Pierre-Maurice-Marie Duhem - Collected Quotes

"A physical theory is not an explanation. It is a system of mathematical propositions, deduced from a small number of principles, which aim to represent as simply, as completely, and as exactly as possible a set of experimental laws. […] Thus a true theory is not a theory which gives an explanation of physical appearances in conformity with reality; it is a theory which represents in a satisfactory manner a group of experimental laws. A false theory is not an attempt at an explanation based on assumptions contrary to reality; it is a ·group of propositions which do not agree with the experimental laws. Agreement with experiment is the sole criterion of truth for a physical theory." (Pierre-Maurice-Marie Duhem, "La théorie physique. Son objet, sa structure", 1906)

"Imagine the forehead of a bull, with the protuberances from which the horns and ears start, and with the collars hollowed out between these protuberances; but elongate these horns and ears without limit so that they extend to infinity; then you will have one of the surfaces we wish to study. On such a surface geodesics may show many different aspects. There are, first of all, geodesics which close on themselves. There are some also which are never infinitely distant from their starting point even though they never exactly pass through it again; some turn continually around the right horn, others around the left horn, or right ear, or left ear; others, more complicated, alternate, in accordance with certain rules, the turns they describe around one horn with the turns they describe around the other horn, or around one of the ears. Finally, on the forehead of our bull with his unlimited horns and ears there will be geodesics going to infinity, some mounting the right horn, others mounting the left horn, and still others following the right or left ear. [...] If, therefore, a material point is thrown on the surface studied starting from a geometrically given position with a geometrically given velocity, mathematical deduction can determine the trajectory of this point and tell whether this path goes to infinity or not. But, for the physicist, this deduction is forever useless. When, indeed, the data are no longer known geometrically, but are determined by physical procedures as precise as we may suppose, the question put remains and will always remain unanswered." (Pierre-Maurice-Marie Duhem, "La théorie physique. Son objet, sa structure", 1906)

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

"Order, wherever it reigns, brings beauty with it. Theory not only renders the group of physical laws it represents easier to handle, more convenient, and more useful, but also more beautiful." (Pierre-Maurice-Marie Duhem, "La théorie physique. Son objet, sa structure", 1906)

"[...] the aim of physical theory is to become a natural classification, to establish among diverse experimental laws a logical coordination serving as a sort of image and reflection of the true order according to which the realities escaping us are organized." (Pierre-Maurice-Marie Duhem, "La théorie physique. Son objet, sa structure", 1906)

"The distinction between physics, which studies phenomena and their laws, and metaphysics, which seeks to know the essence of matter insofar as it is the cause of phenomena and the basis of laws, is deprived of any foundation. The mind does not start from the knowledge of phenomena to rise to the knowledge of matter; what it can know from the start is the very nature of matter, and thence the explanation of phenomena." (Pierre-Maurice-Marie Duhem, "La théorie physique. Son objet, sa structure", 1906)

"Theory is not solely an economical representation of experimental laws; it is also a classification of these laws. […] theory, by developing the numerous ramifications of the deductive reasoning which connects principles to experimental laws, establishes an order and a classification among these laws. It brings some laws together, closely arranged in the same group; it separates some of the others by placing them in two groups very far apart. Theory gives, so to speak, the table of contents and the chapter headings under which the science to be studied will be methodically divided, and it indicates the laws which are to be arranged under each of these chapters." (Pierre-Maurice-Marie Duhem, "La théorie physique. Son objet, sa structure", 1906)

"Thus, physical theory never gives us the explanation of experimental laws; it never reveals realities hiding under the sensible appearances; but the more complete it becomes, the more we apprehend that the logical order in which theory orders experimental laws is the reflection of an ontological order, the more we suspect that the relations it establishes among the data of observation correspond to real relations among things, and the more we feel that theory tends to be a natural classification." (Pierre-Maurice-Marie Duhem, "La théorie physique. Son objet, sa structure", 1906)

"A physical theory is not an explanation. It is a system of mathematical propositions, deduced from a small number of principles, which aim [...] to represent as simply, as completely, and as exactly as possible a group of experimental laws." (Pierre-Maurice-Marie Duhem, “The Aim and Structure of Physical Theory”, 1908) 

"But, once again, what the physical states as the result of an experiment is not the recital of observed facts, but the interpretation and the transposing of these facts into the ideal, abstract, symbolic world created by the theories he regards as established." (Pierre-Maurice-Marie Duhem, “The Aim and Structure of Physical Theory”, 1908) 

"If the aim of physical theories is to explain experimental laws, theoretical physics is not an autonomous science; it is subordinate to metaphysics." (Pierre-Maurice-Marie DuhemDuhem, "The Aim and Structure of Physical Theory", 1908)

"It is impossible to follow the march of one of the greatest theories of physics, to see it unroll majestically its regular deductions starting from initial hypotheses, to see its consequences represent a multitude of experimental laws down to the smallest detail, without being charmed by the beauty of such a construction, without feeling keenly that such a creation of the human mind is truly a work of art." (Pierre-Maurice-Marie DuhemDuhem, "The Aim and Structure of Physical Theory", 1908)

"[...] physics makes progress because experiment constantly causes new disagreements to break out between laws and facts, and because physicists constantly touch up and modify laws in order that they may more faithfully represent the facts." (Pierre-Maurice-Marie Duhem, "The Aim and Structure of Physical Theory", 1908)

"The laws of physics are therefore provisional in that the symbols they relate too simple to represent reality completely." (Pierre-Maurice-Marie Duhem, "The Aim and Structure of Physical Theory", 1908)

“Order wherever it reigns, brings beauty with it. Theory not only renders the group of physical laws it represents easier to handle, more convenient, and more useful, but also more beautiful.” (Pierre-Maurice-Marie Duhem, “The Aim and Structure of Physical Theory”, 1908) 

"Physics makes progress because experiment constantly causes new disagreements to break out between laws and facts, and because physicists constantly touch up and modify laws in order that they may more faithfully represent the facts."  (Pierre-Maurice-Marie Duhem, “The Aim and Structure of Physical Theory”, 1908) 

"The laws of physics are therefore provisional in that the symbols they relate too simple to represent reality completely." (Pierre-Maurice-Marie Duhem, “The Aim and Structure of Physical Theory”, 1908) 

"The sole purpose of physical theory is to provide a representation and classification of experimental laws; the only test permitting us to judge a physical theory and pronounce it good or bad is the comparison between the consequences of this theory and the experimental laws it has to represent and classify."  (Pierre-Maurice-Marie Duhem, “The Aim and Structure of Physical Theory”, 1908) 

"A theory of physics is not an explanation; it is a system of mathematical oppositions deduced from a small number of principles the aim of which is to represent as simply, as completely, and as exactly as possible, a group of experimental laws." (Pierre-Maurice-Marie Duhem)

"Physics is not a machine one can take apart; one cannot try each piece in isolation and wait, to adjust it, until its solidity has been minutely checked. Physical science is a system that must be taken as a whole. It is an organism no part of which can be made to function without the remotest parts coming into play, some more, some less, but all in some degree." (Pierre-Maurice-Marie Duhem)

On Systems (Unsourced)

"A theory of physics is not an explanation; it is a system of mathematical oppositions deduced from a small number of principles the aim of which is to represent as simply, as completely, and as exactly as possible, a group of experimental laws." (Pierre-Maurice-Marie Duhem)

"Ever since the observation of nature has existed, it has held a vague notion of its ultimate goal as the composition of the colorful multiplicity of phenomena in a uniform system, where possible, in a single formula." (Max Planck)

"Instead of seeking to attain consistency and uniformity of system, as some modern writers have attempted, by banishing this thought of time from the higher Algebra, I seek to attain the same object, by systematically introducing it into the lower or earlier parts of the science." (Sir William R Hamilton)

"Nature, despite her seeming diversity, is always a unity, a whole; and thus, when she manifests herself in any part of that whole, the rest must serve as a basis for that particular manifestation, and the latter must have a relationship to the rest of the system." (Johann Wolfgang von Goethe)

"No man becomes proficient in any science who does not transcend system, and gather up new truth for himself in the boundless field of research." (Andrew P Peabody)

"Physics is not a machine one can take apart; one cannot try each piece in isolation and wait, to adjust it, until its solidity has been minutely checked. Physical science is a system that must be taken as a whole. It is an organism no part of which can be made to function without the remotest parts coming into play, some more, some less, but all in some degree." (Pierre-Maurice-Marie Duhem)

"The closer we get to know the creatures around us, the clearer is the understanding we obtain of the chain of nature, and its harmony and system, according to which all things appear to have been created." (Carl Linnaeus)

