, expressed in mathematical form, are symmetrical. Nonetheless, this breakdown happens. It is the surprise, the paradox, a sort of butterfly effect that can highlight small differences between numbers that are very close to one another in the continuum of real numbers; differences that may evade the experimental interpretation of data, but that may increasingly amplify in the system’s dynamics." (Cristoforo S Bertuglia & Franco Vaio, "Nonlinearity, Chaos, and Complexity: The Dynamics of Natural and Social Systems", 2003)

"If some aspects of the behavior of an organism can be modeled by a finite state machine, then the interaction of the organism with its environment may be such a case in question, if the environment is likewise representable by a finite state machine. In fact, such two-machine interactions constitute a popular paradigm for interpreting the behavior of animals in experimental learning situations, with the usual relaxation of the general complexity of the situation, by chosing for the experimental environment a trivial machine. 'Criterion' in these learning experiments is then said to have been reached by the animal when the experimenter succeeded in transforming the animal from a nontrivial machine into a trivial machine, the result of these experiments being the interaction of just two trivial machines." (Heinz von Foerster, "Understanding Understanding: Essays on Cybernetics and Cognition", 2003)

"[...] because observations are all we have, we take them seriously. We choose hard data and the framework of mathematics as our guides, not unrestrained imagination or unrelenting skepticism, and seek the simplest yet most wide-reaching theories capable of explaining and predicting the outcome of today’s and future experiments." (Brian Greene, "The Fabric of the Cosmos", 2004)

"[…] the laws of physics, carefully constructed after thousands of years of experimentation, are nothing but the laws of harmony one can write down for strings and membranes." (Michio Kaku, "Parallel Worlds: A journey through creation, higher dimensions, and the future of the cosmos", 2004)

"Although nature suggests a pathway to a mathematical description of everything, it has thus far eluded a final or complete grand mathematical synthesis. […] Mathematics is therefore inspired by nature. But it does not have to conduct experimental observations to proceed. The worlds of mathematics and theoretical physics are therefore distinct - they have different 'mission statements'. Whereas theoretical physics maps the properties of the nature we experience, mathematics builds a map of all possible 'natures' that logic permits to exist." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"As the frontiers of science are continually pushed back, and the distance between experimenter and the world widens, the intelligibility of the world demands the construction and manipulation of models. Scientific discourse is often used to convey the information from well-grounded models. Scientific thinking is inescapably modeling and intimately involved with inquiry. ls, is essential for revealing unobserved, but observable, events." (Daniel Rothbart [Ed.], "Modeling: Gateway to the Unknown", 2004)

"Because observations are all we have, we take them seriously. We choose hard data and the framework of mathematics as our guides, not unrestrained imagination or unrelenting skepticism, and seek the simplest yet most wide-reaching theories capable of explaining and predicting the outcome of today's and future experiments." (Brian Greene, "The Fabric of the Cosmos: Space, Time, and the Texture of Reality", 2004)

"Feedback and its big brother, control theory, are such important concepts that it is odd that they usually find no formal place in the education of physicists. On the practical side, experimentalists often need to use feedback. Almost any experiment is subject to the vagaries of environmental perturbations. Usually, one wants to vary a parameter of interest while holding all others constant. How to do this properly is the subject of control theory. More fundamentally, feedback is one of the great ideas developed (mostly) in the last century, with particularly deep consequences for biological systems, and all physicists should have some understanding of such a basic concept." (John Bechhoefer, "Feedback for physicists: A tutorial essay on control", Reviews of Modern Physics Vol. 77, 2005)

"The central limit theorem says that, under conditions almost always satisfied in the real world of experimentation, the distribution of such a linear function of errors will tend to normality as the number of its components becomes large. The tendency to normality occurs almost regardless of the individual distributions of the component errors. An important proviso is that several sources of error must make important contributions to the overall error and that no particular source of error dominate the rest." (George E P Box et al, "Statistics for Experimenters: Design, discovery, and innovation" 2nd Ed., 2005)

"An ecology provides the special formations needed by organizations. Ecologies are: loose, free, dynamic, adaptable, messy, and chaotic. Innovation does not arise through hierarchies. As a function of creativity, innovation requires trust, openness, and a spirit of experimentation - where random ideas and thoughts can collide for re-creation." (George Siemens, "Knowing Knowledge", 2006)

"In science, for a theory to be believed, it must make a prediction - different from those made by previous theories - for an experiment not yet done. For the experiment to be meaningful, we must be able to get an answer that disagrees with that prediction. When this is the case, we say that a theory is falsifiable - vulnerable to being shown false. The theory also has to be confirmable, it must be possible to verify a new prediction that only this theory makes. Only when a theory has been tested and the results agree with the theory do we advance the statement to the rank of a true scientific theory." (Lee Smolin, "The Trouble with Physics", 2006)

"Science fiction is essentially a kind of fiction in which people learn more about how to live in the real world, visiting imaginary worlds unlike our own, in order to investigate by way of pleasurable thought-experiments how things might be done differently." (Brian Stableford, "Space, Time, and Infinity: Essays on Fantastic Literature", 2006)

"This phenomenon, common to chaos theory, is also known as sensitive dependence on initial conditions. Just a small change in the initial conditions can drastically change the long-term behavior of a system. Such a small amount of difference in a measurement might be considered experimental noise, background noise, or an inaccuracy of the equipment." (Greg Rae, "Chaos Theory: A Brief Introduction", 2006)

"Frequentist statistics assumes that there is a 'true' state of the world (e.g. the difference between species in predation probability) which gives rise to a distribution of possible experimental outcomes. The Bayesian framework says instead that the experimental outcome - what we actually saw happen - is the truth, while the parameter values or hypotheses have probability distributions. The Bayesian framework solves many of the conceptual problems of frequentist statistics: answers depend on what we actually saw and not on a range of hypothetical outcomes, and we can legitimately make statements about the probability of different hypotheses or parameter values." (Ben Bolker, "Ecological Models and Data in R", 2007)

