"We do not see nature with our eyes, but without understandings and our hearts." (William Hazlitt, "Thoughts on Taste", 1818)
"Anyone who has passed through the regular gradations of a classical education, and is not made a fool by it, may consider himself as having had a very narrow escape." (William Hazlitt, "Table Talk: Essays On Men And Manners", 1821-1822)
"Learning is, in too many cases, but a foil to common sense; a substitute for true knowledge." (William Hazlitt, "Table Talk; or, Original Essays", 1821-1822)
"This is the test and triumph of originality, not to show us what has never been, and what we may therefore very easily never have dreamt of, but to point out to us what is before our eyes and under our feet, though we have had no suspicion of its existence, for want of sufficient strength of intuition, of determined grasp of mind to seize and retain it." (William Hazlitt, "Table Talk; or, Original Essays", 1821-1822)
"Learning is its own exceeding great reward; and at the period of which we speak, it bore other fruits, not unworthy of it." (William Hazlitt, "The Plain Speaker", 1826)
"The origin of all science is in the desire to know causes; and the origin of all false science and imposture is in the desire to accept false causes rather than none; or, which is the same thing, in the unwillingness to acknowledge our own ignorance." (William Hazlitt, "Burke and the Edinburgh Phrenologists", The Atlas 15, 1829)
"The most important and lasting truths are the most obvious ones. Nature cheats us with her mysteries, one after another, like a juggler with his tricks; but shews us her plain honest face, without our paying for it." (William Hazlitt, "Characteristics: In the Manner of Rochefoucault's Maxims", 1837)
"Rules and models destroy genius and art." (William Hazlitt, "Sketches and Essays", 1839)
Quotes and Resources Related to Mathematics, (Mathematical) Sciences and Mathematicians
22 August 2019
18 August 2019
John Wallis - Collected Quotes
"Indeed, many geometric things can be discovered or elucidated by algebraic principles, and yet it does not follow that algebra is geometrical, or even that it is based on geometric principles (as some would seem to think). This close affinity of arithmetic and geometry comes about, rather, because geometry is, as it were, subordinate to arithmetic, and applies universal principles of arithmetic to its special objects." (John Wallis, "Mathesis Universalis", 1657)
"We have before had occasion (in the Solution of some Quadratick and Cubick Equations) to make mention of Negative Squares, and Imaginary Roots, (as contradistinguished to what they call Real Roots, whether affirmative or Negative) […].These ‘Imaginary’ Quantities (as they are commonly called) arising from ‘Supposed’ Root of a Negative Square, (when they happen) are reputed to imply that the Case proposed is Impossible." (John Wallis, "A Treatise of Algebra, Both Historical and Practical", 1673)
"[…] whereas Nature, in propriety of Speech, doth not admit more than Three (Local) Dimensions, (Length, Breadth and Thickness, in Lines, Surfaces and Solids) it may justly seem improper to talk of a Solid (of three Dimensions) drawn into a Fourth, Fifth, Sixth, or further Dimension." (John Wallis, "Treatise of Algebra", 1685)
"According to this Method [of indivisibles], a Line is considered as consisting of an Innumerable Multitude of Points: A Surface, of Lines […]: A Solid, of Plains, or other Surfaces […]. Now this is not to be so understood, as if those Lines (which have no breadth) could fill up a Surface; or those Plains or Surfaces, (which have no thickness) could complete a Solid. But by such Lines are to be understood, small Surfaces, (of such a length, but very narrow) […]." (John Wallis, "Treatise of Algebra", 1685)
"These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative Square (when they happen,) are reputed to imply that the Case proposed is Impossible. And so indeed it is, as to the first and strict notion of what is proposed. For it is not possible that any Number (Negative or Affirmative) Multiplied into it- self can produce (for instance) -4. Since that Like Signs (whether + or -) will produce +; and there- fore not -4. But it is also Impossible that any Quantity (though not a Supposed Square) can be Negative. Since that it is not possible that any Magnitude can be Less than Nothing or any Number Fewer than None. Yet is not that Supposition(of Negative Quantities,) either Unuseful or Absurd; when rightly understood. And though, as to the bare Algebraick Notation, it import a Quantity less than nothing. Yet, when it comes to a Physical Application, it denotes as Real a Quantity as if the Sign were +; but to be interpreted in a contrary sense." (John Wallis, "Treatise of Algebra", 1685)
"Where, by the way, we may observe a great difference between the proportion of Infinite to Finite, and, of Finite to Nothing. For 1/∞, that which is a part infinitely small, may, by infinite Multiplication, equal the whole: But 0/1 , that which is Nothing can by no Multiplication become equal to Something." (John Wallis, "Treatise of Algebra", 1685)
"These Exponents they call Logarithms, which are Artificial Numbers, so answering to the Natural Numbers, as that the addition and Subtraction of these, answers to the Multiplication and Division of the Natural Numbers. By this means, (the Tables being once made) the Work of Multiplication and Division is performed by Addition and Subtraction; and consequently that of Squaring and Cubing, by Duplication and Triplication; and that of Extracting the Square and Cubic Root, by Bisection and Trisection; and the like in the higher Powers." (John Wallis, "Of Logarithms, Their Invention and Use", 1685)
"We have before had occasion (in the Solution of some Quadratick and Cubick Equations) to make mention of Negative Squares, and Imaginary Roots, (as contradistinguished to what they call Real Roots, whether affirmative or Negative) […].These ‘Imaginary’ Quantities (as they are commonly called) arising from ‘Supposed’ Root of a Negative Square, (when they happen) are reputed to imply that the Case proposed is Impossible." (John Wallis, "A Treatise of Algebra, Both Historical and Practical", 1673)
"[…] whereas Nature, in propriety of Speech, doth not admit more than Three (Local) Dimensions, (Length, Breadth and Thickness, in Lines, Surfaces and Solids) it may justly seem improper to talk of a Solid (of three Dimensions) drawn into a Fourth, Fifth, Sixth, or further Dimension." (John Wallis, "Treatise of Algebra", 1685)
"According to this Method [of indivisibles], a Line is considered as consisting of an Innumerable Multitude of Points: A Surface, of Lines […]: A Solid, of Plains, or other Surfaces […]. Now this is not to be so understood, as if those Lines (which have no breadth) could fill up a Surface; or those Plains or Surfaces, (which have no thickness) could complete a Solid. But by such Lines are to be understood, small Surfaces, (of such a length, but very narrow) […]." (John Wallis, "Treatise of Algebra", 1685)
"These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative Square (when they happen,) are reputed to imply that the Case proposed is Impossible. And so indeed it is, as to the first and strict notion of what is proposed. For it is not possible that any Number (Negative or Affirmative) Multiplied into it- self can produce (for instance) -4. Since that Like Signs (whether + or -) will produce +; and there- fore not -4. But it is also Impossible that any Quantity (though not a Supposed Square) can be Negative. Since that it is not possible that any Magnitude can be Less than Nothing or any Number Fewer than None. Yet is not that Supposition(of Negative Quantities,) either Unuseful or Absurd; when rightly understood. And though, as to the bare Algebraick Notation, it import a Quantity less than nothing. Yet, when it comes to a Physical Application, it denotes as Real a Quantity as if the Sign were +; but to be interpreted in a contrary sense." (John Wallis, "Treatise of Algebra", 1685)
"Where, by the way, we may observe a great difference between the proportion of Infinite to Finite, and, of Finite to Nothing. For 1/∞, that which is a part infinitely small, may, by infinite Multiplication, equal the whole: But 0/1 , that which is Nothing can by no Multiplication become equal to Something." (John Wallis, "Treatise of Algebra", 1685)
"These Exponents they call Logarithms, which are Artificial Numbers, so answering to the Natural Numbers, as that the addition and Subtraction of these, answers to the Multiplication and Division of the Natural Numbers. By this means, (the Tables being once made) the Work of Multiplication and Division is performed by Addition and Subtraction; and consequently that of Squaring and Cubing, by Duplication and Triplication; and that of Extracting the Square and Cubic Root, by Bisection and Trisection; and the like in the higher Powers." (John Wallis, "Of Logarithms, Their Invention and Use", 1685)
Carl G Jung - Collected Quotes
"Intuition does not denote something contrary to reason, but something outside of the province of reason." (Carl G Jung, "Psychological types: or, The psychology of individuation", 1926)
"The creation of something new is not accomplished by the intellect but by the play instinct acting from inner necessity. The creative mind plays with the objects it loves. " (Carl G Jung, "Psychological types: or, The psychology of individuation", 1926)
"By a symbol I do not mean an allegory or a sign, but an image that describes in the best possible way the dimly discerned nature of the spirit. A symbol does not define or explain; it points beyond itself to a meaning that is darkly divined yet still beyond our grasp, and cannot be adequately expressed in the familiar words of our language." (Carl G Jung, "The Structure And Dynamics Of The Psyche", 1960)
"Language, in its origin and essence, is simply a system of signs or symbols that denote real occurrences or their echo in the human soul." (Carl G Jung, "The Structure And Dynamics Of The Psyche", 1960)
"Once we give serious consideration to the hypothesis of the unconscious, it follows that our view of the world can be but a provisional one; for if we effect so radical an alteration in the subject of perception and cognition as this dual focus implies, the result must be a world view very different from any known before." (Carl G Jung, "The Structure And Dynamics Of The Psyche", 1960)
"Synchronistic phenomena prove the simultaneous occurrence of meaningful equivalences in heterogenous, causally unrelated processes; in other words, they prove that a content perceived by an observer can, at the same time, be represented by an outside event, without any causal connection. From this it follows either that the psyche cannot be localized in time, or that space is relative to the psyche." (Carl G Jung, "The Structure And Dynamics Of The Psyche", 1960)
"We want to have certainties and no doubts - results and no experiments - without even seeing that certainties can arise only through doubt and results only thorough experiment." (Carl G Jung, "The Structure And Dynamics Of The Psyche", 1960)
"Myth is more individual and expresses life more precisely than does science. Science works with concepts of averages which are far too general to do justice to the subjective variety of an individual life." (Carl G Jung, "Memories, Dreams, Reflections", 1963)
"A dream that is not understood remains a mere occurrence; understood it becomes a living experience." (Carl G Jung, "The Practice of Psychotherapy", 1966)
"Madness is a special form of the spirit and clings to all teachings and philosophies, but even more to daily life, since life itself is full of craziness and at bottom utterly illogical. Man strives toward reason only so that he can make rules for himself. Life itself has no rules. That is its mystery and its unknown law. What you call knowledge is an attempt to impose something comprehensible on life." (Carl G Jung, "Liber Novus", 2009)
"Continuous creation is to be thought of not only as a series of successive acts of creation, but also as the eternal presence of the one creative act." (Carl G Jung)
"Error is just as important a condition of life as truth." (Carl G Jung)
"Intuition is perception via the unconscious that brings forth ideas, images, new possibilities and ways out of blocked situations." (Carl G Jung)
"It would be simple enough, if only simplicity were not the most difficult of all things." (Carl G Jung)
"Science is not indeed a perfect instrument, but it is a superb and invaluable tool that works harm only when it is taken as an end in itself." (Carl G Jung)
"Science is the art of creating suitable illusions which the fool believes or argues against, but the wise man enjoys for their beauty or their ingenuity, without being blind to the fact that they are human veils and curtains concealing the abysmal darkness of the unknowable." (Carl G Jung)
"The meaning and design of a problem seem not to lie in its solution, but in our working at it incessantly." (Carl G Jung)
"The squaring of the circle is a stage on the way to the unconscious, a point of transition leading to a goal lying as yet unformulated beyond it. It is one of those paths to the centre." (Carl G Jung)
"Ultimate truth, if there be such a thing, demands the concert of many voices." (Carl G Jung)
"The creation of something new is not accomplished by the intellect but by the play instinct acting from inner necessity. The creative mind plays with the objects it loves. " (Carl G Jung, "Psychological types: or, The psychology of individuation", 1926)
