"It is true that of far the greater part of things, we must content ourselves with such knowledge as description may exhibit, or analogy supply; but it is true likewise, that these ideas are always incomplete, and that at least, till we have compared them with realities, we do not know them to be just. As we see more, we become possessed of more certainties, and consequently gain more principles of reasoning, and found a wider base of analogy." (Samuel Johnson, 1825)
"Such is the tendency of the human mind to speculation, that on the least idea of an analogy between a few phenomena, it leaps forward, as it were, to a cause or law, to the temporary neglect of all the rest; so that, in fact, almost all our principal inductions must be regarded as a series of ascents and descents, and of conclusions from a few cases, verified by trial on many." (Sir John Herschel, "A Preliminary Discourse on the Study of Natural Philosophy" , 1830)
"Whilst chemical pursuits exalt the understanding, they do not depress the imagination or weaken genuine feeling; whilst they give the mind habits of accuracy, by obliging it to attend to facts, they likewise extend its analogies; and, though conversant with the minute forms of things, they have for their ultimate end the great and magnificent objects of nature." (Sir Humphry Davy, "Consolations in Travel, or the Last Days of a Philosopher", 1830)
"Science is nothing but the finding of analogy, identity, in the most remote parts." (Ralph W Emerson, 1837)
"It is frequently analogy which guides the experienced to what are called good guesses." (Francis W Newman, "Lectures on Logic", 1838)
"On the whole, Analogy is to be regarded as a step towards satisfactory proof, much in advance of first presumptions, if skillfully applied; though if the excessive vagueness of the word like be not checked, arguments from analogy may be of the wildest and silliest kind." (Francis W Newman, "Lectures on Logic", 1838)
"To reason from analogy is often dangerous, but to illustrate by a fanciful analogy is sometimes a means by which we light an idea, as it were, into the understanding of another." (Anna B Jameson, "Studies, Stories, and Memoirs", 1838)
“The invention of what we may call primary or fundamental notation has been but little indebted to analogy, evidently owing to the small extent of ideas in which comparison can be made useful. But at the same time analogy should be attended to, even if for no other reason than that, by making the invention of notation an art, the exertion of individual caprice ceases to be allowable. Nothing is more easy than the invention of notation, and nothing of worse example and consequence than the confusion of mathematical expressions by unknown symbols. If new notation be advisable, permanently or temporarily, it should carry with it some mark of distinction from that which is already in use, unless it be a demonstrable extension of the latter.” (Augustus De Morgan, “Calculus of Functions” Encyclopaedia of Pure Mathematics, 1847)
Showing posts with label completeness. Show all posts
Showing posts with label completeness. Show all posts
13 December 2019
07 March 2019
Mental Models VIII (Limitations II)
“This is the greatest degree of impoverishment; the [mental] image, deprived little by little of its own characteristics, is nothing more than a shadow. […] Being dependent on the state of the brain, the image undergoes change like all living substance, - it is subject to gains and losses, especially losses. But each of the foregoing three classes has its use for the inventor. They serve as material for different kinds of imagination - in their concrete form, for the mechanic and the artist; in their schematic form, for the scientist and for others.” (Théodule-Armand Ribot, “Essay on the Creative Imagination”, 1900)
“It is by abstraction that one can separate movements, knowledge, and affectivity. And the analysis is, here, so far from being a real dismemberment that it is given only as probable. One can never effectively reduce an [mental] image to its elements, for the reason that an image, like all other psychic syntheses, is something more and different from the sum of its elements. […] We will always go from image to image. Comprehension is a movement which is never-ending, it is the reaction of the mind to an image by another image, to this one by another image and so on, in principle to infinity. “(Jean-Paul Sartre, “The Imaginary: A phenomenological psychology of the imagination”, 1940)
“Speaking without metaphor we have to declare that we are here faced with one of these typical antinomies caused by the fact that we have not yet succeeded in elaborating a fairly understandable outlook on the world without retiring our own mind, the producer of the world picture, from it, so that mind has no place in it. The attempt to press it into it, after all, necessarily produces some absurdities.” (Erwin Schrödinger, „Mind and Matter: the Tarner Lectures”, 1956)