"Another feature of Bourbaki is that it rejects intuition of any kind. Bourbaki books tend not to contain explanations, examples, or heuristics. One of the main messages of the present book is that we record mathematics for posterity in a strictly rigorous, axiomatic fashion. This is the mathematician’s version of the reproducible experiment with control used by physicists and biologists and chemists. But we learn mathematics, we discover mathematics, we create mathematics using intuition and trial and error. We draw pictures. Certainly, we try things and twist things around and bend things to try to make them work. Unfortunately, Bourbaki does not teach any part of this latter process." (Steven G Krantz,"The Proof is in the Pudding", 2007)

"Any physical theory is always provisional, in the sense that it is only a hypothesis: you can never prove it. No matter how many times the results of experiments agree with some theory, you can never be sure that the next time the result will not contradict the theory." (Stephen Hawking, "A Briefer History of Time: The Science Classic Made More Accessible", 2007)

"[...] when a theoretical model is said to represent certain phenomena, there is indeed reference to a matching, namely between parts of the theoretical models and the relevant data models - both of them abstract entities. Note now, the crucial word in this sentence: the punch comes in the word 'relevant'. There is nothing in an abstract structure itself that can determine that it is the relevant data model, to be matched by the theory. A particular data model is relevant because it was constructed on the basis of results gathered in a certain way, selected by specific criteria of relevance, on certain occasions, in a practical experimental or observational setting, designed for that purpose." (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"A chess hypothesis is basically the equivalent to drawing up a strategic plan. Experimentation in chess is equivalent to the moves that are found to carry out each plan. Throughout the history of chess, both the plans (the hypotheses) as well as the moves (the experiments) have been evolving (thanks to results from the practice of the game and from analyses), and this knowledge is the patrimony of professional players." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

On Experiments (1990-1999)

"A meaningful physical discussion always requires an operational background. Either this is provided by an existing theory, or you have to give it yourself by the sufficiently explicit description of an experiment that can, at least in principle, be performed." (David Ruelle, "Chance and Chaos", 1991)

"Hard though the scientists of mental imagery try, they cannot get around the fact that the representations they deal with are like pictures. […] The methods have to assume, and the experiments continually corroborate, that having imagery is somehow like perceptual seeing, and that it is somehow like seeing pictures. […] The minimal reason for this assumption is that people do naturally talk of seeing pictures before their mind’s eye." (Eva T H Brann, "The World of Imagination", 1991)

"Quantum mechanics, like other physical theories, consists of a mathematical part, and an operational part that tells you how a certain piece of physical reality is described by the mathematics. Both the mathematical and the operational aspects of quantum mechanics are straightforward and involve no logical paradoxes. Furthermore, the agreement between theory and experiment is as good as one can hope for. Nevertheless, the new mechanics has given rise to many controversies, which involve its probabilistic aspect, the relation of its operational concepts with those of classical mechanics […]" (David Ruelle, "Chance and Chaos", 1991)

"Scientists use mathematics to build mental universes. They write down mathematical descriptions - models - that capture essential fragments of how they think the world behaves. Then they analyse their consequences. This is called 'theory'. They test their theories against observations: this is called 'experiment'. Depending on the result, they may modify the mathematical model and repeat the cycle until theory and experiment agree. Not that it's really that simple; but that's the general gist of it, the essence of the scientific method." (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"A fundamental difference between religious and scientific thought is that the received beliefs in religion are ultimately based on revelations or pronouncements, usually by some long dead prophet or priest.[...] Dogma is interpreted by a caste of priests and is accepted by the multitude on faith or under duress. In contrast, the statements of science are derived from the data of observations and experiment, and from the manipulation of these data according to logical and often mathematical procedures." (John A Moore, "Science as a Way of Knowing: The Foundations of Modern Biology", 1993)

"Submission to the experimental data is the golden rule that dominates any scientific discipline." (Maurice Allais, [speech] 1993)

"Clearly, science is not simply a matter of observing facts. Every scientific theory also expresses a worldview. Philosophical preconceptions determine where facts are sought, how experiments are designed, and which conclusions are drawn from them." (Nancy R Pearcey & Charles B. Thaxton, "The Soul of Science: Christian Faith and Natural Philosophy", 1994)

"Clearly, science is not simply a matter of observing facts. Every scientific theory also expresses a worldview. Philosophical preconceptions determine where facts are sought, how experiments are designed, and which conclusions are drawn from them." (Nancy R Pearcey & Charles B. Thaxton, "The Soul of Science: Christian Faith and Natural Philosophy", 1994)

"The principle of science, the definition, almost, is the following: The test of all knowledge is experiment. Experiment is the sole judge of scientific ‘truth’." (Richard Feynman, "Six Easy Pieces", 1994)

"The sequence for the understanding of mathematics may be: intuition, trial, error, speculation, conjecture, proof. The mixture and the sequence of these events differ widely in different domains, but there is general agreement that the end product is rigorous proof – which we know and can recognize, without the formal advice of the logicians. […] Intuition is glorious, but the heaven of mathematics requires much more. Physics has provided mathematics with many fine suggestions and new initiatives, but mathematics does not need to copy the style of experimental physics. Mathematics rests on proof - and proof is eternal." (Saunders Mac Lane, "Reponses to …", Bulletin of the American Mathematical Society Vol. 30 (2), 1994)

"Having a scientific outlook means being willing to divest yourself of a pet hypothesis, whether it relates to easy self-help improvements, homeopathy, graphology, spontaneous generation, or any other concept, when the data produced by a carefully designed experiment contradict that hypothesis. Retaining a belief in a hypothesis that cannot be supported by data is the hallmark of both the pseudoscientist and the fanatic. Often the more deeply held the hypothesis, the more reactionary is the response to nonsupportive data." (Michael Zimmerman, "Science, Nonscience, and Nonsense: Approaching Environmental Literacy", 1995)

"Probability theory is an ideal tool for formalizing uncertainty in situations where class frequencies are known or where evidence is based on outcomes of a sufficiently long series of independent random experiments. Possibility theory, on the other hand, is ideal for formalizing incomplete information expressed in terms of fuzzy propositions." (George Klir, "Fuzzy sets and fuzzy logic", 1995)

"Schematic diagrams are more abstract than pictorial drawings, showing symbolic elements and their interconnection to make clear the configuration and/or operation of a system." (Ernest O Doebelin, "Engineering experimentation: planning, execution, reporting", 1995)

