On Rigor (2010-2019)

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

"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: The Changing Nature of Mathematical Proof", 2010)

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

"Mathematics is so useful because physical scientists and engineers have the good sense to largely ignore the 'religious' fanaticism of professional mathematicians, and their insistence on so-called rigor, that in many cases is misplaced and hypocritical, since it is based on axioms" that are completely fictional, i. e. those that involve the so-called infinity." (Doron Zeilberger, "Doron Zeilberger's 126th Opinion", 2012)

"Whether information comes in a quantitative or qualitative flavor is not as important as how you use it. [...] The key to making a good forecast […] is not in limiting yourself to quantitative information. Rather, it’s having a good process for weighing the information appropriately. […] collect as much information as possible, but then be as rigorous and disciplined as possible when analyzing it. [...] Many times, in fact, it is possible to translate qualitative information into quantitative information." (Nate Silver, "The Signal and the Noise: Why So Many Predictions Fail-but Some Don't", 2012)

"A conceptual model is a framework that is initially used in research to outline the possible courses of action or to present an idea or thought. When a conceptual model is developed in a logical manner, it will provide a rigor to the research process. (N Elangovan & R Rajendran, "Conceptual Model: A Framework for Institutionalizing the Vigor in Business Research", 2015)

"Definitions are part of the bedrock of mathematical writing and thinking. Mathematics is almost unique among the sciences—not to mention other disciplines - in insisting on strictly rigorous definitions of terminology and concepts. Thus we must state our definitions as succinctly and comprehensibly as possible. Definitions should not hang the reader up, but should instead provide a helping hand as well as encouragement for the reader to push on. (Steven G Krantz, "A Primer of Mathematical Writing" 2nd Ed., 2016)

"Mathematics is a fascinating discipline that calls for creativity, imagination, and the mastery of rigorous standards of proof." (John Meier & Derek Smith, "Exploring Mathematics: An Engaging Introduction to Proof", 2017)

"Mathematical rigour is the thing that enables mathematicians to agree with one another about what is and isn’t correct, rather than just having arguments about competing theories and never coming to a conclusion. Mathematics is based on the rules of logic, the idea being that if you only use objects that behave strictly according to the rules of logic, then as long as you only strictly apply the rules of logic, no disagreements can ever arise."(Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"Samples give us estimates of something, and they will almost always deviate from the true number by some amount, large or small, and that is the margin of error. […] The margin of error does not address underlying flaws in the research, only the degree of error in the sampling procedure. But ignoring those deeper possible flaws for the moment, there is another measurement or statistic that accompanies any rigorously defined sample: the confidence interval. (Daniel J Levitin, "Weaponized Lies", 2017)

On Rigor (2000-2009)

"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)

"Most physical systems, particularly those complex ones, are extremely difficult to model by an accurate and precise mathematical formula or equation due to the complexity of the system structure, nonlinearity, uncertainty, randomness, etc. Therefore, approximate modeling is often necessary and practical in real-world applications. Intuitively, approximate modeling is always possible. However, the key questions are what kind of approximation is good, where the sense of 'goodness' has to be first defined, of course, and how to formulate such a good approximation in modeling a system such that it is mathematically rigorous and can produce satisfactory results in both theory and applications." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)

"The fuzzy set theory is taking the same logical approach as what people have been doing with the classical set theory: in the classical set theory, as soon as the two-valued characteristic function has been defined and adopted, rigorous mathematics follows; in the fuzzy set case, as soon as a multi-valued characteristic function (the membership function) has been chosen and fixed, a rigorous mathematical theory can be fully developed." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)

"In essence, mathematicians wanted to prove two things: 1.Mathematics is consistent: Mathematics contains no internal contradictions. There are no slips of reason or ambiguities. No matter from what direction we approach the edifice of mathematics, it will always display the same rigor and truth. 2.Mathematics is complete: No mathematical truths are left hanging. Nothing needs adding to the system. Mathematicians can prove every theorem with total rigor so that nothing is excluded from the overall system." (F David Peat, "From Certainty to Uncertainty", 2002)

"Pure mathematics was characterized by an obsession with proof, rigor, beauty, and elegance, and sought its foundations in the disembodied worlds of logic or intuition. Far from being coextensive with physics, pure mathematics could be ‘applied’ only after it had been made foundationally secure by the purists." (Andrew Warwick,"Masters of Theory: Cambridge and the rise of mathematical physics", 2003)

"When you’re trying to prove something, it helps to know it’s true. That gives you the confidence you need to keep searching for a rigorous proof." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"Elegance and simplicity should remain important criteria in judging mathematics, but the applicability and consequences of a result are also important, and sometimes these criteria conflict. I believe that some fundamental theorems do not admit simple elegant treatments, and the proofs of such theorems may of necessity be long and complicated. Our standards of rigor and beauty must be sufficiently broad and realistic to allow us to accept and appreciate such results and their proofs. As mathematicians we will inevitably use such theorems when it is necessary in the practice our trade; our philosophy and aesthetics should reflect this reality." (Michael Aschbacher,"Highly complex proofs and implications", 2005)

"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)

"The ever-present rigorous proof is both a science and an art." (Edward B. Burger, Zero To Infinity: A History of Numbers", 2007)

"As students, we learned mathematics from textbooks. In textbooks, mathematics is presented in a rigorous and logical way: definition, theorem, proof, example. But it is not discovered that way. It took many years for a mathematical subject to be understood well enough that a cohesive textbook could be written. Mathematics is created through slow, incremental progress, large leaps, missteps, corrections, and connections." (Richard S Richeson,"Eulers Gem: The Polyhedron Formula and the birth of Topology", 2008)

"The concept of symmetry (invariance) with its rigorous mathematical formulation and generalization has guided us to know the most fundamental of physical laws. Symmetry as a concept has helped mankind not only to define ‘beauty’ but also to express the ‘truth’. Physical laws tries to quantify the truth that appears to be ‘transient’ at the level of phenomena but symmetry promotes that truth to the level of ‘eternity’. (Vladimir G Ivancevic & Tijana T Ivancevic, "Quantum Leap", 2008)

"Therefore, mathematical ecology does not deal directly with natural objects. Instead, it deals with the mathematical objects and operations we offer as analogs of nature and natural processes. These mathematical models do not contain all information about nature that we may know, but only what we think are the most pertinent for the problem at hand. In mathematical modeling, we have abstracted nature into simpler form so that we have some chance of understanding it. Mathematical ecology helps us understand the logic of our thinking about nature to help us avoid making plausible arguments that may not be true or only true under certain restrictions. It helps us avoid wishful thinking about how we would like nature to be in favor of rigorous thinking about how nature might actually work. (John Pastor, "Mathematical Ecology of Populations and Ecosystems", 2008)

"When in the sciences or techniques one states that a certain problem is unsolvable, a rigorous demonstration of such unsolvability is required. And when a scientist submits an article to publication, the least that its referees demand is that it be intelligible. Why? Because rational beings long for understanding and because only clear statements are susceptible to be put to examination to verify whether they are true or false. In the Humanities it is the same, or it should be, but it is not always so. (Mario Bunge, "Xenius, Platón y Manolito", La Nación, 2008)

On Rigor (1990-1999)

"Regarding stability, the state trajectories of a system tend to equilibrium. In the simplest case they converge to one point (or different points from different initial states), more commonly to one (or several, according to initial state) fixed point or limit cycle(s) or even torus(es) of characteristic equilibrial behaviour. All this is, in a rigorous sense, contingent upon describing a potential, as a special summation of the multitude of forces acting upon the state in question, and finding the fixed points, cycles, etc., to be minima of the potential function. It is often more convenient to use the equivalent jargon of 'attractors' so that the state of a system is 'attracted' to an equilibrial behaviour. In any case, once in equilibrial conditions, the system returns to its limit, equilibrial behaviour after small, arbitrary, and random perturbations." (Gordon Pask, "Different Kinds of Cybernetics", 1992)

"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)

"Many pages have been expended on polemics in favor of rigor over intuition, or of intuition over rigor. Both extremes miss the point: the power of mathematics lies precisely in the combination of intuition and rigor." (Ian Stewart,"Concepts of Modern Mathematics", 1995)