"The laws of science are not viewed in independence of one another. Together they form a vast body or system of laws, with each law fitting into a system including many other laws, each mutually reinforcing the others. The laws that scientists are most loathe to abandon are those that form such an integral part of a system of laws that the abandonment of the one law would require the abandonment or alteration of a larger number of other laws in the system. Thus an observation that directly confirms one law indirectly confirms a group of laws, because of the interconnection of the laws in a system […] whether or not something is called a law, then, depends to a large extent on how deeply embedded it is in a wider system of laws." (John Hospers & Grace M Hopper)

"The present state of the system of nature is evidently a consequence of what is in the preceding moment, and if we conceive of an intelligence which at a given instant knew all the forces acting in nature and the position of every object in the universe - if endowed with a brain sufficiently vast to make all necessary calculations - could describe with a single formula the motions of the largest astronomical bodies and those of the smallest atoms. To such an intelligence, nothing would be uncertain; the future, like the past, would be an open book." (Pierre-Simon Laplace)

"What is the inner secret of mathematical power? Briefly stated, it is that mathematics discloses the skeletal outlines of all closely articulated relational systems. For this purpose mathematics uses the language of pure logic with its score or so of symbolic words, which, in its important forms of expression, enables the mind to comprehend systems of relations otherwise completely beyond its power. These forms are creative discoveries which, once made, remain permanently at our disposal. By means of them the scientific imagination is enabled to penetrate ever more deeply into the rationale of the universe about us." (George D Birkhoff)

"When matter is becoming disturbed by non-equilibrium conditions it organizes itself, it wakes up. It happens that our world is a non-equilibrium system." (Ilya Prigogine)

On Systems (1825-1849)

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

"§ 6. It is impossible for the human mind, itself a finite creation, to regard nature, whether her powers or her productions are considered, in the light of the whole manifestation of an infinite power, but only as parts or fragments of such manifestation. But to comprehend these as one whole, that is, as an eternal and immutable yet ever varying body, or, as innumerable forms of one highest whole, is the end. of all disquisition, the sum of which we call a System.
§ 7. A system contains within itself the seeds of some more complete evolution, but it does not admit of arbitrary alterations. Not that any absolute system can ever be contrived; for I am by no means of the opinion of those who expect that a system is to be as unchangeable as if it were petrified.
 § 8. If nature be closely pursued, a system is called Natural; if this Ariadnean thread be not followed, it is called Artificial or factitious.
§ 9. A system of nature proceeding from subjects of the most simple organization to such as are more perfect, or from the circumference to the centre, is called a Mathematical System.
§ 10. A system of nature which takes for the basis of its arrangement the order of development of individuals is called Physiological.
§ 11. Philosophical systems do not depend upon individual productions which are subject to continual variation, but upon eternal and unchangeable ideas. These always proceed from the centre to the circumference, or from the most perfect productions to those of a lower order.” (John Lindley, "Some Account of the Spherical and Numerical System of Nature o/M. Elias Fries", ‘Philosophical magazine: a journal of theoretical, experimental and applied physics’ Vol. 68, 1826)

"Of all the natural sciences, astronomy is that which presents the longest series of discoveries. The first appearance of the heavens is indeed far removed from that enlarged view, by which we comprehend at the present day, the past and future states of the system of the world." (Pierre-Simon Laplace," The Systems of the World" Vol. 1, 1830) 

"[…] the conduct of war branches out in almost all directions and has no definite limits; while any system, any model, has the finite nature of a synthesis. An irreconcilable conflict exists between this type of theory and actual practice." (Carl von Clausewitz, "On War", 1832)

"The function of theory is to put all this in systematic order, clearly and comprehensively, and to trace each action to an adequate, compelling cause. […] Theory should cast a steady light on all phenomena so that we can more easily recognize and eliminate the weeds that always spring from ignorance; it should show how one thing is related to another, and keep the important and the unimportant separate. If concepts combine of their own accord to form that nucleus of truth we call a principle, if they spontaneously compose a pattern that becomes a rule, it is the task of the theorist to make this clear." (Carl von Clausewitz, "On War", 1832)

"No occupation is more worthy of an intelligent and enlightened mind, than the study of Nature and natural objects; and whether we labour to investigate the structure and function of the human system, whether we direct our attention to the classification and habits of the animal kingdom, or prosecute our researches in the more pleasing and varied field of vegetable life, we shall constantly find some new object to attract our attention, some fresh beauties to excite our imagination, and some previously undiscovered source of gratification and delight." (Sir Joseph Paxton, "A Practical Treatise on the Cultivation of the Dahlia", 1838)

"Thought once awakened does not again slumber; unfolds itself into a System of Thought; grows, in man after man, generation after generation, - till its full stature is reached, and such System of Thought can grow no farther, and must give place to another." (Thomas Carlyle, “The Hero As Divinity”, [lecture, in "On Heroes, Hero-Worship and the Heroic in History: Six Lectures", 1857) 1840)

"The determination of the average man is not merely a matter of speculative curiosity; it may be of the most important service to the science of man and the social system. It ought necessarily to precede every other inquiry into social physics, since it is, as it were, the basis. The average man, indeed, is in a nation what the centre of gravity is in a body; it is by having that central point in view that we arrive at the apprehension of all the phenomena of equilibrium and motion." (Adolphe Quetelet, "A Treatise on Man and the Development of his Faculties", 1842)

"The framing of hypotheses is, for the enquirer after truth, not the end, but the beginning of his work. Each of his systems is invented, not that he may admire it and follow it into all its consistent consequences, but that he may make it the occasion of a course of active experiment and observation. And if the results of this process contradict his fundamental assumptions, however ingenious, however symmetrical, however elegant his system may be, he rejects it without hesitation. He allows no natural yearning for the offspring of his own mind to draw him aside from the higher duty of loyalty to his sovereign, Truth, to her he not only gives his affections and his wishes, but strenuous labour and scrupulous minuteness of attention." (William Whewell, "Philosophy of the Inductive Sciences" Vol. 2, 1847)

Gregory Bateson - Collected Quotes

"Whenever we pride ourselves upon finding a newer, stricter way of thought or exposition; whenever we start insisting too hard upon ‘operationalism’ or symbolic logic or any other of these very essential systems of tramlines, we lose something of the ability to think new thoughts. And equally, of course, whenever we rebel against the sterile rigidity of formal thought and exposition and let our ideas run wild, we likewise lose. As I see it, the advances in scientific thought come from a combination of loose and strict thinking, and this combination is the most precious tool of science." (Gregory Bateson, "Culture Contact and Schismogenesis", 1935)

"In order to proceed with abstraction, the organism must be exposed to a sufficient number of events which contain the same factors. Only then is a person equipped to cope with the most frequent happenings that he may encounter." (Gregory Bateson, "Communication: The Social Matrix of Psychiatry", 1951)

"If a man achieves or suffers change in premises which are deeply embedded in his mind, he will surely find that the results of that change will ramify throughout his whole universe." (Gregory Bateson, "Steps to an Ecology of Mind", 1972)

"In no system which shows mental characteristics can any part have unilateral control over the whole. In other words, the mental characteristics of the system are imminent, not in some part, but in the system as a whole." (Gregory Bateson, "Steps to an Ecology of Mind", 1972)

"No organism can afford to be conscious of matters with which it could deal at unconscious levels. Broadly, we can afford to sink those sorts of knowledge which continue to be true regardless of changes in the environment, but we must maintain in an accessible place all those controls of behavior which must be modified for every instance. The economics of the system, in fact, pushes organisms toward sinking into the unconscious those generalities of relationship which remain permanently true and toward keeping within the conscious the pragmatic of particular instances." (Gregory Bateson, "Steps to an Ecology of Mind", 1972)

"Still more astonishing is that world of rigorous fantasy we call mathematics." (Gregory Bateson, "Steps to an Ecology of Mind", 1972)

"We say the map is different from the territory. But what is the territory? Operationally, somebody went out with a retina or a measuring stick and made representations which were then put on paper. What is on the paper map is a representation of what was in the retinal representation of the man who made the map; and as you push the question back, what you find is an infinite regress, an infinite series of maps. The territory never gets in at all. […] Always, the process of representation will filter it out so that the mental world is only maps of maps, ad infinitum." (Gregory Bateson, "Steps to an Ecology of Mind", 1972)

"What we mean by information - the elementary unit of information - is a difference which makes a difference, and it is able to make a difference because the neural pathways along which it travels and is continually transformed are themselves provided with energy. The pathways are ready to be triggered. We may even say that the question is already implicit in them." (Gregory Bateson, "Steps to an Ecology of Mind", 1972)

"Information is any difference that makes a difference."(Gregory Bateson, "Mind and Nature: A necessary unity", 1979)