"Some people derive satisfaction from accumulating data, whereas others are content to dream and leave experiments to colleagues. Still others flit from flower to flower rather than learning more and more about one situation. The difference in approach is a matter of temperament, and we all must understand our own strengths. All workers ultimately contribute to the matrix of facts, ideas, understandings, techniques, and visions that we know as science." (Arthur J Birch, "To See the Obvious", 1995)

"When Physicists speak of 'beauty' in their theories, they really mean that their theory possesses at least two essential features: 1. A unifying symmetry 2. The ability to explain vast amounts of experimental data with the most economical mathematical expressions." (Michio Kaku, "Hyperspace", 1995)

"Factoring big numbers is a strange kind of mathematics that closely resembles the experimental sciences, where nature has the last and definitive word. […] as with the experimental sciences, both rigorous and heuristic analyses can be valuable in understanding the subject and moving it forward. And, as with the experimental sciences, there is sometimes a tension between pure and applied practitioners." (Carl B Pomerance, "A Tale of Two Sieves", The Notices of the American Mathematical Society 43, 1996)

"Science focuses on the study of the natural world. It seeks to describe what exists. Focusing on problem finding, it studies and describes problems in its various domains. The humanities focus on understanding and discussing the human experience. In design, we focus on finding solutions and creating things and systems of value that do not yet exist.   The methods of science include controlled experiments, classification, pattern recognition, analysis, and deduction. In the humanities we apply analogy, metaphor, criticism, and (e)valuation. In design we devise alternatives, form patterns, synthesize, use conjecture, and model solutions." (Béla H Bánáthy, "Designing Social Systems in a Changing World", 1996)

"Science is distinguished not for asserting that nature is rational, but for constantly testing claims to that or any other affect by observation and experiment." (Timothy Ferris, "The Whole Shebang: A State-of-the Universe’s Report", 1996)

"The methods of science include controlled experiments, classification, pattern recognition, analysis, and deduction. In the humanities we apply analogy, metaphor, criticism, and (e)valuation. In design we devise alternatives, form patterns, synthesize, use conjecture, and model solutions." (Béla H Bánáthy, "Designing Social Systems in a Changing World", 1996)

"Trust is the core of human relationships, of gregariousness among men. Friendship, a puzzle to the syllogistic and critical mentality, is not based on experiments or tests of another person's qualities but on trust. It is not critical knowledge but a risk of the heart which initiates affection and preserves loyalty in our fellow men." (Abraham J Heschel, "Moral Grandeur and Spiritual Audacity: Essays", 1997)

"[...] the definitive property of good theory is predictiveness. Those theories endure that are precise in the predictions they make across many phenomena and whose predictions are easiest to test by observation and experiment." (Edward O Wilson, "Consilience: The Unity of Knowledge", 1998)

"The everyday usage of 'theory' is for an idea whose outcome is as yet undetermined, a conjecture, or for an idea contrary to evidence. But scientists use the word in exactly the opposite sense. [In science] 'theory' [...] refers only to a collection of hypotheses and predictions that is amenable to experimental test, preferably one that has been successfully tested. It has everything to do with the facts." (Tony Rothman & George Sudarshan, "Doubt and Certainty: The Celebrated Academy: Debates on Science, Mysticism, Reality, in General on the Knowable and Unknowable", 1998)

"The reason why a 'crude', experimental approach is not adequate for determining mathematical truth lies in the nature of what mathematics is and is intended to be. Though its roots lie in the physical world, mathematics is a precise and idealized discipline. The 'points', 'lines', 'planes', and other ideal constructs of mathematics have no exact counterpart in reality. What the mathematician does is to take a totally abstract, idealized view of the world, and reason with his abstractions in an entirely precise and rigorous fashion." (Keith Devlin, "Mathematics: The New Golden Age", 1998)

"A mathematician experiments, amasses information, makes a conjecture, finds out that it does not work, gets confused and then tries to recover. A good mathematician eventually does so – and proves a theorem." (Steven Krantz, "Conformal Mappings", American Scientist Vol. 87 (5), 1999)

"No matter how beautiful the whole model may be, no matter how naturally it all seems to hang together now, if it disagrees with experiment, then it is wrong." (John Gribbin,"Almost Everyone’s Guide to Science", 1999)

"Even the most elegant and beautiful physical theory may disappear without a trace if not confirmed by experiment, while, as a rule, a theorem, once proved, remains in mathematics forever." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

"I have no idea whether the properties of the universe as we know it are fundamental or emergent, but I believe that the mere possibility of the latter should give the string theorists pause, for it would imply that more than one set of microscopic equations is consistent with experiment - so that we are blind to these equations until better experiments are designed - and also that the true nature of the microscopic equations is irrelevant to our world." (Robert B Laughlin, "Fractional quantization", Reviews of Modern Physics vol. 71 (4), [Nobel lecture] 1999)

On Experiments (1980-1989)

"Like pictures, images seem to depict information about interval spatial extents. The scanning experiments support the claim that portions of images depict corresponding portions of the represented objects, and that the spatial relations between portions of the image index the spatial relations between the corresponding portions of the imaged objects." (Stephen Kosslyn," Image and Mind", 1980)

"[…] there is an external world which can in principle be exhaustively described in scientific language. The scientist, as both observer and language-user, can capture the external facts of the world in propositions that are true if they correspond to the facts and false if they do not. Science is ideally a linguistic system in which true propositions are in one-to-one relation to facts, including facts that are not directly observed because they involve hidden entities or properties, or past events or far distant events. These hidden events are described in theories, and theories can be inferred from observation, that is, the hidden explanatory mechanism of the world can be discovered from what is open to observation. Man as scientist is regarded as standing apart from the world and able to experiment and theorize about it objectively and dispassionately." (Mary B Hesse, "Revolutions and Reconstructions in the Philosophy of Science", 1980)

"The purpose of an experiment is to answer questions. The truth of this seems so obvious, that it would not be worth emphasizing were it not for the fact that the results of many experiments are interpreted and presented with little or no reference to the questions that were asked in the first place." (Thomas M Little, "Interpretation and presentation of results", Hortscience 16, 1981)

"In all scientific fields, theory is frequently more important than experimental data. Scientists are generally reluctant to accept the existence of a phenomenon when they do not know how to explain it. On the other hand, they will often accept a theory that is especially plausible before there exists any data to support it." (Richard Morris, 1983)

