On Machines XX - Between Fiction and Science Fiction

"The humans have a curious force they call ambition. It drives them, and, through them, it drives us. This force which keeps them active, we lack. Perhaps, in time, we machines will acquire it." (John Wyndham, "The Lost Machine", 1932)

"There are so many disadvantages in human construction which do not occur in us machines. [...] Some little thing here or there breaks - they stop working and then, in a short time, they are decomposing. Had he been a machine, like myself, I could have mended him, replaced the broken parts and made him as good as new, but with these animal structures one is almost helpless." (John Wyndham, "The Lost Machine", 1932)

"There’s an affinity between men and the machines they make. They make them out of their own brains, really, a sort of mental conception and gestation, and the result responds to the mind that created them, and to all human minds that understand and manipulate them." (Catherine L Moore, "NoWoman Born", 1944)

"The machinery of civilization was a living body, with organismic Man as its brain." (Walter M Miller Jr., "Way of a Rebel", 1954)

"If a machine had broken down, it would have been quickly replaced. But who can replace a man?" (Brian W Aldiss, ‘‘Who Can Replace a Man?", 1958)

"The study of thinking machines teaches us more about the brain than we can learn by introspective methods. Western man is externalizing himself in the form of gadgets." (William S Burroughs, "Naked Lunch", 1959)

"That perfected machines may one day succeed us is, I remember, an extremely commonplace notion on Earth. It prevails not only among poets and romantics but in all classes of society. Perhaps it is because it is so widespread, born spontaneously in popular imagination, that it irritates scientific minds. Perhaps it is also for this very reason that it contains a germ of truth. Only a germ: Machines will always be machines; the most perfected robot, always a robot." (Pierre Boulle, "Planet of the Apes", 1963)

"Once men turned their thinking over to machines in the hope that this would set them free. But that only permitted other men with machines to enslave them." (Frank Herbert, "Dune", 1965)

"These machines had become old and worn-out, had begun making mistakes; therefore they began to seem almost human." (Philip K Dick & Ray Nelson,"The Ganymede Takeover", 1967)

"A humanoid robot is like any other machine; it can fluctuate between being a benefit and a hazard very rapidly." (Philip K. Dick, "Do Androids Dream of Electric Sheep?", 1968)

"Man has reached the stage where he evolves through his machines." (Gene Wolfe, "Alien Stones", 1972)

"There was no easy way to heaven, or nirvana, or whatever it was that the faithful sought. Merit was acquired solely by one’s own efforts, not with the aid of machines. An interesting doctrine, and one containing much truth; but there were also times when only machines could do the job." (Arthur C Clarke, "The Fountains of Paradise", 1979)

"The dreams of people are in the machines, a planet network of active imaginations hooked into their made-up, make-believe worlds. Artificial reality is taking over; it has its own children." (Storm Constantine, "Immaculate", 1991)

Bernard Bolzano - Collected Quotes

"Every word in language serves to designate an idea and some of them even complete propositions. Therefore, it is only natural to suppose that each idea is composed of at least as many parts as there are words in its expression." (Bernard Bolzano, "Wissenschaftslehre" ["Theory of Science"], 1837)

"One very important genus of complex ideas that we encounter everywhere are those in which the idea of collection (Inbegriff ) appears. There are many types of the latter [...] I must first determine with more precision the concept I associate with the word collection. I use this word in the same sense as it is used in the common usage and thus understand by a collection of certain things exactly the same as what one would express by the words: a combination (Verbindung) or association (Vereinigung) of these things, a gathering (Zusammensein) of the latter, a whole (Ganzes) in which they occur as parts (Teile). Hence the mere idea of a collection does not allow us to determine in which order and sequence the things that are put together appear or, indeed, whether there is or can be such an order. [...] A collection, it seems to me, is nothing other than something complex (das Zusammengesetztheit hat)." (Bernard Bolzano, "Wissenschaftslehre" ["Theory of Science"], 1837)

"The most plausible way to [conceive of the relation between grounding and causality] is that somehow those truths that state the existence of the properties of a cause be considered as the ground, and those that concern the existence and the properties of the effect be considered as the consequence." (Bernard Bolzano, "Wissenschaftslehre" ["Theory of Science"] ,1837)

"[a set is] an embodiment of the idea or concept which we conceive when we regard the arrangement of its parts as a matter of indifference." (Bernard Bolzano, 1847)

"Already within the domain of those things which do not have any pretension of reality, but only of possibility, there indisputably are sets that are infinite. The set of propositions and truths in themselves is infinite, as one can easily see." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"But rather they are able, in spite of that relationship between them that is the same for both of them, to have a relationship of inequality in their pluralities, so that one of them can be presented as a whole, of which the other is a part. An equality of these pluralities may only be concluded if some other reason is added, such as that both multitudes have exactly the same determining grounds, e.g. they have exactly the same way of being formed." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"Even in the realm of things which do not claim actuality, and do not even claim possibility, there exist beyond dispute sets which are infinite. The set of all ‘absolute propositions and truths' is easily seen to be infinite." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"Even with the examples of the infinite considered so far it could not escape our notice that not all infinite multitudes are to be regarded as equal to one another in respect of their plurality, but that some of them are greater (or smaller) than others, i.e. another multitude is contained as a part in one multitude (or on the contrary one multitude occurs in another as a mere part).This also is a claim which sounds to many paradoxical." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"If they [mathematicians] find a quantity greater than any finite number of the assumed units, they call it infinitely great; if they find one so small that its every finite multiple is smaller than the unit, they call it infinitely small; nor do they recognise any other kind of infinitude than these two, together with the quantities derived from them as being infinite to a higher order of greatness or smallness, and thus based after all on the same idea." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"[...] from that circumstance alone we are not allowed to conclude that both sets, if they are infinite, are equal to each other with respect to the multiplicity of their parts (that is, if we abstract from all differences between them); [...] Equality of those multiplicities can only be inferred when some other reason is added, for instance that both sets have absolutely equal grounds of determination, i.e., that their mode of formation is absolutely equal." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"Space and time - which again do not belong to the domain of the actual, though they can be determinations of the actual - form a very important category of infinitely great quantities." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"The existence of infinite sets, at least with non-actual members, is something which I now regard as sufficiently proved and defended; as also, that the set of all absolute truths is an infinite set." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"The very word in finite shows that we put the infinite into contrast with the merely finite. Again, the derivation of the former name from the latter betrays the additional fact that we consider the idea of the infinite to arise from the idea of the finite by, and only by, the adjunction of a new element; for such in fact is the abstract idea of negation." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"Therefore both multitudes have one and the same plurality, as one can also say, equal plurality. Obviously this conclusion becomes void as soon as the multitude of things in A is an infinite multitude, for now not only do we never reach, by counting, the last thing in A, but rather, by virtue of the definition of an infinite multitude, in itself there is no last thing in A, i.e. however many have already been designated, there are always others to designate." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"By a 'Satz an sich' [senternce in itself] I mean any statement whatever to the effect that something is or is not, irrespective of whether the statement be true or false, irrespective of whether any person ever formulated it in words, and even irrespective of whether it ever entered into any mind as a thought."  (Bernard Bolzano)