"Let's not pretend that mental phenomena can be mapped on to the characteristics of billiard balls." (Gregory Bateson, "Mind and Nature: A Necessary Unity", 1979)

"Science sometimes improves hypothesis and sometimes disproves them. But proof would be another matter and perhaps never occurs except in the realms of totally abstract tautology. We can sometimes say that if such and such abstract suppositions or postulates are given, then such and such abstract suppositions or postulates are given, then such and such must follow absolutely. But the truth about what can be perceived or arrived at by induction from perception is something else again." (Gregory Bateson, "Mind and Nature: A Necessary Unity", 1979)

"The map is not the territory, and the name is not the thing named." (Gregory Bateson, "Mind and Nature: A Necessary Unity", 1979)

"The world partly becomes - comes to be - how it is imagined." (Gregory Bateson, "Mind and Nature: A necessary unity", 1979)

"Numbers are the product of counting. Quantities are the product of measurement. This means that numbers can conceivably be accurate because there is a discontinuity between each integer and the next. Between two and three there is a jump. In the case of quantity there is no such jump, and because jump is missing in the world of quantity it is impossible for any quantity to be exact. You can have exactly three tomatoes. You can never have exactly three gallons of water. Always quantity is approximate." (Gregory Bateson, "Number is Different from Quantity", CoEvolution Quarterly, 1978)

"Prediction can never be absolutely valid and therefore science can never prove some generalization or even test a single descriptive statement and in that way arrive at final truth." (Gregory Bateson, Mind and Nature: A necessary unity", 1988)

Henry D Thoreau - Collected Quotes

"Every part of nature teaches that the passing away of one life is the making room for another. The oak dies down to the ground, leaving within its rind a rich virgin mold, which will impart a vigorous life to an infant forest." (Henry D Thoreau, 1837)

"This curious world which we inhabit is more wonderful than it is convenient, more beautiful than it is useful; it is more to be admired than to be used." (Henry D Thoreau, 1837)

"All perception of truth is the detection of an analogy [...]" (Henry D Thoreau, 1851)

"If we knew all the laws of Nature, we should need only one fact, or the description of one actual phenomenon, to infer all the particular results at that point. Now we know only a few laws, and our result is vitiated, not, of course, by any confusion or irregularity in Nature, but by our ignorance of essential elements in the calculation. Our notions of law and harmony are commonly confined to those instances which we detect; but the harmony which results from a far greater number of seemingly conflicting, but really concurring, laws, which we have not detected, is still more wonderful. The particular laws are as our points of view, as to the traveler, a mountain outline varies with every step, and it has an infinite number of profiles, though absolutely but one form. Even when cleft or bored through it is not comprehended in its entireness." (Henry D Thoreau, "Walden; or, Life in the Woods", 1854)

"He is not a true man of science who does not bring some sympathy to his studies, and expect to learn something by behavior as well as by application. It is childish to rest in the discovery of mere coincidences, or of partial and extraneous laws. The study of geometry is a petty and idle exercise of the mind if it is applied to no larger system than the starry one." (Henry D Thoreau, "A Week on the Concord and Merrimack Rivers", 1862)

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

"The eye which can appreciate the naked and absolute beauty of a scientifi c truth is far more rare than that which is attracted by a moral one. Few detect the morality in the former, or the science in the latter." (Henry D Thoreau, "A Week on the Concord and Merrimack Rivers", 1862)

"Our books of science, as they improve in accuracy, are in danger of losing the freshness and vigor and readiness to appreciate the real laws of Nature, which is a marked merit in the ofttimes false theories of the ancients." (Henry D Thoreau, "A Week on the Concord and Merrimack Rivers", 1873)

"The most distinct and beautiful statement of any truth [in science] must take at last the mathematical form. We might so simplify the rules of moral philosophy, as well as of arithmetic, that one formula would express them both." (Henry D Thoreau, "A Week on the Concord and Merrimack Rivers", 1873)

"The poet uses the results of science and philosophy, and generalizes their widest deductions." (Henry D Thoreau, "A Week on the Concord and Merrimack Rivers", 1873)

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

"The study of geometry is a petty and idle exercise of the mind, if it is applied to no larger system than the starry one. Mathematics should be mixed not only with physics but with ethics; that is mixed mathematics." (Henry D Thoreau, "A Week on the Concord and Merrimack Rivers", 1862)

"There is a chasm between knowledge and ignorance which the arches of science can never span." (Henry D Thoreau, "A Week on the Concord and Merrimack Rivers", 1862)
"Every man has to learn the points of the compass again as often as he awakes, whether from sleep or any abstraction." (Henry D Thoreau)
 
"I do believe in simplicity. It is astonishing as well as sad, how many trivial affairs even the wisest thinks he must attend to in a day; how singular an affair he thinks he must omit. When the mathematician would solve a difficult problem, he first frees the equation of all incumbrances, and reduces it to its simplest terms. So simplify the problem of life, distinguish the necessary and the real. Probe the earth to see where your main roots run." (Henry D Thoreau)
 

On Intuition (1700-1799)

“Without the sensuous faculty no object would be given to us, without understanding no object would be thought. Thoughts without content are void, intuitions without conceptions, blind.” Immanuel Kant, “Critique of Pure Reason“, 1781)

“The discovery of truth by slow, progressive meditation is talent. Intuition of the truth, not preceded by perceptible meditation, is genius.” (Johann K Lavater, 1787)

"Pure mathematics can never deal with the possibility, that is to say, with the possibility of ani ntuition answering to the conceptions of the things. Hence it cannot touch the question of causea nd effect, and consequently, all the finality there observed must always be regarded simply as formal, and never as a physical end.” (Immanuel Kant, "The Critique of Judgement", 1790)

“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)

Kurt Gödel - Collected Quotes

"The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules. [...] One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems. It will be shown below that this is not the case, that on the contrary there are in the two systems mentioned relatively simple problems in the theory of integers that cannot be decided on the basis of the axioms." (Kurt Gödel, "On Formally Undecidable Propositions of Principia Mathematica and Related Systems", (1931)

"Classes and concepts may, however, also be conceived as real objects, namely classes as 'pluralities of things' or as structures consisting of a plurality of things and concepts as the properties and relations of things existing independently of our definitions and constructions. It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence. They are in the same sense necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions." (Kurt Gödel, "The Philosophy of Bertrand Russell", 1944)

"But, despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don't see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception, which induces us to build up physical theories and to expect that future sense perception will agree with them and, moreover, to believe that a question not decidable now has meaning and may be decided in future." (Kurt Gödel, "What is Cantor’s Continuum problem?", American Mathematical Monthly 54, 1947)

"On the basis of what has been proved so far, it remains possible that there may exist (and even be empirically discoverable) a theorem-proving machine which in fact is equivalent to mathematical intuition, but cannot be proved to be so, nor even be proved to yield only correct theorems of finitary number theory." (Kurt Gödel, 1951)

"Non-standard analysis frequently simplifies substantially the proofs, not only of elementary theorems, but also of deep results. This is true, e.g., also for the proof of the existence of invariant subspaces for compact operators, disregarding the improvement of the result; and it is true in an even higher degree in other cases. This state of affairs should prevent a rather common misinterpretation of non-standard analysis, namely the idea that it is some kind of extravagance or fad of mathematical logicians. Nothing could be farther from the truth. Rather, there are good reasons to believe that non-standard analysis, in some version or other, will be the analysis of the future." (Kurt Gödel, "Remark on Non-standard Analysis", 1974)

"Either mathematics is too big for the human mind, or the human mind is more than a machine." (Kurt Gödel)

Aldous L Huxley - Collected Quotes

"Great scientific discoveries have been made by men seeking to verify quite erroneous theories about the nature of things." (Aldous L Huxley, "Life and Letters and the London Mercury" Vol. 1, 1928)

"Science has ‘explained’ nothing; the more we know the more fantastic the world becomes and the profounder the surrounding darkness." (Aldous L Huxley, "Along the Road: Notes and Essays of a Tourist", 1928)

"I admit that mathematical science is a good thing. But excessive devotion to it is a bad thing." (Aldous L Huxley, [Interview with J. W. N. Sullivan, Contemporary Mind], 1934)

"Technological progress has merely provided us with more efficient means for going backwards." (Aldous L Huxley, "Ends and Means: An Inquiry Into the Nature of Ideals and Into Methods", 1937)