"[…] nature at the quantum level is not a machine that goes its inexorable way. Instead what answer we get depends on the question we put, the experiment we arrange, the registering device we choose. We are inescapably involved in bringing about that which appears to be happening." (John A Wheeler & Wojciech H Zurek, "Quantum Theory and Measurement", 1983)

"Theoretical scientists, inching away from the safe and known, skirting the point of no return, confront nature with a free invention of the intellect. They strip the discovery down and wire it into place in the form of mathematical models or other abstractions that define the perceived relation exactly. The now-naked idea is scrutinized with as much coldness and outward lack of pity as the naturally warm human heart can muster. They try to put it to use, devising experiments or field observations to test its claims. By the rules of scientific procedure it is then either discarded or temporarily sustained. Either way, the central theory encompassing it grows. If the abstractions survive they generate new knowledge from which further exploratory trips of the mind can be planned. Through the repeated alternation between flights of the imagination and the accretion of hard data, a mutual agreement on the workings of the world is written, in the form of natural law." (Edward O Wilson, "Biophilia", 1984)

"Mathematics is not a deductive science - that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork. You want to find out what the facts are, and what you do is in that respect similar to what a laboratory technician does. Possibly philosophers would look on us mathematicians the same way as we look on the technicians, if they dared." (Paul R Halmos, "I Want to be a Mathematician: An Automathography", 1985)

"The only touchstone for empirical truth is experiment and observation." (Heinz Pagels, "Perfect Symmetry: The Search for the Beginning of Time", 1985)

"A mechanistic model has the following advantages: 1. It contributes to our scientific understanding of the phenomenon under study. 2. It usually provides a better basis for extrapolation (at least to conditions worthy of further experimental investigation if not through the entire range of all input variables). 3. It tends to be parsimonious (i. e, frugal) in the use of parameters and to provide better estimates of the response." (George E P Box, "Empirical Model-Building and Response Surfaces", 1987)

"Although science literally means ‘knowledge’, the scientific attitude is concerned much more with rational perception through the mind and with testing such perceptions against actual fact, in the form of experiments and observations." (David Bohm & F David Peat, "Science, Order, and Creativity", 1987)

"Any physical theory is always provisional, in the sense that it is only a hypothesis: you can never prove it. No matter how many times the results of experiments agree with some theory, you can never be sure that the next time the result will not contradict the theory." (Stephen Hawking, "A Brief History of Time", 1988)

"A first analysis of experimental results should, I believe, invariably be conducted using flexible data analytical techniques - looking at graphs and simple statistics - that so far as possible allow the data to 'speak for themselves'. The unexpected phenomena that such a approach often uncovers can be of the greatest importance in shaping and sometimes redirecting the course of an ongoing investigation." (George Box, "Signal to Noise Ratios, Performance Criteria, and Transformations", Technometrics 30, 1988)

"The submission to observed or experimental data is the golden rule which dominates any scientific discipline. Any theory whatever, if it is not verified by empirical evidence, has no scientific value and should be rejected. This is true, for example, of the contemporary theories of general economic equilibrium." (Maurice Allais, "An Outline of My Main Contributions to Economic Science", [Noble lecture] 1988)

"As a practical matter, mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, mathematics relies on logic rather than observation as its standard of truth, yet employs observation, simulation, and even experimentation as a means of discovering truth." (National Research Council, "Everybody Counts", 1989)

"Neural computing is the study of cellular networks that have a natural property for storing experimental knowledge. Such systems bear a resemblance to the brain in the sense that knowledge is acquired through training rather than programming and is retained due to changes in node functions. The knowledge takes the form of stable states or cycles of states in the operation of the net. A central property of such nets is to recall these states or cycles in response to the presentation of cues." (Igor Aleksander & Helen Morton, "Neural computing architectures: the design of brain-like machines", 1989)

"Statistics is a tool. In experimental science you plan and carry out experiments, and then analyse and interpret the results. To do this you use statistical arguments and calculations. Like any other tool - an oscilloscope, for example, or a spectrometer, or even a humble spanner - you can use it delicately or clumsily, skillfully or ineptly. The more you know about it and understand how it works, the better you will be able to use it and the more useful it will be." (Roger Barlow, "Statistics: A Guide to the Use of Statistical Methods in the Physical Sciences", 1989)

"Some methods, such as those governing the design of experiments or the statistical treatment of data, can be written down and studied. But many methods are learned only through personal experience and interactions with other scientists. Some are even harder to describe or teach. Many of the intangible influences on scientific discovery - curiosity, intuition, creativity - largely defy rational analysis, yet they are often the tools that scientists bring to their work." (Committee on the Conduct of Science, "On Being a Scientist", 1989)

On Experiments (1970-1979)

"Accordingly there are two main types of science, exact science [...] and empirical science [...] seeking laws which are generalizations from particular experiences and are verifiable (or, more strictly, 'probabilities') only by observation and experiment." (Errol E Harris, "Hypothesis and Perception: The Roots of Scientific Method", 1970)

"[...] in probability theory we are faced with situations in which our intuition or some physical experiments we have carried out suggest certain results. Intuition and experience lead us to an assignment of probabilities to events. As far as the mathematics is concerned, any assignment of probabilities will do, subject to the rules of mathematical consistency." (Robert Ash, "Basic probability theory", 1970)

"In relation to any experiment we may speak of this hypothesis as the null hypothesis," and it should be noted that the null hypothesis is never proved or established, but is possibly disproved, in the course of experimentation. Every experiment may be said to exist only in order to give the facts a chance of disproving the null hypothesis." (Sir Ronald A Fisher, "The Design of Experiments", 1971)

"[…] no isolated experiment, however significant in itself, can suffice for the experimental demonstration of any natural phenomenon; for the ‘one chance in a million’ will undoubtedly occur, with no less and no more than its appropriate frequency, however surprised we may be that it should occur to us." (Sir Ronald A Fisher, "The Design of Experiments", 1971)

"Science consists simply of the formulation and testing of hypotheses based on observational evidence; experiments are important where applicable, but their function is merely to simplify observation by imposing controlled conditions." (Henry L Batten, "Evolution of the Earth", 1971)

"Statistical procedure and experimental design are only two different aspects of the same whole, and that whole is the logical requirements of the complete process of adding to natural knowledge by experimentation." (Sir Ronald A Fisher, "The Design of Experiments", 1971)