"Scientists reach their conclusions for the damnedest of reasons: intuition, guesses, redirections after wild-goose chases, all combing with a dollop of rigorous observation and logical reasoning to be sure […] This messy and personal side of science should not be disparaged, or covered up, by scientists for two major reasons. First, scientists should proudly show this human face to display their kinship with all other modes of creative human thought […] Second, while biases and references often impede understanding, these mental idiosyncrasies may also serve as powerful, if quirky and personal, guides to solutions. (Stephen J Gould, "Dinosaur in a Haystack: Reflections in natural history", 1995)

"Empirical evidence can never establish mathematical existence - nor can the mathematician's demand for existence be dismissed by the physicist as useless rigor. Only a mathematical existence proof can ensure that the mathematical description of a physical phenomenon is meaningful." (Richard Courant, "The Parsimonious Universe, Stefan Hildebrandt & Anthony Tromba", 1996)

"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)

"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)

"But as often happens with rigorous theorems in physics, the more serious the conclusions which follow from proven assertions, the more carefully one must examine the initial premises." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

"Mathematicians are more like classical composers, typically working within a much tighter framework, reluctant to go to the next step until all previous ones have been established with due rigor." (Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory", 1999)

"Physicists are more like avant-garde composers, willing to bend traditional rules and brush the edge of acceptability in the search for solutions. Mathematicians are more like classical composers, typically working within a much tighter framework, reluctant to go to the next step until all previous ones have been established with due rigor. Each approach has its advantages as well as drawbacks; each provides a unique outlet for creative discovery. Like modern and classical music, it’s not that one approach is right and the other wrong – the methods one chooses to use are largely a matter of taste and training." (Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory", 1999)

On Rigor (1980-1989)

"When a mathematician asks himself why some result should hold, the answer he seeks is some intuitive understanding. In fact, a rigorous proof means nothing to him if the result doesn’t make sense intuitively." (Morris Kline, "Mathematics: The Loss of Certainty", 1980)

"Rigorous proofs are the hallmark of mathematics, they are an essential part of mathematics’ contribution to general culture." (George Polya,"Mathematical Discovery", 1981)

"It's difficult to be rigorous about whether a machine really 'knows', 'thinks', etc., because we're hard put to define these things. We understand human mental processes only slightly better than a fish understands swimming. (John McCarthy, "The Little Thoughts of Thinking Machines", Psychology Today, 1983)

"[…] how completely inadequate it is to limit the history of mathematics to the history of what has been formalized and made rigorous. The unrigorous and the contradictory play important parts in this history." (Philip J Davis & Rueben Hersh, "The Mathematical Experience", 1985)

"We who are heirs to three recent centuries of scientific development can hardly imagine a state of mind in which many mathematical objects were regarded as symbols of spiritual truths or episodes in sacred history. Yet, unless we make this effort of imagination, a fraction of the history of mathematics is incomprehensible." (Philip J Davis & Rueben Hersh, "The Mathematical Experience", 1985)

"Mathematical rigor is the clarification of the reasoning used in mathematics. Usually, mathematics first arises in some particular situation, and as the demand for rigor becomes apparent more careful definitions of what is being reasoned about are required, and a closer examination of the numerous 'hidden assumptions' is made." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"Science has become a social method of inquiring into natural phenomena, making intuitive and systematic explorations of laws which are formulated by observing nature, and then rigorously testing their accuracy in the form of predictions. The results are then stored as written or mathematical records which are copied and disseminated to others, both within and beyond any given generation. As a sort of synergetic, rigorously regulated group perception, the collective enterprise of science far transcends the activity within an individual brain." (Lynn Margulis & Dorion Sagan, "Microcosmos", 1986)

"A great many problems are easier to solve rigorously if you know in advance what the answer is." (Ian Stewart, "From Here to Infinity", 1987)

"The most abstract conservation laws of physics come into their being in describing equilibrium in the most extreme conditions. They are the most rigorous conservation laws, the last to break down. The more extreme the conditions, the fewer the conserved structures. [...] In a deep sense, we understand the interior of the sun better that the interior of the earth, and the early stages of the big bang best of all." (Frank Wilczek, "Longing for the Harmonies: Themes and Variations from Modern Physics", 1987)

"Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems - general and specific statements - can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences." (Sir Erik C Zeeman,"Private Games", 1988)

"However, mathematics is not and cannot be anything more than a tool, and all my work rests on the conviction that, in its use, the only two really fruitful stages in the scientific approach are, firstly, a thorough examination of the initial hypotheses; and secondly, a discussion of the meaning and empirical relevance of the results obtained. What remains is but tautological calculation, which is of interest only to the mathematician, and the mathematical rigour of the reasoning can never justify a theory based on postulates if these postulates do not correspond to the true nature of the observed phenomena." (Maurice Allais, "An Outline of My Main Contributions to Economic Science", [Noble lecture] 1988)

[…] the chain of possible combinations of the encounter can be studied as such, as an order which subsists in its rigor, independently of all subjectivity. Through cybernetics, the symbol is embodied in the apparatus - with which it is not to be confused, the apparatus being just its support. And it is embodied in it in a literally trans-subjective way. (Jacques Lacan, 1988)

"Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems - general and specific statements - can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences." (Sir Erik C Zeeman, "Private Games", 1988)

On Rigor (1970-1979)

"The real problem which confronts mathematics is not that of rigour, but the problem of the development of ‘meaning’, of the ‘existence’of mathematical objects.'' (René F Thom, "Modern mathematics, does it exist?", 1972)

"Early scientific thinking was holistic, but speculative - the modern scientific temper reacted by being empirical, but atomistic. Neither is free from error, the former because it replaces factual inquiry with faith and insight, and the latter because it sacrifices coherence at the altar of facticity. We witness today another shift in ways of thinking: the shift toward rigorous but holistic theories. This means thinking in terms of facts and events in the context of wholes, forming integrated sets with their own properties and relationships." (Ervin László, "Introduction to Systems Philosophy", 1972)

"The fact that we have to consider more refined explanations - namely, those of science - to predict the change of phenomena shows that the determinism of the change of forms is not rigorous, and that the same local situation can give birth to apparently different outcomes under the influence of unknown or unobservable factors." (René F Thom, "Structural Stability and Morphogenesis", 1972)

"Heuristic reasoning is good in itself. What is bad is to mix up heuristic reasoning with rigorous proof. What is worse is to sell heuristic reasoning for rigorous proof." (George Pólya, "How to Solve It", 1973)

"In mathematics the problem of the essence of proof has been thoroughly worked out and every mathematician must master the methods of demonstrative reasoning. Appropriate rules have been established for this purpose. These rules and the concepts of rigour and exactitude of reasoning vary from century to century, and at the present time every mathematician knows the level of rigour of modern mathematics." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"Mathematics is the sole avenue for learning how to reason via proof. On the other hand, one must also learn how to conjecture.[…] In a rigorous case of demonstrative reasoning, the main thing is to be able to distinguish proof from conjecture, justified proof from an unjustified attempt." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"[…] when a mathematician demands rigorously logical proof about any assertion, he does so not for his own pleasure but to verify the facts, which might easily appear to us to be obvious but which, when verified, prove to lie erroneous." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"Many pages have been expended on polemics in favor of rigor over intuition, or of intuition over rigor. Both extremes miss the point: the power of mathematics lies precisely in the combination of intuition and rigor." (Ian Stewart, "Concepts of Modern Mathematics", 1975)

"[…] the distinction between rigorous thinking and more vague ‘imaginings’; even in mathematics itself, all is not a question of rigor, but rather, at the start, of reasoned intuition and imagination, and, also, repeated guessing. After all, most thinking is a synthesis or juxtaposition of advances along a line of syllogisms - perhaps in a continuous and persistent ‘forward'’ movement, with searching, so to speak ‘sideways’, in directions which are not necessarily present from the very beginning and which I describe as ‘sending out exploratory patrols’ and trying alternative routes." (Stanislaw M Ulam,"Adventures of a Mathematician", 1976)

On Rigor (1960-1969)

"Science is usually understood to depict a universe of strict order and lawfulness, of rigorous economy - one whose currency is energy, convertible against a service charge into a growing common pool called entropy." (Paul A Weiss,"Organic Form: Scientific and Aesthetic Aspects", 1960)

"The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there never was any other object for it." (George Polya,"Mathematical Discovery", 1962)

