“Most people think of science as a series of steps forged in concrete, but it’s not. It’s a puzzle, and not all of the pieces will ever be firmly in place. When you’re able to fit some of the together, to see an answer, it’s thrilling.” (Nora Roberts, “Homeport”, 1998)
"Each of nature's patterns is a puzzle, nearly always a deep one. Mathematics is brilliant at helping us to solve puzzles. It is a more or less systematic way of digging out the rules and structures that lie behind some observed pattern or regularity, and then using those rules and structures to explain what's going on." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)
“Everything in nature is a puzzle until it finds its solution in man, who solves it in some way with God, and so completes the circle of creation. “ (Theodore T Munger, “The Appeal to Life”, 1891)
“The art of science is knowing which observations to ignore and which are the key to the puzzle.” (Edward W Kolb, “Blind Watchers of the Sky”, 1996)
“In her manifold opportunities Nature has thus helped man to polish the mirror of [man’s] mind, and the process continues. Nature still supplies us with abundance of brain-stretching theoretical puzzles and we eagerly tackle them; there are more worlds to conquer and we do not let the sword sleep in our hand; but how does it stand with feeling? Nature is beautiful, gladdening, awesome, mysterious, wonderful, as ever, but do we feel it as our forefathers did?” (Sir John A Thomson, “The System of Animate Nature”, 1920)
“Where chaos begins, classical science stops. […] The irregular side of nature, the discontinuous and erratic side these have been puzzles to science, or worse, monstrosities.” (James Gleick, “Chaos”, 1987)
“The sciences have started to swell. Their philosophical basis has never been very strong. Starting as modest probing operations to unravel the works of God in the world, to follow its traces in nature, they were driven gradually to ever more gigantic generalizations. Since the pieces of the giant puzzle never seemed to fit together perfectly, subsets of smaller, more homogeneous puzzles had to be constructed, in each of which the fit was better.” (Erwin Chargaff, “Voices in the Labyrinth”, 1975)
“It is an outcome of faith that nature - as she is perceptible to our five senses - takes the character of such a well formulated puzzle.” (Albert Einstein)
“Mathematics began to seem too much like puzzle solving. Physics is puzzle solving, too, but of puzzles created by nature, not by the mind of man.” (Maria Goeppert-Mayer)
Quotes and Resources Related to Mathematics, (Mathematical) Sciences and Mathematicians
18 October 2017
14 October 2017
On Equations VI (Figurative Equations I)
“If equations are trains threading the landscape of numbers, then no train stops at pi.” (Richard Preston)
“Science is a differential equation. Religion is a boundary condition.” (Alan Turing)
“Equations are more important to me, because politics is for the present, but an equation is something for eternity.” (Albert Einstein)
“The idea that the world exists is like adding an extra term to an equation that doesn’t belong there." (Marvin Minsky)
“Life is and will ever remain an equation incapable of solution, but it contains certain known factors.” (Nikola Tesla, "A Machine to End War”, 1937)
"What truly is logic? Who decides reason? […] It's only in the mysterious equations of love that any logical reasons can be found." (John F Nash Jr)
“Math is the language of the universe. So the more equations you know, the more you can converse with the cosmos.” (Neil deGrasse Tyson)
“An engineer thinks that his equations are an approximation to reality. A physicist thinks reality is an approximation to his equations. A mathematician doesn’t care.” (Anon)
“Words are a pretty fuzzy substitute for mathematical equations.” (Isaac Asimov)
“Science is a differential equation. Religion is a boundary condition.” (Alan Turing)
“Equations are more important to me, because politics is for the present, but an equation is something for eternity.” (Albert Einstein)
“The idea that the world exists is like adding an extra term to an equation that doesn’t belong there." (Marvin Minsky)
“Life is and will ever remain an equation incapable of solution, but it contains certain known factors.” (Nikola Tesla, "A Machine to End War”, 1937)
"What truly is logic? Who decides reason? […] It's only in the mysterious equations of love that any logical reasons can be found." (John F Nash Jr)
“Math is the language of the universe. So the more equations you know, the more you can converse with the cosmos.” (Neil deGrasse Tyson)
“An engineer thinks that his equations are an approximation to reality. A physicist thinks reality is an approximation to his equations. A mathematician doesn’t care.” (Anon)
“Words are a pretty fuzzy substitute for mathematical equations.” (Isaac Asimov)
On Equations V (Nature II)
“The business of concrete mathematics is to discover the equations which express the mathematical laws of the phenomenon under consideration […]” (Auguste Comte, “Positive Philosophy”, 1851)
”The aim of research is the discovery of the equations which subsist between the elements of phenomena.” (Ernst Mach, 1898)
"A mathematician is not a man who can readily manipulate figures; often he cannot. He is not even a man who can readily perform the transformations of equations by the use of calculus. He is primarily an individual who is skilled in the use of symbolic logic on a high plane, and especially he is a man of intuitive judgment in the choice of the manipulative processes he employs." (Vannevar Bush, "As We May Think", 1945)
”The aim of research is the discovery of the equations which subsist between the elements of phenomena.” (Ernst Mach, 1898)
"A mathematician is not a man who can readily manipulate figures; often he cannot. He is not even a man who can readily perform the transformations of equations by the use of calculus. He is primarily an individual who is skilled in the use of symbolic logic on a high plane, and especially he is a man of intuitive judgment in the choice of the manipulative processes he employs." (Vannevar Bush, "As We May Think", 1945)
“Why are the equations from different phenomena so similar? We might say: ‘It is the underlying unity of nature.’ But what does that mean? What could such a statement mean? It could mean simply that the equations are similar for different phenomena; but then, of course, we have given no explanation. The underlying unity might mean that everything is made out of the same stuff, and therefore obeys the same equations.” (Richard P Feynman, “Lecture Notes on Physics”, Vol. III, 1964)
“The method of guessing the equation seems to be a pretty effective way of guessing new laws. This shows again that mathematics is a deep way of expressing nature, and any attempt to express nature in philosophical principles, or in seat-of-the-pants mechanical feelings, is not an efficient way.” (Richard Feynman, “The Character of Physical Law”, 1965)
"If it should turn out that the whole of physical reality can be described by a finite set of equations, I would be disappointed. I would feel that the Creator had been uncharacteristically lacking in imagination.” (Freeman J Dyson, “Infinite in All Directions”, 1988)
“Being able to solve mathematical equations is useless if you don’t understand what the equation represents in real life.” (Robert S Root-Bernstein, “Discovering”, 1989)
"Chaos theory revealed that simple nonlinear systems could behave in extremely complicated ways, and showed us how to understand them with pictures instead of equations. Complexity theory taught us that many simple units interacting according to simple rules could generate unexpected order. But where complexity theory has largely failed is in explaining where the order comes from, in a deep mathematical sense, and in tying the theory to real phenomena in a convincing way. For these reasons, it has had little impact on the thinking of most mathematicians and scientists."