"[...] it seems self-evident that mathematics is not likely to be much help in discovering laws of nature. If a mathematician wants to make a contribution on this (and I admit it is the highest) level, he will have to master so much experimental material and train himself to think in a way so different from the one he has been accustomed to that he will, in effect, cease to be a mathematician." (Mark Kac, "On Applying Mathematics: Reflections and Examples", Quarterly of Applied Mathematics, 1972)

"In moving from conjecture to experimental data, (D), experiments must be designed which make best use of the experimenter's current state of knowledge and which best illuminate his conjecture. In moving from data to modified conjecture, (A), data must be analyzed so as to accurately present information in a manner which is readily understood by the experimenter." (George E P Box & George C Tjao, "Bayesian Inference in Statistical Analysis", 1973)

"Statistical methods are tools of scientific investigation. Scientific investigation is a controlled learning process in which various aspects of a problem are illuminated as the study proceeds. It can be thought of as a major iteration within which secondary iterations occur. The major iteration is that in which a tentative conjecture suggests an experiment, appropriate analysis of the data so generated leads to a modified conjecture, and this in turn leads to a new experiment, and so on." (George E P Box & George C Tjao, "Bayesian Inference in Statistical Analysis", 1973)

"An experiment is a failure only when it also fails adequately to test the hypothesis in question, when the data it produces don't prove anything one way or the other." (Robert M Pirsig, "Zen and the Art of Motorcycle Maintenance", 1974)

"A hypothesis is empirical or scientific only if it can be tested by experience. […] A hypothesis or theory which cannot be, at least in principle, falsified by empirical observations and experiments does not belong to the realm of science." (Francisco J Ayala, "Biological Evolution: Natural Selection or Random Walk", American Scientist, 1974)

"In the province of the mind, what one believes to be true is true or becomes true, within certain limits to be found experientially and experimentally. These limits are further beliefs to be transcended. In the mind, there are no limit. […] In the province of connected minds, what the network believes to be true, either is true or becomes true within certain limits to be found experientially and experimentally. These limits are further beliefs to be transcended. In the network's mind there are no limits." (John C Lilly, "The Human Biocomputer", 1974)

"Mathematical statistics does not only study procedures for analysing experimental findings but also elaborates methods for taking decisions under conditions of uncertainty, the uncertainty being such as is characterized by statistical stability." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"The point is that every experiment involves an error, the magnitude of which is not known beforehand and it varies from one experiment to another. For this reason, no matter what finite number of experiments have been carried out, the arithmetic mean of the values obtained will contain an error. Of course, if the experiments are conducted under identical conditions and the errors are random errors, then the error of the mean will diminish as the number of experiments is increased, but it cannot be reduced to zero for a finite number of experiments. […] The choice of entities for an experiment must be perfectly random, so that even an apparently inessential cause could not lead to erroneous conclusions." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"We know the laws of trial and error, of large numbers and probabilities. We know that these laws are part of the mathematical and mechanical fabric of the universe, and that they are also at play in biological processes. But, in the name of the experimental method and out of our poor knowledge, are we really entitled to claim that everything happens by chance, to the exclusion of all other possibilities?" (Albert Claude, "The Coming Age of the Cell", Nobel Prize Lecture] 1974)

"Of course, we know the laws of trial and error, of large numbers and probabilities. We know that these laws are part of the mathematical and mechanical fabric of the universe, and that they are also at play in biological processes. But, in the name of the experimental method and out of our poor knowledge, are we really entitled to claim that everything happens by chance, to the exclusion of all other possibilities?" (Albert Claude,"The Coming of Age of the Cell", Science, 1975)

"Taking experimental results and observations for granted and putting the burden of proof on the theory means taking the observational ideology for granted without having ever examined it." (Paul K Feyerabend,"Against Method: Outline of an Anarchistic Theory of Knowledge", 1975)

"We must start with scientific fundamentals, and that means with the data of experiments and not with assumed axioms predicated only upon the misleading nature of that which only superficially seems to be obvious. It is the consensus of great scientists that science is the attempt to set in order the facts of experience." (R Buckminster Fuller, "Synergetics: Explorations in the Geometry of Thinking", 1975)

"A man in daily muddy contact with field experiments could not be expected to have much faith in any direct assumption of independently distributed normal errors." (George E P Box, "Science and Statistics", Journal of the American Statistical Association 71, 1976)

"It is argued that principle of causality is fundamental to human thinking, and it has been observed experimentally that this assumption leads to complex hypothesis formation by human subjects attempting to solve comparatively simple problems involving a causal randomly generated events." (Brian R Gaines, "On the Complexity of Causal Models", Transactions on Systems, Man and Cybernetics, 1976)

"Programming systems can, of course, be built without plan and without knowledge, let alone understanding, of the deep structural issues involved, just as houses, cities, systems of dams, and national economic policies can be similarly hacked together. As a system so constructed begins to get large, however, it also becomes increasingly unstable. When one of its subfunctions fails in an unanticipated way, it may be patched until the manifest trouble disappears. But since there is no general theory of the whole system, the system itself can be only a more or less chaotic aggregate of subsystems whose influence on one another's behavior is discoverable only piecemeal and by experiment. The hacker spends part of his time at the console piling new subsystems onto the structure he has already built - he calls them 'new features' - and the rest of his time in attempts to account for the way in which substructures already in place misbehave. That is what he and the computer converse about." (Joseph Weizenbaum, "Computer power and human reason: From judgment to calculation" , 1976)

"The essential function of a hypothesis consists in the guidance it affords to new observations and experiments, by which our conjecture is either confirmed or refuted." (Ernst Mach, "Knowledge and Error: Sketches on the Psychology of Enquiry", 1976)

"Thinking in words, consciousness is behavior, experiment is measurement." (Celia Green, "The Decline and Fall of Science", 1976)

"Modern physics is not experimental physics because it applies apparatus to the questioning of nature. Rather the reverse is true. Because physics, indeed already as pure theory, sets nature up to exhibit itself as a coherence of forces calculable in advance, it therefore orders its experiments precisely for the purpose of asking whether and how nature reports itself when set up in this way." (Martin Heidegger, "The Question Concerning Technology and other Essays", 1977)