"But both managed to understand mathematics and to make a 'fair' number of contributions to the subject. Rigorous proof is not nearly so important as proving the worth of what we are teaching; and most teachers, instead of being concerned about their failure to be sufficiently rigorous, should really be concerned about their failure to provide a truly intuitive approach. The general principle, then, is that the rigor should be suited to the mathematical age of the student and not to the age of mathematics." (Morris Kline, "Mathematics: A Cultural Approach", 1962)

"Rigorous proofs are the hallmark of mathematics, they are an essential part of mathematics’ contribution to general culture." (George Pólya, "Mathematical Discovery", 1962)

"The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there never was any other object for it." (George Pólya, "Mathematical Discovery", 1962)

"The formulation of a hypothesis carries with it an obligation to test it as rigorously as we can command skills to do so." (Peter Medawar, "Hypothesis and Imagination", 1963)

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

"[…] mathematics is a science whose concepts are too breakable, too dry, too precisely limited. The disciplines of life and society, of human thinking, are fluid disciplines, with some flexibility, with concepts that are not clearly defined, but which are able to include things less strictly delimited than a mathematical definition does it." (Grigore C Moisil, 1968)

"Unfortunately, we are not in a position to give a rigorous definition of the fundamental concept of the theory : the concept of set. Of course, we could say that a set is a collection, a union, an ensemble, a family, a system, a class, etc. But this would not be a mathematical definition, but rather a misuse of the multitude of words available in the English language." (Naum Ya. Vilenkin, "Stories about Sets", 1968)

"The machine rules. Human life is rigorously controlled by it, dominated by the terribly precise will of mechanisms. These creatures of man are exacting. They are now reacting on their creators, making them like themselves. They want well-trained humans; they are gradually wiping out the differences between men, fitting them into their own orderly functioning, into the uniformity of their own regimes. They are thus shaping humanity for their own use, almost in their own image." (Paul A Valéry, "Fairy Tales for Computers", 1969)

On Rigor (1925-1949)

"[…] mathematics, accessible in its full depth only to the very few, holds a quite peculiar position amongst the creation of the mind. It is a science of the most rigorous kind, like logic but more comprehensive and very much fuller; it is a true art, along with sculpture and music, as needing the guidance of inspiration and as developing under great conventions of form […]" (Oswald Spengler, "The Decline of the West" Vol. 1, 1926)

"As the objects of abstract geometry cannot be totally grasped by space intuition, a rigorous proof in abstract geometry can never be based only on intuition, but it must be founded on logical deduction from valid and precise axioms. Nevertheless intuition maintains, also in precision geometry, its irreplaceable value that cannot be substituted by logical considerations. Intuition helps us to construct a proof and to gain an overview, it is, moreover, a source of inventions and new mental connections." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"In order to regain in a rigorously defined function those properties that are analogous to those ascribed to an empirical curve with respect to slope and curvature (first and higher difference quotients), we need not only to require that the function is continuous and has a finite number of maxima and minima in a finite interval, but also assume explicitly that it has the first and a series of higher derivatives (as many as one will want to use)." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"Life doesn't have the time to wait for rigor." (Paul Valéry, "L'idee fixe" ["The Fix Idea"], 1932)

"Geometry, then, is the application of strict logic to those properties of space and figure which are self-evident, and which therefore cannot be disputed. But the rigor of this science is carried one step further; for no property, however evident it may be, is allowed to pass without demonstration, if that can be given. The question is therefore to demonstrate all geometrical truths with the smallest possible number of assumptions." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1943)

"Heuristic reasoning is good in itself. What is bad is to mix up heuristic reasoning with rigorous proof. What is worse is to sell heuristic reasoning for rigorous proof." (George Pólya, "How to Solve It", 1945)

"In mathematics as in the physical sciences we may use observation and induction to discover general laws. But there is a difference. In the physical sciences, there is no higher authority than observation and induction but In mathematics there is such an authority: rigorous proof." (George Pólya, "How to solve it", 1945)

"The rules of algebra show that the square of any number, whether positive or negative, is a positive number: therefore, to speak of the square root of a negative number is mere absurdity. Now, Cardan deliberately commits that absurdity and begins to calculate on such 'imaginary' quantities. One would describe this as pure madness; and yet the whole development of algebra and analysis would have been impossible without that fundament - which, of course, was, in the nineteenth century, established on solid and rigorous bases. It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one." (Jacque S Hadamard, "An Essay on the Psychology of Invention in the Mathematical Field", 1945)

"For, in mathematics or symbolic logic, reason can crank out the answer from the symboled equations - even a calculating machine can often do so - but it cannot alone set up the equations. Imagination resides in the words which define and connect the symbols - subtract them from the most aridly rigorous mathematical treatise and all meaning vanishes." (Ralph W Gerard, "The Biological Basis of Imagination", American Thought, 1947)

"Mathematics is an infinity of flexibles forcing pure thought into a cosmos. It is an arc of austerity cutting realms of reason with geodesic grandeur. Mathematics is crystallized clarity, precision personified, beauty distilled and rigorously sublimated." (Cletus O Oakley, "Mathematics", The American Mathematical Monthly, 1949)

"Mathematics is one component of any plan for liberal education. Mother of all the sciences, it is a builder of the imagination, a weaver of patterns of sheer thought, an intuitive dreamer, a poet. The study of mathematics cannot be replaced by any other activity that will train and develop man's purely logical faculties to the same level of rationality. Through countless dimensions, riding high the winds of intellectual adventure and filled with the zest of discovery, the mathematician tracks the heavens for harmony and eternal verity. There is not wholly unexpected surprise, but surprise nevertheless, that mathematics has direct application to the physical world about us. For mathematics, in a wilderness of tragedy and change, is a creature of the mind, born to the cry of humanity in search of an invariant reality, immutable in substance, unalterable with time. Mathematics is an infinity of flexibles forcing pure thought into a cosmos. It is an arc of austerity cutting realms of reason with geodesic grandeur. Mathematics is crystallized clarity, precision personified, beauty distilled and rigorously sublimated. The life of the spirit is a life of thought; the ideal of thought is truth; everlasting truth is the goal of mathematics." (Cletus O Oakley, "Mathematics", The American Mathematical Monthly, 1949)

On Rigor (1950-1959)

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

"The usefulness of observation and measurement in testing economic theories arises because the theorems of economics are supposed to relate to the actual world. [...] Any economic theorem rigorously deduced from given postulates may be regarded as a hypothesis about the actual world which experience may show to be false." (Richard Stone, "The Role of Measurement in Economics", 1951)

"Rigor is to the mathematician what morality is to man. It does not consist in proving everything, but in maintaining a sharp distinction between what is assumed and what is proved, and in endeavoring to assume as little as possible at every stage." (André Weil, Mathematical Teaching in Universities", The American Mathematical Monthly Vol. 61 (1), 1954)

"[…] no branch of mathematics competes with projective geometry in originality of ideas, coordination of intuition in discovery and rigor in proof, purity of thought, logical finish, elegance of proofs and comprehensiveness of concepts. The science born of art proved to be an art." (Morris Kline,"Projective Geometry", Scientific America Vol. 192 (1), 1955)

"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)

"Two qualifications of this observation are in order, First, it is by no means true that every ‘open system’ is able to exhibit equifinality. ‘Equifinality’ as a property of all open systems has never been rigorously defined. If one assumes a rigorous definition in terms of a unique steady state, then, of course, this is realizable only on paper […]" (Joseph M Yoffey, "Homeostatic Mechanisms", 1958)

"With the idol of certainty (including that of degrees of imperfect certainty or probability) there falls one of the defences of obscurantism which bar the way of scientific advance. For the worship of this idol hampers not only the boldness of our questions, but also the rigor and the integrity of our tests. The wrong view of science betrays itself in the craving to be right; for it is not his possession of knowledge, of irrefutable truth, that makes the man of science, but his persistent and recklessly critical quest for truth. (Karl R Popper, "The Logic of Scientific Discovery", 1959)

On Rigor (1875-1899)