"By a symbol I do not mean an allegory or a sign, but an image that describes in the best possible way the dimly discerned nature of the spirit. A symbol does not define or explain; it points beyond itself to a meaning that is darkly divined yet still beyond our grasp, and cannot be adequately expressed in the familiar words of our language." (Carl G Jung, "The Structure And Dynamics Of The Psyche", 1960)
"Language, in its origin and essence, is simply a system of signs or symbols that denote real occurrences or their echo in the human soul." (Carl G Jung, "The Structure And Dynamics Of The Psyche", 1960)
"Once we give serious consideration to the hypothesis of the unconscious, it follows that our view of the world can be but a provisional one; for if we effect so radical an alteration in the subject of perception and cognition as this dual focus implies, the result must be a world view very different from any known before." (Carl G Jung, "The Structure And Dynamics Of The Psyche", 1960)
"Synchronistic phenomena prove the simultaneous occurrence of meaningful equivalences in heterogenous, causally unrelated processes; in other words, they prove that a content perceived by an observer can, at the same time, be represented by an outside event, without any causal connection. From this it follows either that the psyche cannot be localized in time, or that space is relative to the psyche." (Carl G Jung, "The Structure And Dynamics Of The Psyche", 1960)
"We want to have certainties and no doubts - results and no experiments - without even seeing that certainties can arise only through doubt and results only thorough experiment." (Carl G Jung, "The Structure And Dynamics Of The Psyche", 1960)
"Myth is more individual and expresses life more precisely than does science. Science works with concepts of averages which are far too general to do justice to the subjective variety of an individual life." (Carl G Jung, "Memories, Dreams, Reflections", 1963)
"A dream that is not understood remains a mere occurrence; understood it becomes a living experience." (Carl G Jung, "The Practice of Psychotherapy", 1966)
"Madness is a special form of the spirit and clings to all teachings and philosophies, but even more to daily life, since life itself is full of craziness and at bottom utterly illogical. Man strives toward reason only so that he can make rules for himself. Life itself has no rules. That is its mystery and its unknown law. What you call knowledge is an attempt to impose something comprehensible on life." (Carl G Jung, "Liber Novus", 2009)
"Continuous creation is to be thought of not only as a series of successive acts of creation, but also as the eternal presence of the one creative act." (Carl G Jung)
"Error is just as important a condition of life as truth." (Carl G Jung)
"Intuition is perception via the unconscious that brings forth ideas, images, new possibilities and ways out of blocked situations." (Carl G Jung)
"It would be simple enough, if only simplicity were not the most difficult of all things." (Carl G Jung)
"Science is not indeed a perfect instrument, but it is a superb and invaluable tool that works harm only when it is taken as an end in itself." (Carl G Jung)
"Science is the art of creating suitable illusions which the fool believes or argues against, but the wise man enjoys for their beauty or their ingenuity, without being blind to the fact that they are human veils and curtains concealing the abysmal darkness of the unknowable." (Carl G Jung)
"The meaning and design of a problem seem not to lie in its solution, but in our working at it incessantly." (Carl G Jung)
"The squaring of the circle is a stage on the way to the unconscious, a point of transition leading to a goal lying as yet unformulated beyond it. It is one of those paths to the centre." (Carl G Jung)
"Ultimate truth, if there be such a thing, demands the concert of many voices." (Carl G Jung)
15 August 2019
Theoni Pappas - Collected Quotes
"To experience the joy of mathematics is to realize mathematics is not some isolated subject that has little relationship to the things around us other than to frustrate us with unbalanced check books and complicated computations. Few grasp the true nature of mathematics - so entwined in our environment and in our lives." (Theoni Pappas, "The Joy of Mathematics: Discovering Mathematics All Around You", 1986)
"Mathematics is more than doing calculations, more than solving equations, more than proving theorems, more than doing algebra, geometry or calculus, more than a way of thinking. Mathematics is the design of a snowflake, the curve of a palm frond, the shape of a building, the joy of a game, the frustration of a puzzle, the crest of a wave, the spiral of a spider's web. It is ancient and yet new. Mathematics is linked to so many ideas and aspects of the universe." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)
"Perhaps mathematicians' fascination with pi over the centuries can be likened to the drive that motivates mountain climbers to attempt an ascent." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)
"The chaos theory will require scientists in all fields to, develop sophisticated mathematical skills, so that they will be able to better recognize the meanings of results. Mathematics has expanded the field of fractals to help describe and explain the shapeless, asymmetrical find randomness of the natural environment." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)
"Statistics is a very powerful and persuasive mathematical tool. People put a lot of faith in printed numbers. It seems when a situation is described by assigning it a numerical value, the validity of the report increases in the mind of the viewer. It is the statistician's obligation to be aware that data in the eyes of the uninformed or poor data in the eyes of the naive viewer can be as deceptive as any falsehoods." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)
"It is not surprising to find many mathematical ideas interconnected or linked. The expansion of mathematics depends on previously developed ideas. The formation of any mathematical system begins with some undefined terms and axioms (assumptions) and proceeds from there to definitions, theorems, more axioms and so on. But history points out this is not necessarily the route that creativity" (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)
"Throughout the evolution of mathematics, problems have acted as catalysts in the discovery and development of mathematical ideas. In fact, the history of mathematics can probably be traced by studying the problems that mathematicians have tried to solve over the centuries. It is almost disheartening when an old problem is finally solved, for it will no longer be around to challenge and stimulate mathematical thought." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)
"Mathematics is more than doing calculations, more than solving equations, more than proving theorems, more than doing algebra, geometry or calculus, more than a way of thinking. Mathematics is the design of a snowflake, the curve of a palm frond, the shape of a building, the joy of a game, the frustration of a puzzle, the crest of a wave, the spiral of a spider's web. It is ancient and yet new. Mathematics is linked to so many ideas and aspects of the universe." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)
"Perhaps mathematicians' fascination with pi over the centuries can be likened to the drive that motivates mountain climbers to attempt an ascent." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)
"The chaos theory will require scientists in all fields to, develop sophisticated mathematical skills, so that they will be able to better recognize the meanings of results. Mathematics has expanded the field of fractals to help describe and explain the shapeless, asymmetrical find randomness of the natural environment." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)
"Statistics is a very powerful and persuasive mathematical tool. People put a lot of faith in printed numbers. It seems when a situation is described by assigning it a numerical value, the validity of the report increases in the mind of the viewer. It is the statistician's obligation to be aware that data in the eyes of the uninformed or poor data in the eyes of the naive viewer can be as deceptive as any falsehoods." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)
"It is not surprising to find many mathematical ideas interconnected or linked. The expansion of mathematics depends on previously developed ideas. The formation of any mathematical system begins with some undefined terms and axioms (assumptions) and proceeds from there to definitions, theorems, more axioms and so on. But history points out this is not necessarily the route that creativity" (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)
"Throughout the evolution of mathematics, problems have acted as catalysts in the discovery and development of mathematical ideas. In fact, the history of mathematics can probably be traced by studying the problems that mathematicians have tried to solve over the centuries. It is almost disheartening when an old problem is finally solved, for it will no longer be around to challenge and stimulate mathematical thought." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)
"When looking at the end result of any statistical analysis, one must be very cautious not to over interpret the data. Care must be taken to know the size of the sample, and to be certain the method forg athering information is consistent with other samples gathered. […] No one should ever base conclusions without knowing the size of the sample and how random a sample it was. But all too often such data is not mentioned when the statistics are given - perhaps it is overlooked or even intentionally omitted." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)
"Just as mathematical objects do not precisely describe things in our world, so traditional logic cannot be perfectly applied to the real world and real-world situations." (Theoni Pappas, "The Magic of Mathematics: Discovering the spell of mathematics", 1994)
"Objects in nature have provided and do provide models for stimulating mathematical discoveries. Nature has a way of achieving an equilibrium and an exquisite balance in its creations. The key to understanding the workings of nature is with mathematics and the sciences. [.] Mathematical tools provide a means by which we try to understand, explain, and copy natural phenomena. One discovery leads to the next." (Theoni Pappas, "The Magic of Mathematics: Discovering the spell of mathematics", 1994)
"Just as mathematical objects do not precisely describe things in our world, so traditional logic cannot be perfectly applied to the real world and real-world situations." (Theoni Pappas, "The Magic of Mathematics: Discovering the spell of mathematics", 1994)
"Objects in nature have provided and do provide models for stimulating mathematical discoveries. Nature has a way of achieving an equilibrium and an exquisite balance in its creations. The key to understanding the workings of nature is with mathematics and the sciences. [.] Mathematical tools provide a means by which we try to understand, explain, and copy natural phenomena. One discovery leads to the next." (Theoni Pappas, "The Magic of Mathematics: Discovering the spell of mathematics", 1994)
John L Casti - Collected Quotes
"[…] a complex system is incomprehensible unless we can simplify it by using alternative levels of description." (John L Casti, "On System Complexity: Identification, Measurement, and Management" [in "Complexity, Language, and Life: Mathematical Approaches"] 1986)
"Coping with complexity involves the creation of faithful models of not only the system to be managed. but also of the management system itself." (John L Casti, "On System Complexity: Identification, Measurement, and Management" [in "Complexity, Language, and Life: Mathematical Approaches"] 1986)
"Simple systems generally involve a small number of components. with self-interaction dominating the mutual interaction of the variables. […] Besides involving only a few variables. simple systems generally have very few feedback/feedforward loops. Such loops enable the system to restructure. or at least modify. the interaction pattern of its variables. thereby opening-up the possibility of a wider range of potential behavior patterns." (John L Casti, "On System Complexity: Identification, Measurement, and Management" [in "Complexity, Language, and Life: Mathematical Approaches"] 1986)
"Since most understanding and virtually all control is based upon a model (mental, mathematical, physical, or otherwise) of the system under study, the simplification imperative translates into a desire to obtain an equivalent, but reduced, representation of the original model of the system. This may involve omitting some of the original variables, aggregating others, ignoring weak couplings, regarding slowly changing variables as constants, and a variety of other subterfuges. All of these simplification techniques are aimed at reducing the degrees of freedom that the system has at its disposal to interact with its environment. A theory of system complexity would give us knowledge as to the limitations of the reduction process."
"The failure of individual subsystems to be sufficiently
adaptive to changing environments results in the subsystems forming a
collective association that, as a unit, is better able to function in new
circumstances. Formation of such an association is a structural change; the
behavioral role of the new conglomerate is a junctional change; both types of
change are characteristic of the formation of hierarchies."