"My special pleasure in mathematics rested particularly on its purely speculative part." (Bernard Bolzano)

Set Theory II

"[...] from that circumstance alone we are not allowed to conclude that both sets, if they are infinite, are equal to each other with respect to the multiplicity of their parts (that is, if we abstract from all differences between them); [...] Equality of those multiplicities can only be inferred when some other reason is added, for instance that both sets have absolutely equal grounds of determination, i.e., that their mode of formation is absolutely equal." (Bernard Bolzano, "Paradoxien des Unedlichen", 1851)

"If two well-defined manifolds M and N can be coordinated with each other univocally and completely, element by element (which, if possible in one way, can always happen in many others), we shall employ in the sequel the expression, that those manifolds have the same power or, also, that they are equivalent." (Georg Cantor, "Ein Beitrag zur Mannigfaltigkeitslehre", 1878)

"I say that a manifold (a collection, a set) of elements that belong to any conceptual sphere is well-defined, when on the basis of its definition and as a consequence of the logical principle of excluded middle it must be regarded as internally determined, both whether an object pertaining to the same conceptual sphere belongs or not as an element to the manifold, and whether two objects belonging to the set are equal to each other or not, despite formal differences in the ways of determination." (Georg Cantor, "Ober unendliche, lineare Punktmannichfaltigkeiten", 1879)

"By a manifold or a set I understand in general every Many that can be thought of as One, i.e., every collection of determinate elements which can be bound up into a whole through a law, and with this I believe to define something that is akin to the Platonic εἷδος [form] or ἷδεα [idea]." (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883) 

"If we now notice that all of the numbers previously obtained and their next successors fulfill a certain condition, [that the set of their predecessors is denumerable,] then this condition offers itself, if it is imposed as a requirement on all numbers to be formed next, as a new third principle [...] which I shall call principle of restriction or limitation and which, as I shall show, yields the result that the second number-class (II) defined with its assistance not only has a higher power than [the first number-class] (I), but precisely the next higher, that is, the second power." (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883) 

"In calling arithmetic (algebra, analysis) just a part of logic, I declare already that I take the number-concept to be completely independent of the ideas or intuitions of space and time, that I see it as an immediate product of the pure laws of thought." (Richard Dedekind, "Was sind und was sollen die Zahlen?", 1888)

"A set is formed by the grouping together of single objects into a whole. A set is a plurality thought of as a unit. If these or similar statements were set down as definitions, then it could be objected with good reason that they define idem per idemi or even obscurum per obscurius. However, we can consider them as expository, as references to a primitive concept, familiar to us all, whose resolution into more fundamental concepts would perhaps be neither competent nor necessary." (Felix Hausdorff, "Set Theory", 1962)

"The axiom of choice has many important consequences in set theory. It is used in the proof that every infinite set has a denumerable subset, and in the proof that every set has at least one well-ordering. From the latter, it follows that the power of every set is an aleph. Since any two alephs are comparable, so are any two transfinite powers of sets. The axiom of choice is also essential in the arithmetic of transfinite numbers." (R Bunn, "Developments in the Foundations of Mathematics, 1870-1910", 1980)

"A set is a Many that allows itself to be thought of as a One." (Georg Cantor)

"An infinite set is one that can be put into a one-to-one correspondence with a proper subset of itself." (Georg Cantor)

Set Theory I

"[a set is] an embodiment of the idea or concept which we conceive when we regard the arrangement of its parts as a matter of indifference." (Bernard Bolzano, 1847)

"Since the examination of consistency is a task that cannot be avoided, it appears necessary to axiomatize logic itself and to prove that number theory and set theory are only parts of logic. This method was prepared long ago (not least by Frege’s profound investigations); it has been most successfully explained by the acute mathematician and logician Russell. One could regard the completion of this magnificent Russellian enterprise of the axiomatization of logic as the crowning achievement of the work of axiomatization as a whole." (David Hilbert, "Axiomatisches Denken" ["Axiomatic Thinking"], [address] 1917)

"It seems clear that [set theory] violates against the essence of the continuum, which, by its very nature, cannot at all be battered into a single set of elements. Not the relationship of an element to a set, but of a part to a whole ought to be taken as a basis for the analysis of a continuum." (Hermann Weyl, "Reimanns geometrische Ideen, ihre Auswirkungen und ihre Verknüpfung mit der Gruppentheorie", 1925)

"To say that mathematics in general has been reduced to logic hints at some new firming up of mathematics at its foundations. This is misleading. Set theory is less settled and more conjectural than the classical mathematical superstructure than can be founded upon it." (Willard van Orman Quine, "Elementary Logic", 1941)

"The emphasis on mathematical methods seems to be shifted more towards combinatorics and set theory - and away from the algorithm of differential equations which dominates mathematical physics." (John von Neumann & Oskar Morgenstern, "Theory of Games and Economic Behavior", 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)

"Categorical algebra has developed in recent years as an effective method of organizing parts of mathematics. Typically, this sort of organization uses notions such as that of the category G of all groups. [...] This raises the problem of finding some axiomatization of set theory - or of some foundational discipline like set theory - which will be adequate and appropriate to realizing this intent. This problem may turn out to have revolutionary implications vis-`a-vis the accepted views of the role of set theory." (Saunders Mac Lane, "Categorical algebra and set-theoretic foundations", 1967)

"In set theory, perhaps more than in any other branch of mathematics, it is vital to set up a collection of symbolic abbreviations for various logical concepts. Because the basic assumptions of set theory are absolutely minimal, all but the most trivial assertions about sets tend to be logically complex, and a good system of abbreviations helps to make otherwise complex statements."  (Keith Devlin, "Sets, Functions, and Logic: An Introduction to Abstract Mathematics", 1979)

"Set theory is peculiarly important [...] because mathematics can be exhibited as involving nothing but set-theoretical propositions about set-theoretical entities." (David M Armstrong, "A Combinatorial Theory of Possibility", 1989)

"At the basis of the distance concept lies, for example, the concept of convergent point sequence and their defined limits, and one can, by choosing these ideas as those fundamental to point set theory, eliminate the notions of distance." (Felix Hausdorff)

On Symbols (1970-1979)

"Any mathematician endowed with a modicum of intellectual honesty will recognise then that in each of his proofs he is capable of giving a meaning to the symbols he uses." (René Thom, "Modern mathematics, does it exist?", 1972)

"Mathematics in itself, as I say, is independent of experience. It begins with the free choice of symbols, to which are freely assigned properties, and it then proceeds to deduce the necessary rational implications of those properties." (Herbert Dingle, "Science at the Crossroads", 1972)

"A physical theory is assigned a literal and objective interpretation by assigning every one of its referential primitive symbols a physical object - entity, property, relation, or event - rather than a mental picture or a human operation." (Mario Bunge, "Philosophy of Physics", 1973)

"The mind reproduces itself by transmitting its symbols to other intermediaries, human and mechanical, than the particular brain that first assembled them." (Lewis Mumford, "Interpretations and Forecasts 1922-1972", 1973)