"Most scientific theories, however, are ephemeral. Exceptions will likely be found that invalidate a theory in one or more of its tenets. These can then stimulate a new round of research leading either to a more comprehensive theory or perhaps to a more restrictive (i.e., more precisely defined) theory. Nothing is ever completely finished in science; the search for better theories is endless. The interpretation of a scientific experiment should not be extended beyond the limits of the available data. In the building of theories, however, scientists propose general principles by extrapolation beyond available data. When former theories have been shown to be inadequate, scientists should be prepared to relinquish the old and embrace the new in their never-ending search for better solutions. It is unscientific, therefore, to claim to have 'proof of the truth' when all that scientific methodology can provide is evidence in support of a theory." (William D Stansfield, "The Science of Evolution", 1977)

"The interpretation of a scientific experiment should not be extended beyond the limits of the available data. In the building of theories, however, scientists propose general principles by extrapolation beyond available data. When former theories have been shown to be inadequate, scientists    should be prepared to relinquish the old and embrace the new in their never-ending search for better solutions. It is unscientific, therefore, to claim to have 'proof of the truth' when all that scientific methodology can provide is evidence in support of a theory." (William D Stansfield, "The Science of Evolution", 1977)

"Nature is trying very hard to make us succeed, but nature does not depend on us. We are not the only experiment." (R Buckminster Fuller, [Interview in the Minneapolis Tribune] 1978)

"Part of the art and skill of the engineer and of the experimental physicist is to create conditions in which certain events are sure to occur." (Eugene P Wigner, "Symmetries and Reflections", 1979)

"The conceptual framework of quantum mechanics, supported by massive volumes of experimental data, forces contemporary physicists to express themselves in a manner that sounds, even to the uninitiated, like the language of mystics." (Gary Zukav, "The Dancing Wu Li Masters", 1979)

"There is another fundamental difference between the old physics and the new physics. The old phvsics assumes that there is an external world which exists apart from us. It further assumes that we can observe measure and speculate about the external world without changing it. According to the old physics the external world is indifferent to us and to our needs. [...] The new physics, quantum mechanics, tells us clearly that it is not possible to observe reality without changing it. If we observe a certain particle collision experiment, not only do we have no way of proving that the result would have been the same if we had not been watching it, all that we know indicates that it would not have been the same, because the result that we got was affected by the fact that we were looking for it." (Gary Zukav, "The Dancing Wu Li Masters", 1979)

On Experiments (1960-1969)

"We want to have certainties and no doubts - results and no experiments - without even seeing that certainties can arise only through doubt and results only thorough experiment." (Carl G Jung, "The Structure And Dynamics Of The Psyche", 1960)

"When evaluating the reliability and generality of data, it is often important to know the aims of the experimenter. When evaluating the importance of experimental results, however, science has a trick of disregarding the experimenter's rationale and finding a more appropriate context for the data than the one he proposed." (Murray Sidman, "Tactics of Scientific Research", 1960)

"Model-making, the imaginative and logical steps which precede the experiment, may be judged the most valuable part of scientific method because skill and insight in these matters are rare. Without them we do not know what experiment to do. But it is the experiment which provides the raw material for scientific theory. Scientific theory cannot be built directly from the conclusions of conceptual models." (Herbert G Andrewartha,"Introduction to the Study of Animal Population", 1961)

"The first [principle], is that a mathematical theory can only he developed axiomatically in a fruitful way when the student has already acquired some familiarity with the corresponding material - a familiarity gained by working long enough with it on a kind of experimental, or semiexperimental basis, i.e. with constant appeal to intuition. The other principle [...] is that when logical inference is introduced in some mathematical question, it should always he presented with absolute honesty - that is, without trying to hide gaps or flaws in the argument; any other way, in my opinion, is worse than giving no proof at all." (Jean Dieudonné, "Thinking in School Mathematics", 1961)

"The functional validity of a working hypothesis is not a priori certain, because often it is initially based on intuition. However, logical deductions from such a hypothesis provide expectations (so called prognoses) as to the circumstances under which certain phenomena will appear in nature. Such a postulate or working hypothesis can then be substantiated by additional observations or by experiments especially arranged to test details. The value of the hypothesis is strengthened if the observed facts fit the expectation within the limits of permissible error." (R Willem van Bemmelen, "The Scientific Character of Geology", The Journal of Geology Vol 69 (4), 1961)

"Both the uncertainty principle and the negentropy principle of information make Laplace's scheme [of exact determinism] completely unrealistic. The problem is an artificial one; it belongs to imaginative poetry, not to experimental science." (Léon Brillouin, "Science and Information Theory" 2nd Ed., 1962)

"The physical sciences are used to 'praying over' their data, examining the same data from a variety of points of view. This process has been very rewarding, and has led to many extremely valuable insights. Without this sort of flexibility, progress in physical science would have been much slower. Flexibility in analysis is often to be had honestly at the price of a willingness not to demand that what has already been observed shall establish, or prove, what analysis suggests. In physical science generally, the results of praying over the data are thought of as something to be put to further test in another experiment, as indications rather than conclusions." (John W Tukey, "The Future of Data Analysis", The Annals of Mathematical Statistics, Vol. 33 (1), 1962)

"Observation, reason, and experiment make up what we call the scientific method." (Richard Feynman, "Mainly mechanics, radiation, and heat", 1963)

"[…] it is more important to have beauty in one's equations that to have them fit experiment. […] It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress." (Paul Dirac, Scientific American, 1963)

"A theory with mathematical beauty is more likely to be correct than an ugly one that fits some experimental data. God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe." (Paul Dirac, Scientific American, 1963)

"[…] it is more important to have beauty in one's equations that to have them fit experiment. […] It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress." (Paul A M Dirac, "The Evolution of the Physicist’s Picture of Nature ", Scientific American, 1963)

"Mathematical statistics provides an exceptionally clear example of the relationship between mathematics and the external world. The external world provides the experimentally measured distribution curve; mathematics provides the equation (the mathematical model) that corresponds to the empirical curve. The statistician may be guided by a thought experiment in finding the corresponding equation." (Marshall J Walker, "The Nature of Scientific Thought", 1963)

"Observation, reason, and experiment make up what we call the scientific method." (Richard Feynman, Mainly mechanics, radiation, and heat", 1963)