"[…] a model is a mathematical representation of the modeler's reality, a way of capturing some aspects of a particular reality within the framework of a mathematical apparatus that provides us with a means for exploring the properties of the reality mirrored in the model." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)
"Basically, the point of making models is to be able to bring a measure of order to our experiences and observations, as well as to make specific predictions about certain aspects of the world we experience." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)
"Experiencing the world ultimately comes down to the recognition of boundaries: self/non-self, past/future, inside/outside, subject/object, and so forth. And so it is in mathematics, too, where we are continually called upon to make distinctions: solvable/unsolvable, computable/uncomputable, linear/nonlinear and other categorical distinctions involving the identification of boundaries. In particular, in geometry we characterize the boundaries of especially important figures by giving them names like circles, triangles, ellipses, and polygons. But when it comes to using these kinds of boundaries to describe the natural world, these simple geometrical shapes fail us completely: mountains are not cones, clouds are not spheres, and rivers are not straight lines." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)
"Great mathematics seldom comes from idle speculation about abstract spaces and symbols. More often than not it is motivated by definite questions arising in the worlds of nature and humans." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)
"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. These symbols are then strung together in accordance with a set of rules expressed in a special language, the language of mathematics." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)
"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. These symbols are then strung together in accordance with a set of rules expressed in a special language, the language of mathematics." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)
"Reliable information processing requires the existence of a good code or language, i.e., a set of rules that generate information at a given hierarchical level, and then compress it for use at a higher cognitive level. To accomplish this, a language should strike an optimum balance between variety (stochasticity) and the ability to detect and correct errors (memory)."(John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)
"Skewness is a measure of symmetry. For example, it's zero for the bell-shaped normal curve, which is perfectly symmetric about its mean. Kurtosis is a measure of the peakedness, or fat-tailedness, of a distribution. Thus, it measures the likelihood of extreme values." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)
"System theorists know that it's easy to couple simple-to-understand systems into a ‘super system’ that's capable of displaying behavioral modes that cannot be seen in any of its constituent parts. This is the process called ‘emergence’." (John L Casti, [interview with Austin Allen], 2012)
"[…] the complexity of a given system is always determined relative to another system with which the given system interacts. Only in extremely special cases, where one of these reciprocal interactions is so much weaker than the other that it can be ignored, can we justify the traditional attitude regarding complexity as an intrinsic property of the system itself." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)
"The idea of one description of a system bifurcating from
another also provides the key to begin unlocking one of the most important, and
at the same time perplexing, problems of system theory: characterization of the
complexity of a system."
"The key to making discontinuity emerge from smoothness is the observation that the overall behavior of both static and dynamical systems is governed by what's happening near the critical points. These are the points at which the gradient of the function vanishes. Away from the critical points, the Implicit Function Theorem tells us that the behavior is boring and predictable, linear, in fact. So it's only at the critical points that the system has the possibility of breaking out of this mold to enter a new mode of operation. It's at the critical points that we have the opportunity to effect dramatic shifts in the system's behavior by 'nudging' lightly the system dynamics, one type of nudge leading to a limit cycle, another to a stable equilibrium, and yet a third type resulting in the system's moving into the domain of a 'strange attractor'. It's by these nudges in the equations of motion that the germ of the idea of discontinuity from smoothness blossoms forth into the modern theory of singularities, catastrophes and bifurcations, wherein we see how to make discontinuous outputs emerge from smooth inputs." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)
"To function effectively, the system scientist must know a considerable amount about the natural world AND about mathematics, without being an expert in either field. This is clearly a prescription for career disaster in today's world of ultra-high specialization." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)"Virtually all mathematical theorems are assertions about the
existence or nonexistence of certain entities. For example, theorems assert the
existence of a solution to a differential equation, a root of a polynomial, or the
nonexistence of an algorithm for the Halting Problem. A platonist is one who
believes that these objects enjoy a real existence in some mystical realm
beyond space and time. To such a person, a mathematician is like an explorer
who discovers already existing things. On the other hand, a formalist is one
who feels we construct these objects by our rules of logical inference, and
that until we actually produce a chain of reasoning leading to one of these
objects they have no meaningful existence, at all."
"We can think of the evolution of a cellular automaton as a
pattern-recognition process, in which all initial configurations in the basin
of attraction of a particular attractor are thought of as instances of some
pattern with the attractor being the 'archetype' of this pattern.
Thus, the evolution of the different state trajectories toward this attractor
constitutes recognition of the pattern." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)
"What is usually left unsaid is an account of the equally
great failures of science, failures that most scientists fervently wish would
simply curl up into a little ball, roll off into a corner and disappear, much
like the now mythical ether. Perhaps the greatest failure of this sort in
classical physics is the inability to give any sort of coherent account of the
puzzling phenomenon of turbulence. […] The central difficulty in giving a
mathematical account of turbulence is the lack of any single scale of length
appropriate to the description of the phenomenon. Intuitively - and by
observation - turbulent flow involves nested eddies of all scales, ranging from
the macroscopic down to the molecular. So any mathematical description of the
process must take all these different scales into account. This situation is
rather similar to the problem of phase transitions, where length scales ranging
from the correlation length, which approaches infinity at the transition
temperature, down to the atomic scale all play an important role in the overall
transition process."
"When all the mathematical smoke clears away, Godel's message
is that mankind will never know the final secret of the universe by rational
thought alone. It's impossible for human beings to ever formulate a complete description
of the natural numbers. There will always be arithmetic truths that escape our
ability to fence them in using the tools, tricks and subterfuges of rational
analysis."
"[…] a rule for choosing an action is termed a strategy. If the rule says to always take the same action, it's called a pure strategy; otherwise, the strategy is called mixed. A solution to a game is simply a strategy for each player that gives each of them the best possible payoff, in the sense of being a regret-free choice." (John L Casti, "Five Golden Rules", 1995)
"Allowing more than two players into the game and/or
postulating payoff structures in which one player's gain does not necessarily
equal the other player's loss brings us much closer to the type of games played
in real life. Unfortunately, it's generally the case that the closer you get to
the messiness of the real world, the farther you move from the stylized and
structured world of mathematics. Game theory is no exception." (John L Casti,
"Five Golden Rules", 1995)
"Mathematics is about theorems: how to find them; how to prove them; how to generalize them; how to use them; how to understand them. […] But great theorems do not stand in isolation; they lead to great theories. […] And great theories in mathematics are like great poems, great paintings, or great literature: it takes time for them to mature and be recognized as being 'great'." (John L Casti, "Five Golden Rules", 1995)
"Catastrophe theory is a local theory, telling us what a function looks like in a small neighborhood of a critical point; it says nothing about what the function may be doing far away from the singularity. Yet most of the applications of the theory [...] involve extrapolating these rock-solid, local results to regions that may well be distant in time and space from the singularity." (John L Casti, "Five Golden Rules", 1995)
"Probably the most important reason that catastrophe theory received as much popular press as it did in the mid-1970s is not because of its unchallenged mathematical elegance, but because it appears to offer a coherent mathematical framework within which to talk about how discontinuous behaviors - stock market booms and busts or cellular differentiation, for instance - might emerge as the result of smooth changes in the inputs to a system, things like interest rates in a speculative market or the diffusion rate of chemicals in a developing embryo. These kinds of changes are often termed bifurcations, and playa central role in applied mathematical modeling. Catastrophe theory enables us to understand more clearly how - and why - they occur." (John L Casti, "Five Golden Rules", 1995)
"Since geometry is the mathematical idealization of space, a natural way to organize its study is by dimension. First we have points, objects of dimension O. Then come lines and curves, which are one-dimensional objects, followed by two-dimensional surfaces, and so on. A collection of such objects from a given dimension forms what mathematicians call a 'space'. And if there is some notion enabling us to say when two objects are 'nearby' in such a space, then it's called a topological space." (John L Casti, "Five Golden Rules", 1995)
"[...] there is no area of mathematics where thinking abstractly has paid more handsome dividends than in topology, the study of those properties of geometrical objects that remain unchanged when we deform or distort them in a continuous fashion without tearing, cutting, or breaking them." (John L Casti, "Five Golden Rules", 1995)
"So the strategy of mixing the choices with equal likelihood is an equilibrium point for the game, in the same sense that the minimax point is an equilibrium for a game having a saddle point. Thus, using a strategy that randomizes their choices, Max and Min can each announce his or her strategy to the other without the opponent being able to exploit this information to get a larger average payoff for himself or herself." (John L Casti, "Five Golden Rules", 1995)
"The goal of catastrophe theory is to classify smooth functions with degenerate critical points, just as Morse's Theorem gives us a complete classification for Morse functions. The difficulty, of course, is that there are a lot more ways for critical points to 'go bad' than there are for them to stay 'nice'. Thus, the classification problem is much harder for functions having degenerate critical points, and has not yet been fully carried out for all possible types of degeneracies. Fortunately, though, we can obtain a partial classification for those functions having critical points that are not too bad. And this classification turns out to be sufficient to apply the results to a wide range of phenomena like the predator-prey situation sketched above, in which 'jumps' in the system's biomass can occur when parameters describing the process change only slightly." (John L Casti, "Five Golden Rules", 1995)
"The Minimax Theorem applies to games in which there are just two players and for which the total payoff to both parties is zero, regardless of what actions the players choose. The advantage of these two properties is that with two players whose interests are directly opposed we have a game of pure competition, which allows us to define a clear-cut mathematical notion of rational behavior that leads to a single, unambiguous rule as to how each player should behave." (John L Casti, "Five Golden Rules", 1995)
"When it comes to modeling processes that are manifestly governed by nonlinear relationships among the system components, we can appeal to the same general idea. Calculus tells us that we should expect most systems to be 'locally' flat; that is, locally linear. So a conservative modeler would try to extend the word 'local' to hold for the region of interest and would take this extension seriously until it was shown to be no longer valid." (John L Casti, "Five Golden Rules", 1995)
"When we examine the modeling literature, its most striking aspect is the predominance of 'flat' linear models. Why is this the case? After all, from a singularity theory viewpoint these linear objects are mathematical rarities. On mathematical grounds we should certainly not expect to see them put forth as credible representations of reality. Yet they are. And the reason is simple: linearity is a neutral assumption that leads to mathematically tractable models. So unless there is good reason to do otherwise, why not use a linear model?" (John L Casti, "Five Golden Rules", 1995)
"The fruitful generalization in mathematics often involves starting from a commonsense concept such as a point on a line. A mathematical framework is then developed within which the particular example of a point in space is seen to be just a very special case of a much broader structure, say a point in three-dimensional space. Further generalizations then show this new structure itself to be only a special case of an even broader framework, the notion of a point in a space of n dimensions. And so it goes, one generalization piled atop another, each element leading to a deeper understanding of how the original object fits into a bigger picture." (John L Casti, "Five More Golden Rules : Knots, Codes, Chaos, and Other Great Theories of 20th Century Mathematics", 2000)
"A large part of Turing's genius was to show that the very primitive type of abstract computing machine he invented is actually the most general type of computer imaginable. In fact, every real-life computer that's ever been built is just a special case that materially embodies the machine that Turing dreamed up." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)
"By common consensus in the mathematical world, a good proof displays
three essential characteristics: a good proof is (1) convincing, (2) surveyable,
and (3) formalizable. The first requirement means simply that most
mathematicians believe it when they see it. […] Most mathematicians and
philosophers of mathematics demand more than mere plausibility, or even belief.
A proof must be able to be understood, studied, communicated, and verified by
rational analysis. In short, it must be surveyable. Finally, formalizability
means we can always find a suitable formal system in which an informal proof
can be embedded and fleshed out into a formal proof."