"Whether or not a given conceptual model or representation of a physical system happens to be picturable, is irrelevant to the semantics of the theory to which it eventually becomes attached. Picturability is a fortunate psychological occurrence, not a scientific necessity. Few of the models that pass for visual representations are picturable anyhow. For one thing, the model may be and usually is constituted by imperceptible items such as unextended particles and invisible fields. True, a model can be given a graphic representation - but so can any idea as long as symbolic or conventional diagrams are allowed. Diagrams, whether representational or symbolic, are meaningless unless attached to some body of theory. On the other hand theories are in no need of diagrams save for psychological purposes. Let us then keep theoretical models apart from visual analogues."  (Mario Bunge, "Philosophy of Physics", 1973)

"The unconscious reveals its meaning imaginatively, through symbols and images, and it speaks [...] a basically mythological language." (Morton Kelsey, "Myth, History & Faith", 1974)

"Models are not intended to either reflect or construct a single objective reality. Rather, their purpose is to simulate some aspect of a possible reality. In NLP, for instance, it is not important whether or not a model is 'true' , but rather that it is 'useful' . In fact, all models can be perceived as symbolic or metaphoric, as opposed to reflective of reality. Whether the description being used is metaphorical or literal, the usefulness of a model depends on the degree to which it allows us to move effectively to the next step in the sequence of transformations connecting deeper structures and surface structures. Instead of 'constructing' reality, models establish a set of functions that serve as a tool or a bridge between deep structures and surface structures. It is this bridge that forms our 'understanding' of reality and allows us to generate new experiences and expressions of reality." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)

"Symbols, formulae and proofs have another hypnotic effect. Because they are not immediately understood, they, like certain jokes, are suspected of holding in some sort of magic embrace the secret of the universe, or at least some of its more hidden parts." (Scott Buchanan, "Poetry and Mathematics", 1975)

"Symbol and myth do bring into awareness infantile, archaic dreads and similar primitive psychic content. This is their regressive aspect. But they also bring out new meaning, new forms, and disclose a reality that was literally not present before, a reality that is not merely subjective but has a second pole which is outside ourselves. This is the progressive side of symbol and myth." (Rollo May, "The Courage to Create", 1975) 

"The most pervasive paradox of the human condition which we see is that the processes which allow us to survive, grow, change, and experience joy are the same processes which allow us to maintain an impoverished model of the world - our ability to manipulate symbols, that is, to create models. So the processes which allow us to accomplish the most extraordinary and unique human activities are the same processes which block our further growth if we commit the error of mistaking the model of the world for reality." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)

"Whenever the Eastern mystics express their knowledge in words - be it with the help of myths, symbols, poetic images or paradoxical statements-they are well aware of the limitations imposed by language and 'linear' thinking. Modern physics has come to take exactly the same attitude with regard to its verbal models and theories. They, too, are only approximate and necessarily inaccurate. They are the counterparts of the Eastern myths, symbols and poetic images, and it is at this level that I shall draw the parallels. The same idea about matter is conveyed, for example, to the Hindu by the cosmic dance of the god Shiva as to the physicist by certain aspects of quantum field theory. Both the dancing god and the physical theory are creations of the mind: models to describe their authors' intuition of reality." (Fritjof Capra, "The Tao of Physics: An Exploration of the Parallels Between Modern Physics and Eastern Mysticism", 1975)

"The chief difficulty of modern theoretical physics resides not in the fact that it expresses itself almost exclusively in mathematical symbols, but in the psychological difficulty of supposing that complete nonsense can be seriously promulgated and transmitted by persons who have sufficient intelligence of some kind to perform operations in differential and integral calculus […]" (Celia Green, "The Decline and Fall of Science", 1976)

"[…] all our symbols have the same purpose; words are merely the symbols we use most commonly. The function of words in human thought is to stand for things which are not present to the senses, and allow the mind to manipulate them - things, concepts, ideas, everything that does not have a physical reality in front of us now." (Jacob Bronowski, "The Imaginative Mind in Art", 1978) 

"[Human consciousness] depends wholly on our seeing the outside world in such categories. And the problems of consciousness arise from putting reconstitution beside internalization, from our also being able to see ourselves as if we were objects in the outside world. That is in the very nature of language; it is impossible to have a symbolic system without it." (Jacob Bronowski, "The origins of knowledge and imagination", 1978)

"In set theory, perhaps more than in any other branch of mathematics, it is vital to set up a collection of symbolic abbreviations for various logical concepts. Because the basic assumptions of set theory are absolutely minimal, all but the most trivial assertions about sets tend to be logically complex, and a good system of abbreviations helps to make otherwise complex statements."  (Keith Devlin, "Sets, Functions, and Logic: An Introduction to Abstract Mathematics", 1979)

On Symbols (1980-1989)

"[…] mathematics is not just a symbolism, a set of conventions for the use of special, formal vocabularies, but is intimately connected with the structure of rational thought, with reasoning practices. [...] mathematics is not just a language, and of refusing the foundationalist move of trying to reduce mathematics to logic, instead seeing mathematics as providing rational frameworks for science, is to set science against a background of rational structures and rational methods which itself has a built-in dynamics. The rational framework of science is itself historically conditioned, for it changes with developments in mathematics." (Mary Tiles, "Bachelard: Science and Objectivity", 1984)

"Scientific laws give algorithms, or procedures, for determining how systems behave. The computer program is a medium in which the algorithms can be expressed and applied. Physical objects and mathematical structures can be represented as numbers and symbols in a computer, and a program can be written to manipulate them according to the algorithms. When the computer program is executed, it causes the numbers and symbols to be modified in the way specified by the scientific laws. It thereby allows the consequences of the laws to be deduced." (Stephen Wolfram, "Computer Software in Science and Mathematics", 1984)

"A computer is an interpreted automatic formal system - that is to say, a symbol-manipulating machine." (John Haugeland, "Artificial intelligence: The very idea", 1985)

"We who are heirs to three recent centuries of scientific development can hardly imagine a state of mind in which many mathematical objects were regarded as symbols of spiritual truths or episodes in sacred history. Yet, unless we make this effort of imagination, a fraction of the history of mathematics is incomprehensible.” (Philip J Davis & Rueben Hersh, “The Mathematical Experience”, 1985)

"Artificial intelligence is based on the assumption that the mind can be described as some kind of formal system manipulating symbols that stand for things in the world. Thus it doesn't matter what the brain is made of, or what it uses for tokens in the great game of thinking. Using an equivalent set of tokens and rules, we can do thinking with a digital computer, just as we can play chess using cups, salt and pepper shakers, knives, forks, and spoons. Using the right software, one system (the mind) can be mapped onto the other (the computer)." (George Johnson, "Machinery of the Mind: Inside the New Science of Artificial Intelligence", 1986)

"Meaning does not reside in the mathematical symbols. It resides in the cloud of thought enveloping these symbols. It is conveyed in words; these assign meaning to the symbols." (Marvin Chester, "Primer of Quantum Mechanics", 1987)

"[…] the chain of possible combinations of the encounter can be studied as such, as an order which subsists in its rigor, independently of all subjectivity. Through cybernetics, the symbol is embodied in the apparatus - with which it is not to be confused, the apparatus being just its support. And it is embodied in it in a literally trans-subjective way." (Jacques Lacan, 1988)