“Equations seem like treasures, spotted in the rough by some discerning individual, plucked and examined, placed in the grand storehouse of knowledge, passed on from generation to generation. This is so convenient a way to present scientific discovery, and so useful for textbooks, that it can be called the treasure-hunt picture of knowledge.” (Robert P Crease, “The Great Equations”, 2009)
On Equations IV: Unknowns I
"Equations are the mathematician's way of working out the value of some unknown quantity from circumstantial evidence. ‘Here are some known facts about an unknown number: deduce the number.’ An equation, then, is a kind of puzzle, centered upon a number. We are not told what this number is, but we are told something useful about it. Our task is to solve the puzzle by finding the unknown number." (Ian Stewart, “Why Beauty Is Truth”, 2007)
“No equation, however impressive and complex, can arrive at the truth if the initial assumptions are incorrect.” (Arthur C Clarke, “Profiles of the Future”, 1973)
”It is sometimes said that the great discovery of the nineteenth century was that the equations of nature were linear, and the great discovery of the twentieth century is that they are not.” (Thomas W Körner, “Fourier Analysis”, 1988)
”Without the clear understanding that equations in physical science always have hidden limitations, we cannot expect to interpret or apply them successfully.” (Duane H D Roller, “Foundations of Modern Physical Science”, 1950)
“Being able to solve mathematical equations is useless if you don’t understand what the equation represents in real life.” (Robert S Root-Bernstein, “Discovering”, 1989)
"It often happens that understanding of the mathematical nature of an equation is impossible without a detailed understanding of its solution." (Freeman J Dyson)
“It is important to remember that the physical interpretation of the mathematical notions occurring in a physical theory must be compatible with the equations of the theory.” (Andrzej Trautman)
“I consider that I understand an equation when I can predict the properties of its solutions, without actually solving it.” (Paul A M Dirac)
“It would seem that more than function itself, simplicity is the deciding factor in the aesthetic equation. One might call the process beauty through function and simplification.” (Raymond Loewy)
On Equations III (Sound and Language I)
“Poetry is a sort of inspired mathematics, which gives us equations, not for abstract figures, triangles, squares, and the like, but for the human emotions. If one has a mind which inclines to magic rather than science, one will prefer to speak of these equations as spells or incantations; it sounds more arcane, mysterious, recondite. “ (Ezra Pound, “The Spirit of Romance”, 1910)
“Translating mathematics into ordinary language is like translating music. It cannot be done. One could describe in detail a sheet of music and tell the shape of each note and where it is placed on the staff, but that would not convey any idea of how it would sound when played. So, too, I suppose that even the most complicated equation could be described in common words, but it would be so verbose and involved that nobody could get the sense of it.” (Edwin E Slosson, “Chats on Science”, 1924)
"There is probably no one word which is more closely associated in everyone's mind with the mathematician than the word equation. The reason for this is easy to find. In the language of mathematics the word 'equation' plays the same role as that played by the word 'sentence' in a spoken and written language. Now the sentence is the unit for the expression of thought; the equation is the unit for the expression of a mathematical idea."
“There is nothing that can be said by mathematical symbols and relations which cannot also be said by words. The converse, however, is false. Much that can be and is said by words cannot successfully be put into equations, because it is nonsense.” (Clifford Truesdell, “Six Lectures on Modern Natural Philosophy”, 1966)
”Just as one can appreciate the beauty of a Beethoven quartet without being able to read a note of music, it is possible to learn about the scope and power and, yes, beauty of a scientific explanation of nature without solving equations.” (Gilbert Shapiro, “Physics Without Math”, 1979)
”How can it be that writing down a few simple and elegant formulae, like short poems governed by strict rules such as those of the sonnet or the waka, can predict universal regularities of Nature? Perhaps we see equations as simple because they are easily expressed in terms of mathematical notation already invented at an earlier stage of development of the science, and thus what appears to us as elegance of description really reflects the interconnectedness of Nature’s laws at different levels.” (Murray Gell-Mann, 1969)
“[…] equations are like poetry: They speak truths with a unique precision, convey volumes of information in rather brief terms, and often are difficult for the uninitiated to comprehend.” (Michael Guillen, “Five Equations That Changed the World”, 1995)
"When you get to know them, equations are actually rather friendly. They are clear, concise, sometimes even beautiful. The secret truth about equations is that they are a simple, clear language for describing certain ‘recipes’ for calculating things." (Ian Stewart, “Why Beauty Is Truth”, 2007)
"If you don’t read poetry how the hell can you solve equations?" (Harvey Jackins)
11 October 2017
On Proofs II
"The folly of mistaking a paradox for a discovery, a metaphor for a proof, a torrent of verbiage for a spring of capital truths, and oneself for an oracle, is inborn in us." (Paul Valéry, "Introduction to the Method of Leonardo da Vinci", 1895)
"It is by logic that we prove, but by intuition that we discover." (Henri Poincaré, “Science and Method”, 1908)
"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)
"A proof only becomes a proof after the social act of ‘accepting it as a proof’." (Yu I Manin, "A Course in Mathematical Logic", 1977)
"A proof in science does more than eliminate doubt. It eliminates inconsistencies and provides the underlying logical basis of the statement." (Edward Teller, “The Pursuit of Simplicity”, 1981)
"A proof only becomes a proof after the social act of 'accepting it as a proof'" (Yuri I Manin, "Provable and Unprovable", 1982)
"A math lecture without a proof is like a movie without a love scene." (Hendrik Lenstra, 2002)
“The more powerful the mathematical tools used to prove a result, the shorter that proof might be expected to be […]” (Julian Havil, “Nonplussed!”, 2007
On Proofs I
“There is no other scientific or analytical discipline that uses proof as readily and routinely as does mathematics. This is the device that makes theoretical mathematics special: the tightly knit chain of reasoning, following strict logical rules, that leads inexorably to a particular conclusion. It is proof that is our device for establishing the absolute and irrevocable truth of statements […].” (Steven G Krantz, “The Proof is in the Pudding”, 2007)
“There are two aspects of proof to be borne in mind. One is that it is our lingua franca. It is the mathematical mode of discourse. It is our tried-and true methodology for recording discoveries in a bullet-proof fashion that will stand the test of time. The second, and for the working mathematician the most important, aspect of proof is that the proof of a new theorem explains why the result is true. In the end what we seek is new understanding, and ’proof’ provides us with that golden nugget.” (Steven G Krantz, “The Proof is in the Pudding”, 2007)
“Rigorous proofs are the hallmark of mathematics, they are an essential part of mathematics’ contribution to general culture.” (George Polya, “Mathematical Discovery”, 1981)
“An intuitive proof allows you to understand why the theorem must be true; the logic merely provides firm grounds to show that it is true.” (Ian Stewart, “Concepts of Modern Mathematics”, 1975)
“A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.” (G H Hardy, “A Mathematician’s Apology”, 1940)
“The best proofs in mathematics are short and crisp like epigrams, and the longest have swings and rhythms that are like music.” (Scott Buchanan, “Poetry and Mathematics”, 1975)
”[…] a mathematician is more anonymous than an artist. While we may greatly admire a mathematician who discovers a beautiful proof, the human story behind the discovery eventually fades away and it is, in the end, the mathematics itself that delights us.” (Timothy Gowers, “Mathematics”, 2002)
“Some people believe that a theorem is proved when a logically correct proof is given; but some people believe a theorem is proved only when the student sees why it is inevitably true.” (Wesley R Hamming, “Coding and Information Theory”, 1980)
“Proofs knit the fabric of mathematics together, and if a single thread is weak, the entire fabric may unravel.” (Ian Stewart, “Nature’s Numbers”, 1995)