"In order for there to be a variable quantity in some mathematical study, the domain of its variability must strictly speaking be known beforehand through a definition. However, this domain cannot itself be something variable, since otherwise each fixed support for the study would collapse. Thus this domain is a definite, actually infinite set of values. Hence each potential infinite, if it is rigorously applicable mathematically, presupposes an actual infinite." (Georg Cantor, "Über die verschiedenen Ansichten in Bezug auf die actualunendlichen Zahlen" ["Over the different views with regard to the actual infinite numbers"], 1886)

"It always seems to me absurd to speak of a complete proof, or of a theorem being rigorously demonstrated. An incomplete proof is no proof, and a mathematical truth not rigorously demonstrated is not demonstrated at all." (James J Sylvester, "On certain inequalities related to prime numbers", Nature Vol. 38, 1888)

"Mathematics renders its best service through the immediate furthering of rigorous thought and the spirit of invention." (Johann F Herbart, "Mathematischer Lehrplan fur Realschulen", 1890)

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

"Geometry, then, is the application of strict logic to those properties of space and figure which are self-evident, and which therefore cannot be disputed. But the rigor of this science is carried one step further; for no property, however evident it may be, is allowed to pass without demonstration, if that can be given. The question is therefore to demonstrate all geometrical truths with the smallest possible number of assumptions." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1898)

On Rigor (1850-1874)

"Few will deny that even in the first scientific instruction in mathematics the most rigorous method is to be given preference over all others. Especially will every teacher prefer a consistent proof to one which is based on fallacies or proceeds in a vicious circle, indeed it will be morally impossible for the teacher to present a proof of the latter kind consciously and thus in a sense deceive his pupils. Notwithstanding these objectionable so-called proofs, so far as the foundation and the development of the system is concerned, predominate in our textbooks to the present time. Perhaps it will be answered, that rigorous proof is found too difficult for the pupil’s power of comprehension. Should this be anywhere the case, - which would only indicate some defect in the plan or treatment of the whole, - the only remedy would be to merely state the theorem in a historic way, and forego a proof with the frank confession that no proof has been found which could be comprehended by the pupil; a remedy which is ever doubtful and should only be applied in the case of extreme necessity. But this remedy is to be preferred to a proof which is no proof, and is therefore either wholly unintelligible to the pupil, or deceives him with an appearance of knowledge which opens the door to all superficiality and lack of scientific method." (Hermann G Grassmann, "Stücke aus dem Lehrbuche der Arithmetik", 1861)

"Consider an arbitrary figure in general position, indeterminate in the sense that it can be chosen from all such figures without upsetting the laws, conditions, and connections among the different parts of the system; suppose that given these data we have found one or more relations or properties, metric or descriptive, of that figure using the usual obvious inference (i.e., in a way regarded in certain cases as the only rigorous argument). Is it not obvious that if, preserving these very data, one begins to change the initial figure by insensible steps, or applies to some parts of the figure an arbitrary continuous motion, then is it not obvious that the properties and relations established for the initial system remain applicable to subsequent states of this system provided that one is mindful of particular changes, when, say, certain magnitudes vanish, change direction or sign, and so on - changes which one can always anticipate a priori on the basis of reliable rules." (Jean V Poncelet,"Treatise on Projective Properties of Figures", 1865)

"If one would see how a science can be constructed and developed to its minutest details from a very small number of intuitively perceived axioms, postulates, and plain definitions, by means of rigorous, one would almost say chaste, syllogism, which nowhere makes use of surreptitious or foreign aids, if one would see how a science may thus be constructed one must turn to the elements of Euclid." (Hermann Hankel, Theorie der Complexen Zahlensysteme", 1867)

"[...] very often the laws derived by physicists from a large number of observations are not rigorous, but approximate." (Augustin-Louis Cauchy, "Sept leçons de physique" ["Seven lessons of Physics"], Bureau du Journal Les Mondes, 1868)

"As in the experimental sciences, truth cannot be distinguished from error as long as firm principles have not been established through the rigorous observation of facts." (Louis Pasteur, "Étude sur la maladie des vers à soie", 1870)

"The leading characteristic of algebra is that of operation on relations. This also is the leading characteristic of Thought. Algebra cannot exist without values, nor Thought without Feelings. The operations are so many blank forms till the values are assigned. Words are vacant sounds, ideas are blank forms, unless they symbolize images and sensations which are their values. Nevertheless it is rigorously true, and of the greatest importance, that analysts carry on very extensive operations with blank forms, never pausing to supply the symbols with values until the calculation is completed; and ordinary men, no less than philosophers, carry on long trains of thought without pausing to translate their ideas (words) into images." (George H Lewes "Problems of Life and Mind", 1873)

On Rigor (1900-1924)

"[…] it is an error to believe that rigor in the proof is the enemy of simplicity." (David Hilbert, [Paris International Congress] 1900)

"[...] it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This requirement of logical deduction by means of a finite number of processes is simply the requirement of rigor in reasoning." (David Hilbert, "Mathematical Problems", 1900)

"The very possibility of mathematical science seems an insoluble contradiction. If this science is only deductive in appearance, from whence is derived that perfect rigour which is challenged by none? If, on the contrary, all the propositions which it enunciates may be derived in order by the rules of formal logic, how is it that mathematics is not reduced to a gigantic tautology? The syllogism can teach us nothing essentially new, and if everything must spring from the principle of identity, then everything should be capable of being reduced to that principle." (Henri Poincaré, "Science and Hypothesis", 1901)

"Besides it is an error to believe that rigour is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigor forces us to find out simpler methods of proof." (David Hilbert,"Mathematical Problems", Bulletin of the American Mathematical Society Vol. 8, 1902)

"If, to go further, we [...] attribute to matter the infinitely granular structure that is in the spirit of atomic theory, our power to apply to reality the rigorous mathematical concept of continuity will greatly decrease." (Jean-Baptiste Perrin, 1906)

"Indeed, a mathematical deduction is of no use to the physicist so long as it is limited to asserting that a given rigorously true proposition has for its consequence the rigorous accuracy of some such other proposition. To be useful to the physicist, it must still be proved that the second proposition remains approximately exact when the first is only approximately true. And even that does not suffice. The range of these two approximations must be delimited; it is necessary to fix the limits of error which can be made in the result when the degree of precision of the methods of measuring the data is known; it is necessary to define the probable error that can be granted the data when we wish to know the result within a definite degree of approximation." (Pierre-Maurice-Marie Duhem, "La théorie physique. Son objet, sa structure", 1906)

"Much of the skill of the true mathematical physicist and of the mathematical astronomer consists in the power of adapting methods and results carried out on an exact mathematical basis to obtain approximations sufficient for the purposes of physical measurements. It might perhaps be thought that a scheme of Mathematics on a frankly approximative basis would be sufficient for all the practical purposes of application in Physics, Engineering Science, and Astronomy, and no doubt it would be possible to develop, to some extent at least, a species of Mathematics on these lines. Such a system would, however, involve an intolerable awkwardness and prolixity in the statements of results, especially in view of the fact that the degree of approximation necessary for various purposes is very different, and thus that unassigned grades of approximation would have to be provided for. Moreover, the mathematician working on these lines would be cut off from the chief sources of inspiration, the ideals of exactitude and logical rigour, as well as from one of his most indispensable guides to discovery, symmetry, and permanence of mathematical form. The history of the actual movements of mathematical thought through the centuries shows that these ideals are the very life-blood of the science, and warrants the conclusion that a constant striving toward their attainment is an absolutely essential condition of vigorous growth. These ideals have their roots in irresistible impulses and deep-seated needs of the human mind, manifested in its efforts to introduce intelligibility in certain great domains of the world of thought." (Ernest W Hobson, [address] 1910)

 "At difficult times like this, the only salvation is an enthusiasm for science and elevated thinking, and from all sciences, mathematics, through its precise problems, through its rigorous proofs, gives the most important and immediate rewarding and serves then as a solid foundation for any other theoretical or applied profession." (Gheorghe Ţiţeica, "Gazeta Matematica", ["Mathematical Gazette"] XXXVI, 1916)  

"The ideal of thought is rigor; mathematics is the name that usage employs to designate, not attainment of the ideal, for it cannot be attained, but its devoted pursuit and close approximation. (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)

"The rigor of mathematics is not absolute - absolute rigor is an ideal, to be, like other ideals, aspired unto, forever approached, but never quite attained, for such attainment would mean that every possibility of error or indetermination, however slight, had been eliminated from idea, from symbol, and from argumentation. (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)