"Western culture’s world-view appears to be dominated by material objects. […] One of the ways mathematics has gained its power is through the activity of objectivising the abstractions from reality. Through its symbols (letters, numerals, figures) mathematics has taught people how to deal with abstract entities, as if they were objects." (Alan J Bishop, "Mathematics education in its cultural context", Educational Studies in Mathematics 19, 1988)

"People who have a casual interest in mathematics may get the idea that a topologist is a mathematical playboy who spends his time making Möbius bands and other diverting topological models. If they were to open any recent textbook in topology, they would be surprised. They would find page after page of symbols, seldom relieved by a picture or diagram." (Martin Gardner, "Hexaflexagons and Other Mathematical Diversions", 1988)

On Symbols (1990-1999)

"When a person has learned a symbolic system well enough to use it, she has established a portable self-contained world within the mind." (Mihaly Csikszentmihalyi, "Flow", 1990)

"Mathematics […] is mired in a language of symbols foreign to most of us, [it] explores regions of the infinitesimally small and the infinitely large that elude words, much less understanding." (Robert Kanigel, "The Man Who Knew Infinity", 1991)

"Great mathematics seldom comes from idle speculation about abstract spaces and symbols. More often than not it is motivated by definite questions arising in the worlds of nature and humans." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"Mathematical modeling is about rules - the rules of reality. What distinguishes a mathematical model from, say, a poem, a song, a portrait or any other kind of ‘model’, is that the mathematical model is an image or picture of reality painted with logical symbols instead of with words, sounds or watercolors." (John Casti, "Reality Rules", 1992)

"Scientific claims or statements are inexact and provisional. They depend on dozens of simplifying assumptions and on a particular choice of words and symbols and on 'all other things being equal'." (Bart Kosko, "Fuzzy Thinking: The new science of fuzzy logic", 1993)

"The insight at the root of artificial intelligence was that these 'bits' (manipulated by computers) could just as well stand as symbols for concepts that the machine would combine by the strict rules of logic or the looser associations of psychology." (Daniel Crevier, "AI: The tumultuous history of the search for artificial intelligence", 1993)

"Above all, words must be recognized as symbolic pointers to truth, not objective containers of truth." (John S Spong, "Resurrection: Myth or Reality?", 1994) 

"[...] images are probably the main content of our thoughts, regardless of the sensory modality in which they are generated and regardless of whether they are about a thing or a process involving things; or about words or other symbols, in a given language, which correspond to a thing or process. Hidden behind those images, never or rarely knowable by us, there are indeed numerous processes that guide the generation and deployment of those images in space and time. Those processes utilize rules and strategies embodied in dispositional representations. They are essential for our thinking but are not a content of our thoughts.” (Antonio R Damasio, “Descartes' Error. Emotion, Reason, and the Human Brain”, 1994)

"Just as music comes alive in the performance of it, the same is true of mathematics. The symbols on the page have no more to do with mathematics than the notes on a page of music. They simply represent the experience." (Keith Devlin, "Mathematics: The Science of Patterns", 1994)

"Every phenomenon on earth is symbolic, and each symbol is an open gate through which the soul, if it is ready, can enter into the inner part of the world, where you and I and day and night are all one." (Hermann Hesse, "The Fairy Tales of Hermann Hesse", 1995)

“Mathematics is not the study of an ideal, preexisting nontemporal reality. Neither is it a chess-like game with made-up symbols and formulas. Rather, it is the part of human studies which is capable of achieving a science-like consensus, capable of establishing reproducible results. The existence of the subject called mathematics is a fact, not a question. This fact means no more and no less than the existence of modes of reasoning and argument about ideas which are compelling an conclusive, ‘noncontroversial when once understood’." (Philip J Davis & Rueben Hersh, “The Mathematical Experience”, 1995)

"Schematic diagrams are more abstract than pictorial drawings, showing symbolic elements and their interconnection to make clear the configuration and/or operation of a system." (Ernest O Doebelin, "Engineering experimentation: planning, execution, reporting", 1995)

"The logic of the emotional mind is associative; it takes elements that symbolize a reality, or trigger a memory of it, to be the same as that reality. That is why similes, metaphors and images speak directly to the emotional mind." (Daniel Goleman, "Emotional Intelligence", 1996)

“In many ways, the mathematical quest to understand infinity parallels mystical attempts to understand God. Both religions and mathematics attempt to express the relationships between humans, the universe, and infinity. Both have arcane symbols and rituals, and impenetrable language. Both exercise the deep recesses of our mind and stimulate our imagination. Mathematicians, like priests, seek ‘ideal’, immutable, nonmaterial truths and then often try to apply theses truth in the real world.” (Clifford A Pickover, "The Loom of God: Mathematical Tapestries at the Edge of Time", 1997)

"Reality contains not only evidence, but also the means (such as our minds, and our artefacts) of understanding it. There are mathematical symbols in physical reality. The fact that it is we who put them there does not make them any less physical." (David Deutsch, "The Fabric of Reality", 1997)

"Meaning is conferred not by a one-to-one correspondence of a symbol with some external concept or object, but by the relationships between the structural components of the system itself." (Paul Cilliers, "Complexity and Postmodernism: Understanding Complex Systems", 1998)

"A formal system consists of a number of tokens or symbols, like pieces in a game. These symbols can be combined into patterns by means of a set of rules which defines what is or is not permissible (e.g. the rules of chess). These rules are strictly formal, i.e. they conform to a precise logic. The configuration of the symbols at any specific moment constitutes a ‘state’ of the system. A specific state will activate the applicable rules which then transform the system from one state to another. If the set of rules governing the behaviour of the system are exact and complete, one could test whether various possible states of the system are or are not permissible." (Paul Cilliers, "Complexity and Postmodernism: Understanding Complex Systems", 1998)

“Cultural archetypes are the unconscious models that help us make sense of the world: they are the myths, narratives, images, symbols, and files into which we organize the data of our life experience” (Clotaire Rapaille, “Cultural Imprints”, Executive Excellence Vol. 16 (10), 1999)

"In broad terms, a mental model is to be understood as a dynamic symbolic representation of external objects or events on the part 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)

On Symbols (2000-2009)

"The motion of the mind is conveyed along a cloud of meaning. There is this paradox that we get to meaning only when we strip the meaning from symbols." (David Berlinski, "The Advent of the Algorithm: The Idea that Rules the World", 2000)

"A symbol is a mental representation regarding the internal reality referring to its object by a convention and produced by the conscious interpretation of a sign. In contrast to signals, symbols may be used every time if the receiver has the corresponding representation. Symbols also relate to feelings and thus give access not only to information but also to the communicator’s motivational and emotional state. The use of symbols makes it possible for the organism using it to evoke in the receiver the same response it evokes in himself. To communicate with symbols is to use a language." (Lars Skyttner, "General Systems Theory: Ideas and Applications", 2001)

"In the definition of meaning, it is assumed that both the source and receiver have previously coded (and stored) signals of the same or similar referents, such that the messages may have meaning and relate to behaviour. That is, the used symbols must have the same signification for both sender and receiver. If not, the receiver will create a different mental picture than intended by the transmitter. Meaning is generated by individuals in a process of social interaction with a more or less common environment. It is a relation subsisting within a field of experience and appears as an emergent property of a symbolic representation when used in culturally accepted interaction. The relation between the symbolic representation and its meaning is random. Of this, however, the mathematical theory has nothing to say. If human links in the chain of communication are missing, of course no questions of meaning will arise." (Lars Skyttner, "General Systems Theory: Ideas and Applications", 2001)