“An intuitive proof allows you to understand why the theorem must be true; the logic merely provides firm grounds to show that it is true.” (Ian Stewart, “Concepts of Modern Mathematics”, 1995)
“There are two aspects of proof to be borne in mind. One is that it is our lingua franca. It is the mathematical mode of discourse. It is our tried-and true methodology for recording discoveries in a bullet-proof fashion that will stand the test of time. The second, and for the working mathematician the most important, aspect of proof is that the proof of a new theorem explains why the result is true. In the end what we seek is new understanding, and ’proof’ provides us with that golden nugget.” (Steven G Krantz, “The Proof is in the Pudding”, 2007)
“Rigorous proofs are the hallmark of mathematics, they are an essential part of mathematics’ contribution to general culture.” (George Polya, “Mathematical Discovery”, 1981)
“An intuitive proof allows you to understand why the theorem must be true; the logic merely provides firm grounds to show that it is true.” (Ian Stewart, “Concepts of Modern Mathematics”, 1975)
“A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.” (G H Hardy, “A Mathematician’s Apology”, 1940)
“The best proofs in mathematics are short and crisp like epigrams, and the longest have swings and rhythms that are like music.” (Scott Buchanan, “Poetry and Mathematics”, 1975)
”[…] a mathematician is more anonymous than an artist. While we may greatly admire a mathematician who discovers a beautiful proof, the human story behind the discovery eventually fades away and it is, in the end, the mathematics itself that delights us.” (Timothy Gowers, “Mathematics”, 2002)
“Some people believe that a theorem is proved when a logically correct proof is given; but some people believe a theorem is proved only when the student sees why it is inevitably true.” (Wesley R Hamming, “Coding and Information Theory”, 1980)
“Proofs knit the fabric of mathematics together, and if a single thread is weak, the entire fabric may unravel.” (Ian Stewart, “Nature’s Numbers”, 1995)
“An intuitive proof allows you to understand why the theorem must be true; the logic merely provides firm grounds to show that it is true.” (Ian Stewart, “Concepts of Modern Mathematics”, 1995)
10 October 2017
Mathematical Proofs – Definitions
“A proof is a construction that can be looked over, reviewed, verified by a rational agent. We often say that a proof must be perspicuous or capable of being checked by hand. It is an exhibition, a derivation of the conclusion, and it needs nothing outside itself to be convincing. The mathematician surveys the proof in its entirety and thereby comes to know the conclusion.” (Thomas Tymoczko, “The Four Color Problems”, Journal of Philosophy , Vol. 76, 1979)
“A proof tells us where to concentrate our doubts. […] An elegantly executed proof is a poem in all but the form in which it is written.” (Morris Kline)
“A proof is any completely convincing argument.” (Errett Bishop)
“A proof is a description, like driving instructions.” (Arie Hinkins, “Proofs of the Cantor-Bernstein Theorem”, 2013)
“A proof in mathematics is a psychological device for convincing some person, or some audience, that a certain mathematical assertion is true. The structure, and the language used, in formulating that proof will be a product of the person creating it; but it also must be tailored to the audience that will be receiving it and evaluating it. Thus there is no ‘unique’ or ‘right’ or ‘best’ proof of any given result. A proof is part of a situational ethic.” (Steven G Krantz, “The Proof is in the Pudding”, 2007)
“[…] a proof is a device of communication. The creator or discoverer of this new mathematical result wants others to believe it and accept it.” (Steven G Krantz, “The Proof is in the Pudding”, 2007)
“Heuristically, a proof is a rhetorical device for convincing someone else that a mathematical statement is true or valid.” (Steven G Krantz, “The Proof is in the Pudding”, 2007)
“[…] proof is central to what modern mathematics is about, and what makes it reliable and reproducible.” (Steven G Krantz, “The Proof is in the Pudding”, 2007)
“A proof in logic and mathematics is, traditionally, a deductive argument from some given assumptions to a conclusion. Proofs are meant to present conclusive evidence in the sense that the truth of the conclusion should follow necessarily from the truth of the assumptions. Proofs must be, in principle, communicable in every detail, so that their correctness can be checked.” (Sara Negri & Jan von Plato, “Proof Analysis”, 2011)
“A mathematical proof is a sequence of sentences that convey a mathematical argument.” (Donald Bindner & Martin Erickson, “A Student’s Guide to the Study, Practice and Tools of Modern Mathematics”, 2011)
“A theorem is simply a sentence expressing something true; a proof is just an explanation of why it is true.” (Matthias Beck & Ross Geoghegan, “The Art Of Proof”, 2011)
“Proof is an idol before whom the pure mathematician tortures himself. In physics we are generally content to sacrifice before the lesser shrine of Plausibility.” (Sir Arthur S Eddington)
“A proof tells us where to concentrate our doubts. […] An elegantly executed proof is a poem in all but the form in which it is written.” (Morris Kline)
“A proof is any completely convincing argument.” (Errett Bishop)
“A proof is a description, like driving instructions.” (Arie Hinkins, “Proofs of the Cantor-Bernstein Theorem”, 2013)
“A proof in mathematics is a psychological device for convincing some person, or some audience, that a certain mathematical assertion is true. The structure, and the language used, in formulating that proof will be a product of the person creating it; but it also must be tailored to the audience that will be receiving it and evaluating it. Thus there is no ‘unique’ or ‘right’ or ‘best’ proof of any given result. A proof is part of a situational ethic.” (Steven G Krantz, “The Proof is in the Pudding”, 2007)
“[…] a proof is a device of communication. The creator or discoverer of this new mathematical result wants others to believe it and accept it.” (Steven G Krantz, “The Proof is in the Pudding”, 2007)
“Heuristically, a proof is a rhetorical device for convincing someone else that a mathematical statement is true or valid.” (Steven G Krantz, “The Proof is in the Pudding”, 2007)
“[…] proof is central to what modern mathematics is about, and what makes it reliable and reproducible.” (Steven G Krantz, “The Proof is in the Pudding”, 2007)
“A proof in logic and mathematics is, traditionally, a deductive argument from some given assumptions to a conclusion. Proofs are meant to present conclusive evidence in the sense that the truth of the conclusion should follow necessarily from the truth of the assumptions. Proofs must be, in principle, communicable in every detail, so that their correctness can be checked.” (Sara Negri & Jan von Plato, “Proof Analysis”, 2011)
“A mathematical proof is a sequence of sentences that convey a mathematical argument.” (Donald Bindner & Martin Erickson, “A Student’s Guide to the Study, Practice and Tools of Modern Mathematics”, 2011)
“A theorem is simply a sentence expressing something true; a proof is just an explanation of why it is true.” (Matthias Beck & Ross Geoghegan, “The Art Of Proof”, 2011)
“Proof is an idol before whom the pure mathematician tortures himself. In physics we are generally content to sacrifice before the lesser shrine of Plausibility.” (Sir Arthur S Eddington)
09 October 2017
On Metaphors I
“A metaphor is not an ornament. It is an organ of perception. Through metaphors, we see the world as one thing or another.” (Neil Postman)
“[…] an idea is a feat of association, and the height of it is a good metaphor.” (Robert Frost)
“Human thinking depends on metaphor. We understand new and complex things in relation to the things we already know […] once you pick a metaphor it will guide your thinking.” (Jonathan Haidt)
“[…] key metaphors help determine what and how we perceive and how we think about our perceptions.” (Meyer H Abrams)
“Metaphor is pervasive in everyday life, not just in language but in thought and action. Our ordinary conceptual system, in terms of which we both think and act, is fundamentally metaphorical in nature.” (George Lakoff)
"Metaphor [...] may be said to be the algebra of language." (Charles C Colton, "Lacon", 1820)
“The metaphor never goes very far, anymore than a curve can long be confused with its tangent.” (Henri Bergson, “A World of Ideas”, 1989)
“The price of metaphor is eternal vigilance.” (Arturo Rosenblueth & Norbert Wiener)
“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)
“Thought is metaphoric, and proceeds by comparison, and the metaphors of language derive therefrom.” (Ivor A Richards) [Link]
“[…] an idea is a feat of association, and the height of it is a good metaphor.” (Robert Frost)