“Mental models are fuzzy, incomplete, and imprecisely stated. Furthermore, within a single individual, mental models change with time, even during the flow of a single conversation. The human mind assembles a few relationships to fit the context of a discussion. As debate shifts, so do the mental models. Even when only a single topic is being discussed, each participant in a conversation employs a different mental model to interpret the subject. Fundamental assumptions differ but are never brought into the open. […] A mental model may be correct in structure and assumptions but, even so, the human mind - either individually or as a group consensus - is apt to draw the wrong implications for the future.” (Jay W Forrester, “Counterintuitive Behaviour of Social Systems”, Technology Review, 1971)
“At present, no complete account can be given - one may as well ask for an inventory of the entire products of the human imagination - and indeed such an account would be premature, since mental models are supposed to be in people's heads, and their exact constitution is an empirical question. Nevertheless, there are three immediate constraints on possible models. […] 1. The principle of computability: Mental models, and the machinery for constructing and interpreting them, are computable. […] 2. The principle of finitism: A mental model must be finite in size and cannot directly represent an infinite domain. […] 3. The principle of constructivism: A mental model is constructed from tokens arranged in a particular structure to represent a state of affairs.” (Philip Johnson-Laird, “Mental Models” 1983)
"Almost every aspect of our lives is shaped in some way by how we make sense of the world. Our thinking and our actions are affected by the mental models we hold. These models define our limits or open our opportunities. Despite their power and pervasiveness, these models are usually virtually invisible to us. We don't realize they are there at all.” (Robert Gunther et al, “The Power of Impossible Thinking: Transform the Business of Your Life and the Life of Your Business”, 2004)
“Humans have difficulty perceiving variables accurately […]. However, in general, they tend to have inaccurate perceptions of system states, including past, current, and future states. This is due, in part, to limited ‘mental models’ of the phenomena of interest in terms of both how things work and how to influence things. Consequently, people have difficulty determining the full implications of what is known, as well as considering future contingencies for potential systems states and the long-term value of addressing these contingencies. ” (William B. Rouse, “People and Organizations: Explorations of Human-Centered Design”, 2007)
“[…] we cannot accurately assess both what a mental model is and what it is becoming because the act of assessment affects the model.” (William B. Rouse, “People and Organizations: Explorations of Human-Centered Design”, 2007) [see Werner Heisenberg’s principle]
“Mental models are problematic because they typically operate unconsciously. This means that they influence behavior and structure thinking in ways that individuals do not recognize and therefore cannot easily articulate. This makes certain kinds of exploratory conversations difficult or impossible, and causes even reasonable propositions to be rejected out of hand.” (Kim Erwin, “Communicating The New: Methods to Shape and Accelerate Innovation”, 2013)
“A mental model is not necessarily founded on facts or complete understanding of reality. Let's be honest, most of our mental models are flawed in many ways, and that's perfectly normal. They work because they are fast and simple and not because they are a complete representation of the reality. […] The most important thing about a person's mental model is that it's simplified and very limited compared to what it models.” (Peter W Szabo, “User Experience Mapping”, 2017)
See also:
Mental Models VIII – More on their Limits
Mental Models I, II, III, V, VI, VII
“It is by abstraction that one can separate movements, knowledge, and affectivity. And the analysis is, here, so far from being a real dismemberment that it is given only as probable. One can never effectively reduce an [mental] image to its elements, for the reason that an image, like all other psychic syntheses, is something more and different from the sum of its elements. […] We will always go from image to image. Comprehension is a movement which is never-ending, it is the reaction of the mind to an image by another image, to this one by another image and so on, in principle to infinity. “(Jean-Paul Sartre, “The Imaginary: A phenomenological psychology of the imagination”, 1940)
“Speaking without metaphor we have to declare that we are here faced with one of these typical antinomies caused by the fact that we have not yet succeeded in elaborating a fairly understandable outlook on the world without retiring our own mind, the producer of the world picture, from it, so that mind has no place in it. The attempt to press it into it, after all, necessarily produces some absurdities.” (Erwin Schrödinger, „Mind and Matter: the Tarner Lectures”, 1956)