"The suggestion is that the function of the brain and nervous system and sense organs is in the main eliminative and not productive. Each person is at each moment capable of remembering all that has ever happened to him and of perceiving everything that is happening everywhere in the universe. The function of the brain and the nervous system is to protect us from being overwhelmed and confused by this mass of largely useless and irrelevant knowledge, by shutting out most of what we should otherwise perceive or remember at any moment, and leaving only that very small and special selection which is likely to be practically useful. According to such a theory, each one of us is potentially Mind at Large. [...] To make biological survival possible, Mind at Large has to be funneled through the reducing valve of the brain and nervous system. What comes out the other end is a measly trickle of the kind of consciousness which will help us to stay alive on the surface of this particular planet. To formulate and express the contents of this reduced awareness, man has invented and endlessly elaborated upon those symbol-systems and implicit philosophies which we call languages. Every individual is at once the beneficiary and the victim of the linguistic tradition into which he has been born - the beneficiary inasmuch as language gives access to the accumulated record of other people's experience, the victim insofar as it confirms in him the belief that reduced awareness is the only awareness, and as it bedevils his sense of reality, so that he i s all too apt to take his concepts for I data, his words for actual things." (Aldous Huxley, "The Doors of Perception", 1954)

"Science is the reduction of the bewildering diversity of unique events to manageable uniformity within one of a number of symbol systems, and technology is the art of using these symbol systems so as to control and organize unique events. Scientific observation is always a viewing of things through the refracting medium of a symbol system, and technological praxis is always handling of things in ways that some symbol system has dictated. Education in science and technology is essentially education on the symbol level." (Aldous L Huxley, "Essay", Daedalus, 1962)

"For Science in its totality, the ultimate goal is the creation of a monistic system in which - on the symbolic level and in terms of the inferred components of invisibility and intangibly fine structure — the world’s enormous multiplicity is reduced to something like unity, and the endless successions of unique events of a great many different kinds get tidied and simplified into a single rational order. Whether this goal will ever be reached remains to be seen. Meanwhile we have the various sciences, each with its own system coordinating concepts, its own criterion of explanation." (Aldous Huxley, "Literature and Science", 1963)

"Science is a matter of disinterested observation, patient ratiocination within some system of logically correlated concepts. In real-life conflicts between reason and passion the issue is uncertain. Passion and prejudice are always able to mobilize their forces more rapidly and press the attack with greater fury; but in the long run (and often, of course, too late) enlightened self-interest may rouse itself, launch a counterattack and win the day for reason." (Aldous L Huxley, "Literature and Science", 1963)

"It takes a certain amount of intelligence and imagination to realize the extraordinary queerness and mysteriousness of the world in which we live." (Aldous L Huxley)

On Systems (1875-1899)

"In every physical science we have carefully to distinguish between the facts which form its subject-matter and the theories by which we attempt to explain these facts, and group them in our scientific systems." (Josiah P Cooke, "The New Chemistry", 1876)

"There is a maxim which is often quoted, that ‘The same causes will always produce the same effects.’ To make this maxim intelligible we must define what we mean by the same causes and the same effects, since it is manifest that no event ever happens more that once, so that the causes and effects cannot be the same in all respects. [...] There is another maxim which must not be confounded with that quoted at the beginning of this article, which asserts ‘That like causes produce like effects’. This is only true when small variations in the initial circumstances produce only small variations in the final state of the system. In a great many physical phenomena this condition is satisfied; but there are other cases in which a small initial variation may produce a great change in the final state of the system, as when the displacement of the ‘points’ causes a railway train to run into another instead of keeping its proper course." (James C Maxwell, "Matter and Motion", 1876)

"Science is the systematic classification of experience." (George H Lewes, "The Physical Basis of Mind", 1877)

"The very genius of the common geometry as a method of reasoning - a system of investigation - is, that it is but a series of observations. The figure being before the eye in actual representation, or before the mind in conception, is so closely scrutinized, that all its distinctive features are perceived; auxiliary lines are drawn (the imagination leading in this), and a new series of inspections is made; and thus, by means of direct, simple observations, the investigation proceeds." (Edward Olney, "Mathematics", The Cyclopedia of Education, 1877)

"As long as the training of a naturalist enables him to trace the action only of a particular material system, without giving him the power of dealing with the general properties of all such systems, he must proceed by the method so often described in histories of science - he must imagine model after model of hypothetical apparatus, till he finds one which will do the required work. If this apparatus should afterwards be found capable of accounting for many of the known phenomena, and not demonstrably inconsistent with any of them, he is strongly tempted to conclude that his hypothesis is a fact, at least until an equally good rival hypothesis has been invented." (James C Maxwell, "Tait’s Thermodynamics", Nature Vol. XVII (431), 1878)

"It is very different to make a practical system and to introduce it. A few experiments in the laboratory would prove the practicability of system long before it could be brought into general use." (Thomas Edison, 1879)

"Science, while it penetrates deeply the system of things about us, sees everywhere, in the dim limits of vision, the word mystery." (James D Dana, "Corals and Coral Islands", 1879)

"[…] not only a knowledge of the ideas that have been accepted and cultivated by subsequent teachers is necessary for the historical understanding of a science, but also that the rejected and transient thoughts of the inquirers, nay even apparently erroneous notions, may be very important and very instructive. The historical investigation of the development of a science is most needful, lest the principles treasured up in it become a system of half-understood prescripts, or worse, a system of prejudices." (Ernst Mach, "The Science of Mechanics", 1883)

"The easiest and surest way of acquiring facts is to learn them in groups, in systems, and systematized knowledge is science. You can very often carry two facts fastened together more easily than one by itself, as a house-maid can carry two pails of water with a hoop more easily than one without it." (Oliver W Holmes, "Medical Essays", 1883)

"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)

"Science is a match that man has just got alight. He thought he was in a room - in moments of devotion, a temple - and that his light would be reflected from and display walls inscribed with wonderful secrets and pillars carved with philosophical systems wrought into harmony. It is a curious sensation, now that the preliminary splutter is over and the flame burns up clear, to see his hands and just a glimpse of himself and the patch he stands on visible, and around him, in place of all that human comfort and beauty he anticipated - darkness still." (Herbert G Wells, "The Rediscovery of the Unique", The Fortnightly Review, 1891)

"In every symmetrical system every deformation that tends to destroy the symmetry is complemented by an equal and opposite deformation that tends to restore it. […] One condition, therefore, though not an absolutely sufficient one, that a maximum or minimum of work corresponds to the form of equilibrium, is thus applied by symmetry." (Ernst Mach, "The Science of Mechanics: A Critical and Historical Account of Its Development", 1893)

"Society is not a mere sum of individuals. Rather, the system formed by their association represents a specific reality which has its own characteristics. [...] The group thinks, feels, and acts quite differently from the way in which its members would were they isolated. If, then, we begin with the individual, we shall be able to understand nothing of what takes place in the group." (Émile Durkheim, "The Rules of Sociological Method", 1895)

"An act cannot be defined by the end sought by the actor, for an identical system of behaviour may be adjustable to too many different ends without altering its nature." (Émile Durkheim, "Suicide: A Study in Sociology", 1897)

"Certainly, if a system moves under the action of given forces and its initial conditions have given values in the mathematical sense, its future motion and behavior are exactly known. But, in astronomical problems, the situation is quite different: the constants defining the motion are only physically known, that is with some errors; their sizes get reduced along the progresses of our observing devices, but these errors can never completely vanish." (Jacques Hadamard, "Les surfaces à courbures opposées et leurs lignes géodésiques", Journal de mathématiques pures et appliqués 5e (4), 1898)

"The ordinary logic has a great deal to say about genera and species, or in our nineteenth century dialect, about classes. Now a class is a set of objects compromising all that stand to one another in a special relation of similarity. But where ordinary logic talks of classes the logic of relatives talks of systems. A system is a set of objects compromising all that stands to one another in a group of connected relations. Induction according to ordinary logic rises from the contemplation of a sample of a class to that of a whole class; but according to the logic of relatives it rises from the contemplation of a fragment of a system to the envisagement of the complete system." (Charles S Peirce, "Cambridge Lectures on Reasoning and the Logic of Things: Detached Ideas on Vitally Important Topics", 1898)

Jean L R Agassiz - Collected Quotes

"I may say that here, as in most cases where the operations of nature interfere with the designs of man, it is not by a direct intervention on our part that we may remedy the difficulties, but rather by a precise knowledge of [nature’s] causes, which may enable us, if not to check, at least to avoid the evil consequences." (Jean L R Agassiz, "Annual Report of the Superintendent of the Coast Survey, Showing the Progress of that Work During the Year Ending November", 1851)

"As long as men inquire, they will find opportunities to know more upon these topics than those who have gone before them, so inexhaustibly rich is nature in the innermost diversity of her treasures of beauty, order, and intelligence." (Jean L R Agassiz, “Essay on Classification”, 1859)