"Generally speaking, there are three grades of proof in mathematics. The first, or highest quality type of proof, is one that incorporates why and how the result is true, not simply that it is so. […] Second-grade proofs content themselves with showing that their conclusion is true, by relying on the law of the excluded middle. Thus, they assume that the conclusion they want to demonstrate is false and then derive a contradiction from this assumption. In polite company, these are often termed "nonconstructive proofs," since they lack the how and why. […] Finally, there is the third order, or lowest grade, of proof. In these situations, the idea of proof degenerates into mere verification, in which a (usually) large number of cases are considered separately and verified, one by one, very often by a computer." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)
"Somehow mathematicians seem to long for more than just results from their proofs; they want insight." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)
"That a proof must be convincing is part of the anthropology
of mathematics, providing the key to understanding
mathematics as a human activity. We invoke the logic of mathematics when we
demand that every informal proof be capable of being formalized within the
confines of a definite formal system. Finally, the epistemology of mathematics
comes into play with the requirement that a proof be surveyable. We can't
really say that we have created a genuine piece of knowledge unless it can be
examined and verified by others; there are no private truths in mathematics."
"The core of a decision problem is always to find a single
method that can be applied to each question, and that will always give the
correct answer for each individual problem."
"The double periodicity of the torus is fairly obvious: the
circle that goes around the torus in the 'long' direction around the
rim, together with the circle that goes around it through the hole in the
center. And just as periodic functions can be defined on a circle, doubly
periodic functions can be defined on a torus."
"The general idea of a model is to provide a concrete example of a mathematical framework that satisfies the axioms and relations of an abstract mathematical theory." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)
"The real raison d'etre for the mathematician's existence is simply to solve problems. So what mathematics really consists of is problems and solutions. And it is the "good" problems, the ones that challenge the greatest minds for decades, if not centuries, that eventually become enshrined as mathematical mountaintops." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)
"Traditionally, mathematical truths have been considered to be a priori truths, either in the sense that they are truths that would be true in any possible universe, or in the sense that they are truths whose validity is independent of our sensory impressions." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)
"[…] Turing machines are definitely not machines in the everyday sense of being material devices. Rather they are "paper computers," completely specified by their programs. Thus, when we use the term machine in what follows, the reader should read program or algorithm (i.e., software) and put all notions of hardware out of sight and out of mind." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)
"What's important about the Turing machine from a theoretical
point of view is that it represents a formal mathematical object. So with the invention
of the Turing machine, for the first time we had a well-defined notion of what
it means to compute something."
"[…] accept that X-events will occur. That is simply a fact
of life. So prepare for them as you’d prepare for any other life-changing, but
inherently unpredictable, event. This means remaining adaptive and open to new
possibilities, creating a life with as many degrees of freedom in it as
possible by educating yourself to be as self-sufficient as you can, and not
letting hope be replaced by fear and despair."
"[…] according to the bell-shaped curve the likelihood of a
very-large-deviation event (a major outlier) located in the striped region
appears to be very unlikely, essentially zero. The same event, though, is
several thousand times more likely if it comes from a set of events obeying a
fat-tailed distribution instead of the bell-shaped one."
"[…] both rarity and impact have to go into any meaningful characterization of how black any particular [black] swan happens to be." (John L Casti, "X-Events: The Collapse of Everything", 2012)
"[...] complexity overload is the precipitating cause of X-events. That overload may show up as unmanageable stress or pressure in a single system, be it a society, a corporation, or even an individual. The X-event that reduces the pressure then ranges from a societal collapse to a corporate bankruptcy to a nervous breakdown. […] You must add and subtract complexity judiciously
throughout the entire system in order to bring the imbalances back into line."
"Due to the problem of predicting outlier events, they are
not usually factored into the design of systems."
"[…] events will always occur that cannot be foreseen by following a chain of logical deductive reasoning. Successful prediction requires intuitive leaps and/or information that is not part of the original data available." (John L Casti, "X-Events: The Collapse of Everything", 2012)
"Forecasting models […] ordinarily are based only on past data, which is generally a tiny sample of the total range of possible outcomes. The problem is that those 'experts' who develop the models often come to believe they have mapped the entire space of possible system behaviors, which could not be further from the truth. Worse yet, when outliers do crop up, they are often discounted as 'once in a century' events and are all but ignored in planning for the future. […] the world is much more unpredictable than we’d like to believe."(John L Casti, "X-Events: The Collapse of Everything", 2012)
"Generally speaking, the best solution for solving a
complexity mismatch is to simplify the system that’s too complex rather than 'complexify' the simpler system."
"If you want a system - economic, social, political, or otherwise - to operate at a high level of efficiency, then you have to optimize its operation in such a way that its resilience is dramatically reduced to unknown - and possibly unknowable - shocks and/or changes in its operating environment. In other words, there is an inescapable price to be paid in efficiency in order to gain the benefits of adaptability and survivability in a highly uncertain environment. There is no escape clause!" (John L Casti, "X-Events: The Collapse of Everything", 2012)
"Sustainability is a delicate balancing act calling upon us to remain on the narrow path between organization and chaos, simplicity and complexity." (John L Casti, "X-Events: The Collapse of Everything", 2012)
"The first path of increasing complexity via innovation often faces limits as to how much complexity can be added or reduced in a given system. This is because if you change the complexity level in one place, a compensating change in the opposite direction generally occurs somewhere else." (John L Casti, "X-Events: The Collapse of Everything", 2012)
"[…] the law of requisite complexity […] states that in order
to fully regulate/control a system, the complexity of the controller has to be
at least as great as the complexity of the system that’s being controlled. To
put it in even simpler terms, only complexity can destroy complexity." (John L
Casti, "X-Events: The Collapse of Everything", 2012)
"What is and isn’t complex depends to a large degree not only on a target system, but also on the system(s) the target interacts with, together with the overall context in which the interacting systems are embedded." (John L Casti, "X-Events: The Collapse of Everything", 2012)
"A good analogy is stretching a rubber band. You can stretch and stretch and even feel the tension increase in the muscles in your hands and arms as the gap from one end of the band to the other widens. But at some point you reach the limits of elasticity of the band and it snaps. The same thing happens with human systems." (John L Casti)
"Reality is a wave function traveling both backward and forward in time." (John L Casti)
"Reality is a wave function traveling both backward and forward in time." (John L Casti)
13 August 2019
Francis Galton - Collected Quotes
"Exercising the right of occasional suppression and slight modification, it is truly absurd to see how plastic a limited number of observations become, in the hands of men with preconceived ideas." (Francis Galton, "Meteorographica" 1863)
"A visual image is the most perfect form of mental representation wherever the shape, position, and relations of objects in space are concerned. It is of importance in every handicraft and profession where design is required." (Francis Galton, "Mental Imagery" [in "Inquiries into Human Faculty and Development"] 1883)
"In the highest minds a descriptive word is sufficient to evoke crowds of shadowy associations, each striving to manifest itself. When they differ so much from one another as to be unfitted for combination into a single idea, there will be a conflict, each being prevented by the rest from obtaining sole possession of the field of consciousness. There could, therefore, be no definite imagery so long as the aggregate of all the pictures that the word suggested of objects presenting similar aspects, reduced to the same size, and accurately superposed, resulted in a blur; but a picture would gradually evolve as qualifications were added to the word, and it would attain to the distinctness and vividness of a generic image long before the word had been so restricted as to be individualised. If the intellect be slow, though correct in its operations, the associations will be few, and the generalised image based on insufficient data. If thevisualising power be faint, the generalised image will be indistinct." (Francis Galton, "Mental Imagery" [in "Inquiries into Human Faculty and Development"] 1883)
"The furniture of a man’s mind chiefly consists of his recollections and the bonds that unite them. As all this is the fruit of experience, it must differ greatly in different minds according to their individual experiences." (Francis Galton, "Associations" [in "Inquiries into Human Faculty and Development"] 1883)
"The object of statistical science is to discover methods of condensing information concerning large groups of allied facts into brief and compendious expressions suitable for discussion. The possibility of doing this is based on the constancy and continuity with which objects of the same species are found to vary." (Sir Francis Galton, "Inquiries into Human Faculty and Its Development, Statistical Methods", 1883)
"The place where the image appears to lie, differs much. Most persons see it in an indefinable sort of way, others see it in front of the eye, others at a distance corresponding to reality. There exists a power which is rare naturally, but can, I believe, be acquired without much difficulty, of projecting a mental picture upon a piece of paper, and of holding it fast there, so that it can be outlined with a pencil." (Francis Galton, "Mental Imagery" [in "Inquiries into Human Faculty and Development"] 1883)
"I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the ‘Law of Frequency of Error’. The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along." (Sir Francis Galton, "Natural Inheritance", 1889)
"It is always well to retain a clear geometric view of the facts when we are dealing with statistical problems, which abound with dangerous pitfalls, easily overlooked by the unwary, while they are cantering gaily along upon their arithmetic." (Sir Francis Galton, "Natural Inheritance", 1889)
"It is difficult to understand why statisticians commonly limit their inquiries to Averages, and do not revel in more comprehensive views. […] An Average is but a solitary fact, whereas if a single other fact be added to it, an entire Normal Scheme, which nearly corresponds to the observed one, starts potentially into existence. Some people hate the very name of statistics, but I find them full of beauty and interest. Whenever they are not brutalised, but delicately handled by the higher methods, and are warily interpreted, their power of dealing with complicated phenomena is extraordinary. They are the only tools by which an opening can be cut through the formidable thicket of difficulties that bars the path of those who pursue the Science of man." (Sir Francis Galton, "Natural Inheritance", 1889)
"It is difficult to understand why statisticians commonly limit their inquiries to Averages, and do not revel in more comprehensive views. Their souls seem as dull to the charm of variety as that of the native of one of our flat English counties, whose retrospect of Switzerland was that, if its mountains could be thrown into its lakes, two nuisances would be got rid of at once. An Average is but a solitary fact, whereas if a single other fact be added to it, an entire Normal Scheme, which nearly corresponds to the observed one, starts potentially into existence." (Sir Francis Galton, "Natural Inheritance", 1889)
"[Statistics] are the only tools by which an opening can be cut through the formidable thicket of difficulties that bars the path of those who pursue the Science of man." (Sir Francis Galton, "Natural Inheritance", 1889)
"It is now beginning to be generally understood, even by merely practical statisticians, that there is truth in the theory that all variability is much the same kind." (Francis Galton, "Kinship and Correlation", North American Review Vol. 150 (11), 1890)
"It had appeared from observation, and it was fully confirmed by this theory, that such a thing existed as an 'Index of Correlation', that is to say, a fraction, now commonly written T, that connects with close approximation every value of the deviation on the part of the subject, with the average of all the associated deviations of the Relative [...]" (Francis Galton, "Memories of My Life", 1908)
"General impressions are never to be trusted. Unfortunately when they are of long standing they become fixed rules of life and assume a prescriptive right not to be questioned. Consequently those who are not accustomed to original inquiry entertain a hatred and horror of statistics. They cannot endure the idea of submitting sacred impressions to cold-blooded verification. But it is the triumph of scientific men to rise superior to such superstitions, to desire tests by which the value of beliefs may be ascertained, and to feel sufficiently masters of themselves to discard contemptuously whatever may be found untrue." (Sir Francis Galton)
"It is notorious that the same discovery is frequently made simultaneously and quite independently, by different persons. […] It would seem, that discoveries are usually made when the time is ripe for them - that is to say, when the ideas from which they naturally flow are fermenting in the minds of many men." (Sir Francis Galton)
"[Statistics are] the only tools by which an opening can be cut through the formidable thicket of difficulties that bars the path of those who pursue the Science of Man." (Sir Ronald Galton)
11 August 2019
Abraham de Moivre - Collected Quotes
"And thus in all cases it will be found, that although Chance produces Irregularities, still the odds will be infinitely great that in the process of time, those Irregularities will bear no proportion to the recurrency of that Order which naturally results from ORIGINAL DESIGN." (Abraham de Moivre, "The Doctrine of Chances", 1718)
"If the obtaining of any Sum requires the happening of several Events that are independent on each other, then the Value of the Expectation of that Sum is found by multiplying together the several Probabilities of happening, and again multiplying the product by the Value of the Sum expected." (Abraham de Moivre, "The Doctrine of Chances", 1718)
"Further, the same Arguments which explode the Notion of Luck, may, on the other side, be useful in some Cases to establish a due comparison between Chance and Design: We may imagine Chance and Design to be, as it were, in Competition with each other, for the production of some sorts of Events, and many calculate what Probability there is, that those Events should be rather be owing to the one than to the other." (Abraham de Moivre, "The Doctrine of Chances", 1718)
"The Fractions which represent the Probabilities of happening and failing, being added together, their Sum will always be equal to Unity, since the Sum of their Numerators will be equal to their common Denominator : now it being a certainty that an Event will either happen or fail, it follows that Certainty, which may be conceived under the notion of an infinitely great degree of Probability, is fitly represented by Unity." (Abraham de Moivre, "The Doctrine of Chances", 1718)
"The probability of an Event is greater, or less, according to the number of Chances by which it may Happen, compar’d with the number of all the Chances, by which it may either Happen or Fail. […] Therefore, if the Probability of Happening and Failing are added together, the Sum will always be equal to Unit." (Abraham De Moivre, "The Doctrine of Chances", 1718)
"The Risk of losing any Sum is the reverse of Expectation; and the true measure of it is, the product of the Sum adventured multiplied by the Probability of the Loss." (Abraham de Moivre, "The Doctrine of Chances", 1718)
"Two Events are independent, when they have no connexion one with the other, and that the happening of one neither forwards nor obstructs the happening of the other.