“Human thinking depends on metaphor. We understand new and complex things in relation to the things we already know […] once you pick a metaphor it will guide your thinking.” (Jonathan Haidt)
“[…] key metaphors help determine what and how we perceive and how we think about our perceptions.” (Meyer H Abrams)
“Metaphor is pervasive in everyday life, not just in language but in thought and action. Our ordinary conceptual system, in terms of which we both think and act, is fundamentally metaphorical in nature.” (George Lakoff)
"Metaphor [...] may be said to be the algebra of language." (Charles C Colton, "Lacon", 1820)
“The metaphor never goes very far, anymore than a curve can long be confused with its tangent.” (Henri Bergson, “A World of Ideas”, 1989)
“The price of metaphor is eternal vigilance.” (Arturo Rosenblueth & Norbert Wiener)
“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)
“Thought is metaphoric, and proceeds by comparison, and the metaphors of language derive therefrom.” (Ivor A Richards) [Link]
Mathematical Models I
"The physical object cannot be determined by axioms and definitions. It is a thing of the real world, not an object of the logical world of mathematics. Offhand it looks as if the method of representing physical events by mathematical equations is the same as that of mathematics. Physics has developed the method of defining one magnitude in terms of others by relating them to more and more general magnitudes and by ultimately arriving at 'axioms', that is, the fundamental equations of physics. Yet what is obtained in this fashion is just a system of mathematical relations. What is lacking in such system is a statement regarding the significance of physics, the assertion that the system of equations is true for reality." (Hans Reichenbach, "The Theory of Relativity and A Priori Knowledge", 1920)
"If the system exhibits a structure which can be represented by a mathematical equivalent, called a mathematical model, and if the objective can be also so quantified, then some computational method may be evolved for choosing the best schedule of actions among alternatives. Such use of mathematical models is termed mathematical programming." (George Dantzig, "Linear Programming and Extensions", 1959)
“In fact, the construction of mathematical models for various fragments of the real world, which is the most essential business of the applied mathematician, is nothing but an exercise in axiomatics.” (Marshall Stone, cca 1960)
"[...] sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work - that is, correctly to describe phenomena from a reasonably wide area. Furthermore, it must satisfy certain aesthetic criteria - that is, in relation to how much it describes, it must be rather simple.” (John von Neumann, “Method in the physical sciences”, 1961)
“Mathematical statistics provides an exceptionally clear example of the relationship between mathematics and the external world. The external world provides the experimentally measured distribution curve; mathematics provides the equation (the mathematical model) that corresponds to the empirical curve. The statistician may be guided by a thought experiment in finding the corresponding equation.” (Marshall J Walker, “The Nature of Scientific Thought”, 1963)
“A mathematical model is any complete and consistent set of mathematical equations which are designed to correspond to some other entity, its prototype. The prototype may be a physical, biological, social, psychological or conceptual entity, perhaps even another mathematical model." (Rutherford Aris, "Mathematical Modelling", 1978)
“Theoretical scientists, inching away from the safe and known, skirting the point of no return, confront nature with a free invention of the intellect. They strip the discovery down and wire it into place in the form of mathematical models or other abstractions that define the perceived relation exactly. The now-naked idea is scrutinized with as much coldness and outward lack of pity as the naturally warm human heart can muster. They try to put it to use, devising experiments or field observations to test its claims. By the rules of scientific procedure it is then either discarded or temporarily sustained. Either way, the central theory encompassing it grows. If the abstractions survive they generate new knowledge from which further exploratory trips of the mind can be planned. Through the repeated alternation between flights of the imagination and the accretion of hard data, a mutual agreement on the workings of the world is written, in the form of natural law.” (Edward O Wilson, “Biophilia”, 1984)
“The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing?” (Stephen Hawking, "A Brief History of Time", 1988)
“Mathematical modeling is about rules - the rules of reality. What distinguishes a mathematical model from, say, a poem, a song, a portrait or any other kind of ‘model’, is that the mathematical model is an image or picture of reality painted with logical symbols instead of with words, sounds or watercolors.” (John L Casti, "Reality Rules, The Fundamentals", 1992)
“Nature behaves in ways that look mathematical, but nature is not the same as mathematics. Every mathematical model makes simplifying assumptions; its conclusions are only as valid as those assumptions. The assumption of perfect symmetry is excellent as a technique for deducing the conditions under which symmetry-breaking is going to occur, the general form of the result, and the range of possible behaviour. To deduce exactly which effect is selected from this range in a practical situation, we have to know which imperfections are present” (Ian Stewart & Martin Golubitsky, “Fearful Symmetry”, 1992)
“In fact, the construction of mathematical models for various fragments of the real world, which is the most essential business of the applied mathematician, is nothing but an exercise in axiomatics.” (Marshall Stone, cca 1960)
"[...] sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work - that is, correctly to describe phenomena from a reasonably wide area. Furthermore, it must satisfy certain aesthetic criteria - that is, in relation to how much it describes, it must be rather simple.” (John von Neumann, “Method in the physical sciences”, 1961)
“Mathematical statistics provides an exceptionally clear example of the relationship between mathematics and the external world. The external world provides the experimentally measured distribution curve; mathematics provides the equation (the mathematical model) that corresponds to the empirical curve. The statistician may be guided by a thought experiment in finding the corresponding equation.” (Marshall J Walker, “The Nature of Scientific Thought”, 1963)
“A mathematical model is any complete and consistent set of mathematical equations which are designed to correspond to some other entity, its prototype. The prototype may be a physical, biological, social, psychological or conceptual entity, perhaps even another mathematical model." (Rutherford Aris, "Mathematical Modelling", 1978)
“Theoretical scientists, inching away from the safe and known, skirting the point of no return, confront nature with a free invention of the intellect. They strip the discovery down and wire it into place in the form of mathematical models or other abstractions that define the perceived relation exactly. The now-naked idea is scrutinized with as much coldness and outward lack of pity as the naturally warm human heart can muster. They try to put it to use, devising experiments or field observations to test its claims. By the rules of scientific procedure it is then either discarded or temporarily sustained. Either way, the central theory encompassing it grows. If the abstractions survive they generate new knowledge from which further exploratory trips of the mind can be planned. Through the repeated alternation between flights of the imagination and the accretion of hard data, a mutual agreement on the workings of the world is written, in the form of natural law.” (Edward O Wilson, “Biophilia”, 1984)
“The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing?” (Stephen Hawking, "A Brief History of Time", 1988)
“Mathematical modeling is about rules - the rules of reality. What distinguishes a mathematical model from, say, a poem, a song, a portrait or any other kind of ‘model’, is that the mathematical model is an image or picture of reality painted with logical symbols instead of with words, sounds or watercolors.” (John L Casti, "Reality Rules, The Fundamentals", 1992)
“Nature behaves in ways that look mathematical, but nature is not the same as mathematics. Every mathematical model makes simplifying assumptions; its conclusions are only as valid as those assumptions. The assumption of perfect symmetry is excellent as a technique for deducing the conditions under which symmetry-breaking is going to occur, the general form of the result, and the range of possible behaviour. To deduce exactly which effect is selected from this range in a practical situation, we have to know which imperfections are present” (Ian Stewart & Martin Golubitsky, “Fearful Symmetry”, 1992)