"Are our systems the inventions of naturalists, or only their reading of the Book of Nature? and can that book have more than one reading? If these classifications are not mere inventions, if they are not an attempt to classify for our own convenience the objects we study, then they are thoughts which, whether we detect them or not, are expressed in Nature, - then Nature is the work of thought, the production of intelligence carried out according to plan, therefore premeditated, - and in our study of natural objects we are approaching the thoughts of the Creator, reading His conceptions, interpreting a system that is His and not ours." (Jean L R Agassiz, "Methods of Study in Natural History", 1863)

"[...] it must be for truth’s sake, and not even for the sake of its usefulness to humanity, that the scientific man studies Nature." (Jean L R Agassiz, "Methods of Study in Natural History", 1863)

"The education of a naturalist now consists chiefly in learning how to compare." (Jean L R Agassiz, "Methods of Study in Natural History", 1863)

"[...] the time has come when scientific truth must cease to be the property of the few, when it must be woven into the common life of the world; for we have reached the point where the results of science touch the very problem of existence, and all men listen for the solving of that mystery." (Jean L R Agassiz, "Methods of Study in Natural History", 1863)

"Philosophers and theologians have yet to learn that a physical fact is as sacred as a moral principle." (Jean L R Agassiz, "Evolution and Permanence of Type", The Atlantic Monthly, 1874)

"Facts are stupid things until brought into connection with some general law." (Jean L R Agassiz)

"Lay aside all conceit. Learn to read the book of nature for yourself. Those who have succeeded best have followed for years some slim thread which has once in a while broadened out and disclosed some treasure worth a life-long search." (Jean L R Agassiz)

"The only true scientific system must be one in which the thought, the intellectual structure, rises out of, and is based upon, facts." (Jean L R Agassiz)

"The study of nature is an intercourse with the highest mind. You should never trifle with nature. At the lowest her works are the works of the highest powers - the highest something, in whatever way we may look at it." (Jean L R Agassiz)

On Systems (2010-2019)

"Abstract formulations of simply stated concrete ideas are often the result of efforts to create idealized models of complex systems. The models are 'idealized' in the sense that they retain only the most fundamental properties of the original systems. The vocabulary is chosen to be as inclusive as possible so that research into the model reveals facts about a wide variety of similar systems. Unfortunately, it is often the case that over time the connection between a model and the systems on which it was based is lost, and the interested reader is faced with something that looks as if it were created to be deliberately complicated - deliberately confusing - but the original intention was just the opposite. Often, the model was devised to be simpler and more transparent than any of the systems on which it was based." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"When some systems are stuck in a dangerous impasse, randomness and only randomness can unlock them and set them free." (Nassim N Taleb, "Antifragile: Things That Gain from Disorder", 2012) 

"Complex systems defy intuitive solutions. Even a third-order, linear differential equation is unsolvable by inspection. Yet, important situations in management, economics, medicine, and social behavior usually lose reality if simplified to less than fifth-order nonlinear dynamic systems. Attempts to deal with nonlinear dynamic systems using ordinary processes of description and debate lead to internal inconsistencies. Underlying assumptions may have been left unclear and contradictory, and mental models are often logically incomplete. Resulting behavior is likely to be contrary to that implied by the assumptions being made about' underlying system structure and governing policies." (Jay W. Forrester, "Modeling for What Purpose?", The Systems Thinker Vol. 24 (2), 2013)

"Each systems archetype embodies a particular theory about dynamic behavior that can serve as a starting point for selecting and formulating raw data into a coherent set of interrelationships. Once those relationships are made explicit and precise, the "theory" of the archetype can then further guide us in our data-gathering process to test the causal relationships through direct observation, data analysis, or group deliberation." (Daniel H Kim, "Systems Archetypes as Dynamic Theories", The Systems Thinker Vol. 24 (1), 2013)

"Models are present in everything we do. One does not have a family or corporation in one's head. Instead, one has observations about those systems. Such observations and assumptions constitute mental models, which are then used as the basis for action. System dynamics models have little impact unless they change the way people perceive a situation. They must relate to and improve mental models if they are to fill an effective role." (Jay W. Forrester, "Modeling for What Purpose?", The Systems Thinker Vol. 24 (2), 2013)

"Systems archetypes thus provide a good starting theory from which we can develop further insights into the nature of a particular system. The diagram that results from working with an archetype should not be viewed as the "truth," however, but rather a good working model of what we know at any point in time." (Daniel H Kim, "Systems Archetypes as Dynamic Theories", The Systems Thinker Vol. 24 (1), 2013)

"System dynamics models have little impact unless they change the way people perceive a situation. A model must help to organize information in a more understandable way. A model should link the past to the present by showing how present conditions arose, and extend the present into persuasive alternative futures under a variety of scenarios determined by policy alternatives. In other words, a system dynamics model, if it is to be effective, must communicate with and modify the prior mental models. Only people's beliefs - that is, their mental models - will determine action. Computer models must relate to and improve mental models if the computer models are to fill an effective role." (Jay W. Forrester, "Modeling for What Purpose?", The Systems Thinker Vol. 24 (2), 2013)

"System meaning is informed by the circumstances and factors that surround the system. The contextual axiom's propositions are those which bound the system by providing guidance that enables an investigator to understand the set of external circumstances or factors that enable or constrain a particular system. The contextual axiom has three principles: (1) holism, (2) darkness, and (3) complementarity." (Patrick Hester & Kevin Adams," Systemic Thinking: Fundamentals for Understanding Problems and Messes", 2014)

"This spontaneous emergence of order at critical points of instability, which is often referred to simply as 'emergence', is one of the hallmarks of life. It has been recognized as the dynamic origin of development, learning, and evolution. In other words, creativity - the generation of new forms - is a key property of all living systems." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

"The proper measure of a philosophical system or a scientific theory is not the degree to which it anticipated modern thought, but its degree of success in treating the philosophical and scientific problems of its own day." (Steven Weinberg, "To Explain the World: The Discovery of Modern Science", 2015)

"[…] the role that symmetry plays is not confined to material objects. Symmetries can also refer to theories and, in particular, to quantum theory. For if the laws of physics are to be invariant under changes of reference frames, the set of all such transformations will form a group. Which transformations and which groups depends on the systems under consideration." (William H Klink & Sujeev Wickramasekara, "Relativity, Symmetry and the Structure of Quantum Theory I: Galilean quantum theory", 2015)

"Science, at its core, is simply a method of practical logic that tests hypotheses against experience. Scientism, by contrast, is the worldview and value system that insists that the questions the scientific method can answer are the most important questions human beings can ask, and that the picture of the world yielded by science is a better approximation to reality than any other." (John Michael Greer, "After Progress: Reason and Religion at the End of the Industrial Age", 2015) 

"A worldview consists of observations of the individual and other people with respect to the self, time and space, the natural and the supernatural and the sacred and profane. […] Beliefs about the world do not reside in the human mind in chaotic disorder; rather they form a latent system. A worldview cannot, however, be viewed as a well-organised network of cognitive models or a static collection of values; instead it should be regarded as the product of a process shaped by historical, cultural and social perspectives and contexts." (Helena Helve, "A longitudinal perspective on worldviews, values and identities", 2016)

"Although cascading failures may appear random and unpredictable, they follow reproducible laws that can be quantified and even predicted using the tools of network science. First, to avoid damaging cascades, we must understand the structure of the network on which the cascade propagates. Second, we must be able to model the dynamical processes taking place on these networks, like the flow of electricity. Finally, we need to uncover how the interplay between the network structure and dynamics affects the robustness of the whole system." (Albert-László Barabási, "Network Science", 2016)

"The goal of a system dynamics approach is to understand how a dynamic pattern of behaviour is generated by a system and to find leverage points within the system structure that have the potential to change the problematic trend to a more desirable one. The key steps in a system dynamics approach are identifying one or more trends that characterise the problem, describing the structure of the system generating the behaviour and finding and testing leverage points in the system to change the problematic behaviour. System dynamics is an appropriate modelling approach for sustainability questions because of the long-term perspective and feedback dynamics inherent in such questions." (Bilash K Bala et al, "System Dynamics: Modelling and Simulation", 2017)

"[…] the system boundary should encompass that portion of the whole system which includes all the important and relevant variables to address the problem and the purpose of policy analysis and design. The scope of the study should be clearly stated in order to identify the causes of the problem for clear understanding of the problem and policies for solving the problem in the short run and long run." (Bilash K Bala et al, "System Dynamics: Modelling and Simulation", 2017)

"A system governed by a deterministic theory can only evolve along a single trajectory - namely, that dictated by its laws and initial conditions; all other trajectories are excluded. Symmetry principles, on the other hand, fit the freedom-inducing model. Rather than distinguishing what is excluded from what is bound to happen, these principles distinguish what is excluded from what is possible. In other words, although they place restrictions on what is possible, they do not usually determine a single trajectory." (Yemima Ben-Menahem, "Causation in Science", 2018)