"A person thinking in the nonverbal mode is actually thinking with the meaning of the language in the form of mental pictures of the concepts and ideas it contains. Nonverbal thought doesn't require literacy. An illiterate person can communicate without knowing what the symbols look like. [...] Literacy, then, is established as the person learns how the symbols look and becomes able to recognize them as representing certain things or concepts." (Ronald D Davis & Eldon M Braun, "The Gift of Learning", 2003)

"Science does not speak of the world in the language of words alone, and in many cases it simply cannot do so. The natural language of science is a synergistic integration of words, diagrams, pictures, graphs, maps, equations, tables, charts, and other forms of visual and mathematical expression. […] [Science thus consists of] the languages of visual representation, the languages of mathematical symbolism, and the languages of experimental operations." (Jay Lemke, "Teaching all the languages of science: Words, symbols, images and actions", 2003)

"I often told the fanatics of realism that there is no such thing as realism in art: it only exists in the mind of the observer. Art is a symbol, a thing conjuring up reality in our mental image. That is why I don't see any contradiction between abstract and figurative art either." (Antoni Tàpies, "Tàpies, Werke auf Papier 1943 – 2003", 2004)

"A symbol is an object, act, or event that conveys meaning to others. Symbols can be considered a rich, non-verbal language that vibrantly conveys the organization’s important values concerning how people relate to one another and interact with the environment" (Richard L Daft & Dorothy Marcic, "Understanding Management" 5th Ed., 2006)

"But because of the way in which depictions represent, there is a correspondence between parts and spatial relations of the representation and those of the object; this structural mapping, which confers a type of resemblance, underlies the way images convey specific content. In this respect images are like pictures. Unlike words and symbols, depictions are not arbitrarily paired with what they represent." (Stephen Kosslyn et al," The Case for Mental Imagery", 2006)

"Imagination has the creative task of making symbols, joining things together in such a way that they throw new light on each other and on everything around them. The imagination is a discovering faculty, a faculty for seeing relationships, for seeing meanings that are special and even quite new." (Thomas Merton, "Angelic Mistakes: The Art of Thomas Merton", 2006)

"[...] the scientific models of concrete things are symbolic rather than iconic: they are systems of propositions, not pictures. Besides, such models are seldom if ever completely accurate, if only because they involve more or less brutal simplifications, such as pretending that a metallic surface is smooth, a crystal has no impurities, a biopopulation has a single predator, or a market is in equilibrium.  These are all fictions. However, they are stylizations rather than wild fantasies. Hence, introducing and using them to account for real existents does not commit us to fictionism, just as defending the role of experience need not make us empiricists, nor is admitting the role of intuition enough to qualify as intuitionist." (Mario Bunge, "Chasing Reality: Strife over Realism", 2006)

"But notice, a subatomic particle is itself a holon [hole/parts]. And so is a cell. And so is a symbol, and an image, and a concept. What all of those entities are, before they are anything else, is a holon. So the world is not composed of atoms or symbols or cells or concepts. It is composed of holons." (Ken Wilber, "A Brief History of Everything", 2007)

"Language use is a curious behavior, but once the transition to language is made, literature is a likely consequence, since it is linked to the dynamic of the linguistic symbol through the functioning of the imagination." (Russell Berman, "Fiction Sets You Free: Literature, Liberty and Western Culture", 2007)

"Images and pictures […] have played a key role in shaping our scientific picture of the world. […] Carefully constructed families of pictures can act as a calculus all their own. Like any successful systems of symbols, with an appropriate grammar they enlarge the number of things that we can do without consciously thinking." (John D Barrow, "Cosmic Imagery: Key Images in the History of Science", 2008)

"How are we to explain the contrast between the matter-of-fact way in which √-1 and other imaginary numbers are accepted today and the great difficulty they posed for learned mathematicians when they first appeared on the scene? One possibility is that mathematical intuitions have evolved over the centuries and people are generally more willing to see mathematics as a matter of manipulating symbols according to rules and are less insistent on interpreting all symbols as representative of one or another aspect of physical reality. Another, less self-congratulatory possibility is that most of us are content to follow the computational rules we are taught and do not give a lot of thought to rationales." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2009)

"Mathematicians are sometimes described as living in an ideal world of beauty and harmony. Instead, our world is torn apart by inconsistencies, plagued by non sequiturs and, worst of all, made desolate and empty by missing links between words, and between symbols and their referents; we spend our lives patching and repairing it. Only when the last crack disappears are we rewarded by brief moments of harmony and joy." (Alexandre V Borovik, "Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice", 2009)

"Mathematical ideas like number can only be really 'seen' with the 'eyes of the mind' because that is how one 'sees' ideas. Think of a sheet of music which is important and useful but it is nowhere near as interesting, beautiful or powerful as the music it represents. One can appreciate music without reading the sheet of music. Similarly, mathematical notation and symbols on a blackboard are just like the sheet of music; they are important and useful but they are nowhere near as interesting, beautiful or powerful as the actual mathematics (ideas) they represent." (Fiacre O Cairbre, "The Importance of Being Beautiful in Mathematics", IMTA Newsletter 109, 2009)

On Tektology

"Tectology, or the theory of structure in organisms, is the comprehensive science of individuality among living natural bodies, which usually represent an aggregate of individuals of various orders.  The task of organic tectology is therefore to identify and explain organic individuality, i.e. to identify the precise natural laws according to which organic matter individualises itself, and according to which most organisms construct a unified form-complex composed of individuals of various orders." (Ernst  Häckel, "Generelle Morphologie der Organismen" ["General Morphology of Organisms"], 1866)