08 October 2017
Mathematics as Model
“Mathematics is not only the model along the lines of which the exact sciences are striving to design their structure; mathematics is the cement which holds the structure together.” (Tobias Dantzig, “Number: The Language of Science” 4th Ed, 1954)
“Mathematics is a model of exact reasoning, an absorbing challenge to the mind, an esthetic experience for creators and some students, a nightmarish experience to other students, and an outlet for the egotistic display of mental power.” (Morris Kline, “Mathematics and the Physical World”, 1959)
“Mathematics is much more than a language for dealing with the physical world. It is a source of models and abstractions which will enable us to obtain amazing new insights into the way in which nature operates. Indeed, the beauty and elegance of the physical laws themselves are only apparent when expressed in the appropriate mathematical framework.” (Melvin Schwartz)
“Mathematics is a model of exact reasoning, the most precise branch of human knowledge.” (Paul Hartal, Mathematics and Reality, 2010) [Link]
“Mathematics is a model for thinking, for developing scientific structure, for drawing conclusions and for solving problems.” (Kehinde E Adenegan) [Link]
“[…] although mathematical concepts and operations are formulated to represent aspects of the physical world, mathematics is not to be identified with the physical world. However, it tells us a good deal about that world if we are careful to apply it and interpret it properly.” (Morris Kline, “Mathematics for the Nonmathematician”, 1967)
“[…] mathematics is not portraying laws inherent in the design of the universe but is merely providing man-made schemes or models which we can use to deduce conclusions about our world only to the extent that the model is a good idealization.” (Morris Kline, “Mathematics for the Nonmathematician”, 1967)
On Models I
"To use an old analogy – and here we can hardly go except upon analogy – while the building of Nature is growing spontaneously from within, the model of it, which we seek to construct in our descriptive science, can only be constructed by means of scaffolding from without, a scaffolding of hypotheses. While in the real building all is continuous, in our model there are detached parts which must be connected with the rest by temporary ladders and passages, or which must be supported till we can see how to fill in the understructure. To give the hypotheses equal validity with facts is to confuse the temporary scaffolding with the building itself." (John H Poynting, 1899)
"[…] the more you see how strangely Nature behaves, the harder it is to make a model that explains how even the simplest phenomena actually work." (Richard P Feynman, "QED", 1985)
"No matter how beautiful the whole model may be, no matter how naturally it all seems to hang together now, if it disagrees with experiment, then it is wrong." (John Gribbin, "Almost Everyone’s Guide to Science", 1999)
"The purpose of models is not to fit the data but to sharpen the questions." (Samuel Karlin, 1983)
"I never satisfy myself until I can make a mechanical model of a thing. If I can make a mechanical model, I understand it." (Lord William T Kelvin, 1904)
"As we continue the great adventure of scientific exploration our models must often be recast. New laws and postulates will be required, while those that we already have must be broadened, extended and generalized in ways that we are now hardly able to surmise." (Gilbert Newton Lewis, "The Anatomy of Science", 1926)
"One good experiment is worth a thousand models […]; but one good model can make a thousand experiments unnecessary." (Evgenii I Volkov)
"We should always aim toward the economy of thought. It is not enough to give models for imitation. It must be possible to pass beyond these models and, in place of repeating their reasoning at length each time, to sum this in a few words." (Jules H Poincaré, 1909)
"No good model ever accounted for all the facts, since some data was bound to be misleading if not plain wrong." (James Dewey Watson)
"There are no surprising facts, only models that are surprised by facts; and if a model is surprised by the facts, it is no credit to that model." (Eliezer S Yudkowsky, "Quantum Explanations", 2008)
On Models: Models and Reality
"Imagining the unseeable is hard, because imagining means having an image in your mind. And how can you have a mental image of something you have never seen? Like perception itself, the models of science are embedded inextricably in the current worldview we call culture." (K C Cole, "First You Build a Cloud", 1999)
"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)
"To make progress in understanding all this, we probably need to begin with simplified (oversimplified?) models and ignore the critics’ tirade that the real world is more complex. The real world is always more complex, which has the advantage that we shan’t run out of work." (John Ball, "Memes as Replicators", Ethology and Sociobiology, Vol. 5, No. 3, 1984)
"We can only study Nature through our senses – that is […] we can only study the model of Nature that our senses enable our minds to construct; we cannot decide whether that model, consistent though it be, represents truly the real structure of Nature; whether, indeed, there be any Nature as an ultimate reality behind its phenomena." (Sir William C Dampier, "The Recent Development of Physical Science", 1904)
"The model of the natural world we build in our minds by such a process will forever be inadequate, just a little cathedral in the mountains. Still it is better than no model at all." (Timothy Ferris, "The Red Limit", 1977)
"A model, like a novel, may resonate with nature, but it is not a ‘real’ thing. Like a novel, a model may be convincing – it may ‘ring true’ if it is consistent with our experience of the natural world. But just as we may wonder how much the characters in a novel are drawn from real life and how much is artifice, we might ask the same of a model: How much is based on observation and measurement of accessible phenomena, how much is convenience? Fundamentally, the reason for modeling is a lack of full access, either in time or space, to the phenomena of interest." (K. Belitz, Science, Vol. 263, 1944)
"The different sciences are not even parts of a whole; they are but different aspects of a whole, which essentially has nothing in it corresponding to the divisions we make; they are, so to speak, sections of our model of Nature in certain arbitrary planes, cut in directions to suit our convenience." (Sir William Cecil Dampier, "The Recent Development of Physical Science", 1904)
"Confronted with the mystery of the Universe, we are driven to ask if the model our minds have framed at all corresponds with the reality; if, indeed, there be any reality behind the image." (Sir William Cecil Dampier, "The Recent Development of Physical Science", 1904)
"Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality." (Nikola Tesla, "Radio Power Will Revolutionize the World", Modern Mechanics and Inventions, 1934)
"The fact that [the model] is an approximation does not necessarily detract from its usefulness because models are approximations. All models are wrong, but some are useful." (George Box, 1987)
"When evaluating a model, at least two broad standards are relevant. One is whether the model is consistent with the data. The other is whether the model is consistent with the ‘real world’." (Kenneth A Bollen, "Structural Equations with Latent Variables", 1989)
"The model is only a suggestive metaphor, a fiction about the messy and unwieldy observations of the real world. In order for it to be persuasive, to convey a sense of credibility, it is important that it not be too complicated and that the assumptions that are made be clearly in evidence. In short, the model must be simple, transparent, and verifiable." (Edward Beltrami, "Mathematics for Dynamic Modeling", 1987)
"The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work" (John Von Neumann, "Method in the Physical Sciences", 1955)
"The laws of nature 'discovered' by science are merely mathematical or mechanical models that describe how nature behaves, not why, nor what nature 'actually' is. Science strives to find representations that accurately describe nature, not absolute truths. This fact distinguishes science from religion." (George Ogden Abell)
"Models of the real world are not always easy to formulate because of the richness, variety, and ambiguity that exists in the real world or because of our ambiguous understanding of it." (George Dantzig & Mukund N Thapa, "Linear Programming" Vol I, 1997)
"You never change things by fighting the existing reality. To change something, build a new model that makes the existing model obsolete." (R Buckminster Fuller)