"The relationship of math to the real world has been a conundrum for philosophers for centuries, but it is also an inspiration for poets. The patterns of mathematics inhabit a liminal space - they were initially derived from the natural world and yet seem to exist in a separate, self-contained system standing apart from that world. This makes them a source of potential metaphor: mapping back and forth between the world of personal experience and the world of mathematical patterns opens the door to novel connections." (Alice Major, "Mapping from e to Metaphor", 2018) 

On Systems (2000-2009)

"I propose a new concept based on an interpretation of ecosystems: sympoietic systems. These are complex, self-organizing but collectively producing, boundaryless systems. A subsequent distinction between sympoietic and autopoietic systems is discussed. This distinction arises from defining a difference between three key system characteristics: 1) autopoietic systems have self-defined boundaries, sympoietic systems do not; 2) autopoietic systems are self-produced, sympoietic systems are collectively produced; and, 3) autopoietic systems are organizationally closed, sympoietic systems are organizationally ajar." (Beth Dempster, "Sympoietic and Autopoietic Systems: A New Distinction for Self-Organizing Systems". 2000)

"Following the traditional classification in the field of control systems, a system that describes the input-output behavior in a way similar to a mathematical mapping without involving a differential operator or equation is called a static system. In contrast, a system described by a differential operator or equation is called a dynamic system." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)

"Formulation of a mathematical model is the first step in the process of analyzing the behaviour of any real system. However, to produce a useful model, one must first adopt a set of simplifying assumptions which have to be relevant in relation to the physical features of the system to be modelled and to the specific information one is interested in. Thus, the aim of modelling is to produce an idealized description of reality, which is both expressible in a tractable mathematical form and sufficiently close to reality as far as the physical mechanisms of interest are concerned." (Francois Axisa, "Discrete Systems" Vol. I, 2001)

"Although the detailed moment-to-moment behavior of a chaotic system cannot be predicted, the overall pattern of its 'random' fluctuations may be similar from scale to scale. Likewise, while the fine details of a chaotic system cannot be predicted one can know a little bit about the range of its 'random' fluctuation." (F David Peat, "From Certainty to Uncertainty", 2002)

"Most physical processes in the real world are nonlinear. It is our abstraction of the real world that leads us to the use of linear systems in modeling these processes. These linear systems are simple, understandable, and, in many situations, provide acceptable simulations of the actual processes. Unfortunately, only the simplest of linear processes and only a very small fraction of the nonlinear having verifiable solutions can be modeled with linear systems theory. The bulk of the physical processes that we must address are, unfortunately, too complex to reduce to algorithmic form - linear or nonlinear. Most observable processes have only a small amount of information available with which to develop an algorithmic understanding. The vast majority of information that we have on most processes tends to be nonnumeric and nonalgorithmic. Most of the information is fuzzy and linguistic in form." (Timothy J Ross & W Jerry Parkinson, "Fuzzy Set Theory, Fuzzy Logic, and Fuzzy Systems", 2002)

"Nature normally hates power laws. In ordinary systems all quantities follow bell curves, and correlations decay rapidly, obeying exponential laws. But all that changes if the system is forced to undergo a phase transition. Then power laws emerge-nature's unmistakable sign that chaos is departing in favor of order. The theory of phase transitions told us loud and clear that the road from disorder to order is maintained by the powerful forces of self-organization and is paved by power laws. It told us that power laws are not just another way of characterizing a system's behavior. They are the patent signatures of self-organization in complex systems." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"One of the key insights of the systems approach has been the realization that the network is a pattern that is common to all life. Wherever we see life, we see networks." (Fritjof Capra, "The Hidden Connections: A Science for Sustainable Living", 2002)

"What is a mathematical model? One basic answer is that it is the formulation in mathematical terms of the assumptions and their consequences believed to underlie a particular ‘real world’ problem. The aim of mathematical modeling is the practical application of mathematics to help unravel the underlying mechanisms involved in, for example, economic, physical, biological, or other systems and processes." (John A Adam, "Mathematics in Nature", 2003)

"According to a 'sociological' view of mathematics, a system, in general, should be able to do whatever is permitted by the laws governing it: the normal state of anarchy is chaos! From this point of view, we should expect that, in the absence of conservation laws, typical motions should be dense in the space available to them; Kolomogorov’s theorem denies this, saying that when the laws are relaxed a bit, the majority of motions stay 'pretty much' where they were, as if in fear of a non-existent police force." (John H Hubbard, "The KAM Theorem", 2004)

"All models are mental projections of our understanding of processes and feedbacks of systems in the real world. The general approach is that models are as good as the system upon which they are based. Models should be designed to answer specific questions and only incorporate the necessary details that are required to provide an answer." (Hördur V Haraldsson & Harald U Sverdrup, "Finding Simplicity in Complexity in Biogeochemical Modelling", 2004)

"Complexity is the characteristic property of complicated systems we don’t understand immediately. It is the amount of difficulties we face while trying to understand it. In this sense, complexity resides largely in the eye of the beholder - someone who is familiar with s.th. often sees less complexity than someone who is less familiar with it. [...] A complex system is created by evolutionary processes. There are multiple pathways by which a system can evolve. Many complex systems are similar, but each instance of a system is unique." (Jochen Fromm, The Emergence of Complexity, 2004)

"Group theory is a powerful tool for studying the symmetry of a physical system, especially the symmetry of a quantum system. Since the exact solution of the dynamic equation in the quantum theory is generally difficult to obtain, one has to find other methods to analyze the property of the system. Group theory provides an effective method by analyzing symmetry of the system to obtain some precise information of the system verifiable with observations." (Zhong-Qi Ma & Xiao-Yan Gu, "Problems and Solutions in Group Theory for Physicists", 2004)

"In complexity thinking the darkness principle is covered by the concept of incompressibility… The concept of incompressibility suggests that the best representation of a complex system is the system itself and that any representation other than the system itself will necessarily misrepresent certain aspects of the original system." (Kurt Richardson, "Systems theory and complexity: Part 1", Emergence: Complexity & Organization Vol.6 (3), 2004)

"Naturalism is the view that the physical world is a self-contained system that works by blind, unbroken natural laws. Naturalism doesn’t come right out and say there’s nothing beyond nature. Rather, it says that nothing beyond nature could have any conceivable relevance to what happens in nature. Naturalism’s answer to theism is not atheism but benign neglect. People are welcome to believe in God, though not a God who makes a difference in the natural order." (William A Dembski, "The Design Revolution: Answering the Toughest Questions About Intelligent Design", 2004)

 "[…] some systems […] are very sensitive to their starting conditions, so that a tiny difference in the initial ‘push’ you give them causes a big difference in where they end up, and there is feedback, so that what a system does affects its own behavior." (John Gribbin, "Deep Simplicity", 2004)

"Technology can relieve the symptoms of a problem without affecting the underlying causes. Faith in technology as the ultimate solution to all problems can thus divert our attention from the most fundamental problem - the problem of growth in a finite system - and prevent us from taking effective action to solve it." (Donella H Meadows & Dennis L Meadows, "The Limits to Growth: The 30 Year Update", 2004)

"The basic concept of complexity theory is that systems show patterns of organization without organizer (autonomous or self-organization). Simple local interactions of many mutually interacting parts can lead to emergence of complex global structures. […] Complexity originates from the tendency of large dynamical systems to organize themselves into a critical state, with avalanches or 'punctuations' of all sizes. In the critical state, events which would otherwise be uncoupled became correlated." (Jochen Fromm, "The Emergence of Complexity", 2004)

"What do people do today when they don’t understand 'the system'? They try to assign responsibility to someone to fix the problem, to oversee 'the system', to coordinate and control what is happening. It is time we recognized that 'the system' is how we work together. When we don’t work together effectively putting someone in charge by its very nature often makes things worse, rather than better, because no one person can understand 'the system' well enough to be responsible. We need to learn how to improve the way we work together, to improve 'the system' without putting someone in charge, in order to make things work." (Yaneer Bar-Yam, "Making Things Work: Solving Complex Problems in a Complex World", 2004)

"A conceptual model is a mental image of a system, its components, its interactions. It lays the foundation for more elaborate models, such as physical or numerical models. A conceptual model provides a framework in which to think about the workings of a system or about problem solving in general. An ensuing operational model can be no better than its underlying conceptualization." (Henry N Pollack, "Uncertain Science … Uncertain World", 2005)