Two Events are dependent, when they are so connected together as that the Probability of cither's happening is altered by the happening of the other." (Abraham de Moivre, "The Doctrine of Chances", 1718)
"If the obtaining of any Sum requires the happening of several Events that are independent on each other, then the Value of the Expectation of that Sum is found by multiplying together the several Probabilities of happening, and again multiplying the product by the Value of the Sum expected." (Abraham de Moivre, "The Doctrine of Chances", 1718)
"Further, the same Arguments which explode the Notion of Luck, may, on the other side, be useful in some Cases to establish a due comparison between Chance and Design: We may imagine Chance and Design to be, as it were, in Competition with each other, for the production of some sorts of Events, and many calculate what Probability there is, that those Events should be rather be owing to the one than to the other." (Abraham de Moivre, "The Doctrine of Chances", 1718)
"The Fractions which represent the Probabilities of happening and failing, being added together, their Sum will always be equal to Unity, since the Sum of their Numerators will be equal to their common Denominator : now it being a certainty that an Event will either happen or fail, it follows that Certainty, which may be conceived under the notion of an infinitely great degree of Probability, is fitly represented by Unity." (Abraham de Moivre, "The Doctrine of Chances", 1718)
"The probability of an Event is greater, or less, according to the number of Chances by which it may Happen, compar’d with the number of all the Chances, by which it may either Happen or Fail. […] Therefore, if the Probability of Happening and Failing are added together, the Sum will always be equal to Unit." (Abraham De Moivre, "The Doctrine of Chances", 1718)
"The Risk of losing any Sum is the reverse of Expectation; and the true measure of it is, the product of the Sum adventured multiplied by the Probability of the Loss." (Abraham de Moivre, "The Doctrine of Chances", 1718)
"Two Events are independent, when they have no connexion one with the other, and that the happening of one neither forwards nor obstructs the happening of the other.
Two Events are dependent, when they are so connected together as that the Probability of cither's happening is altered by the happening of the other." (Abraham de Moivre, "The Doctrine of Chances", 1718)
07 August 2019
Bernard de Fontenelle - Collected Quotes
"Grant a mathematician but one minute principle, he immediately draws a consequence from it, to which you must necessarily assent; and from this consequence another, till he leads you so far (whether you will or no) that you have much ado to believe all he has proved, and what you have already assented to." (Bernard Le Bovier de Fontenelle, "Conversations on the Plurality of Worlds", 1686)
"The universe is but a watch on a larger scale; all its motions depending on determined laws and mutual relation of its parts." (Bernard Le Bovier de Fontenelle, "Conversations on the Plurality of Worlds", 1686)
"We are under obligation to the ancients for having exhausted all the false theories that could be formed." (Bernard le Bovier de Fontenelle, "Conversations on the Plurality of Worlds", 1686)
"We do not yet pretend to have discovered all things, or that what we have discovered can receive no addition; and therefore, pray let us agree, there are yet many things to be done in the ages to come." (Bernard Le Bovier de Fontenelle, "Conversations on the Plurality of Worlds", 1686)
"Nothing proves more clearly that the mind seeks truth, and nothing reflects more glory upon it, than the delight it takes, sometimes in spite of itself, in the driest and thorniest researches of algebra." (Bernard de Fontenelle, "Histoire du Renouvellement de l'Académie des Sciences", 1708)
"From this it follows that the idea of positive or negative is added to those magnitudes which are contrary in some way. […] All contrariness or opposition suffices for the idea of positive or negative. […] Thus every positive or negative magnitude does not have just its numerical being, by which it is a certain number, a certain quantity, but has in addition its specific being, by which it is a certain Thing opposite to another. I say opposite to another, because it is only by this opposition that it attains a specific being (Bernard le Bouyer de Fontenelle, "Éléments de la géométrie de l'Infini", 1727)
"The calculus is to mathematics no more than what experiment is to physics, and all the truths produced solely by the calculus can be treated as truths of experiment." (Bernard Le Bovier de Fontenelle)
"There is in mathematics, so to speak, only what we have placed there, only the clearest ideas that the human mind can form of magnitude, compared with one another and combined in an infinity of different ways, while Nature could well have used in the construction of the universe some mechanics that escapes us entirely." (Bernard Le Bovier de Fontenelle)
"The universe is but a watch on a larger scale; all its motions depending on determined laws and mutual relation of its parts." (Bernard Le Bovier de Fontenelle, "Conversations on the Plurality of Worlds", 1686)
"We are under obligation to the ancients for having exhausted all the false theories that could be formed." (Bernard le Bovier de Fontenelle, "Conversations on the Plurality of Worlds", 1686)
"We do not yet pretend to have discovered all things, or that what we have discovered can receive no addition; and therefore, pray let us agree, there are yet many things to be done in the ages to come." (Bernard Le Bovier de Fontenelle, "Conversations on the Plurality of Worlds", 1686)
"Nothing proves more clearly that the mind seeks truth, and nothing reflects more glory upon it, than the delight it takes, sometimes in spite of itself, in the driest and thorniest researches of algebra." (Bernard de Fontenelle, "Histoire du Renouvellement de l'Académie des Sciences", 1708)
"From this it follows that the idea of positive or negative is added to those magnitudes which are contrary in some way. […] All contrariness or opposition suffices for the idea of positive or negative. […] Thus every positive or negative magnitude does not have just its numerical being, by which it is a certain number, a certain quantity, but has in addition its specific being, by which it is a certain Thing opposite to another. I say opposite to another, because it is only by this opposition that it attains a specific being (Bernard le Bouyer de Fontenelle, "Éléments de la géométrie de l'Infini", 1727)
"The calculus is to mathematics no more than what experiment is to physics, and all the truths produced solely by the calculus can be treated as truths of experiment." (Bernard Le Bovier de Fontenelle)
"There is in mathematics, so to speak, only what we have placed there, only the clearest ideas that the human mind can form of magnitude, compared with one another and combined in an infinity of different ways, while Nature could well have used in the construction of the universe some mechanics that escapes us entirely." (Bernard Le Bovier de Fontenelle)
David P Ruelle - Collected Quotes
"Due to this sensitivity any uncertainty about seemingly insignificant digits in the sequence of numbers which defines an initial condition, spreads with time towards the significant digits, leading to chaotic behavior. Therefore there is a change in the information we have about the state of the system. This change can be thought of as a creation of information if we consider that two initial conditions that are different but indistinguishable (within a certain precision), evolve into distinguishable states after a finite time." (David Ruelle, "Chaotic Evolution and Strange Attractors: The statistical analysis of time series for deterministic nonlinear systems", 1989)
"In a real experiment the noise present in a signal is usually considered to be the result of the interplay of a large number of degrees of freedom over which one has no control. This type of noise can be reduced by improving the experimental apparatus. But we have seen that another type of noise, which is not removable by any refinement of technique, can be present. This is what we have called the deterministic noise. Despite its intractability it provides us with a way to describe noisy signals by simple mathematical models, making possible a dynamical system approach to the problem of turbulence." (David Ruelle, "Chaotic Evolution and Strange Attractors: The statistical analysis of time series for deterministic nonlinear systems", 1989)
"In fact, in all those cases in which the initial state is
given with limited precision (if we assume that the space-time is continuous
this is always the case because a generic point turns out to be completely
specified only by an infinite amount of information, for example by an infinite
string of numbers), we can observe a situation in which, when time becomes
large, two trajectories emerge from the 'same' initial point. So, even though
there is a deterministic situation from a mathematical point of view (the uniqueness
theorem for ordinary differential equations is not in question), nevertheless
the exponential growth of errors makes the time evolution self-independent from
its past history and then nondeterministic in any practical sense."