On Simplicity III (Complexity vs Simplicity I)
“[…] the simplicity of nature which we at present grasp is really the result of infinite complexity; and that below the uniformity there underlies a diversity whose depths we have not yet probed, and whose secret places are still beyond our reach.” (William Spottiswoode, 1879)
“The aim of science is always to reduce complexity to simplicity.” (William James, “The Principles of Psychology”, 1890)
”The central task of a natural science is to make the wonderful commonplace: to show that complexity, correctly viewed, is only a mask for simplicity; to find pattern hidden in apparent chaos.” (Herbert A Simon, “The Sciences of the Artificial”, 1969)
“The aim of science is to seek the simplest explanations of complex facts. We are apt to fall into the error of thinking that the facts are simple because simplicity is the goal of our quest. The guiding motto in the life of every natural philosopher should be, ‘Seek simplicity and distrust it.’” (Alfred N Whitehead, “The Concept of Nature”, 1919)
“The beauty of physics lies in the extent which seemingly complex and unrelated phenomena can be explained and correlated through a high level of abstraction by a set of laws which are amazing in their simplicity.” (Melvin Schwartz, “In Principles of Electrodynamics”, 1972)
“Any intelligent fool can make things bigger, more complex, and more violent. It takes a touch of genius — and a lot of courage to move in the opposite direction.” (Ernst F Schumacher, “Small is Beautiful”, 1973)
"Simplicity does not precede complexity, but follows it." (Alan Perlis, “Epigrams on Programming”, 1982)
“[…] it is important to emphasize the value of simplicity and elegance, for complexity has a way of compounding difficulties.” (Fernando J Corbato, “On Building Systems That Will Fail”, 1991)
"Nature is capable of building complex structures by processes of self-organization; simplicity begets complexity." (Victor J Stenger, "God: The Failed Hypothesis", 2010)
“Complexity is the prodigy of the world. Simplicity is the sensation of the universe. Behind complexity, there is always simplicity to be revealed. Inside simplicity, there is always complexity to be discovered.” (Gang Yu, “in Data Warehousing in the Age of Big Data”, 2013)
“The aim of science is always to reduce complexity to simplicity.” (William James, “The Principles of Psychology”, 1890)
”The central task of a natural science is to make the wonderful commonplace: to show that complexity, correctly viewed, is only a mask for simplicity; to find pattern hidden in apparent chaos.” (Herbert A Simon, “The Sciences of the Artificial”, 1969)
“The aim of science is to seek the simplest explanations of complex facts. We are apt to fall into the error of thinking that the facts are simple because simplicity is the goal of our quest. The guiding motto in the life of every natural philosopher should be, ‘Seek simplicity and distrust it.’” (Alfred N Whitehead, “The Concept of Nature”, 1919)
“The beauty of physics lies in the extent which seemingly complex and unrelated phenomena can be explained and correlated through a high level of abstraction by a set of laws which are amazing in their simplicity.” (Melvin Schwartz, “In Principles of Electrodynamics”, 1972)
“Any intelligent fool can make things bigger, more complex, and more violent. It takes a touch of genius — and a lot of courage to move in the opposite direction.” (Ernst F Schumacher, “Small is Beautiful”, 1973)
"Simplicity does not precede complexity, but follows it." (Alan Perlis, “Epigrams on Programming”, 1982)
“[…] it is important to emphasize the value of simplicity and elegance, for complexity has a way of compounding difficulties.” (Fernando J Corbato, “On Building Systems That Will Fail”, 1991)
"Nature is capable of building complex structures by processes of self-organization; simplicity begets complexity." (Victor J Stenger, "God: The Failed Hypothesis", 2010)
“Complexity is the prodigy of the world. Simplicity is the sensation of the universe. Behind complexity, there is always simplicity to be revealed. Inside simplicity, there is always complexity to be discovered.” (Gang Yu, “in Data Warehousing in the Age of Big Data”, 2013)
05 October 2017
On Simplicity II
“The aim of science is not things themselves, as the dogmatists in their simplicity imagine, but the relation between things. (Henri Poincaré, “Science and Hypothesis”, 1905)
”Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness.” (Eric T Bell, “The Queen of the Sciences”, 1938)
“The greatest mathematics has the simplicity and inevitableness of supreme poetry and music, standing on the borderland of all that is wonderful in Science, and all that is beautiful in Art.” (Herbert W Turnbull)
“One of the principal objects of theoretical research in any department of knowledge is to find the point of view from which the subject appears in its greatest simplicity.” (J Willard Gibbs)
“Discoveries are not generally made in the order of their scientific arrangement: their connexions and relations are made out gradually; and it is only when the fermentation of invention has subsided that the whole clears into simplicity and order. “ (William Whewell)
”It is not merely the truth of science that makes it beautiful, but its simplicity.” (Walker Percy, “Signposts in a Strange Land”, 1991)
”The awkward richness of possibilities seems to shatter any possible coherent theory of simplicity…” (Lawrence B Slobodkin)
”It is often the scientist’s experience that he senses the nearness of truth when such connections are envisioned. A connection is a step toward simplification, unification. Simplicity is indeed often the sign of truth and a criterion of beauty.” (Mahlon B Hoagland, “Toward the Habit of Truth”, 1990)
”It would be simple enough, if only simplicity were not the most difficult of all things.” (Carl G Jung)
“[…] it is only through Mathematics that we can thoroughly understand what true science is. Here alone can we find in the highest degree simplicity and severity of scientific law, and such abstraction as the human mind can attain.” (Auguste Comte)
“Science attempts to find logic and simplicity in nature. Mathematics attempts to establish order and simplicity in human thought.” (Edward Teller, “The Pursuit of Simplicity”, 1980)
”Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness.” (Eric T Bell, “The Queen of the Sciences”, 1938)
“The greatest mathematics has the simplicity and inevitableness of supreme poetry and music, standing on the borderland of all that is wonderful in Science, and all that is beautiful in Art.” (Herbert W Turnbull)
“One of the principal objects of theoretical research in any department of knowledge is to find the point of view from which the subject appears in its greatest simplicity.” (J Willard Gibbs)
“Discoveries are not generally made in the order of their scientific arrangement: their connexions and relations are made out gradually; and it is only when the fermentation of invention has subsided that the whole clears into simplicity and order. “ (William Whewell)
”It is not merely the truth of science that makes it beautiful, but its simplicity.” (Walker Percy, “Signposts in a Strange Land”, 1991)
”The awkward richness of possibilities seems to shatter any possible coherent theory of simplicity…” (Lawrence B Slobodkin)
”It is often the scientist’s experience that he senses the nearness of truth when such connections are envisioned. A connection is a step toward simplification, unification. Simplicity is indeed often the sign of truth and a criterion of beauty.” (Mahlon B Hoagland, “Toward the Habit of Truth”, 1990)
”It would be simple enough, if only simplicity were not the most difficult of all things.” (Carl G Jung)
“[…] it is only through Mathematics that we can thoroughly understand what true science is. Here alone can we find in the highest degree simplicity and severity of scientific law, and such abstraction as the human mind can attain.” (Auguste Comte)
“Science attempts to find logic and simplicity in nature. Mathematics attempts to establish order and simplicity in human thought.” (Edward Teller, “The Pursuit of Simplicity”, 1980)
On Simplicity I (Nature’s Simplicity I)
”[…] it is astonishing and incredible to us, but not to Nature; for she performs with utmost ease and simplicity things which are even infinitely puzzling to our minds, and what is very difficult for us to comprehend is quite easy for her to perform.” (Galileo Galilei, "Dialog Concerning the Two World Systems", 1630)
"The supreme Being is everywhere; but He is not equally visible everywhere. Let us seek Him in the simplest things, in the most fundamental laws of Nature, in the universal rules by which movement is conserved, distributed or destroyed; and let us not seek Him in phenomena that are merely complex consequences of these laws."