"It should therewith be remembered that as mathematics studies neutral complexes, mathematical thinking is an organizational process and hence its methods, as well as the methods of all other sciences and those of any practice, fall within the province of a general tektology. Tektology is a unique science which must not only work out its own methods by itself but must study them as well; therefore it is the completion of the cycle of sciences." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"Mathematics abstracts from all the particular properties of the elements hidden behind its schemata. This is achieved by mathematics with the help of indifferent symbols, like numbers or letters. Tektology must do likewise. Its generalizations should abstract from the concreteness of elements whose organizational relationships they express, and conceal this concreteness behind indifferent symbols." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"Tektology must discover what modes of organization are observed in nature and human activities; then generalize and systemize these modes; further it should explain them, that is, elaborate abstract schemes of their tendencies and regularities; finally, based on these schemes it must determine the directions of organizational modes development and elucidate their role in the economy of world processes. This general plan is similar to the plan of any other science but the object studied differs essentially. Tektology deals with the organizational experience not of some particular branch but with that of all of them in the aggregate; to put it in other words, tektology embraces the material of all the other sciences, as well as of all the vital practices from which those sciences arose, but considers this material only in respect of methods, i.e. everywhere it takes an interest in the mode of the organization of this material."  (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"The methods of tektology, as is seen, combine the abstract symbolism of mathematics and the experimental character Of the natural sciences. Furthermore, the very formulation of its problems, the very treatment of organizedness by tektology, as has been elucidated, should stick to the social historical viewpoint. And whatever the subject matter, or the content, of tektology , it embraces the whole world of experience. So tektology is really a universal science by its methods and its content."  (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"[…] there is a special relationship, a profound affinity between mathematics and tektology. Mathematical laws do not refer to a particular area of natural phenomena, as the laws of the other, special, sciences do, but to each and all phenomena, considered merely in their quantitative aspect; mathematics is in its own way universal, like tektology."  (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"Two divisions are distinguished in all natural sciences - 'statics' which deals with forms in equilibrium, and 'dynamics' which deals with the same forms, as well as their motion, in the process of change. […] Statics always evolves earlier than dynamics, the former being then reconstructed under the influence of the latter. The relationship between mathematics and tektology is seen to be similar: one represents the standpoint of organizational statics and the other - that of organizational dynamics. The latter standpoint is the more general, for equilibrium is only a particular case of motion, and in essence, is just an ideal case resulting from changes which are completely equal but quite opposite in direction." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"We shall call this universal organizational science the 'Tektology'. The literal translation of this word from the Greek is 'the theory of construction'. 'Construction' is the most generaI and suitable synonym for the modern concept of 'organization'. [...] The aim of tektology is to systematize organizational experience; this science is clearly empirical and should draw its conclusions by way of induction." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"For tektology the unity of experience is not 'discovered', but actively created by organizational means: ‘philosophers wanted to explain the world, but the main point is it change it’ said the greater precursor of organizational science, Karl Marx. The explanation of organizational forms and methods by tektology is directed not to a contemplation of their unity, but to a practical mastery over them." (Alexander Bogdanov, "Tektology: The Universal Organizational Science", 1922)

"Tektology must clarify the modes of organization that are perceived to exist in nature and human activity; then it must generalize and systematize these modes; further it must explain them, that is, propose abstract schemes of their tendencies and laws; finally, based on these schemes, determine the direction of organizational methods and their role in the universal process. This general plan is similar to the plan of any natural science; but the objective of tektology is basically different. Tektology deals with organizational experiences not of this or that specialized field, but of all these fields together. In other words, tektology embraces the subject matter of all the other sciences and of all the human experience giving rise to these sciences, but only from the aspect of method, that is, it is interested only in the modes of organization of this subject matter." (Alexander Bogdanov, "Tektology: The Universal Organizational Science", 1922)

"The strength of an organization lies in precise coordination of its parts, in strict correspondence of various mutually connected functions. This coordination is maintained through constant growth in tektological variety, but not without bounds […] there comes a moment when the parts of the whole become too differentiated in their organization and their resistance to the surrounding environment weakens. This leads sooner or later to disorganization." (Alexander Bogdanov, "Tektology: The Universal Organizational Science", 1922)

"Tektology was the first attempt in the history of science to arrive at a systematic formulation of the principles of organization operating in living and nonliving systems." (Fritjof Capra, "The Web of Life", 1996)

"Tektology is concerned only with activities, but activities are characterized by the fact that they produce changes. From this point of view it is out of the question to think about a simple and pure 'preservation' of forms, one that would constitute a real absence of changes. Preservation is always only a result of immediately equilibrating each of the appearing changes by another opposing change; it Is a dynamic equilibrium of changes."(Alexander Bogdanov) 

Alexander A Bogdanov - Collected Quotes

"All scientific experience persuades us that the possibility and probability of the successful resolution of problems increases, when they are stated in general form. […] Generalization is at the same time simplification. The problem is reduced to the minimum number of the most recurrent elements; numerous complicating points are extracted and discarded; certainly, the task is thus facilitated; and, constructed in this form, transition to the more specific task is carried out by the reverse inclusion of discarded particular data." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"For thought raised on specialization the most potent objection to the possibility of a universal organizational science is precisely its universality. Is it ever possible that the same laws be applicable to the combination of astronomic worlds and those of biological cells, of living people and the waves of the ether, of scientific ideas and quanta of energy? .. Mathematics provide a resolute and irrefutable answer: yes, it is undoubtedly possible, for such is indeed the case. Two and two homogenous separate elements amount to four such elements, be they astronomic systems or mental images, electrons or workers; numerical structures are indifferent to any element, there is no place here for specificity." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"It should therewith be remembered that as mathematics studies neutral complexes, mathematical thinking is an organizational process and hence its methods, as well as the methods of all other sciences and those of any practice, fall within the province of a general tektology. Tektology is a unique science which must not only work out its own methods by itself but must study them as well; therefore it is the completion of the cycle of sciences." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"Mathematics abstracts from all the particular properties of the elements hidden behind its schemata. This is achieved by mathematics with the help of indifferent symbols, like numbers or letters. Tektology must do likewise. Its generalizations should abstract from the concreteness of elements whose organizational relationships they express, and conceal this concreteness behind indifferent symbols." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"Tektology must discover what modes of organization are observed in nature and human activities; then generalize and systemize these modes; further it should explain them, that is, elaborate abstract schemes of their tendencies and regularities; finally, based on these schemes it must determine the directions of organizational modes development and elucidate their role in the economy of world processes. This general plan is similar to the plan of any other science but the object studied differs essentially. Tektology deals with the organizational experience not of some particular branch but with that of all of them in the aggregate; to put it in other words, tektology embraces the material of all the other sciences, as well as of all the vital practices from which those sciences arose, but considers this material only in respect of methods, i.e. everywhere it takes an interest in the mode of the organization of this material."  (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"Theoretical philosophy aimed to discover the unity of experience, namely, in the form of some universal explanation. It strived to yield a world picture, one which is harmoniously integral and completely understandable." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"The symbols organized by knowledge, or concepts, themselves belong to social nature as its ideological elements. Therefore, by operating upon them, knowledge is able to expand its organizing function much more broadly than labour in its technological operation of real things; and as we have already seen that many things, which are not organized in practice, can be organized by knowledge, i.e. in symbols: where the ingression of things is absent, the ingression of their concepts is still possible. Here ingression becomes universal, all-embracing." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"[…] there is a special relationship, a profound affinity between mathematics and tektology. Mathematical laws do not refer to a particular area of natural phenomena, as the laws of the other, special, sciences do, but to each and all phenomena, considered merely in their quantitative aspect; mathematics is in its own way universal, like tektology."  (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"Two divisions are distinguished in all natural sciences - 'statics' which deals with forms in equilibrium, and 'dynamics' which deals with the same forms, as well as their motion, in the process of change. […] Statics always evolves earlier than dynamics, the former being then reconstructed under the influence of the latter. The relationship between mathematics and tektology is seen to be similar: one represents the standpoint of organizational statics and the other - that of organizational dynamics. The latter standpoint is the more general, for equilibrium is only a particular case of motion, and in essence, is just an ideal case resulting from changes which are completely equal but quite opposite in direction." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"We shall call this universal organizational science the 'Tektology'. The literal translation of this word from the Greek is 'the theory of construction'. 'Construction' is the most generaI and suitable synonym for the modern concept of 'organization'. [...] The aim of tektology is to systematize organizational experience; this science is clearly empirical and should draw its conclusions by way of induction." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"Would it be possible for a 'mental image' , perception or idea, to correspond to a 'physical object', if the parts of the former were not combined in the same order as the parts of the latter? […] The more fully the similarity of two mental images is 'recognized' , i.e., the more elements of both images are brought to identity in the consciousness, the greater the extent they are associated 'by similarity'." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"For tektology the unity of experience is not 'discovered', but actively created by organizational means: ‘philosophers wanted to explain the world, but the main point is it change it’ said the greater precursor of organizational science, Karl Marx. The explanation of organizational forms and methods by tektology is directed not to a contemplation of their unity, but to a practical mastery over them." (Alexander Bogdanov, "Tektology: The Universal Organizational Science", 1922)