“Mental models are fuzzy, incomplete, and imprecisely stated. Furthermore, within a single individual, mental models change with time, even during the flow of a single conversation. The human mind assembles a few relationships to fit the context of a discussion. As debate shifts, so do the mental models. Even when only a single topic is being discussed, each participant in a conversation employs a different mental model to interpret the subject. Fundamental assumptions differ but are never brought into the open. […] A mental model may be correct in structure and assumptions but, even so, the human mind - either individually or as a group consensus - is apt to draw the wrong implications for the future.” (Jay W Forrester, “Counterintuitive Behaviour of Social Systems”, Technology Review, 1971)
“At present, no complete account can be given - one may as well ask for an inventory of the entire products of the human imagination - and indeed such an account would be premature, since mental models are supposed to be in people's heads, and their exact constitution is an empirical question. Nevertheless, there are three immediate constraints on possible models. […] 1. The principle of computability: Mental models, and the machinery for constructing and interpreting them, are computable. […] 2. The principle of finitism: A mental model must be finite in size and cannot directly represent an infinite domain. […] 3. The principle of constructivism: A mental model is constructed from tokens arranged in a particular structure to represent a state of affairs.” (Philip Johnson-Laird, “Mental Models” 1983)
"Almost every aspect of our lives is shaped in some way by how we make sense of the world. Our thinking and our actions are affected by the mental models we hold. These models define our limits or open our opportunities. Despite their power and pervasiveness, these models are usually virtually invisible to us. We don't realize they are there at all.” (Robert Gunther et al, “The Power of Impossible Thinking: Transform the Business of Your Life and the Life of Your Business”, 2004)
“Humans have difficulty perceiving variables accurately […]. However, in general, they tend to have inaccurate perceptions of system states, including past, current, and future states. This is due, in part, to limited ‘mental models’ of the phenomena of interest in terms of both how things work and how to influence things. Consequently, people have difficulty determining the full implications of what is known, as well as considering future contingencies for potential systems states and the long-term value of addressing these contingencies. ” (William B. Rouse, “People and Organizations: Explorations of Human-Centered Design”, 2007)
“[…] we cannot accurately assess both what a mental model is and what it is becoming because the act of assessment affects the model.” (William B. Rouse, “People and Organizations: Explorations of Human-Centered Design”, 2007) [see Werner Heisenberg’s principle]
“Mental models are problematic because they typically operate unconsciously. This means that they influence behavior and structure thinking in ways that individuals do not recognize and therefore cannot easily articulate. This makes certain kinds of exploratory conversations difficult or impossible, and causes even reasonable propositions to be rejected out of hand.” (Kim Erwin, “Communicating The New: Methods to Shape and Accelerate Innovation”, 2013)
“A mental model is not necessarily founded on facts or complete understanding of reality. Let's be honest, most of our mental models are flawed in many ways, and that's perfectly normal. They work because they are fast and simple and not because they are a complete representation of the reality. […] The most important thing about a person's mental model is that it's simplified and very limited compared to what it models.” (Peter W Szabo, “User Experience Mapping”, 2017)
See also:
Mental Models VIII – More on their Limits
Mental Models I, II, III, V, VI, VII
29 December 2018
On Randomness I (Trivia I)
“When a rule is extremely complex, that which conforms to it passes for irregular (random).” (Gottfried Leibniz, “Discourse on Metaphysics”, 1686)
“The tissue of the world is built from necessities and randomness; the intellect of men places itself between both and can control them; it considers the necessity and the reason of its existence; it knows how randomness can be managed, controlled, and used.” (Goethe)
“The very events which in their own nature appear most capricious and uncertain, and which in any individual case no attainable degree of knowledge would enable us to foresee, occur, when considerable numbers are taken into account, with a degree of regularity approaching to mathematical.” (John S Mills, “A System of Logic”, 1862)
“The definition of random in terms of a physical operation is notoriously without effect on the mathematical operations of statistical theory because so far as these mathematical operations are concerned random is purely and simply an undefined term.” (Walter A Shewhart and W. Edwards “Deming, Statistical Method from the Viewpoint of Quality Control”, 1939)
“Perhaps randomness is not merely an adequate description for complex causes that we cannot specify. Perhaps the world really works this way, and many events are uncaused in any conventional sense of the word.” (Stephen Jay Gould,"Hen's Teeth and Horse's Toes”, 1983).