"Men are often led into errors by the love of simplicity, which disposes us to reduce things to few principles, and to conceive a greater simplicity in nature than there really is." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)
“Nature does nothing in vain when less will serve; for Nature is pleased with simplicity and affects not the pomp of superfluous causes.“ (Sir Isaac Newton, “The Mathematical Principles of Natural Philosophy”, Voll. II, 1803)
“[…] 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 S Laplace, "The System of the World", 1809)
”How wonderful it is to me the simplicity of nature when we rightly interpret her laws and how different the convictions which they produce on the mind in comparison with the uncertain conclusions which hypothesis or even theory present.” (Michael Faraday, [letter to A F Svanberg] cca 1854)
”[…] we cannot a priori demand from nature simplicity, nor can we judge what in her opinion is simple.” (Heinrich Hertz, “The Principles of Mechanics Presented in a New Form”, 1894)
“Man’s first glance at the universe discovers only variety, diversity, multiplicity of phenomena. Let that glance be illuminated by science - by the science which brings man closer to God, - and simplicity and unity shine on all sides.” (Louis Pasteur)
”The simplicity of nature is not that which may easily be read, but is inexhaustible. The last analysis can no wise be made.” (Ralph W Emerson)
"Men are often led into errors by the love of simplicity, which disposes us to reduce things to few principles, and to conceive a greater simplicity in nature than there really is." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)
“Nature does nothing in vain when less will serve; for Nature is pleased with simplicity and affects not the pomp of superfluous causes.“ (Sir Isaac Newton, “The Mathematical Principles of Natural Philosophy”, Voll. II, 1803)
“[…] 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 S Laplace, "The System of the World", 1809)
”How wonderful it is to me the simplicity of nature when we rightly interpret her laws and how different the convictions which they produce on the mind in comparison with the uncertain conclusions which hypothesis or even theory present.” (Michael Faraday, [letter to A F Svanberg] cca 1854)
”[…] we cannot a priori demand from nature simplicity, nor can we judge what in her opinion is simple.” (Heinrich Hertz, “The Principles of Mechanics Presented in a New Form”, 1894)
“Man’s first glance at the universe discovers only variety, diversity, multiplicity of phenomena. Let that glance be illuminated by science - by the science which brings man closer to God, - and simplicity and unity shine on all sides.” (Louis Pasteur)
”The simplicity of nature is not that which may easily be read, but is inexhaustible. The last analysis can no wise be made.” (Ralph W Emerson)
03 October 2017
On Theories (Unsourced)
"First a new theory is attacked as absurd; then it is admitted to be true, but obvious and insignificant; finally it is seen to be so important that its adversaries claim they themselves discovered it.” (William James)
“Such is the advantage of a well-constructed language that its simplified notation often becomes the source of profound theories.” (Pierre-Simon Laplace)
"Somewhere out there is a theory that would explain my empirical observations, and this theory has yet to be discovered. Mathematics thrives on such mysteries.” (Henri Darmon)
“A theory is a supposition which we hope to be true, a hypothesis is a supposition which we expect to be useful; fictions belong to the realm of art; if made to intrude elsewhere, they become either make-believes or mistakes.” (G Johnstone Stoney)
"There is nothing more practical than a good theory.” ([attributed to] David Hilbert)
“Theory helps us to bear our ignorance of facts.” (George Santayana)
“What is especially striking and remarkable is that in fundamental physics, a beautiful or elegant theory is more likely to be right than a theory that is inelegant. A theory appears to be beautiful or elegant (or simple, if you prefer) when it can be expressed concisely in terms of mathematics we already have.” (Murray Gell-Mann)
"A mark of a good theory is that it proves even the most trivial results." (Hector Sussmann)
”If the facts do not conform to the theory, they must be disposed of.” (Norman R F Maier)
“First accumulate a mass of Facts: and then construct a Theory.” (Lewis Carroll)
“Such is the advantage of a well-constructed language that its simplified notation often becomes the source of profound theories.” (Pierre-Simon Laplace)
"Somewhere out there is a theory that would explain my empirical observations, and this theory has yet to be discovered. Mathematics thrives on such mysteries.” (Henri Darmon)
“A theory is a supposition which we hope to be true, a hypothesis is a supposition which we expect to be useful; fictions belong to the realm of art; if made to intrude elsewhere, they become either make-believes or mistakes.” (G Johnstone Stoney)
"There is nothing more practical than a good theory.” ([attributed to] David Hilbert)
“Theory helps us to bear our ignorance of facts.” (George Santayana)
“What is especially striking and remarkable is that in fundamental physics, a beautiful or elegant theory is more likely to be right than a theory that is inelegant. A theory appears to be beautiful or elegant (or simple, if you prefer) when it can be expressed concisely in terms of mathematics we already have.” (Murray Gell-Mann)
"A mark of a good theory is that it proves even the most trivial results." (Hector Sussmann)
”If the facts do not conform to the theory, they must be disposed of.” (Norman R F Maier)
“First accumulate a mass of Facts: and then construct a Theory.” (Lewis Carroll)
More on Theories
"It seems to be one of the fundamental features of nature the fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. " (Paul A M Dirac , “The Evolution of the Physicist’s Picture of Nature" , Scientific American, 1963)
“It [a theory] ought to furnish a compass which, if followed, will lead the observer further and further into previously unexplored regions. Whether these regions will be barren or fertile experience alone will decide; but, at any rate, one who is guided in this way will travel onward in a definite direction, and will not wander aimlessly to and fro.” (Sir Joseph J Thomson, “The Corpuscular Theory of Matter”, 1907)
”As soon as we inquire into the reasons for the phenomena, we enter the domain of theory, which connects the observed phenomena and traces them back to a single ‘pure’ phenomena, thus bringing about a logical arrangement of an enormous amount of observational material.” (Georg Joos, “Theoretical Physics”, 1968)
"[...] we and our models are both part of the universe we are describing. Thus a physical theory is self referencing, like in Gödel’s theorem. One might therefore expect it to be either inconsistent or incomplete. The theories we have so far are both inconsistent and incomplete." (Stephen Hawking, “Gödel and the End of the Universe” )
“The scientist who discovers a theory is usually guided to his discovery by guesses; he cannot name a method by means of which he found the theory and can only say that it appeared plausible to him, that he had the right hunch or that he saw intuitively which assumption would fit the facts.” (Hans Reichenbach, “The Rise of Scientific Philosophy”, 1951)
“It is in the nature of theoretical science that there can be no such thing as certainty. A theory is only ‘true’ for as long as the majority of the scientific community maintain the view that the theory is the one best able to explain the observations.” (Jim Baggott, “The Meaning of Quantum Theory”, 1992)
"Science is not about control. It is about cultivating a perpetual condition of wonder in the face of something that forever grows one step richer and subtler than our latest theory about it. It is about reverence, not mastery." (Richard Power, “Gold Bug Variations”, 1993)
”Books on physics are full of complicated mathematical formulae. But thought and ideas, not formulas, are the beginning of every physical theory.” (Leopold Infeld, “The Evolution of Physics”, 1961)
”A discovery in science, or a new theory, even where it appears most unitary and most all-embracing, deals with some immediate element of novelty or paradox within the framework of far vaster, unanalyzed, unarticulated reserves of knowledge, experience, faith, and presupposition. Our progress is narrow: it takes a vast world unchallenged and for granted.” (James R Oppenheimer, “Atom and Void”, 1989)
"Every theory of the course of events in nature is necessarily based on some process of simplification and is to some extent, therefore, a fairy tale." (Sir Napier Shaw, “Manual of Meteorology”, 1932)
“It [a theory] ought to furnish a compass which, if followed, will lead the observer further and further into previously unexplored regions. Whether these regions will be barren or fertile experience alone will decide; but, at any rate, one who is guided in this way will travel onward in a definite direction, and will not wander aimlessly to and fro.” (Sir Joseph J Thomson, “The Corpuscular Theory of Matter”, 1907)
”As soon as we inquire into the reasons for the phenomena, we enter the domain of theory, which connects the observed phenomena and traces them back to a single ‘pure’ phenomena, thus bringing about a logical arrangement of an enormous amount of observational material.” (Georg Joos, “Theoretical Physics”, 1968)
"[...] we and our models are both part of the universe we are describing. Thus a physical theory is self referencing, like in Gödel’s theorem. One might therefore expect it to be either inconsistent or incomplete. The theories we have so far are both inconsistent and incomplete." (Stephen Hawking, “Gödel and the End of the Universe” )
“The scientist who discovers a theory is usually guided to his discovery by guesses; he cannot name a method by means of which he found the theory and can only say that it appeared plausible to him, that he had the right hunch or that he saw intuitively which assumption would fit the facts.” (Hans Reichenbach, “The Rise of Scientific Philosophy”, 1951)
“It is in the nature of theoretical science that there can be no such thing as certainty. A theory is only ‘true’ for as long as the majority of the scientific community maintain the view that the theory is the one best able to explain the observations.” (Jim Baggott, “The Meaning of Quantum Theory”, 1992)