"Thus science must begin with myths, and with the criticism of myths; neither with the collection of observations, nor with the invention of experiments, but with the critical discussion of myths, and of magical techniques and practices." (Karl Popper, "Conjectures and Refutations: The Growth of Scientific Knowledge", 1963)

"Perfect logic and faultless deduction make a pleasant theoretical structure, but it may be right or wrong; the experimenter is the only one to decide, and he is always right." (Léon Brillouin, "Scientific Uncertainty and Information", 1964)

"The trouble with group theory is that it leaves so much unexplained that one would like to explain. It isolates in a beautiful way those aspects of nature that can be understood in terms of abstract symmetry alone. It does not offer much hope of explaining the messier facts of life, the numerical values of particle lifetimes and interaction strengths - the great bulk of quantitative experimental data that is now waiting for explanation. The process of abstraction seems to have been too drastic, so that many essential and concrete features of the real world have been left out of consideration. Altogether group theory succeeds just because its aims are modest. It does not try to explain everything, and it does not seem likely that it will grow into a complete or comprehensive theory of the physical world." (Freeman J Dyson, "Mathematics in the Physical Sciences", Scientific American Vol. 211 (3), 1964)

"No experimental result can ever kill a theory: any theory can be saved from counterinstances either by some auxiliary hypothesis or by a suitable reinterpretation of its terms." (Imre Lakatos, "Falsification and the Methodology of Scientific Research Programmes", 1965)

"After all, without the experiment - either a real one or a mathematical model - there would be no reason for a theory of probability." (Thornton C Fry, "Probability and Its Engineering Uses", 1965)

"Another thing I must point out is that you cannot prove a vague theory wrong. If the guess that you make is poorly expressed and rather vague, and the method that you use for figuring out the consequences is a little vague - you are not sure, and you say, 'I think everything's right because it's all due to so and so, and such and such do this and that more or less, and I can sort of explain how this works' […] then you see that this theory is good, because it cannot be proved wrong! Also if the process of computing the consequences is indefinite, then with a little skill any experimental results can be made to look like the expected consequences." (Richard P Feynman,"The Character of Physical Law", 1965)

"If science is to progress, what we need is the ability to experiment, honestly in reporting the results - the results must be reported without somebody saying what they would like the results to have been - and finally - an important thing - the intelligence to interpret the results. An important point about this intelligence is that it should not be sure ahead of time what must be. It cannot be prejudiced, and say 'That is very unlikely; I don’t like that.'" (Richard P Feynman, "The Character of Physical Law", 1965)

"It is only through refined measurements and careful experimentation that we can have a wider vision. And then we see unexpected things: we see things that are far from what we would guess - far from what we could have imagined. Our imagination is stretched to the utmost, not, as in fiction, to imagine things which are not really there, but just to comprehend those things which are there." (Richard P Feynman, "The Character of Physical Law", 1965)

"No experimental result can ever kill a theory: any theory can be saved from counterinstances either by some auxiliary hypothesis or by a suitable reinterpretation of its terms." (Imre Lakatos, "Falsification and the Methodology of Scientific Research Programmes", 1965)

"Nature does not seem full of circles and triangles to the ungeometrical; rather, mastery of the theory of triangles and circles, and later of conic sections, has taught the theorist, the experimenter, the carpenter, and even the artist to find them everywhere, from the heavenly motions to the pose of a Venus. (Clifford Truesdell, "Six Lectures on Modern Natural Philosophy", 1966)

"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)

"Applying this approach, systems belonging to different scientific disciplines are investigated in their natural forms. On the basis of experimental results, isomorphic relations between different systems are studied and, finally, some general principles applicable for all systems of a certain class are formulated." (George Klir, "An approach to general systems theory", 1969)

"Any theory starts off with an observer or experimenter. He has in mind a collection of abstract models with predictive capabilities. Using various criteria of relevance, he selects one of them. In order to actually make predictions, this model must be interpreted and identified with a real assembly to form a theory. The interpretation may be prescriptive or predictive, as when the model is used like a blueprint for designing a machine and predicting its states. On the other hand, it may be descriptive and predictive as it is when the model is used to explain and predict the behaviour of a given organism." (Gordon Pask, "The meaning of cybernetics in the behavioural sciences", 1969)

"It is not enough to observe, experiment, theorize, calculate and communicate; we must also argue, criticize, debate, expound, summarize, and otherwise transform the information that we have obtained individually into reliable, well established, public knowledge." (John M Ziman, "Information, Communication, Knowledge", Nature Vol. 224 (5217), 1969)

On Experiments (1950-1959)

"An experiment is a question which man asks of nature; one result of the observation is an answer which nature yields to man." (Ferdinand Gonseth, "The Primeval Atom", 1950)

"The first thing to realize about physics […] is its extraordinary indirectness. […] For physics is not about the real world, it is about 'abstractions' from the real world, and this is what makes it so scientific. […] Theoretical physics runs merrily along with these unreal abstractions, but its conclusions are checked, at every possible point, by experiments." (Anthony Standen, "Science is a Sacred Cow", 1950)

"The hypothesis is the principal intellectual instrument in research. Its function is to indicate new experiments and observations and it therefore sometimes leads to discoveries even when not correct itself. We must resist the temptation to become too attached to our hypothesis, and strive to judge it objectively and modify it or discard it as soon as contrary evidence is brought to light. Vigilance is needed to prevent our observations and interpretations being biased in favor of the hypothesis. Suppositions can be used without being believed." (William I B Beveridge, "The Art of Scientific Investigation", 1950)

"[…] no one believes an hypothesis except its originator but everyone believes an experiment except the experimenter." (William I B Beveridge, "The Art of Scientific Investigation", 1950)

"An experiment is a question which man asks of nature; one result of the observation is an answer which nature yields to man." (Ferdinand Gonseth, "The Primeval Atom", 1950)

"The first thing to realize about physics […] is its extraordinary indirectness. […] For physics is not about the real world, it is about 'abstractions' from the real world, and this is what makes it so scientific. […] Theoretical physics runs merrily along with these unreal abstractions, but its conclusions are checked, at every possible point, by experiments." (Anthony Standen, "Science is a Sacred Cow", 1950)