"Tektology must clarify the modes of organization that are perceived to exist in nature and human activity; then it must generalize and systematize these modes; further it must explain them, that is, propose abstract schemes of their tendencies and laws; finally, based on these schemes, determine the direction of organizational methods and their role in the universal process. This general plan is similar to the plan of any natural science; but the objective of tektology is basically different. Tektology deals with organizational experiences not of this or that specialized field, but of all these fields together. In other words, tektology embraces the subject matter of all the other sciences and of all the human experience giving rise to these sciences, but only from the aspect of method, that is, it is interested only in the modes of organization of this subject matter." (Alexander Bogdanov, "Tektology: The Universal Organizational Science", 1922)

"The strength of an organization lies in precise coordination of its parts, in strict correspondence of various mutually connected functions. This coordination is maintained through constant growth in tektological variety, but not without bounds […] there comes a moment when the parts of the whole become too differentiated in their organization and their resistance to the surrounding environment weakens. This leads sooner or later to disorganization." (Alexander Bogdanov, "Tektology: The Universal Organizational Science", 1922)

"Tektology is concerned only with activities, but activities are characterized by the fact that they produce changes. From this point of view it is out of the question to think about a simple and pure 'preservation' of forms, one that would constitute a real absence of changes. Preservation is always only a result of immediately equilibrating each of the appearing changes by another opposing change; it Is a dynamic equilibrium of changes."(Alexander Bogdanov) 

Ernst Häckel - Collected Quotes

 "Tectology, or the theory of structure in organisms, is the comprehensive science of individuality among living natural bodies, which usually represent an aggregate of individuals of various orders.  The task of organic tectology is therefore to identify and explain organic individuality, i.e. to identify the precise natural laws according to which organic matter individualises itself, and according to which most organisms construct a unified form-complex composed of individuals of various orders." (Ernst  Häckel, "Generelle Morphologie der Organismen" ["General Morphology of Organisms"], 1866)

"[…] without the theory of evolution all the big general series of phenomena of organic nature remain completely incomprehensible and inexplicable riddles, while by means of this theory they can be explained simply and consistently. This holds especially true for two complexes of biological phenomena which we now in conclusion wish to single out in a few words. These form the subject of two special branches of physiology which so far have been largely neglected, namely, the ecology and chorology of organisms." (Ernst Häckel, "Generelle Morphologie der Organismen", 1866)

"By ecology we mean the body of knowledge concerning the economy of nature - the investigation of the total relations of the animal both to its inorganic and to its organic environment; including, above all, its friendly and inimical relations with those animals and plants with which it comes directly or indirectly into contact - in a word, ecology is the study of all those complex interrelations referred to by Darwin as the conditions of the struggle for existence." (Ernst Häckel, [lecture] 1869)

"[employment of] exact or mathematical methods […] unfortunately is impossible in most branches of science (particularly in biology), because the empirical foundations are much too imperfect and the present problems much too complicated. Mathematical treatment of these does more harm than good because it gives a deceptive semblance of certainty which is not actually attainable. Part of physiology also involves problems which are difficult or impossible to resolve exactly, and these include the chorology and ecology of plankton." (Ernst Häckel, “Plantonic studies”, 1891)

"Nothing is constant but change! All existence is a perpetual flux of 'being and becoming!' That is the broad lesson of the evolution of the world." (Ernst Häckel, "Wonders of Life", 1904)

"All natural phenomena, without exception, from the motions of the celestial bodies to the growth of plants and the consciousness of men […] are ultimately to be reduced to atomic mechanics." (Ernst Häckel)

"Ecology is] the science of relations between organisms and their environment." (Ernst Häckel)

"The great regard which mathematics enjoys as an exact science in all branches of knowledge is chiefly due to its formal accuracy, and to the possibility of expressing infallibly spatial and time quantities in number and mass." (Ernst Häckel)

"The highest triumph of the human mind, the true knowledge of the most general laws of nature, ought not to remain the private possession of a privileged class of learned men, but ought to become the common property of all mankind."  (Ernst Häckel)

James G Miller - Collected Quotes

"General systems theory is a series of related definitions, assumptions, and postulates about all levels of systems from atomic particles through atoms, molecules, crystals, viruses, cells, organs, individuals, small groups, societies, planets, solar systems, and galaxies. General behavior systems theory is a subcategory of such theory, dealing with living systems, extending roughly from viruses through societies. A significant fact about living things is that they are open systems, with important inputs and outputs. Laws which apply to them differ from those applying to relatively closed systems." (James G Miller, "General behavior systems theory and summary", Journal of Counseling Psychology 3 (2), 1956)

"In the most general mathematical sense, a space is a set of elements which conform to certain postulates." (James G Miller, "Living Systems: Basic Concepts", 1969) 

"My analysis of living systems uses concepts of thermodynamics, information theory, cybernetics, and systems engineering, as well as the classical concepts appropriate to each level. The purpose is to produce a description of living structure and process in terms of input and output, flows through systems, steady states, and feedbacks, which will clarify and unify the facts of life." (James G Miller, "Living Systems: Basic Concepts", 1969)

"All nature is a continuum. The endless complexity of life is organized into patterns which repeat themselves - theme and variations - at each level of system. These similarities and differences are proper concerns for science. From the ceaseless streaming of protoplasm to the many-vectored activities of supranational systems, there are continuous flows through living systems as they maintain their highly organized steady states." (James G Miller, "Living Systems", 1978)

"My presentation of a general theory of living systems will employ two sorts of spaces in which they may exist, physical or geographical space and conceptual or abstract space [...] The characteristics and constraints of physical space affect the action of all concrete systems, living and nonliving [...] Physical space is a common space because it is the only space in which all concrete systems, living and nonliving, exist (though some may exist in other spaces simultaneously). Physical space is shared by all scientific observers, and all scientific data must be collected in it. This is equally true for natural science and behavioral science." (James G Miller, "Living Systems", 1978)

"The most general form of systems theory is a set of logical or mathematical statements about all conceptual systems. A subset of this concerns all concrete systems. A subsubset concerns the very special and very important living systems, i. e., general living systems theory." (James G Miller, "Living systems", 1978) 

"The structure of a system is the arrangement of its subsystems and components in three-dimensional space at a given moment of time. This always changes over time. It may remain relatively fixed for a long period or it may change from moment to moment, depending upon the characteristics of the process in the system. This process halted at any given moment, as when motion is frozen by a high-speed photograph, reveals the three-dimensional spatial arrangement of the system's components as of that instant." (James G Miller, "Living systems", 1978) 