"Art is constructivist in nature, aimed at the deliberate refinement and elaboration of mental models and worldviews. These are the natural products of cognition itself, the outcome of the brain’s tendency to strive for the integration of perceptual and conceptual material over time. […] human culture is essentially a distributed cognitive system within which worldviews and mental models are constructed and shared by the members of a society. Artists are traditionally at the forefront of that process, and have a large influence on our worldviews and mental models." (Mark Turner, "The Artful Mind : cognitive science and the riddle of human creativity", 2006)

"The progress of science requires the growth of understanding in both directions, downward from the whole to the parts and upward from the parts to the whole." (Freeman J Dyson, "The Scientist As Rebel", 2006)

"Effective models require a real world that has enough structure so that some of the details can be ignored. This implies the existence of solid and stable building blocks that encapsulate key parts of the real system’s behavior. Such building blocks provide enough separation from details to allow modeling to proceed."(John H. Miller & Scott E. Page," Complex Adaptive Systems: An Introduction to Computational Models of Social Life", 2007)

"Historically, science has pursued a premise that Nature can be understood fully, its future predicted precisely, and its behavior controlled at will. However, emerging knowledge indicates that the nature of Earth and biological systems transcends the limits of science, questioning the premise of knowing, prediction, and control. This knowledge has led to the recognition that, for civilized human survival, technological society has to adapt to the constraints of these systems." (Nari Narasimhan, "Limitations of Science and Adapting to Nature", Environmental Research Letters, 2007)

"Humans have difficulty perceiving variables accurately […]. However, in general, they tend to have inaccurate perceptions of system states, including past, current, and future states. This is due, in part, to limited ‘mental models’ of the phenomena of interest in terms of both how things work and how to influence things. Consequently, people have difficulty determining the full implications of what is known, as well as considering future contingencies for potential systems states and the long-term value of addressing these contingencies. " (William B. Rouse, "People and Organizations: Explorations of Human-Centered Design", 2007)

"Systemic problems trace back in the end to worldviews. But worldviews themselves are in flux and flow. Our most creative opportunity of all may be to reshape those worldviews themselves. New ideas can change everything." (Anthony Weston, "How to Re-Imagine the World", 2007)

"The system is highly sensitive to some small changes and blows them up into major alterations in weather patterns. This is popularly known as the butterfly effect in that it is possible for a butterfly to flap its wings in São Paolo, so making a tiny change to air pressure there, and for this tiny change to escalate up into a hurricane over Miami. You would have to measure the flapping of every butterfly’s wings around the earth with infinite precision in order to be able to make long-term forecasts. The tiniest error made in these measurements could produce spurious forecasts. However, short-term forecasts are possible because it takes time for tiny differences to escalate."  (Ralph D Stacey, "Strategic Management and Organisational Dynamics: The Challenge of Complexity" 5th Ed. , 2007)

"A characteristic of such chaotic dynamics is an extreme sensitivity to initial conditions (exponential separation of neighboring trajectories), which puts severe limitations on any forecast of the future fate of a particular trajectory. This sensitivity is known as the ‘butterfly effect’: the state of the system at time t can be entirely different even if the initial conditions are only slightly changed, i.e., by a butterfly flapping its wings." (Hans J Korsch et al, „Chaos: A Program Collection for the PC", 2008)

"A system is a set of things – people, cells, molecules, or whatever – interconnected in such a way that they produce their own pattern of behavior over time. […] The system, to a large extent, causes its own behavior." (Donella H Meadows, "Thinking in Systems: A Primer", 2008) 

 "A system, it is said, is a collection of parts together with their relationships that forms a whole that serves a purpose that is meaningful to the system alone, that is, not to its parts or their relationships." (John Boardman & Brian Sauser, "Systems Thinking: Coping with 21st Century Problems", 2008)

"[…] our mental models fail to take into account the complications of the real world - at least those ways that one can see from a systems perspective. It is a warning list. Here is where hidden snags lie. You can’t navigate well in an interconnected, feedback-dominated world unless you take your eyes off short-term events and look for long-term behavior and structure; unless you are aware of false boundaries and bounded rationality; unless you take into account limiting factors, nonlinearities and delays. You are likely to mistreat, misdesign, or misread systems if you don’t respect their properties of resilience, self-organization, and hierarchy." (Donella H Meadows, "Thinking in Systems: A Primer", 2008)

"The addition of new elements or agents to a particular system multiplies exponentially the number of connections or potential interactions among those elements or agents, and hence the number of possible outcomes. This is an important attribute of complexity theory." (Mark Marson, "What Are Its Implications for Educational Change?", 2008)

"Two systems concepts lie at the disposal of the architect to reflect the beauty of harmony: parsimony and variety. The law of parsimony states that given several explanations of a specific phenomenon, the simplest is probably the best. […] On the other hand, the law of requisite variety states that for a system to survive in its environment the variety of choice that the system is able to make must equal or exceed the variety of influences that the environment can impose on the system." (John Boardman & Brian Sauser, "Systems Thinking: Coping with 21st Century Problems", 2008)

"A model is a representation in that it (or its properties) is chosen to stand for some other entity (or its properties), known as the target system. A model is a tool in that it is used in the service of particular goals or purposes; typically these purposes involve answering some limited range of questions about the target system." (Wendy S Parker, "Confirmation and Adequacy-for-Purpose in Climate Modelling", Proceedings of the Aristotelian Society, Supplementary Volumes, Vol. 83, 2009)

On Systems (1990-1999)

"Autopoietic systems, then, are not only self-organizing systems, they not only produce and eventually change their own structures; their self-reference applies to the production of other components as well. This is the decisive conceptual innovation. […] Thus, everything that is used as a unit by the system is produced as a unit by the system itself. This applies to elements, processes, boundaries, and other structures and, last but not least, to the unity of the system itself." (Niklas Luhmann, "The Autopoiesis of Social Systems", 1990)

"The problem with mental models lie not in whether they are right or wrong - by definition, all models are simplifications. The problem with mental models arise when they become implicit - when they exist below the level of our awareness. […] models, if unexamined, limit an organization's range of actions to what is familiar and comfortable. [...] Each person's mental model focuses on different parts of the system. Each emphasizes different cause-effect chains. This makes it virtually impossible for a shared picture of the system as a whole to emerge in normal conversation." (Peter Senge, "The Fifth Discipline”, 1990)

"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)

"Systems philosophy brings forth a reorganization of ways of thinking. It creates a new worldview, a new paradigm of perception and explanation, which is manifested in integration, holistic thinking, purpose-seeking, mutual causality, and process-focused inquiry." (Béla H Bánáthy, "Systems Design of Education", 1991)

"Systems science is a science whose domain of inquiry consists of those properties of systems and associated problems that emanate from the general notion of systemhood." (George Klir, "Facets of Systems Science", 1991)

"Systems theory pursues the scientific exploration and understanding of systems that exist in the various realms of experience, in order to arrive at a general theory of systems: an organized expressing of sets of interrelated concepts and principles that apply to all systems." (Béla H Bánáthy, "Systems Design of Education", 1991)

"The term ‘chaos’ currently has a variety of accepted meanings, but here we shall use it to mean deterministically, or nearly deterministically, governed behavior that nevertheless looks rather random. Upon closer inspection, chaotic behavior will generally appear more systematic, but not so much so that it will repeat itself at regular intervals, as do, for example, the oceanic tides." (Edward N Lorenz, "Chaos, spontaneous climatic variations and detection of the greenhouse effect", 1991)

"Unfortunately, recognizing a system as chaotic will not tell us all that we might like to know. It will not provide us with a means of predicting the future course of the system. It will tell us that there is a limit to how far ahead we can predict, but it may not tell us what this limit is. Perhaps the best advice that chaos ‘theory’ can give us is not to jump at conclusions; unexpected occurrences may constitute perfectly normal behavior." (Edward N Lorenz, "Chaos, spontaneous climatic variations and detection of the greenhouse effect", 1991)

"Mathematics is also seen by many as an analogy. But it is implicitly assumed to be the analogy that never breaks down. Our experience of the world has failed to reveal any physical phenomenon that cannot be described mathematically. That is not to say that there are not things for which such a description is wholly inappropriate or pointless. Rather, there has yet to be found any system in Nature so unusual that it cannot be fitted into one of the strait-jackets that mathematics provides." (John Barrow," Pi in the Sky: Counting, Thinking, and Being", 1992)

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

"Systems, acting dynamically, produce (and incidentally, reproduce) their own boundaries, as structures which are complementary (necessarily so) to their motion and dynamics. They are liable, for all that, to instabilities chaos, as commonly interpreted of chaotic form, where nowadays, is remote from the random. Chaos is a peculiar situation in which the trajectories of a system, taken in the traditional sense, fail to converge as they approach their limit cycles or 'attractors' or 'equilibria'. Instead, they diverge, due to an increase, of indefinite magnitude, in amplification or gain." (Gordon Pask, "Different Kinds of Cybernetics", 1992)