“If you perceive the world as some place where things happen at random - random events over which you have sometimes very little control, sometimes fairly good control, but still random events - well, one has to be able to have some idea of how these things behave. […] People who are not used to statistics tend to see things in data - there are random fluctuations which can sometimes delude them - so you have to understand what can happen randomly and try to control whatever can be controlled. You have to expect that you are not going to get a clean-cut answer. So how do you interpret what you get? You do it by statistics.” (Lucien LeCam, [interview] 1988)
“Events may appear to us to be random, but this could be attributed to human ignorance about the details of the processes involved.” (Brain S Everitt, “Chance Rules”, 1999)
“While in theory randomness is an intrinsic property, in practice, randomness is incomplete information.” (Nassim N Taleb, “The Black Swan”, 2007)
“The fact that randomness requires a physical rather than a mathematical source is noted by almost everyone who writes on the subject, and yet the oddity of this situation is not much remarked.” (Brian Hayes, “Group Theory in the Bedroom”, 2008)
“Randomness might be defined in terms of order - its absence, that is. […] Everything we care about lies somewhere in the middle, where pattern and randomness interlace.” (James Gleick, “The Information: A History, a Theory, a Flood”, 2011)
“The tissue of the world is built from necessities and randomness; the intellect of men places itself between both and can control them; it considers the necessity and the reason of its existence; it knows how randomness can be managed, controlled, and used.” (Goethe)
“The very events which in their own nature appear most capricious and uncertain, and which in any individual case no attainable degree of knowledge would enable us to foresee, occur, when considerable numbers are taken into account, with a degree of regularity approaching to mathematical.” (John S Mills, “A System of Logic”, 1862)
“The definition of random in terms of a physical operation is notoriously without effect on the mathematical operations of statistical theory because so far as these mathematical operations are concerned random is purely and simply an undefined term.” (Walter A Shewhart and W. Edwards “Deming, Statistical Method from the Viewpoint of Quality Control”, 1939)
“Perhaps randomness is not merely an adequate description for complex causes that we cannot specify. Perhaps the world really works this way, and many events are uncaused in any conventional sense of the word.” (Stephen Jay Gould,"Hen's Teeth and Horse's Toes”, 1983).
“If you perceive the world as some place where things happen at random - random events over which you have sometimes very little control, sometimes fairly good control, but still random events - well, one has to be able to have some idea of how these things behave. […] People who are not used to statistics tend to see things in data - there are random fluctuations which can sometimes delude them - so you have to understand what can happen randomly and try to control whatever can be controlled. You have to expect that you are not going to get a clean-cut answer. So how do you interpret what you get? You do it by statistics.” (Lucien LeCam, [interview] 1988)
“Events may appear to us to be random, but this could be attributed to human ignorance about the details of the processes involved.” (Brain S Everitt, “Chance Rules”, 1999)
“While in theory randomness is an intrinsic property, in practice, randomness is incomplete information.” (Nassim N Taleb, “The Black Swan”, 2007)
“The fact that randomness requires a physical rather than a mathematical source is noted by almost everyone who writes on the subject, and yet the oddity of this situation is not much remarked.” (Brian Hayes, “Group Theory in the Bedroom”, 2008)
“Randomness might be defined in terms of order - its absence, that is. […] Everything we care about lies somewhere in the middle, where pattern and randomness interlace.” (James Gleick, “The Information: A History, a Theory, a Flood”, 2011)
03 October 2018
5 Books 10 Quotes IV: On Complex Numbers IV
| Ian Stewart, "Why Beauty Is Truth: The History of Symmetry", 2007 | |
 | “A complex number is just a pair of real numbers, manipulated according to a short list of simple rules. Since a pair of real numbers is surely just as ‘real’ as a single real number, real and complex numbers are equally closely related to reality, and ‘imaginary’ is misleading.” |
| “The complex numbers extend the real numbers by throwing in a new kind of number, the square root of minus one. But the price we pay for being able to take square roots of negative numbers is the loss of order. The complex numbers are a complete system but are spread out across a plane rather than aligned in a single orderly sequence.” | |
| David Mumford, Caroline Series & David Wright, "Indra’s Pearls: The Vision of Felix Klein", 2002 | |
 | “Complex numbers are really not as complex as you might expect from their name, particularly if we think of them in terms of the underlying two dimensional geometry which they describe. Perhaps it would have been better to call them 'nature's numbers'. Behind complex numbers is a wonderful synthesis between two dimensional geometry and an elegant arithmetic in which every polynomial equation has a solution.” |
| “Ordinary numbers have immediate connection to the world around us; they are used to count and measure every sort of thing. Adding, subtracting, multiplying and dividing all have simple interpretations in terms of the objects being counted and measured. When we pass to complex numbers, though, the arithmetic takes on a life of its own. Since -1 has no square root, we decided to create a new number game which supplies the missing piece. By adding in just this one new element √-1. we created a whole new world in which everything arithmetical, miraculously, works out just fine.”