On Rigor (1800-1849)

"[…] we must not measure the simplicity of the laws of nature by our facility of conception; but when those which appear to us the most simple, accord perfectly with observations of the phenomena, we are justified in supposing them rigorously exact." (Pierre-Simon Laplace, "The System of the World", 1809)

"As for methods I have sought to give them all the rigour that one requires in geometry, so as never to have recourse to the reasons drawn from the generality of algebra." (Augustin-Louis Cauchy, "Cours d'analyse", 1821)

"I shall devote all my efforts to bring light into the immense obscurity that today reigns in Analysis. It so lacks any plan or system, that one is really astonished that so many people devote themselves to it - and, still worse, it is absolutely devoid of any rigour." (Niels H Abel, "Oeuvres", 1826)

"In the same way as philosophy loses sight of its true object and appropriate matter, when either it passes into and merges in theology, or meddles with external politics, so also does it mar its proper form when it attempts to mimic the rigorous method of mathematics." (K W Friedrich von Schlegel, "Philosophy of Life", 1828)

"Geometry, then, is the application of strict logic to those properties of space and figure which are self-evident, and which therefore cannot be disputed. But the rigor of this science is carried one step further; for no property, however evident it may be, is allowed to pass without demonstration, if that can be given. The question is therefore to demonstrate all geometrical truths with the smallest possible number of assumptions." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1830)

"So far we have studies how, for each commodity by itself, the law of demand in connection with the conditions of production of that commodity, determines the price of it and regulates the incomes of its producers. We considered as given and invariable the prices of other commodities and the incomes of other producers; but, in reality the economic system is a whole of which the parts are connected and react on each other. An increase in the incomes of the producers of commodity A will affect the demand for commodities Band C, etc., and the incomes of their producers, and, by its reaction will involve a change in the demand for A. It seems, therefore, as if, for a complete and rigorous solution of the problems relative to some parts of the economic system, it were indispensable to take the entire system into consideration. But this would surpass the powers of mathematical analysis and of our practical methods of calculation, even if the values of all the constants could be assigned to them numerically." (Antoine A Cournot, "Researches into the Mathematical Principles of the Theory of Wealth", 1838)

"The process of scientific discovery is cautious and rigorous, not by abstaining from hypothesis, but by rigorously comparing hypotheses with facts, and by resolutely rejecting all which the comparison does not confirm." (William Whewell, "The Philosophy of the Inductive Sciences Founded Upon Their History" Vol. 2, 1840)

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

On Rigor (-1799)

"Thus, joining the rigor of demonstrations in mathematics with the uncertainty of chance, and conciliating these apparently contradictory matters, it can, taking its name from both of them, with justice arrogate the stupefying name: The Mathematics of Chance." (Blaise Pascal, [Address to the Academie Parisienne de Mathematiques] 1654)

 "[…] even if someone refuses to admit infinite and infinitesimal lines in a rigorous metaphysical sense and as real things, he can still use them with confidence as ideal concepts (notions ideales) which shorten his reasoning, similar to what we call imaginary roots in the ordinary algebra, for example, √-2." (Gottfried W Leibniz, [letter to Varignon], 1702)

"It may be observed of mathematicians that they only meddle with such things as are certain, passing by those that are doubtful and unknown. They profess not to know all things, neither do they affect to speak of all things. What they know to be true, and can make good by invincible arguments, that they publish and insert among their theorems. Of other things they are silent and pass no judgment at all, chusing [choosing] rather to acknowledge their ignorance, than affirm anything rashly. They affirm nothing among their arguments or assertions which is not most manifestly known and examined with utmost rigour, rejecting all probable conjectures and little witticisms. They submit nothing to authority, indulge no affection, detest subterfuges of words, and declare their sentiments, as in a Court of Judicature [Justice], without passion, without apology; knowing that their reasons, as Seneca testifies of them, are not brought to persuade, but to compel." (Isaac Barrow, "Mathematical Lectures", 1734)

"Especially when we investigate the general laws of Nature, induction has very great power; & there is scarcely any other method beside it for the discovery of these laws. By its assistance, even the ancient philosophers attributed to all bodies extension, figurability, mobility, & impenetrability; & to these properties, by the use of the same method of reasoning, most of the later philosophers add inertia & universal gravitation. Now, induction should take account of every single case that can possibly happen, before it can have the force of demonstration; such induction as this has no place in establishing the laws of Nature. But use is made of an induction of a less rigorous type ; in order that this kind of induction may be employed, it must be of such a nature that in all those cases particularly, which can be examined in a manner that is bound to lead to a definite conclusion as to whether or no the law in question is followed, in all of them the same result is arrived at; & that these cases are not merely a few. Moreover, in the other cases, if those which at first sight appeared to be contradictory, on further & more accurate investigation, can all of them be made to agree with the law; although, whether they can be made to agree in this way better than in any Other whatever, it is impossible to know directly anyhow. If such conditions obtain, then it must be considered that the induction is adapted to establishing the law." (Roger J Boscovich, "De Lege Continuitatis" ["On the law of continuity"], 1754)

"He who has not made the experiment, or who is not accustomed to require rigorous accuracy from himself, will scarcely believe how much a few hours take from certainty of knowledge, and distinctness of imagery; how the succession of objects will be broken, how separate parts will be confused, and how many particular features and discriminations will be compressed and conglobated into one gross and general idea." (Samuel Johnson, "A Journey to the Western Islands of Scotland", 1775)

"[…] the way in which I have proceeded does not lead to the desired goal, the goal that you declare you have reached, but instead to a doubt of the validity of [Euclidean] geometry. I have certainly achieved results which most people would look upon as proof, but which in my eyes prove almost nothing; if, for example, one can prove that there exists a right triangle whose area is greater than any given number, then I am able to establish the entire system of [Euclidean] geometry with complete rigor. Most people would certainly set forth this theorem as an axiom; I do not do so, though certainly it may be possible that, no matter how far apart one chooses the vertices of a triangle, the triangle's area still stays within a finite bound. I am in possession of several theorems of this sort, but none of them satisfy me." (Carl F Gauss, 1799) [answer to a letter from Farkas Bolyai in which Bolyai claimed to have proved Euclid's fifth postulate]

On Rigor (Unsourced)

"[…] all mathematical cognition has this pecularity: that it must first exhibit its concept in intuitional form. […] Without this, mathematics cannot take a single step. Its judgements are therefore always intuitional, whereas philosophy must make do with discursive judgements from mere concepts. It may illustrate its judgements by means of a visual form, but it can never derive them from such a form." (Immanuel Kant)    The object of mathematical rigor is to sanction and legimize the conquests of intuition, and there never was any other object for it." (Jacques S Hadamard)

"Dirichlet alone, not I, nor Cauchy, nor Gauss knows what a completely rigorous mathematical proof is. Rather we learn it first from him. When Gauss says that he has proved something, it is very clear; when Cauchy says it, one can wager as much pro as con; when Dirichlet says it, it is certain.(Carl G J Jacobi)

"Empirical evidence can never establish mathematical existence - nor can the mathematician's demand for existence be dismissed by the physicist as useless rigor. Only a mathematical existence proof can ensure that the mathematical description of a physical phenomenon is meaningful." (Richard Courant)

"If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. In other words, the most important parts of mathematics stand without a foundation." (Niels H Abel)

"In order to draw any conclusion... it is prudent to wait until more numerous and exact observations have provided a solid foundation on which we may build a rigorous theory." (Joseph L Gay-Lussac)

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

"Indeed, when in the course of a mathematical investigation we encounter a problem or conjecture a theorem, our minds will not rest until the problem is exhaustively solved and the theorem rigorously proved; or else, until we have found the reasons which made success impossible and, hence, failure unavoidable. Thus, the proofs of the impossibility of certain solutions plays a predominant role in modern mathematics; the search for an answer to such questions has often led to the discovery of newer and more fruitful fields of endeavour." (David Hilbert)

"It always seems to me absurd to speak of a complete proof, or of a theorem being rigorously demonstrated. An incomplete proof is no proof, and a mathematical truth not rigorously demonstrated is not demonstrated at all." (James J Sylvester)

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

"Logic today is not only an opportunity for philosophy, but an important instrument which people must learn to use." (Grigore C Moisil)

"[…] mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs." (Felix Klein)