"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)

"[…] the study of natural systems begins and ends with the specification of observables describing such a system, and a characterization of the manner in which these observables are linked." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"The word theory, as used in the natural sciences, doesn’t mean an idea tentatively held for purposes of argument - that we call a hypothesis. Rather, a theory is a set of logically consistent abstract principles that explain a body of concrete facts. It is the logical connections among the principles and the facts that characterize a theory as truth. No one element of a theory [...] can be changed without creating a logical contradiction that invalidates the entire system. Thus, although it may not be possible to substantiate directly a particular principle in the theory, the principle is validated by the consistency of the entire logical structure." (Alan Cromer, "Uncommon Sense: The Heretical Nature of Science", 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)

"A world view is a system of co-ordinates or a frame of reference in which everything presented to us by our diverse experiences can be placed. It is a symbolic system of representation that allows us to integrate everything we know about the world and ourselves into a global picture, one that illuminates reality as it is presented to us within a certain culture. […] A world view is a coherent collection of concepts and theorems that must allow us to construct a global image of the world, and in this way to understand as many elements of our experience as possible.” (Diederick Aerts et al, "World views: From Fragmentation to Integration”, 1994)

"[…] chaos and fractals are part of an even grander subject known as dynamics. This is the subject that deals with change, with systems that evolve in time. Whether the system in question settles down to equilibrium, keeps repeating in cycles, or does something more complicated, it is dynamics that we use to analyze the behavior." (Steven H Strogatz, "Non-Linear Dynamics and Chaos, 1994)

"Mathematicians can and do fill in gaps, correct errors, and supply more detail and more careful scholarship when they are called on or motivated to do so. Our system is quite good at producing reliable theorems that can be backed up. It’s just that the reliability does not primarily come from mathematicians checking formal arguments; it come from mathematicians thinking carefully and critically about mathematical ideas." (William P Thurston, "On Proof and Progress in Mathematics", Bulletin of the American Mathematical Society Vol. 30 (2), 1994)

"The impossibility of constructing a complete, accurate quantitative description of a complex system forces observers to pick which aspects of the system they most wish to understand." (Thomas Levenson, "Measure for Measure: A musical history of science", 1994)

"The prevailing style of management must undergo transformation. A system cannot understand itself. The transformation requires a view from outside. The aim [...] is to provide an outside view - a lens - that I call a system of profound knowledge. It provides a map of theory by which to understand the organizations that we work in." (Dr. W. Edwards Deming, "The New Economics for Industry, Government, Education", 1994)

"Why are nonlinear systems so much harder to analyze than linear ones? The essential difference is that linear systems can be broken down into parts. Then each part can be solved separately and finally recombined to get the answer. This idea allows a fantastic simplification of complex problems, and underlies such methods as normal modes, Laplace transforms, superposition arguments, and Fourier analysis. In this sense, a linear system is precisely equal to the sum of its parts." (Steven H Strogatz, "Non-Linear Dynamics and Chaos, 1994)

"Human mind and culture have developed a formal system of thought for recognizing, classifying, and exploiting patterns. We call it mathematics. By using mathematics to organize and systematize our ideas about patterns, we have discovered a great secret: nature's patterns are not just there to be admired, they are vital clues to the rules that govern natural processes." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Music and higher mathematics share some obvious kinship. The practice of both requires a lengthy apprenticeship, talent, and no small amount of grace. Both seem to spring from some mysterious workings of the mind. Logic and system are essential for both, and yet each can reach a height of creativity beyond the merely mechanical." (Frederick Pratter, "How Music and Math Seek Truth in Beauty", Christian Science Monitor, 1995)

"All systems evolve, although the rates of evolution may vary over time both between and within systems. The rate of evolution is a function of both the inherent stability of the system and changing environmental circumstances. But no system can be stabilized forever. For the universe as a whole, an isolated system, time’s arrow points toward greater and greater breakdown, leading to complete molecular chaos, maximum entropy, and heat death. For open systems, including the living systems that are of major interest to us and that interchange matter and energy with their external environments, time’s arrow points to evolution toward greater and greater complexity. Thus, the universe consists of islands of increasing order in a sea of decreasing order. Open systems evolve and maintain structure by exporting entropy to their external environments." (L Douglas Kiel, "Chaos Theory in the Social Sciences: Foundations and Applications", 1996)

"By irreducibly complex I mean a single system composed of several well-matched, interacting parts that contribute to the basic function, wherein the removal of any one of the parts causes the system to effectively cease functioning. An irreducibly complex system cannot be produced directly (that is, by continuously improving the initial function, which continues to work by the same mechanism) by slight, successive modification of a precursor, system, because any precursors to an irreducibly complex system that is missing a part is by definition nonfunctional." (Michael Behe, "Darwin’s Black Box", 1996)

"Contrary to what happens at equilibrium, or near equilibrium, systems far from equilibrium do not conform to any minimum principle that is valid for functions of free energy or entropy production." (Ilya Prigogine, "The End of Certainty: Time, Chaos, and the New Laws of Nature", 1996) 

"The more we study the major problems of our time, the more we come to realise that they cannot be understood in isolation. They are systemic problems, which means that they are interconnected and interdependent." (Fritjof Capra, "The Web of Life: A New Scientific Understanding of Living Systems", 1996)

"Understanding ecological interdependence means understanding relationships. It requires the shifts of perception that are characteristic of systems thinking - from the parts to the whole, from objects to relationships, from contents to patterns. […] Nourishing the community means nourishing those relationships." (Fritjof Capra, "The Web of Life: A New Scientific Understanding of Living Systems", 1996)

"A dictionary definition of the word ‘complex’ is: ‘consisting of interconnected or interwoven parts’ […] Loosely speaking, the complexity of a system is the amount of information needed in order to describe it. The complexity depends on the level of detail required in the description. A more formal definition can be understood in a simple way. If we have a system that could have many possible states, but we would like to specify which state it is actually in, then the number of binary digits (bits) we need to specify this particular state is related to the number of states that are possible." (Yaneer Bar-Yamm, "Dynamics of Complexity", 1997)

"When the behavior of the system depends on the behavior of the parts, the complexity of the whole must involve a description of the parts, thus it is large. The smaller the parts that must be described to describe the behavior of the whole, the larger the complexity of the entire system. […] A complex system is a system formed out of many components whose behavior is emergent, that is, the behavior of the system cannot be simply inferred from the behavior of its components." (Yaneer Bar-Yamm, "Dynamics of Complexity", 1997)

"Complex systems operate under conditions far from equilibrium. Complex systems need a constant flow of energy to change, evolve and survive as complex entities. Equilibrium, symmetry and complete stability mean death. Just as the flow, of energy is necessary to fight entropy and maintain the complex structure of the system, society can only survive as a process. It is defined not by its origins or its goals, but by what it is doing." (Paul Cilliers,"Complexity and Postmodernism: Understanding Complex Systems", 1998)

"Distributed control means that the outcomes of a complex adaptive system emerge from a process of self-organization rather than being designed and controlled externally or by a centralized body." (Brenda Zimmerman et al, "A complexity science primer", 1998) 

"Is a random outcome completely determined, and random only by virtue of our ignorance of the most minute contributing factors? Or are the contributing factors unknowable, and therefore render as random an outcome that can never be determined? Are seemingly random events merely the result of fluctuations superimposed on a determinate system, masking its predictability, or is there some disorderliness built into the system itself?” (Deborah J Bennett, "Randomness", 1998)

"The notion of system we are interested in may be described generally as a complex of elements or components directly or indirectly related in a network of interrelationships of various kinds, such that it constitutes a dynamic whole with emergent properties." (Walter F. Buckley, "Society: A Complex Adaptive System - Essays in Social Theory", 1998)

"In broad terms, a mental model is to be understood as a dynamic symbolic representation of external objects or events on the par. t of some natural or artificial cognitive system. Mental models are thought to have certain properties which make them stand out against other forms of symbolic representations." (Gert Rickheit & Lorenz Sichelschmidt, "Mental Models: Some Answers, Some Questions, Some Suggestions", 1999)

"The self-similarity of fractal structures implies that there is some redundancy because of the repetition of details at all scales. Even though some of these structures may appear to teeter on the edge of randomness, they actually represent complex systems at the interface of order and disorder."  (Edward Beltrami, "What is Random?: Chaos and Order in Mathematics and Life", 1999)

"The thing the ecologically illiterate don't realize about an ecosystem is that it's a system. A system! A system maintains a certain fluid stability that can be destroyed by a misstep in just one niche." (Frank Herbert, "Dune: House Atreides", 1999)