| |
| Paul J Nahin, "An Imaginary Tale: The History of √-1", 1998 | |
 | “The discovery of complex numbers was the last in a sequence of discoveries that gradually filled in the set of all numbers, starting with the positive integers (finger counting) and then expanding to include the positive rationals and irrational reals, negatives, and then finally the complex.” |
| “When we try to take the square root of -1 (a real number), for example, we suddenly leave the real numbers, and so the reals are not complete with respect to the square root operation. We don’t have to be concerned that something like that will happen with the complex numbers, however, and we won’t have to invent even more exotic numbers (the ‘really complex’!) Complex numbers are everything there is in the two-dimensional plane.” | |
| Jerry R Muir Jr., “Complex Analysis: A Modern First Course in Function Theory”, 2015 | |
 | “Complex analysis should never be underestimated as simply being calculus with complex numbers in place of real numbers and is distinguished from being so at every possible opportunity.” |
| “The upgrade from the real numbers to the complex numbers has both algebraic and analytic motivation. The real numbers are not algebraically complete, meaning there are polynomial equations such as x^2 = −1 with no solutions. The incorporation of √-1 […] is a direct response to this.” | |
| Tobias Dantzig, “Number: The Language of Science”, 1930 | |
 | “[…] extensions beyond the complex number domain are possible only at the expense of the principle of permanence. The complex number domain is the last frontier of this principle. Beyond this either the commutativity of the operations or the rôle which zero plays in arithmetic must be sacrificed.” |
| “And so it was that the complex number, which had its origin in a symbol for a fiction, ended by becoming an indispensable tool for the formulation of mathematical ideas, a powerful instrument for the solution of intricate problems, a means for tracing kinships between remote mathematical disciplines.”
Previous Post <<||>> Next Post See also: More Quotes on Complex Numbers III More Quotes on Complex Numbers II More Quotes on Complex Numbers I Complex Numbers | |
25 December 2017
Models vs Theory
“A theory is just a mathematical model to describe the observations.” (Karl Popper)
"A theory is a purely mental image of how something should be." (Adrian Bejan)
“I am of the opinion that the task of the theory consists in constructing a picture of the external world that exists purely internally and must be our guiding star in all thought and experiment.” (Ludwig E Boltzmann)
"[...] 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, “Godel and the End of the Universe”) [Link]
“With each theory or model, our concepts of reality and of the fundamental constituents of the universe have changed.” (Stephen Hawking & Leonard Mlodinow, “The Grand Design”, 2010)
"A theory has only the alternative of being right or wrong. A model has a third possibility: it may be right, but irrelevant." (Manfred Eigen, 1973)
“Model is used as a theory. It becomes theory when the purpose of building a model is to understand the mechanisms involved in the developmental process. Hence as theory, model does not carve up or change the world, but it explains how change takes place and in what way or manner. This leads to build change in the structures.” (Laxmi Kanta Patnaik, “Model Building in Political Science”, The Indian Journal of Political Science, Vol. 50, No. 2, 1989) [Link]
“A theory is a set of deductively closed propositions that explain and predict empirical phenomena, and a model is a theory that is idealized.” (Jay Odenbaugh, “True Lies: Realism, Robustness, and Models”, Philosophy of Science, Vol. 78, No. 5, 2011) [Link]
“A theory is a good theory if it satisfies two requirements: it must accurately describe a large class of observations on the basis of a model that contains only a few arbitrary elements, and it must make definite predictions about the results of future observations.” (Stephen Hawking, “A Brief History of Time: From Big Bang To Black Holes”, 1988)
"A theory is a purely mental image of how something should be." (Adrian Bejan)
“I am of the opinion that the task of the theory consists in constructing a picture of the external world that exists purely internally and must be our guiding star in all thought and experiment.” (Ludwig E Boltzmann)
"[...] 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, “Godel and the End of the Universe”) [Link]
“With each theory or model, our concepts of reality and of the fundamental constituents of the universe have changed.” (Stephen Hawking & Leonard Mlodinow, “The Grand Design”, 2010)
"A theory has only the alternative of being right or wrong. A model has a third possibility: it may be right, but irrelevant." (Manfred Eigen, 1973)
“Model is used as a theory. It becomes theory when the purpose of building a model is to understand the mechanisms involved in the developmental process. Hence as theory, model does not carve up or change the world, but it explains how change takes place and in what way or manner. This leads to build change in the structures.” (Laxmi Kanta Patnaik, “Model Building in Political Science”, The Indian Journal of Political Science, Vol. 50, No. 2, 1989) [Link]
“A theory is a set of deductively closed propositions that explain and predict empirical phenomena, and a model is a theory that is idealized.” (Jay Odenbaugh, “True Lies: Realism, Robustness, and Models”, Philosophy of Science, Vol. 78, No. 5, 2011) [Link]
“A theory is a good theory if it satisfies two requirements: it must accurately describe a large class of observations on the basis of a model that contains only a few arbitrary elements, and it must make definite predictions about the results of future observations.” (Stephen Hawking, “A Brief History of Time: From Big Bang To Black Holes”, 1988)
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)
03 October 2017
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)
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