"Much of the best mathematical inspiration comes from experience and that it is hardly possible to believe in the existence of an absolute, immutable concept of mathematical rigor, dissociated from all human experience." (John von Neumann)

"Music is architecture translated or transposed from space into time; for in music, besides the deepest feeling, there reigns also a rigorous mathematical intelligence. (Georg W F Hegel)

"Poetry is a form of mathematics, a highly rigorous relationship with words." (Tahar Ben Jelloun)

"Since primes are the basic building blocks of the number universe from which all the other natural numbers are composed, each in its own unique combination, the perceived lack of order among them looked like a perplexing discrepancy in the otherwise so rigorously organized structure of the mathematical world." (H Peter Aleff, "Prime Passages to Paradise")

"[…] the mathematician learns early to accept no fact, to believe no statement, however apparently reasonable or obvious or trivial, until it has been proved, rigorously and totally by a series of steps proceeding from universally accepted first principles." (Alfred Adler)

"The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there never was any other object for it." (Jacques S Hadamard)

"The rigor of science requires that we distinguish well the undraped figure of nature itself from the gay-coloured vesture with which we clothe it at our pleasure." (Heinrich Hertz)

"Today we can say that the abstract beauty of the theory is flanked by the plastic beauty of the curve, a beauty that is astounding. Thus, within this mathematics that is a hundred years old, very elegant from a formal point of view, very beautiful for specialists, there is also a physical beauty that is accessible to everyone. [...] By letting the eye and the hand intervene in the mathematics, not only have we found again the ancient beauty, which remains intact, but we have also discovered a new beauty, hidden and extraordinary. [...] Those who are only concerned with practical applications may perhaps tend not to insist too much on the artistic aspect, because they prefer to entrench themselves in the technicalities that appertain to practical applications. But why should the rigorous mathematician be afraid of beauty? (Benoît B Mandelbrot)

"Undoubtedly, the capstone of every mathematical theory is a convincing proof of all of its assertions. Undoubtedly mathematics inculpates itself when it foregoes convincing proofs. But the mystery of brilliant productivity will always be the posing of new questions, the anticipation of new theorems that make accessible valuable results and connections. Without the creation of new viewpoints, without the statement of new aims, mathematics would soon exhaust itself in the rigor of its logical proofs and begin to stagnate as its substance vanish. Thus, in a sense, mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs." (Felix Klein)

"[…] we must not measure the simplicity of the laws of nature by our facility of conception; but when those which appear to us the most simple, accord perfectly with observations of the phenomena, we are justified in supposing them rigorously exact." (Pierre-Simon Laplace)

"With the exception of the geometric series, there does not exist in all of mathematics a single infinite series whose sum has been determined rigorously." (Niels H Abel)

"You can often hear from non-mathematicians, especially from philosophers, that mathematics consists exclusively in drawing conclusions from clearly stated premises; and that in this process, it makes no difference what these premises signify, whether they are true or fa1se, provided only that they do not contradict one another. But a per. son who has done productive mathematical work will talk quite differently. In fact these people [the non-mathematicians] are thinking only of the crystallized form into which finished mathematica1 theories are finally cast. However, the investigator himself, in mathematics as in every other science, does not work in this rigorous deductive fashion. On the contrary, he makes essential use of his imagination and proceeds inductively aided by heuristic expedients. One can give numerous examples of mathematicians who have discovered theorems of the greatest importance which they were unable to prove. Should one then refuse to recognize this as a great accomplishment and in deference to the above definition insist that this is not mathematics? After all it is an arbitrary thing how the word is to be used, but no judgment of value can deny that the inductive work of the person who first announces the theorem is at least as valuable as the deductive work. of the one who proves it. For both are equally necessary and the discovery is the presupposition of the later conclusion. (Felix Klein)

On Continuity XI (Thought II)

"The function of man’s highest faculty, his reason, consists precisely of the continuous limitation of infinity, the breaking up of infinity into convenient, easily digestible portions - differentials. This is precisely what lends my field, mathematics, its divine beauty." (Yevgeny Zamiatin, "We", 1924)

"Rationality consists [of] the continuous adaptation of our language to our continually expanding world, and metaphor is one of the chief means by which this is accomplished." (Mary B Hesse, "Models and Analogies in Science", 1966)

"Truth is a totality, the sum of many overlapping partial images. History, on the other hand, sacrifices totality in the interest of continuity." (Edmund Leach, "Brain-Twister", 1967)

"[…] the distinction between rigorous thinking and more vague ‘imaginings’; even in mathematics itself, all is not a question of rigor, but rather, at the start, of reasoned intuition and imagination, and, also, repeated guessing. After all, most thinking is a synthesis or juxtaposition of advances along a line of syllogisms - perhaps in a continuous and persistent 'forward' movement, with searching, so to speak ‘sideways’, in directions which are not necessarily present from the very beginning and which I describe as ‘sending out exploratory patrols’ and trying alternative routes." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)

"I shall here present the view that numbers, even whole numbers, are words, parts of speech, and that mathematics is their grammar. Numbers were therefore invented by people in the same sense that language, both written and spoken, was invented. Grammar is also an invention. Words and numbers have no existence separate from the people who use them. Knowledge of mathematics is transmitted from one generation to another, and it changes in the same slow way that language changes. Continuity is provided by the process of oral or written transmission." (Carl Eckart, "Our Modern Idol: Mathematical Science", 1984)

"To form a mental picture of the event, the knowledge developer attempts to integrate his or her perception of the situation with the expert’s perception. That mental picture is then recorded. What happens is a continuous shuttle process; the knowledge developer mentally moves back and forth from the initial impression of the event to the later evaluation of the event. What is finally recorded is the evaluation made during this retrospective period. Because a time lapse can make details of a situation less clear, the information is not always valid." (Elias M Awad, "Knowledge Management", 2003)

"It is from this continuousness of thought and perception that the scientist, like the writer, receives the crucial flash of insight out of which a piece of work is conceived and executed. And the scientist (again like the writer) is grateful when the insight comes, because insight is the necessary catalyst through which the abstract is made concrete, intuition be given language, language provides specificity, and real work can go forward." (Vivian Gornick, "Women in Science: Then and Now", 2009)

On Imagination (1950-1974)

"[…] observation is not enough, and it seems to me that in science, as in the arts, there is very little worth having that does not require the exercise of intuition as well as of intelligence, the use of imagination as well as of information." (Kathleen Lonsdale, "Facts About Crystals", American Scientist Vol. 39 (4), 1951)

"There is always an analogy between nature and the imagination, and possibly poetry is merely the strange rhetoric of that parallel." (Wallace Stevens, "The Necessary Angel", 1951)

"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. (11), 1953)

"The creative act owes little to logic or reason. In their accounts of the circumstances under which big ideas occurred to them, mathematicians have often mentioned that the inspiration had no relation to the work they happened to be doing. Sometimes it came while they were traveling, shaving or thinking about other matters. The creative process cannot be summoned at will or even cajoled by sacrificial offering. Indeed, it seems to occur most readily when the mind is relaxed and the imagination roaming freely." (Morris Kline, Scientific American, 1955)

"Nevertheless, there are three distinct types of paradoxes which do arise in mathematics. There are contradictory and absurd propositions, which arise from fallacious reasoning. There are theorems which seem strange and incredible, but which, because they are logically unassailable, must be accepted even though they transcend intuition and imagination. The third and most important class consists of those logical paradoxes which arise in connection with the theory of aggregates, and which have resulted in a re-examination of the foundations of mathematics." (James R Newman, "The World of Mathematics" Vol. III, 1956)

"The ultimate origin of the difficulty lies in the fact (or philosophical principle) that we are compelled to use the words of common language when we wish to describe a phenomenon, not by logical or mathematical analysis, but by a picture appealing to the imagination. Common language has grown by everyday experience and can never surpass these limits. Classical physics has restricted itself to the use of concepts of this kind; by analysing visible motions it has developed two ways of representing them by elementary processes; moving particles and waves. There is no other way of giving a pictorial description of motions - we have to apply it even in the region of atomic processes, where classical physics breaks down." (Max Born, "Atomic Physics", 1957)