"The hypothesis is the principal intellectual instrument in research. Its function is to indicate new experiments and observations and it therefore sometimes leads to discoveries even when not correct itself. We must resist the temptation to become too attached to our hypothesis, and strive to judge it objectively and modify it or discard it as soon as contrary evidence is brought to light. Vigilance is needed to prevent our observations and interpretations being biased in favor of the hypothesis. Suppositions can be used without being believed." (William I B Beveridge, "The Art of Scientific Investigation", 1950)

"Common sense […] may be thought of as a series of concepts and conceptual schemes which have proved highly satisfactory for the practical uses of mankind. Some of those concepts and conceptual schemes were carried over into science with only a little pruning and whittling and for a long time proved useful. As the recent revolutions in physics indicate, however, many errors can be made by failure to examine carefully just how common sense ideas should be defined in terms of what the experimenter plans to do." (James B Conant, "Science and Common Sense", 1951)

"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)

"Science is an interconnected series of concepts and schemes that have developed as a result of experimentation and observation and are fruitful of further experimentation and observation." (James B Conant, "Science and Common Sense", 1951)

"Being built on concepts, hypotheses, and experiments, laws are no more accurate or trustworthy than the wording of the definitions and the accuracy and extent of the supporting experiments." (Gerald Holton, "Introduction to Concepts and Theories in Physical Science", 1952)

"The older physicist believed in Nature and thought of himself as making experiments to see what She was like. She was there whether he could observe her or not. But the modern physicist thinks first of all of what he observes in his experiments and is not interested in anything that he cannot possibly observe. He looks for relations between his observations and ignores everything else. But he still expresses his results as though they were discoveries of the essence of Nature, because he is so used to this way of speaking that he does not realise that his discoveries no longer conform to it. When they are expressed as the characteristics of a world existing outside us and independently of us, which causes our experience by its impact on our sense organs, these discoveries require such a world to have contradictory properties. Hence, by retaining this form of expression, the physicist finds himself presenting his perfectly rational achievements as though they were nonsensical." (Herbert Dingle, "The Scientific Adventure", British Journal for the Philosophy of Science, 1952)

"All great discoveries in experimental physics have been due to the intuition of men who made free use of models, which were for them not products of the imagination, but representatives of real things." (Max Born, "Physical Reality", Philosophical Quarterly, Vol. 3, No. 11,1953)

"[…] if the aim of physical theories is to explain experimental laws, theoretical physics is not an autonomous science; it is subordinate to metaphysics." (Pierre-Maurice-Marie Duhem,"The Aim and Structure of Physical Theory", 1954)

"Science cannot be based on dogma or authority of any kind, nor on any institution or revelation, unless indeed it be of the Book of Nature that lies open before our eyes. We need not dwell on the processes of acquiring knowledge by observation, experiment, and inductive and deductive reasoning. The study of scientific method both in theory and practice is of great importance. It is inherent in the philosophy that the record may be imperfect and the conceptions erroneous; the potential fallibility of our science is not only acknowledged but also insisted upon." (Sir Robert Robinson, "Science and the Scientist", Nature Vol. 176 (4479), 1955)

"That mathematics is a handmaiden of science is a commonplace; but it is less well understood that experiments stimulate mathematical imagination, aid in the formulation of concepts and shape the direction and emphasis of mathematical studies. One of the most remarkable features of the relationship is the successful use of physical models and experiments to solve problems arising in mathematics. In some cases a physical experiment is the only means of determining whether a solution to a specific problem exists; once the existence of a solution has been demonstrated, it may then be possible to complete the mathematical analysis, even to move beyond the conclusions furnished by the model-a sort of boot-strap procedure. It is interesting to point out that what counts in this action and reaction is as much the 'physical way of thinking', the turning over in imagination of physical events, as the actual doing of the experiment." (James R Newman, "The World of Mathematics" Vol. II, 1956)

"The predictions of physical theories for the most part concern situations where initial conditions can be precisely specified. If such initial conditions are not found in nature, they can be arranged. Such arrangements are considerably easier to realize with inanimate than with animate matter, because the properties of animate matter are much more sensitive to being tampered with than inanimate matter. In particular, living tissue in vitro may behave quite differently than in situ. Controlled biological experiments are, of course, possible, but they are more difficult and their scope is more limited than that of physical experiments. For this reason, biology has had to depend to a greater extent than physics on theories of larger speculative scope, in which reasoning by imaginative analogy plays a more important role." (Anatol Rapoport, "The Search for Simplicity", 1956)

"[...] if we can never actually determine more than one of the two properties (possession of a definite position and of a definite momentum), and if when one is determined we-can make no assertion at all about the other property for the same moment, so far as our experiment goes, then we are not justified in concluding that the 'thing' under examination can actually be described as a particle in the usual sense of the term." (Max Born, "Atomic Physics", 1957)

"It is clear to all that the animal organism is a highly complex system consisting of an almost infinite series of parts connected both with one another and, as a total complex, with the surrounding world, with which it is in a state of equilibrium." (Ivan P Pavlov, "Experimental psychology, and other essays", 1957)

"The well-known virtue of the experimental method is that it brings situational variables under tight control. It thus permits rigorous tests of hypotheses and confidential statements about causation. The correlational method, for its part, can study what man has not learned to control. Nature has been experimenting since the beginning of time, with a boldness and complexity far beyond the resources of science. The correlator’s mission is to observe and organize the data of nature’s experiments." (Lee J Cronbach, "The Two Disciplines of Scientific Psychology", The American Psychologist Vol. 12, 1957)

"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)

"It is perhaps possible to distinguish two different aspects of numeracy […]. On the one hand is an understanding of the scientific approach to the study of phenomena - observation, hypothesis, experiment, verification. On the other hand, there is the need in the modern world to think quantitatively, to realise how far our problems are problems of degree even when they appear as problems of kind." (Sir Geoffrey Crowther, "A Report of the Central Advisory Committee for Education", 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)

"It is perhaps possible to distinguish two different aspects of numeracy […]. On the one hand is an understanding of the scientific approach to the study of phenomena - observation, hypothesis, experiment, verification. On the other hand, there is the need in the modern world to think quantitatively, to realise how far our problems are problems of degree even when they appear as problems of kind." (Sir Geoffrey Crowther, "A Report of the Central Advisory Committee for Education", 1959)