"Now, the main problem with a quasiperiodic theory of turbulence (putting several oscillators together) is the following: when there is a nonlinear coupling between the oscillators, it very often happens that the time evolution does not remain quasiperiodic. As a matter of fact, in this latter situation, one can observe the appearance of a feature which makes the motion completely different from a quasiperiodic one. This feature is called sensitive dependence on initial conditions and turns out to be the conceptual key to reformulating the problem of turbulence." (David Ruelle, "Chaotic Evolution and Strange Attractors: The statistical analysis of time series for deterministic nonlinear systems", 1989)
"Roughly speaking the dimension of a set is the amount of information needed to specify points in it accurately." (David Ruelle, "Chaotic Evolution and Strange Attractors: The statistical analysis of time series for deterministic nonlinear systems", 1989)
"Very often a strange attractor is a fractal object, whose geometric structure is invariant under the time evolution maps." (David Ruelle, "Chaotic Evolution and Strange Attractors: The statistical analysis of time series for deterministic nonlinear systems", 1989)
"A purely psychological approach to science would miss the importance of the comprehensibility of mathematics, and of 'the unreasonable effectiveness of mathematics in the natural sciences'. In fact, some scientists in the 'soft' sciences seem to miss this as well. But mathematicians and physicists know that they deal with a reality that has laws of its own, a reality above our little psychological problems, a reality that is strange, fascinating, and in some sense beautiful." (David Ruelle, "Chance and Chaos", 1991)
"Although a system may exhibit sensitive dependence on initial condition, this does not mean that everything is unpredictable about it. In fact, finding what is predictable in a background of chaos is a deep and important problem. (Which means that, regrettably, it is unsolved.) In dealing with this deep and important problem, and for want of a better approach, we shall use common sense." (David Ruelle, "Chance and Chaos", 1991)
"And you should not think that the mathematical game is arbitrary and gratuitous. The diverse mathematical theories have many relations with each other: the objects of one theory may find an interpretation in another theory, and this will lead to new and fruitful viewpoints. Mathematics has deep unity. More than a collection of separate theories such as set theory, topology, and algebra, each with its own basic assumptions, mathematics is a unified whole." (David Ruelle, "Chance and Chaos", 1991)
"Because mathematical proofs are long, they are also difficult to invent. One has to construct, without making any mistakes, long chains of assertions, and see what one is doing, see where one is going. To see means to be able to guess what is true and what is false, what is useful and what is not. To see means to have a feeling for which definitions one should introduce, and what the key assertions are that will allow one to develop a theory in a natural manner." (David Ruelle, "Chance and Chaos", 1991)"By gluing a mathematical theory on a piece of physical reality we obtain a physical theory. There exist many such theories, covering a great diversity of phenomena. And for a given phenomenon there are usually several different theories. In the better cases one passes from one theory to another one by an approximation (usually an uncontrolled approximation)." (David Ruelle, "Chance and Chaos", 1991)
"First, strange attractors look strange: they are not smooth curves or surfaces but have 'non-integer dimension' - or, as Benoit Mandelbrot puts it, they are fractal objects. Next, and more importantly, the motion on a strange attractor has sensitive dependence on initial condition. Finally, while strange attractors have only finite dimension, the time-frequency analysis reveals a continuum of frequencies." (David Ruelle, "Chance and Chaos", 1991)
"[…] if a system is sufficiently complicated, the time it takes to return near a state already visited is huge (think of the hundred fleas on the checkerboard). Therefore if you look at the system for a moderate amount of time, eternal return is irrelevant, and you had better choose another idealization." (David Ruelle, "Chance and Chaos", 1991)
"If we have several modes, oscillating independently, the motion is, as we saw, not chaotic. Suppose now that we put a coupling, or interaction, between the different modes. This means that the evolution of each mode, or oscillator, at a certain moment is determined not just by the state of this oscillator at that moment, but by the states of the other oscillators as well. When do we have chaos then? Well, for sensitive dependence on initial condition to occur, at least three oscillators are necessary. In addition, the more oscillators there are, and the more coupling there is between them, the more likely you are to see chaos." (David Ruelle, "Chance and Chaos", 1991)
"Because there are regularities in the structure of the universe, and because life can take advantage of it, a new feature of life, which we call intelligence, has slowly emerged." (David Ruelle, "Chance and Chaos", 1991)
"But natural selection does not explain how we came to understand the chemistry of stars, or subtle properties of prime numbers. Natural selection explains only that humans have acquired higher intellectual functions; it cannot explain why so much is understandable about the physical universe, or the abstract world of mathematics." (David Ruelle, "Chance and Chaos", 1991)
"In brief, an algorithm is a systematic way of performing a certain task. […] The algorithmic complexity of a problem depends therefore on the availability of efficient algorithms to handle the problem." (David Ruelle, "Chance and Chaos", 1991)
"In brief, the way we do mathematics is human, very much so. But mathematicians have no doubt that there is a mathematical reality beyond our puny existence. We discover mathematical truth, we do not create it. We ask ourselves what seems to be a natural question and start working on it, and not uncommonly we find the solution (or someone else does). And we know that the answer could not have been different." (David Ruelle, "Chance and Chaos", 1991)
"Mathematicians, like physicists, are pushed by a strong fascination. Research in mathematics is hard, it is intellectually painful even if it is rewarding, and you wouldn't do it without some strong urge." (David Ruelle, "Chance and Chaos", 1991)
"Mathematics has deep unity. More than a collection of separate theories such as set theory, topology, and algebra, each with its own basic assumptions, mathematics is a unified whole. Mathematics is a great kingdom, and that kingdom belongs to those who see." (David Ruelle, "Chance and Chaos", 1991)"Mathematics is not just a collection of formulas and theorems; it also contains ideas. One of the most pervasive ideas in mathematics is that of geometrization. This means, basically, visualization of all kinds of things as points of a space." (David Ruelle, "Chance and Chaos", 1991)
"Quantum mechanics, like other physical theories, consists of a mathematical part, and an operational part that tells you how a certain piece of physical reality is described by the mathematics. Both the mathematical and the operational aspects of quantum mechanics are straightforward and involve no logical paradoxes. Furthermore, the agreement between theory and experiment is as good as one can hope for. Nevertheless, the new mechanics has given rise to many controversies, which involve its probabilistic aspect, the relation of its operational concepts with those of classical mechanics […]" (David Ruelle, "Chance and Chaos", 1991)
"Sometimes the old philosophical problems are clarified by science; sometimes they subvert science. But the questions that are suggested by introspection often remain unanswered, and when the answers come they tend to be intellectually convincing rather than psychologically satisfying." (David Ruelle, "Chance and Chaos", 1991)
"The definition of information was modeled after that of entropy, the latter measuring the amount of randomness present in a system. Why should information be measured by randomness? Simply because by choosing one message in a class of possible messages you dispel the randomness present in that class." (David Ruelle, "Chance and Chaos", 1991)
"The ideas of chaos apply most naturally to time evolutions with 'eternal return'. These are time evolutions of systems that come back again and again to near the same situations. In other words, if the system is in a certain state at a certain time, it will return arbitrarily near the same state at a later time." (David Ruelle, "Chance and Chaos", 1991)
"[...] the mean information of a message is defined as the amount of chance (or randomness) present in a set of possible messages. To see that this is a natural definition, note that by choosing a message, one destroys the randomness present in the variety of possible messages. Information theory is thus concerned, as is statistical mechanics, with measuring amounts of randomness. The two theories are therefore closely related." (David Ruelle, "Chance and Chaos", 1991)
"The problem of meaning is obviously deep and complex. It is tied among other things to the question of how our brain works, and we don't know too much about that. We should thus not wonder that today's science can tackle only some rather superficial aspects of the problem of meaning." (David Ruelle, "Chance and Chaos", 1991)
"[…] the standard theory of chaos deals with time evolutions that come back again and again close to where they were earlier. Systems that exhibit this "eternal return" are in general only moderately complex. The historical evolution of very complex systems, by contrast, is typically one way: history does not repeat itself. For these very complex systems with one-way evolution it is usually clear that sensitive dependence on initial condition is present. The question is then whether it is restricted by regulation mechanisms, or whether it leads to long-term important consequences." (David Ruelle, "Chance and Chaos", 1991)
"The starting point of a mathematical theory consists of a few basic assertions on a certain number of mathematical objects (instead of mathematical objects, we might speak of words or phrases, because in a sense that is what they are). Starting from the basic assumptions one tries, by pure logic, to deduce new assertions, called theorems." (David Ruelle, "Chance and Chaos", 1991)
"The unity of mathematics is due to the logical relation between different mathematical theories. The physical theories, by contrast, need not be logically coherent; they have unity because they describe the same physical reality." (David Ruelle, "Chance and Chaos", 1991)
"The universe has quite a bit of randomness in it, but also quite a bit of structure." (David Ruelle, "Chance and Chaos", 1991)
“This transition from uncertainty to near certainty when we observe long series of events, or large systems, is an essential theme in the study of chance.” (David Ruelle, "Chance and Chaos", 1991)"What causes difficulties is the apparent contradiction between determinism and our free will, introspectively characterized by the fact that several possibilities are open, and we engage our responsibility by choosing one. Introducing chance into the laws of physics does not help us in any way to resolve this contradiction. […] what allows our free will to be a meaningful notion is the complexity of the universe or, more precisely, our own complexity." (David Ruelle, "Chance and Chaos", 1991)
"What is an attractor? It is the set on which the point P, representing the system of interest, is moving at large times (i.e., after so-called transients have died out). For this definition to make sense it is important that the external forces acting on the system be time independent (otherwise we could get the point P to move in any way we like). It is also important that we consider dissipative systems (viscous fluids dissipate energy by self-friction). Dissipation is the reason why transients die out. Dissipation is the reason why, in the infinite-dimensional space representing the system, only a small set (the attractor) is really interesting." (David Ruelle, "Chance and Chaos", 1991)
"What we call intelligence is the activity of the mind and takes place in the brain. Intelligence guides our actions on the basis of what we perceive from the outside universe, and the interpretation of visual messages is therefore part of it." (David Ruelle, "Chance and Chaos", 1991)
"What we now call chaos is a time evolution with sensitive dependence on initial condition. The motion on a strange attractor is thus chaotic. One also speaks of deterministic noise when the irregular oscillations that are observed appear noisy, but the mechanism that produces them is deterministic." (David Ruelle, "Chance and Chaos", 1991)
"To avoid getting mired in mathematical questions beyond human capabilities, perhaps you should stay closer to physics." (David Ruelle, "Conversations on Nonequilibrium Physics With an Extraterrestrial", Physics Today, 2004)
“Human language is a vehicle of truth but also of error, deception, and nonsense. Its use, as in the present discussion, thus requires great prudence. One can improve the precision of language by explicit definition of the terms used. But this approach has its limitations: the definition of one term involves other terms, which should in turn be defined, and so on. Mathematics has found a way out of this infinite regression: it bypasses the use of definitions by postulating some logical relations (called axioms) between otherwise undefined mathematical terms. Using the mathematical terms introduced with the axioms, one can then define new terms and proceed to build mathematical theories. Mathematics need, not, in principle rely on a human language. It can use, instead, a formal presentation in which the validity of a deduction can be checked mechanically and without risk of error or deception.“ (David Ruelle, “The Mathematician's Brain”, 2007)
“Mathematics as done by mathematicians is not just heaping up statements logically deduced from the axioms. Most such statements are rubbish, even if perfectly correct. A good mathematician will look for interesting results. These interesting results, or theorems, organize themselves into meaningful and natural structures, and one may say that the object of mathematics is to find and study these structures.” (David Ruelle, “The Mathematician's Brain”, 2007)
“The beauty of mathematics is that clever arguments give answers to problems for which brute force is hopeless, but there is no guarantee that a clever argument always exists! We just saw a clever argument to prove that there are infinitely many primes, but we don't know any argument to prove that there are infinitely many pairs of twin primes.” (David Ruelle, “The Mathematician's Brain”, 2007)