"In imagination there exists the perfect mystery story. Such a story presents all the essential clews, and compels us to form our own theory of the case. If we follow the plot carefully we arrive at the complete solution for ourselves just before the author’s disclosure at the end of the book. The solution itself, contrary to those of inferior mysteries, does not disappoint us; moreover, it appears at the very moment we expect it." (Leopold Infeld, "The Evolution of Physics", 1961)

"The structures of mathematics and the propositions about them are ways for the imagination to travel and the wings, or legs, or vehicles to take you where you want to go." (Scott Buchanan, "Poetry and Mathematics", 1962)

"That perfected machines may one day succeed us is, I remember, an extremely commonplace notion on Earth. It prevails not only among poets and romantics but in all classes of society. Perhaps it is because it is so widespread, born spontaneously in popular imagination, that it irritates scientific minds. Perhaps it is also for this very reason that it contains a germ of truth. Only a germ: Machines will always be machines; the most perfected robot, always a robot." (Pierre Boulle, "Planet of the Apes", 1963)

"Science begins with the world we have to live in, accepting its data and trying to explain its laws. From there, it moves toward the imagination: it becomes a mental construct, a model of a possible way of interpreting experience." (Northrop Frye, "The Educated Imagination", 1964)

"The imagination equips us to perceive reality when it is not fully materialized." (Mary C Richards, "Centering in Pottery, Poetry, and the Person", 1964)

"[…] 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)

"Fantasies are more than substitutes for unpleasant reality; they are also dress rehearsals, plans. All acts performed in the world begin in the imagination." (Barbara G Harrison, [Ms. Magazine] 1973)

"Equilibrium is a figment of the human imagination." (Kenneth Boulding, Toward a General Social Science, 1974)

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On Method II (Mathematical Method)

"He who seeks for methods without having a definite problem in mind seeks for the most part in vain." (David Hilbert, 1902)

“Statistics may be regarded as (i) the study of populations, (ii) as the study of variation, and (iii) as the study of methods of the reduction of data.” (Sir Ronald A Fisher, “Statistical Methods for Research Worker”, 1925)

“[Statistics] is both a science and an art. It is a science in that its methods are basically systematic and have general application; and an art in that their successful application depends to a considerable degree on the skill and special experience of the statistician, and on his knowledge of the field of application, e.g. economics.” (Leonard H C Tippett, “Statistics”, 1943)

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

“We can scarcely imagine a problem absolutely new, unlike and unrelated to any formerly solved problem; but if such a problem could exist, it would be insoluble. In fact, when solving a problem, we should always profit from previously solved problems, using their result or their method, or the experience acquired in solving them.” (George Polya, 1945)

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

"Algebra reverses the relative importance of the factors in ordinary language. It is essentially a written language, and it endeavors to exemplify in its written structures the patterns which it is its purpose to convey. The pattern of the marks on paper is a particular instance of the pattern to be conveyed to thought. The algebraic method is our best approach to the expression of necessity, by reason of its reduction of accident to the ghost-like character of the real variable.” (Alfred N Whitehead, “Essays in Science and Philosophy”, 1948)

"Real-life phenomena are generally so complicated in relation to the mathematical methods at our disposal that we cannot hope to represent and account for their every characteristic. Consequently, some simplifying hypotheses must be made. The moment that we do this, we are leaving the real world and beginning to make a (mathematical) model." (Peter Lancaster, "Mathematics: Models of the Real World", 1976)

"In the long run, the methods are the important part of the course. It is not enough to know the theory; you should be able to apply it." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"I regarded as quite useless the reading of large treatises of pure analysis: too large a number of methods pass at once before the eyes. It is in the works of application that one must study them; one judges their utility there and appraises the manner of making use of them." (Joseph-Louis de Lagrange)

On Chance (-1699)

"Is it possible, then, for any man to apprehend in advance occurrences for which no cause or reason can be assigned? What do we mean when we employ such terms as luck, fortune, accident, turn of the die, except that we are seeking to describe events which happened and came to pass in such a way that they either might not have happened and come to pass at all or might have happened and come to pass under quite different circumstances? How then can an event be anticipated and predicted which occurs fortuitously and as a result of blind chance and of the spinning of Fortune's wheel?" (Marcus Tullius Cicero, "De Divinatione", 44 BC)

"The universal cause is one thing, a particular cause another. An effect can be haphazard with respect to the plan of the second, but not of the first. For an effect is not taken out of the scope of one particular cause save by another particular cause which prevents it, as when wood dowsed with water, will not catch fire. The first cause, however, cannot have a random effect in its own order, since all particular causes are comprehended in its causality. When an effect does escape from a system of particular causality, we speak of it as fortuitous or a chance happening […]" (Thomas Aquinas, "Summa Theologica", cca. 1266-1273)

"Our wisdom and deliberation for the most part follow the lead of chance." (Michel de Montaigne, "Essays", 1580)

"Men are further beholding […] generally to chance, or anything else, than to logic, for the invention or arts and sciences." (Francis Bacon, "The Advancement of Learning", 1605)

"Even the effects already discovered are due to chance and experiment, rather than to the sciences; for our present sciences are nothing more than peculiar arrangements of matters already discovered, and not methods for discovery or plans for new operations." (Francis Bacon, "Novum Organum", 1620)

"Thus, joining the rigor of demonstrations in mathematics with the uncertainty of chance, and conciliating these apparently contradictory matters, it can, taking its name from both of them, with justice arrogate the stupefying name: The Mathematics of Chance." (Blaise Pascal, [Address to the Academie Parisienne de Mathematiques] 1654)

"As a Foundation to the following Proposition, I shall take Leave to lay down this Self-evident Truth: That any one Chance or Expectation to win any thing is worth just such a Sum, as wou’d procure in the same Chance and Expectation at a fair Lay." (Christiaan Huygens, "De ratiociniis in ludo aleae", 1657)

"According to the doctrine of chance, you ought to put yourself to the trouble of searching for the truth […]." (Blaise Pascal, "Pensées", 1670)

"It is impossible for a Die, with such determin’d force and direction, not to fall on such a determin’d side, only I don’t know the force and direction which makes it fall on such a determin’d side, and therefore I call that Chance, which is nothing but want of Art [...]" (John Arbuthnot, "Of the Laws of Chance", 1692)

"I believe the calculation of the quantity of probability might be improved to a very useful and pleasant speculation, and applied to a great many events which are accidental, besides those of games; only these cases would be infinitely more confused, as depending on chances which the most part of men are ignorant of." (John Arbuthnot, "Of the Laws of Chance", 1692)

On Observation (Unsourced)

"A few observation and much reasoning lead to error; many observations and a little reasoning to truth." (Alexis Carrel)

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

"By observation, facts are distinctly and minutely impressed in the mind; by analogy, similar facts are connected ; by experiment, new facts are discovered ; and, in the progression of knowledge, observation, guided by analogy, leads to experiment, and analogy, confirmed by experiment, becomes scientific truth." (Sir Humphry Davy)

"[…] facts by themselves are silent. Observation discovers nothing directly of the actions of causes, but only of sequences in time." (Alfred Marshall) 

"For although it is certainly true that quantitative measurements are of great importance, it is a grave error to suppose that the whole of experimental physics can be brought under this heading. We can start measuring only when we know what to measure: qualitative observation has to precede quantitative measurement, and by making experimental arrangements for quantitative measurements we may even eliminate the possibility of new phenomena appearing." (Heinrich B G Casimir)

"In the study of Nature conjecture must be entirely put aside, and vague hypothesis carefully guarded against. The study of Nature begins with facts, ascends to laws, and raises itself, as far as the limits of man’s intellect will permit, to the knowledge of causes, by the threefold means of observation, experiment and logical deduction." (Jean Baptiste-Andre Dumas)

"In order to draw any conclusion... it is prudent to wait until more numerous and exact observations have provided a solid foundation on which we may build a rigorous theory." (Joseph L Gay-Lussac) 

"Nothing destroys the powers of general observation quite so much as a life of experimental science." (Herbert G Wells)

"Observation more than books and experience more than persons, are the prime educators." (Amos B Alcott)

"Science descends ever more deeply into the hidden recesses of things, but it must halt at a certain point when questions arise which cannot be settled by means of sense observations. At that point the scientist needs a light which is capable of revealing to him truth which entirely escapes his senses. This light is philosophy." (Pope Pius XII)