"Science is not about control. It is about cultivating a perpetual condition of wonder in the face of something that forever grows one step richer and subtler than our latest theory about it. It is about reverence, not mastery." (Richard Power, “Gold Bug Variations”, 1993)
”Books on physics are full of complicated mathematical formulae. But thought and ideas, not formulas, are the beginning of every physical theory.” (Leopold Infeld, “The Evolution of Physics”, 1961)
”A discovery in science, or a new theory, even where it appears most unitary and most all-embracing, deals with some immediate element of novelty or paradox within the framework of far vaster, unanalyzed, unarticulated reserves of knowledge, experience, faith, and presupposition. Our progress is narrow: it takes a vast world unchallenged and for granted.” (James R Oppenheimer, “Atom and Void”, 1989)
"Every theory of the course of events in nature is necessarily based on some process of simplification and is to some extent, therefore, a fairy tale." (Sir Napier Shaw, “Manual of Meteorology”, 1932)
From Facts to Theory
“[…] ideas may be both novel and important, and yet, if they are incorrect – if they lack the very essential support of incontrovertible fact, they are unworthy of credence. Without this, a theory may be both beautiful and grand, but must be as evanescent as it is beautiful, and as unsubstantial as it is grand.” (George Brewster, “A New Philosophy of Matter”, 1858)
“Perfect readiness to reject a theory inconsistent with fact is a primary requisite of the philosophic mind. But it, would be a mistake to suppose that this candour has anything akin to fickleness; on the contrary, readiness to reject a false theory may be combined with a peculiar pertinacity and courage in maintaining an hypothesis as long as its falsity is not actually apparent.” (William S Jevons, “The Principles of Science”, 1887)
”Scientific facts accumulate rapidly, and give rise to theories with almost equal rapidity. These theories are often wonderfully enticing, and one is apt to pass from one to another, from theory to theory, without taking care to establish each before passing on to the next, without assuring oneself that the foundation on which one is building is secure. Then comes the crash; the last theory
breaks down utterly, and on attempting to retrace our steps to firm ground and start anew, we may find too late that one of the cards, possibly at the very foundation of the pagoda, is either faultily placed or in itself defective, and that this blemish easily remedied if detected in time has, neglected, caused the collapse of the whole structure on whose erection so much skill and perseverance have been spent.” (Arthur M Marshall, 1894)
”Facts are carpet-tacks under the pneumatic tires of theory.” (Austin O’Malley, “Keystones of Thought”, 1918)
“Nothing is more interesting to the true theorist than a fact which directly contradicts a theory generally accepted up to that time, for this is his particular work.” (Max Planck, “A Survey of Physics”, 1925)
“[…] the mere collection of facts, without some basis of theory for guidance and elucidation, is foolish and profitless.” (Gamaliel Bradford, “Darwin”, 1926)
“[…] facts are too bulky to be lugged about conveniently except on the wheels of theory.” (Julian Huxley, “Essays of a Biologist”, 1929)
”[while] the traditional way is to regard the facts of science as something like the parts of a jig-saw puzzle, which can be fitted together in one and only one way, I regard them rather as the tiny pieces of a mosaic, which can be fitted together in many ways. A new theory in an old subject is, for me, a new mosaic pattern made with the pieces taken from an older pattern.” (William H George, “The Scientist in Action”, 1936)
”We can put it down as one of the principles learned from the history of science that a theory is only overthrown by a better theory, never merely by contradictory facts.” (James B Conant, “On Understanding Science”, 1947)
”Without facts we have no science. Facts are to the scientist what words are to the poet. The scientist has a love of facts, even isolated facts, similar to a poet’s love of words. But a collection of facts is not a science any more than a dictionary is poetry. Around his facts the scientist weaves a logical pattern or theory which gives the facts meaning, order and significance.” (Isidor Isaac Rabi, “Faith in Science”, Atlantic Monthly , Vol. 187, 1951)
“The true aim of science is to discover a simple theory which is necessary and sufficient to cover the facts, when they have been purified of traditional prejudices.” (Lancelot L Whyte, “Accent on Form”, 1954)
“Science does not begin with facts; one of its tasks is to uncover the facts by removing misconceptions.” (Lancelot L Whyte, “Accent on Form”, 1954)
“When we meet a fact which contradicts a prevailing theory, we must accept the fact and abandon the theory, even when the theory is supported by great names and generally accepted.” (Claude Bernard, “An Introduction to the Study of Experimental Medicine”, 1957)
”Facts do not ‘speak for themselves’; they are read in the light of theory. Creative thought, in science as much as in the arts, is the motor of changing opinion. Science is a quintessentially human activity, not a mechanized, robot-like accumulation of objective information, leading by laws of logic to inescapable interpretation.” (Stephen J Gould, “Ever Since Darwin”, 1977)
”No theory ever agrees with all the facts in its domain, yet it is not always the theory that is to blame. Facts are constituted by older ideologies, and a clash between facts and theories may be proof of progress.” (Paul K Feyerabend, “Against Method”, 1978)
[Maier’s Law:] ”If the facts do not conform to the theory, they must be disposed of.” (Norman R F Maier)
“First accumulate a mass of Facts: and then construct a Theory.” (Lewis Carroll)
“Perfect readiness to reject a theory inconsistent with fact is a primary requisite of the philosophic mind. But it, would be a mistake to suppose that this candour has anything akin to fickleness; on the contrary, readiness to reject a false theory may be combined with a peculiar pertinacity and courage in maintaining an hypothesis as long as its falsity is not actually apparent.” (William S Jevons, “The Principles of Science”, 1887)
”Scientific facts accumulate rapidly, and give rise to theories with almost equal rapidity. These theories are often wonderfully enticing, and one is apt to pass from one to another, from theory to theory, without taking care to establish each before passing on to the next, without assuring oneself that the foundation on which one is building is secure. Then comes the crash; the last theory
breaks down utterly, and on attempting to retrace our steps to firm ground and start anew, we may find too late that one of the cards, possibly at the very foundation of the pagoda, is either faultily placed or in itself defective, and that this blemish easily remedied if detected in time has, neglected, caused the collapse of the whole structure on whose erection so much skill and perseverance have been spent.” (Arthur M Marshall, 1894)
”Facts are carpet-tacks under the pneumatic tires of theory.” (Austin O’Malley, “Keystones of Thought”, 1918)
“Nothing is more interesting to the true theorist than a fact which directly contradicts a theory generally accepted up to that time, for this is his particular work.” (Max Planck, “A Survey of Physics”, 1925)
“[…] the mere collection of facts, without some basis of theory for guidance and elucidation, is foolish and profitless.” (Gamaliel Bradford, “Darwin”, 1926)
“[…] facts are too bulky to be lugged about conveniently except on the wheels of theory.” (Julian Huxley, “Essays of a Biologist”, 1929)
”[while] the traditional way is to regard the facts of science as something like the parts of a jig-saw puzzle, which can be fitted together in one and only one way, I regard them rather as the tiny pieces of a mosaic, which can be fitted together in many ways. A new theory in an old subject is, for me, a new mosaic pattern made with the pieces taken from an older pattern.” (William H George, “The Scientist in Action”, 1936)
”We can put it down as one of the principles learned from the history of science that a theory is only overthrown by a better theory, never merely by contradictory facts.” (James B Conant, “On Understanding Science”, 1947)
”Without facts we have no science. Facts are to the scientist what words are to the poet. The scientist has a love of facts, even isolated facts, similar to a poet’s love of words. But a collection of facts is not a science any more than a dictionary is poetry. Around his facts the scientist weaves a logical pattern or theory which gives the facts meaning, order and significance.” (Isidor Isaac Rabi, “Faith in Science”, Atlantic Monthly , Vol. 187, 1951)
“The true aim of science is to discover a simple theory which is necessary and sufficient to cover the facts, when they have been purified of traditional prejudices.” (Lancelot L Whyte, “Accent on Form”, 1954)
“Science does not begin with facts; one of its tasks is to uncover the facts by removing misconceptions.” (Lancelot L Whyte, “Accent on Form”, 1954)
“When we meet a fact which contradicts a prevailing theory, we must accept the fact and abandon the theory, even when the theory is supported by great names and generally accepted.” (Claude Bernard, “An Introduction to the Study of Experimental Medicine”, 1957)
”Facts do not ‘speak for themselves’; they are read in the light of theory. Creative thought, in science as much as in the arts, is the motor of changing opinion. Science is a quintessentially human activity, not a mechanized, robot-like accumulation of objective information, leading by laws of logic to inescapable interpretation.” (Stephen J Gould, “Ever Since Darwin”, 1977)
”No theory ever agrees with all the facts in its domain, yet it is not always the theory that is to blame. Facts are constituted by older ideologies, and a clash between facts and theories may be proof of progress.” (Paul K Feyerabend, “Against Method”, 1978)
[Maier’s Law:] ”If the facts do not conform to the theory, they must be disposed of.” (Norman R F Maier)
“First accumulate a mass of Facts: and then construct a Theory.” (Lewis Carroll)
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