"To avoid getting mired in mathematical questions beyond human capabilities, perhaps you should stay closer to physics." (David Ruelle, "Conversations on Nonequilibrium Physics With an Extraterrestrial", Physics Today, 2004)
“Human language is a vehicle of truth but also of error, deception, and nonsense. Its use, as in the present discussion, thus requires great prudence. One can improve the precision of language by explicit definition of the terms used. But this approach has its limitations: the definition of one term involves other terms, which should in turn be defined, and so on. Mathematics has found a way out of this infinite regression: it bypasses the use of definitions by postulating some logical relations (called axioms) between otherwise undefined mathematical terms. Using the mathematical terms introduced with the axioms, one can then define new terms and proceed to build mathematical theories. Mathematics need, not, in principle rely on a human language. It can use, instead, a formal presentation in which the validity of a deduction can be checked mechanically and without risk of error or deception.“ (David Ruelle, “The Mathematician's Brain”, 2007)
“Mathematics as done by mathematicians is not just heaping up statements logically deduced from the axioms. Most such statements are rubbish, even if perfectly correct. A good mathematician will look for interesting results. These interesting results, or theorems, organize themselves into meaningful and natural structures, and one may say that the object of mathematics is to find and study these structures.” (David Ruelle, “The Mathematician's Brain”, 2007)
“The beauty of mathematics is that clever arguments give answers to problems for which brute force is hopeless, but there is no guarantee that a clever argument always exists! We just saw a clever argument to prove that there are infinitely many primes, but we don't know any argument to prove that there are infinitely many pairs of twin primes.” (David Ruelle, “The Mathematician's Brain”, 2007)
05 August 2019
Ludwig von Bertalanffy - Collected Quotes
“Since the fundamental character of the living thing is its organization, the customary investigation of the single parts and processes cannot provide a complete explanation of the vital phenomena. This investigation gives us no information about the coordination of parts and processes. Thus, the chief task of biology must be to discover the laws of biological systems (at all levels of organization). We believe that the attempts to find a foundation for theoretical biology point at a fundamental change in the world picture. This view, considered as a method of investigation, we shall call ‘organismic biology’ and, as an attempt at an explanation, ‘the system theory of the organism’” (Ludwig von Bertalanffy, “Kritische Theorie der Formbildung”, 1928)
"Every organism represents a system, by which term we mean a complex of elements in mutual interaction. From this obvious statement the limitations of the analytical and summative conceptions must follow. First, it is impossible to resolve the phenomena of life completely into elementary units; for each individual part and each individual event depends not only on conditions within itself, but also to a greater or lesser extent on the conditions within the whole, or within superordinate units of which it is a part. Hence the behavior of an isolated part is, in general, different from its behavior within the context of the whole… Secondly, the actual whole shows properties that are absent from its isolated parts." (Ludwig von Bertalanffy, "Problems of Life", 1952)
"The evolution of science is not a movement in an intellectual vacuum; rather it is both an expression and a driving force of the historical process." (Ludwig von Bertalanffy, "Problems of Life: An Evaluation of Modern Biological Thought", 1952)
"Higher, directed forms of energy (e.g., mechanical, electric, chemical) are dissipated, that is, progressively converted into the lowest form of energy, i.e., undirected heat movement of molecules; chemical systems tend toward equilibria with maximum entropy; machines wear out owing to friction; in communication channels, information can only be lost by conversion of messages into noise but not vice versa, and so forth." (Ludwig von Bertalanffy, "Robots, Men and Minds", 1967)
"It is necessary to study not only parts and processes in isolation, but also to solve the decisive problems found in organization and order unifying them, resulting from dynamic interaction of parts, and making the behavoir of the parts different when studied in isolation or within the whole." (Ludwig von Bertalanffy, "General System Theory: Foundations, Development, Applications", 1968)
"Now we are looking for another basic outlook on the world - the world as organization. Such a conception - if it can be substantiated - would indeed change the basic categories upon which scientific thought rests, and profoundly influence practical attitudes. This trend is marked by the emergence of a bundle of new disciplines such as cybernetics, information theory, general system theory, theories of games, of decisions, of queuing and others; in practical applications, systems analysis, systems engineering, operations research, etc. They are different in basic assumptions, mathematical techniques and aims, and they are often unsatisfactory and sometimes contradictory. They agree, however, in being concerned, in one way or another, with ‘systems’, ‘wholes’ or ‘organizations’; and in their totality, they herald a new approach." (Ludwig von Bertalanffy, "General System Theory", 1968)
"Progress is only possible by passing from a state of undifferentiated wholeness to differentiation of parts." (Ludwig von Bertalanffy, "General System Theory", 1968)
"System' is the concept that refers both to a complex of interdependencies between parts, components, and processes, that involves discernible regularities of relationships, and to a similar type of interdependency between such a complex and its surrounding environment." (Talcott Parsons, "Systems Analysis: Social Systems", 1968)
"The properties and modes of action of higher levels are not explicable by the summation of the properties and modes of action of their components taken in isolation. If, however, we know the ensemble of the components and the relations existing between them, then the higher levels are derivable from the components." (Ludwig von Bertalanffy, "System Theory: Foundations, Development, Applications", 1968)
"The system problem is essentially the problem of the limitation of analytical procedures in science. This used to be expressed by half-metaphysical statements, such as emergent evolution or ‘the whole is more than the sum of its parts,’ but has a clear operational meaning." (Ludwig von Bertalanffy, "General System Theory", 1968)
"Thus, there exist models, principles, and laws that apply to generalized systems or their subclasses, irrespective of their particular kind, the nature of their component elements, and the relations or "forces" between them. It seems legitimate to ask for a theory, not of systems of a more or less special kind, but of universal principles applying to systems in general. In this way we postulate a new discipline called General System Theory. Its subject matter is the formulation and derivation of those principles which are valid for ‘systems’ in general." (Ludwig von Bertalanffy, „General System Theory: Foundations, Development, Applications", 1968)
"Every organism represents a system, by which term we mean a complex of elements in mutual interaction. From this obvious statement the limitations of the analytical and summative conceptions must follow. First, it is impossible to resolve the phenomena of life completely into elementary units; for each individual part and each individual event depends not only on conditions within itself, but also to a greater or lesser extent on the conditions within the whole, or within superordinate units of which it is a part. Hence the behavior of an isolated part is, in general, different from its behavior within the context of the whole… Secondly, the actual whole shows properties that are absent from its isolated parts." (Ludwig von Bertalanffy, "Problems of Life", 1952)
"The evolution of science is not a movement in an intellectual vacuum; rather it is both an expression and a driving force of the historical process." (Ludwig von Bertalanffy, "Problems of Life: An Evaluation of Modern Biological Thought", 1952)
"Higher, directed forms of energy (e.g., mechanical, electric, chemical) are dissipated, that is, progressively converted into the lowest form of energy, i.e., undirected heat movement of molecules; chemical systems tend toward equilibria with maximum entropy; machines wear out owing to friction; in communication channels, information can only be lost by conversion of messages into noise but not vice versa, and so forth." (Ludwig von Bertalanffy, "Robots, Men and Minds", 1967)
"It is necessary to study not only parts and processes in isolation, but also to solve the decisive problems found in organization and order unifying them, resulting from dynamic interaction of parts, and making the behavoir of the parts different when studied in isolation or within the whole." (Ludwig von Bertalanffy, "General System Theory: Foundations, Development, Applications", 1968)
"Now we are looking for another basic outlook on the world - the world as organization. Such a conception - if it can be substantiated - would indeed change the basic categories upon which scientific thought rests, and profoundly influence practical attitudes. This trend is marked by the emergence of a bundle of new disciplines such as cybernetics, information theory, general system theory, theories of games, of decisions, of queuing and others; in practical applications, systems analysis, systems engineering, operations research, etc. They are different in basic assumptions, mathematical techniques and aims, and they are often unsatisfactory and sometimes contradictory. They agree, however, in being concerned, in one way or another, with ‘systems’, ‘wholes’ or ‘organizations’; and in their totality, they herald a new approach." (Ludwig von Bertalanffy, "General System Theory", 1968)
"Progress is only possible by passing from a state of undifferentiated wholeness to differentiation of parts." (Ludwig von Bertalanffy, "General System Theory", 1968)
"System' is the concept that refers both to a complex of interdependencies between parts, components, and processes, that involves discernible regularities of relationships, and to a similar type of interdependency between such a complex and its surrounding environment." (Talcott Parsons, "Systems Analysis: Social Systems", 1968)
"The properties and modes of action of higher levels are not explicable by the summation of the properties and modes of action of their components taken in isolation. If, however, we know the ensemble of the components and the relations existing between them, then the higher levels are derivable from the components." (Ludwig von Bertalanffy, "System Theory: Foundations, Development, Applications", 1968)
"The system problem is essentially the problem of the limitation of analytical procedures in science. This used to be expressed by half-metaphysical statements, such as emergent evolution or ‘the whole is more than the sum of its parts,’ but has a clear operational meaning." (Ludwig von Bertalanffy, "General System Theory", 1968)
"Thus, there exist models, principles, and laws that apply to generalized systems or their subclasses, irrespective of their particular kind, the nature of their component elements, and the relations or "forces" between them. It seems legitimate to ask for a theory, not of systems of a more or less special kind, but of universal principles applying to systems in general. In this way we postulate a new discipline called General System Theory. Its subject matter is the formulation and derivation of those principles which are valid for ‘systems’ in general." (Ludwig von Bertalanffy, „General System Theory: Foundations, Development, Applications", 1968)
"Today our main problem is that of organized complexity. Concepts like those of organization, wholeness, directiveness, teleology, control, self-regulation, differentiation and the like are alien to conventional physics. However, they pop up everywhere in the biological, behavioural and social sciences, and are, in fact, indispensable for dealing with living organisms or social groups. Thus, a basic problem posed to modern science is a general theory of organization." (Ludwig von Bertalanff, "General System Theory" , 1968)
"We completely agree that description by differential equations is not only a clumsy but, in principle, inadequate way to deal with many problems of organization." (Ludwig von Bertalanffy, „General System Theory: Foundations, Development, Applications", 1968)
"While we can conceive of a sum [or aggregate] as being composed gradually, a system as a total of parts with its [multiplicative] interrelations has to be conceived of as being composed instantly." (Ludwig von Bertalanffy, "General System Theory", 1968)
"You cannot sum up the behavior of the whole from the isolated parts, and you have to take into account the relations between the various subordinate systems which are super-ordinated to them in order to understand the behavior of the parts." (Ludwig von Bertalanffy, "General System Theory", 1968)
"The characteristic of the organism is first that it is more than the sum of its parts and second that the single processes are ordered for the maintenance of the whole." (Ludwig von Bertalanffy)
"What in the whole denotes a causal equilibrium process, appears for the part as a teleological event." (Ludwig von Bertalanffy)
"We completely agree that description by differential equations is not only a clumsy but, in principle, inadequate way to deal with many problems of organization." (Ludwig von Bertalanffy, „General System Theory: Foundations, Development, Applications", 1968)
"While we can conceive of a sum [or aggregate] as being composed gradually, a system as a total of parts with its [multiplicative] interrelations has to be conceived of as being composed instantly." (Ludwig von Bertalanffy, "General System Theory", 1968)
"The characteristic of the organism is first that it is more than the sum of its parts and second that the single processes are ordered for the maintenance of the whole." (Ludwig von Bertalanffy)
"What in the whole denotes a causal equilibrium process, appears for the part as a teleological event." (Ludwig von Bertalanffy)
03 August 2019
Ludwig Boltzmann - Collected Quotes
"Since a given system can never of its own accord go over into another equally probable state but into a more probable one, it is likewise impossible to construct a system of bodies that after traversing various states returns periodically to its original state, that is a perpetual motion machine." (Ludwig Boltzmann, "'The Second Law of Thermodynamics", [Address to a Formal meeting of the Imperial Academy of Science], 1886)