"Science does more than collect facts; it makes sense of them. Great scientists are virtuosi of the art of discovering the meaning of what otherwise might seem barren observations." (Theodosius Dobzhansky)

"Science is the observation of things possible." (Leonardo da Vinci)

"The aim of every science is foresight (prevoyance). For the laws of established observation of phenomena are generally employed to foresee their succession. All men, however little advanced make true predictions, which are always based on the same principle, the knowledge of the future from the past." (Auguste Compte)

"The art of concluding from experience and observation consists in evaluating probabilities, in estimating if they are high or numerous enough to constitute proof. This type of calculation is more complicated and more difficult than one might think. It demands a great sagacity generally above the power of common people." (Benjamin Franklin)

"The art of observation and that of experimentation are very distinct. In the first case, the fact may either proceed from logical reasons or be mere good fortune; it is sufficient to have some penetration and a sense of truth in order to profit by it. But the art of experimentation leads from the first to the last link of the chain, without hesitation and without a blank, making successive use of Reason, which suggests an alternative, and of Experience, which decides on it, until, starting from a faint glimmer, the full blaze of light is reached." (Jean Baptiste-Andre Dumas)

"The experiment serves two purposes, often independent one from the other: it allows the observation of new facts, hitherto either unsuspected, or not yet well defined; and it determines whether." (René J Dubos)

"The progress of science requires more than new data; it needs novel frameworks and contexts. And where do these fundamentally new views of the world arise? They are not simply discovered by pure observation; they require new modes of thought. And where can we find them, if old modes do not even include the right metaphors? The nature of true genius must lie in the elusive capacity to construct these new modes from apparent darkness. The basic chanciness and unpredictability of science must also reside in the inherent difficulty of such a task." (Stephen J Gould)

"The seemingly useless or trivial observation made by one worker leads on to a useful observation by another; and so science advances, ‘creeping on from point to point’." (Silvanus P Thompson)

"There is a great difference between the spirit of Mathematics and the spirit of Observation. In the former, the principles are palpable, but remote from common use; so that from want of custom it is not easy to turn our head in that direction; but if it be thus turned ever so little, the principles are seen fully confessed, and it would argue a mind incorrigibly false to reason inconsequentially on principles so obtrusive that it is hardly possible to overlook them." (Blaise Pascal)

"Thinking by analogy is not to be despised. Analogy has this merit, that it does not settle things - does not pretend to be conclusive. On the other hand, that induction is pernicious which, with a preconceived end in view, and working right forward for only that,  drags in its train a number of unsifted observations, both false and true." (Johann Wolfgang von Goethe)

"Throughout science there is a constant alternation between periods when a particular subject is in a state of order, with all known data falling neatly into their places, and a state of puzzlement and confusion, when new observations throw all neatly arranged ideas into disarray." (Sir Hermann Bondi, "Astronomy and the Physical Sciences", 1966)

"We are convinced that exactitude in experiments is less the outcome of faithful observation of the divisions of an instrument than of exactitude of method." (Joseph L Gay-Lussac)

"[…] we must not measure the simplicity of the laws of nature by our facility of conception; but when those which appear to us the most simple, accord perfectly with observations of the phenomena, we are justified in supposing them rigorously exact." (Pierre-Simon Laplace)

On Observation (1850-1874)

"The Laws of Nature are merely truths or generalized facts, in regard to matter, derived by induction from experience, observation, arid experiment. The laws of mathematical science are generalized truths derived from the consideration of Number and Space." (Charles Davies, "The Logic and Utility of Mathematics", 1850) 

"Every science consists in the coordination of facts; if the different observations were entirely isolated, there would be no science." (Auguste Comte, "Philosophy of Mathematics", 1851)

"In the fields of observation chance favors only the prepared mind." (Louis Pasteur, [Inaugural lecture], 1854)

"The progress of science has always been the result of a close interplay between our concepts of the universe and our observations on nature. The former can only evolve out of the latter and yet the latter is also conditioned greatly by the former. Thus, in our exploration of nature, the interplay between our concepts and our observations may sometimes lead to totally unexpected aspects among already familiar phenomena." (Tsung Dao Lee, [Nobel lecture for award received] 1957)

"When a power of nature, invisible and impalpable, is the subject of scientific inquiry, it is necessary, if we would comprehend its essence and properties, to study its manifestations and effects. For this purpose simple observation is insufficient, since error always lies on the surface, whilst truth must be sought in deeper regions." (Justus von Liebig," Familiar Letters on Chemistry", 1859) 

"Observation is so wide awake, and facts are being so rapidly added to the sum of human experience, that it appears as if the theorizer would always be in arrears, and were doomed forever to arrive at imperfect conclusion; but the power to perceive a law is equally rare in all ages of the world, and depends but little on the number of facts observed." (Henry D Thoreau, "A Week on the Concord and Merrimack Rivers", 1862)

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

"Exercising the right of occasional suppression and slight modification, it is truly absurd to see how plastic a limited number of observations become, in the hands of men with preconceived ideas." (Francis Galton, "Meteorographica" 1863)

"An anticipative idea or an hypothesis is, then, the necessary starting point for all experimental reasoning. Without it, we could not make any investigation at all nor learn anything; we could only pile up sterile observations. If we experiment without a preconceived idea, we should move at random […]" (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Men who have excessive faith in their theories or ideas are not only ill prepared for making discoveries; they also make very poor observations." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Observation is a passive science, experimentation an active science." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Only within very narrow boundaries can man observe the phenomena which surround him; most of them naturally escape his senses, and mere observation is not enough." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Facts can be accurately known to us only by the most rigid observation and sustained and scrutinizing skepticism […]" (James A Froude, Short Studies on Great Subjects Vol. 2, 1867)

"Only by taking infinitesimally small units for observation (the differential of history, that is, the individual tendencies of men) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history."  (Lev N Tolstoy, “War and Peace”, 1867)

"Most, if not all, of the great ideas of modern mathematics have had their origin in observation." (James J Sylvester, A Plea for the Mathematician, Nature Vol. 1, 1869)

"As in the experimental sciences, truth cannot be distinguished from error as long as firm principles have not been established through the rigorous observation of facts." (Louis Pasteur, "Étude sur la maladie des vers à soie", 1870)

"As the prerogative of Natural Science is to cultivate a taste for observation, so that of Mathematics is, almost from the starting point, to stimulate the faculty of invention." (James J Sylvester, "A Plea for the Mathematician", Nature Vol. 1, 1870)

"[Mathematics] is that [subject] which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation." (Thomas H Huxley, "Lay Sermons, Addresses and Reviews", 1870)

"[…] wrong hypotheses, rightly worked from, have produced more useful results than unguided observation." (Augustus de Morgan, "A Budget of Paradoxes", 1872)

"A law of nature, however, is not a mere logical conception that we have adopted as a kind of memoria technical to enable us to more readily remember facts. We of the present day have already sufficient insight to know that the laws of nature are not things which we can evolve by any speculative method. On the contrary, we have to discover them in the facts; we have to test them by repeated observation or experiment, in constantly new cases, under ever-varying circumstances; and in proportion only as they hold good under a constantly increasing change of conditions, in a constantly increasing number of cases with greater delicacy in the means of observation, does our confidence in their trustworthiness rise." (Hermann von Helmholtz, "Popular Lectures on Scientific Subjects", 1873)

"Hence, even in the domain of natural science the aid of the experimental method becomes indispensable whenever the problem set is the analysis of transient and impermanent phenomena, and not merely the observation of persistent and relatively constant objects." (Wilhelm Wundt, "Principles of Physiological Psychology", 1874)

"Mathematics is a science of Observation, dealing with reals, precisely as all other sciences deal with reals. It would be easy to show that its Method is the same: that, like other sciences, having observed or discovered properties, which it classifies, generalises, co-ordinates and subordinates, it proceeds to extend discoveries by means of Hypothesis, Induction, Experiment and Deduction." (George H Lewes, "Problems of Life and Mind: The Method of Science and its Application", 1874)

"When we merely note and record the phenomena which occur around us in the ordinary course of nature we are said to observe. When we change the course of nature by the intervention of our will and muscular powers, and thus produce unusual combinations and conditions of phenomena, we are said to experiment. […] an experiment differs from a mere observation in the fact that we more or less influence the character of the events which we observe." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)