“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)
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On Thresholds (From Fiction to Science-Ficttion)
"For many men that stumble at the threshold Are well foretold that danger lurks within." (William Shakespeare, "King Henry th...