"[...] 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; that is in completing, as it were, the thinking process and carrying out globally what on a small scale occurs within us whenever we form an idea." (Ludwig E Boltzmann, "On the Significance of Theories", 1890)
"One is almost tempted to assert that quite apart from its intellectual mission, theory is the most practical thing conceivable, the quintessence of practice as it were, since the precision of its conclusions cannot be reached by any routine of estimating or trial and error; although given the hidden ways of theory, this will hold only for those who walk them with complete confidence." (Ludwig E Boltzmann, "On the Significance of Theories", 1890)
"Most surprising and far-reaching analogies revealed themselves between apparently quite disparate natural processes. It seemed that nature had built the most various things on exactly the same pattern; or, in the dry words of the analyst, the same differential equations hold for the most various phenomena. (Ludwig Boltzmann, "On the methods of theoretical physics", 1892)
"Every hypothesis must derive indubitable results from mechanically well-defined assumptions by mathematically correct methods." (Ludwig Boltzmann, "Certain Questions of the Theory of Gasses", Nature Vol. 51 (1322), 1895)
"Experience teaches that one will be led to new discoveries almost exclusively by means of special mechanical models." (Ludwig Boltzmann, "Lectures on Gas Theory", 1896)
"All our ideas and concepts are only internal pictures, or if spoken, combinations of sounds. The task of our thinking is so to use and combine them that by their means we always most readily hit upon the correct actions and guide others likewise. In this, metaphysics follows the most down-to-earth and practical point of view, so that extremes meet. The conceptual signs that we form thus exist only within us, we cannot measure external phenomena by the standard of our ideas. We can therefore pose such formal questions as whether only matter exists and force is a property of it, or whether force exists independently of matter or conversely whether matter is a product of force but none of these questions are significant since all these concepts are only mental pictures whose purpose is to represent phenomena correctly." (Ludwig Boltzmann, 1899)
"[…] no theory can be objective, actually coinciding with nature, but rather that each theory is only a mental picture of phenomena, related to them as sign is to designatum. From this it follows that it cannot be our task to find an absolutely correct theory but rather a picture that is, as simple as possible and that represents phenomena as accurately as possible. One might even conceive of two quite different theories both equally simple and equally congruent with phenomena, which therefore in spite of their difference are equally correct. (Ludwig Boltzmann, "On the development of the methods of theoretical physics", 1899)
"What, then, is meant by having perfectly correct understanding of a mechanism? Everybody knows that the practical criterion for this consists in being able to handle it correctly. However, I go further and assert that this is the only tenable definition of understanding a mechanism. (Ludwig Boltzmann, "On the principles of mechanics", 1902)
"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)
"Mathematics and music! the most glaring possible opposites of human thought! and yet connected, mutually sustained! It is as if they would demonstrate the hidden consensus of all the actions of our mind, which in the revelations of genius makes us forefeel unconscious utterances of a mysteriously active intelligence." (Ludwig Boltzmann)
"Since in the differential equations of mechanics themselves there is absolutely nothing analogous to the second law of thermodynamics the latter can be mechanically represented only by means of assumptions regarding initial conditions." (Ludwig Boltzmann)
"[...] 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; that is in completing, as it were, the thinking process and carrying out globally what on a small scale occurs within us whenever we form an idea." (Ludwig E Boltzmann, "On the Significance of Theories", 1890)
"One is almost tempted to assert that quite apart from its intellectual mission, theory is the most practical thing conceivable, the quintessence of practice as it were, since the precision of its conclusions cannot be reached by any routine of estimating or trial and error; although given the hidden ways of theory, this will hold only for those who walk them with complete confidence." (Ludwig E Boltzmann, "On the Significance of Theories", 1890)
"Most surprising and far-reaching analogies revealed themselves between apparently quite disparate natural processes. It seemed that nature had built the most various things on exactly the same pattern; or, in the dry words of the analyst, the same differential equations hold for the most various phenomena. (Ludwig Boltzmann, "On the methods of theoretical physics", 1892)
"Every hypothesis must derive indubitable results from mechanically well-defined assumptions by mathematically correct methods." (Ludwig Boltzmann, "Certain Questions of the Theory of Gasses", Nature Vol. 51 (1322), 1895)
"Experience teaches that one will be led to new discoveries almost exclusively by means of special mechanical models." (Ludwig Boltzmann, "Lectures on Gas Theory", 1896)
"All our ideas and concepts are only internal pictures, or if spoken, combinations of sounds. The task of our thinking is so to use and combine them that by their means we always most readily hit upon the correct actions and guide others likewise. In this, metaphysics follows the most down-to-earth and practical point of view, so that extremes meet. The conceptual signs that we form thus exist only within us, we cannot measure external phenomena by the standard of our ideas. We can therefore pose such formal questions as whether only matter exists and force is a property of it, or whether force exists independently of matter or conversely whether matter is a product of force but none of these questions are significant since all these concepts are only mental pictures whose purpose is to represent phenomena correctly." (Ludwig Boltzmann, 1899)
"[…] no theory can be objective, actually coinciding with nature, but rather that each theory is only a mental picture of phenomena, related to them as sign is to designatum. From this it follows that it cannot be our task to find an absolutely correct theory but rather a picture that is, as simple as possible and that represents phenomena as accurately as possible. One might even conceive of two quite different theories both equally simple and equally congruent with phenomena, which therefore in spite of their difference are equally correct. (Ludwig Boltzmann, "On the development of the methods of theoretical physics", 1899)
"What, then, is meant by having perfectly correct understanding of a mechanism? Everybody knows that the practical criterion for this consists in being able to handle it correctly. However, I go further and assert that this is the only tenable definition of understanding a mechanism. (Ludwig Boltzmann, "On the principles of mechanics", 1902)
"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)
"Mathematics and music! the most glaring possible opposites of human thought! and yet connected, mutually sustained! It is as if they would demonstrate the hidden consensus of all the actions of our mind, which in the revelations of genius makes us forefeel unconscious utterances of a mysteriously active intelligence." (Ludwig Boltzmann)
"Since in the differential equations of mechanics themselves there is absolutely nothing analogous to the second law of thermodynamics the latter can be mechanically represented only by means of assumptions regarding initial conditions." (Ludwig Boltzmann)
William E Deming - Collected Quotes
“It is important to realize that it is not the one measurement, alone, but its relation to the rest of the sequence that is of interest.” (William E Deming, “Statistical Adjustment of Data”, 1938)
“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 & William E Deming, “Statistical Method from the Viewpoint of Quality Control”, 1939)
"Scientific data are not taken for museum purposes; they are taken as a basis for doing something. If nothing is to be done with the data, then there is no use in collecting any. The ultimate purpose of taking data is to provide a basis for action or a recommendation for action. The step intermediate between the collection of data and the action is prediction." (William E Deming, "On a Classification of the Problems of Statistical Inference", Journal of the American Statistical Association Vol. 37 (218), 1942)
”Experience without theory teaches nothing.” (William E Deming, “Out of the Crisis”, 1986)
“It is important to realize that it is not the one measurement, alone, but its relation to the rest of the sequence that is of interest.” (William E Deming, “Statistical Adjustment of Data”, 1943)
“Sampling is the science and art of controlling and measuring the reliability of useful statistical information through the theory of probability.” (William E Deming, “Some Theory of Sampling”, 1950)
“Experience by itself teaches nothing [...] Without theory, experience has no meaning. Without theory, one has no questions to ask. Hence without theory there is no learning.” (William E Deming, “The New Economics for Industry, Government, Education”, 1993)
“Knowledge is theory. We should be thankful if action of management is based on theory. Knowledge has temporal spread. Information is not knowledge. The world is drowning in information but is slow in acquisition of knowledge. There is no substitute for knowledge.” (William E Deming, “The New Economics for Industry, Government, Education”, 1993)
“What is a system? A system is a network of interdependent components that work together to try to accomplish the aim of the system. A system must have an aim. Without an aim, there is no system. The aim of the system must be clear to everyone in the system. The aim must include plans for the future. The aim is a value judgment.” (William E Deming, “The New Economics for Industry, Government, Education”, 1993)
"The only useful function of a statistician is to make predictions, and thus to provide a basis for action." (William E Deming)
On Worldviews (1700-1899)
"The largest views are not always the clearest, and that he who is short-sighted will be obliged to draw the object nearer, and may, perhaps, by a close and narrow survey, discern that which had escaped far better eyes." (George Berkeley, "A Treatise Concerning the Principles of Human Knowledge", 1710)
"It is not therefore the business of philosophy, in our present situation in the universe, to attempt to take in at once, in one view, the whole scheme of nature; but to extend, with great care and circumspection, our knowledge, by just steps, from sensible things, as far as our observations or reasonings from them will carry us, in our enquiries concerning either the greater motions and operations of nature, or her more subtile and hidden works." (Colin Maclaurin, "An Account of Sir Isaac Newton's Philosophical Discoveries, in Four Books", 1748)
“If the human mind is nonetheless to be able even to think the given infinite without contradiction, it must have within itself a power that is supersensible, whose idea of the noumenon cannot be intuited but can yet be regarded as the substrate underlying what is mere appearance, namely, our intuition of the world [worldview]." (Immanuel Kant, “Critique of Judgment”, 1790)
"It is not therefore the business of philosophy, in our present situation in the universe, to attempt to take in at once, in one view, the whole scheme of nature; but to extend, with great care and circumspection, our knowledge, by just steps, from sensible things, as far as our observations or reasonings from them will carry us, in our enquiries concerning either the greater motions and operations of nature, or her more subtile and hidden works." (Colin Maclaurin, "An Account of Sir Isaac Newton's Philosophical Discoveries, in Four Books", 1748)
“If the human mind is nonetheless to be able even to think the given infinite without contradiction, it must have within itself a power that is supersensible, whose idea of the noumenon cannot be intuited but can yet be regarded as the substrate underlying what is mere appearance, namely, our intuition of the world [worldview]." (Immanuel Kant, “Critique of Judgment”, 1790)
"The diversity of languages is not a diversity of signs and sounds but a diversity of views of the world." (Wilhelm von Humboldt, 1820)
“So it happens at times that a person believes that he has a world-view, but that there is yet one particular phenomenon that is of such a nature that it baffles the understanding, and that he explains differently and attempts to ignore in order not to harbor the thought that this phenomenon might overthrow the whole view, or that his reflection does not possess enough courage and resolution to penetrate the phenomenon with his world-view.” (Søren Kierkegaard, 1844)
“The last change in our point of view gives the whole world a pictorial air.” (Ralph W Emerson, “Miscellanies, Embracing Nature, Addresses, and Lectures”, 1866)
“The mutual interdependence of thought and word illuminates clearly the truth that languages are not really means for representing already known truths, but are rather instruments for discovering previously unrecognised ones. The differences between languages are not those of sounds and signs but those of differing worldviews […] objective truth always rises from the entire energy of subjective individuality.” (Wilhelm von Humboldt)
“So it happens at times that a person believes that he has a world-view, but that there is yet one particular phenomenon that is of such a nature that it baffles the understanding, and that he explains differently and attempts to ignore in order not to harbor the thought that this phenomenon might overthrow the whole view, or that his reflection does not possess enough courage and resolution to penetrate the phenomenon with his world-view.” (Søren Kierkegaard, 1844)
“The last change in our point of view gives the whole world a pictorial air.” (Ralph W Emerson, “Miscellanies, Embracing Nature, Addresses, and Lectures”, 1866)
“The mutual interdependence of thought and word illuminates clearly the truth that languages are not really means for representing already known truths, but are rather instruments for discovering previously unrecognised ones. The differences between languages are not those of sounds and signs but those of differing worldviews […] objective truth always rises from the entire energy of subjective individuality.” (Wilhelm von Humboldt)
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Alexander von Humboldt - Collected Quotes
"Whatever relates to extent and quantity may be represented by geometrical figures. Statistical projections which speak to the senses w...