"The term system has a number of meanings. There are systems of numbers and of equations, systems of value and of thought, systems of law, solar systems, organic systems, management systems, command and control systems, electronic systems, even the Union Pacific Railroad system. The meanings of 'system' are often confused. The most general, however, is: A system is a set of interacting units with relationships among them. The word 'set' implies that the units have some common properties. These common properties are essential if the units are to interact or have relationships. The state of each unit is constrained by, conditioned by, or dependent on the state of other units. The units are coupled. Moreover, there is at least one measure of the sum of its units which is larger than the sum of that measure of its units." (James G Miller, "Living systems", 1978) 

"It [Living Systems Theory (LST)] involves observing and measuring important relationships between inputs and outputs of the total system and identifying the structures that perform each of the sub‐system processes. […] The flows of relevant matter, energy, and information through the system and the adjustment processes of subsystems and the total system are also examined. The status and function of the system are analyzed and compared with what is average or normal for that type of system. If the system is experiencing a disturbance in some steady state, an effort is made to discover the source of the strain and correct it." (James G Miller & Jessie L Miller, "Applications of living systems theory", Systemic Practice and Action Research 8, 1995)

"The universe of all things that exist may be understood as a universe of systems where a system is defined as any set of related and interacting elements. This concept is primitive and powerful and has been used increasingly over the last half-century to organize knowledge in virtually all domains of interest to investigators. As human inventions and social interactions grow more complex, general conceptual frameworks that integrate knowledge among different disciplines studying those emerging systems grow more important. Living systems theory (LST) instructs integrative research among biological and social sciences and related academic disciplines." (G A Swanson & James G Miller, "Living Systems Theory", 2013)

Living Systems Theory III

"In a biological or social system each holon must assert its individuality in order to maintain the system's stratified order, but it must also submit to the demands of the whole in order to make the system viable. These two tendencies are opposite but complementary. In a healthy system - an individual, a society, or an ecosystem - there is a balance between integration and self-assertion. This balance is not static but consists of a dynamic interplay between the two complementary tendencies, which makes the whole system flexible and open to change." (Fritjof Capra, "The Turning Point: Science, Society, and the Turning Culture", 1982)

"Living systems are organized in such a way that they form multileveled structures, each level consisting of subsystems which are wholes in regard to their parts, and parts with respect to the larger wholes." (Fritjof Capra, "The Turning Point: Science, Society, and the Turning Culture", 1982)

"The autonomy of living systems is characterized by closed, recursive organization. [...] A system's highest order of recursion or feedback process defines, generates, and maintains the autonomy of a system. The range of deviation this feedback seeks to control concerns the organization of the whole system itself. If the system should move beyond the limits of its own range of organization it would cease to be a system. Thus, autonomy refers to the maintenance of a systems wholeness. In biology, it becomes a definition of what maintains the variable called living." (Bradford P Keeney, "Aesthetics of Change", 1983)

"Living systems are never in equilibrium. They are inherently unstable. They may seem stable, but they're not. Everything is moving and changing. In a sense, everything is on the edge of collapse. Michael Crichton, "Jurassic Park", 1990)

"Systems thinking is a discipline for seeing wholes. It is a framework for seeing interrelationships rather than things, for seeing patterns of change rather than static 'snapshots'. It is a set of general principles- distilled over the course of the twentieth century, spanning fields as diverse as the physical and social sciences, engineering, and management. [...] During the last thirty years, these tools have been applied to understand a wide range of corporate, urban, regional, economic, political, ecological, and even psychological systems. And systems thinking is a sensibility for the subtle interconnectedness that gives living systems their unique character." (Peter Senge, "The Fifth Discipline", 1990)

"Living systems exist in the solid regime near the edge of chaos, and natural selection achieves and sustains such a poised state." (Stuart Kauffman, "The Origins of Order: Self-organization and selection in evolution", 1993) 

"It [Living Systems Theory (LST)] involves observing and measuring important relationships between inputs and outputs of the total system and identifying the structures that perform each of the [20] sub‐system processes. […] The flows of relevant matter, energy, and information through the system and the adjustment processes of subsystems and the total system are also examined. The status and function of the system are analyzed and compared with what is average or normal for that type of system. If the system is experiencing a disturbance in some steady state, an effort is made to discover the source of the strain and correct it." (James G Miller & Jessie L Miller, "Applications of living systems theory", Systemic Practice and Action Research 8, 1995)

"According to the systems view, the essential properties of an organism, or living system, are properties of the whole, which none of the parts have. They arise from the interactions and relationships among the parts. These properties are destroyed when the system is dissected, either physically or theoretically, into isolated elements. Although we can discern individual parts in any system, these parts are not isolated, and the nature of the whole is always different from the mere sum of its parts." (Fritjof Capra, "The Web of Life", 1996)

Living Systems Theory IV

"This spontaneous emergence of order at critical points of instability is one of the most important concepts of the new understanding of life. It is technically known as self-organization and is often referred to simply as ‘emergence’. 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. And since emergence is an integral part of the dynamics of open systems, we reach the important conclusion that open systems develop and evolve. Life constantly reaches out into novelty." (Fritjof  Capra, "The Hidden Connections", 2002)

"The science of cybernetics is not about thermostats or machines; that characterization is a caricature. Cybernetics is about purposiveness, goals, information flows, decision-making control processes and feedback (properly defined) at all levels of living systems." (Peter Corning, "Synergy, Cybernetics, and the Evolution of Politics", 2005) 

"When defining living systems, the term dynamic equilibrium is essential. It does not imply something which is steady or stable. On the contrary, it is a floating state characterized by invisible movements and preparedness for change. To be in dynamic equilibrium is adapting adjustment to balance. Homeostasis stands for the sum of all control functions creating the state of dynamic equilibrium in a healthy organism. It is the ability of the body to maintain a narrow range of internal conditions in spite of environmental changes." (Lars Skyttner, "General Systems Theory: Problems, Perspective, Practice", 2005)

"The universe of all things that exist may be understood as a universe of systems where a system is defined as any set of related and interacting elements. This concept is primitive and powerful and has been used increasingly over the last half-century to organize knowledge in virtually all domains of interest to investigators. As human inventions and social interactions grow more complex, general conceptual frameworks that integrate knowledge among different disciplines studying those emerging systems grow more important. Living systems theory (LST) instructs integrative research among biological and social sciences and related academic disciplines." (G A Swanson & James G Miller, "Living Systems Theory", 2013)

"All living systems are networks of smaller components, and the web of life as a whole is a multilayered structure of living systems nesting within other living systems - networks within networks." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

"Deep ecology does not separate humans - or anything else-from the natural environment. It sees the world not as a collection of isolated objects, but as a network of phenomena that are fundamentally interconnected and interdependent. Deep ecology recognizes the intrinsic value of all living beings and views humans as just one particular strand in the web of life." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 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)

Living Systems Theory II

"A cognitive system is a system whose organization defines a domain of interactions in which it can act with relevance to the maintenance of itself, and the process of cognition is the actual (inductive) acting or behaving in this domain. Living systems are cognitive systems, and living as a process is a process of cognition. This statement is valid for all organisms, with and without a nervous system." (Humberto R Maturana, "Biology of Cognition", 1970)

"A living system, due to its circular organization, is an inductive system and functions always in a predictive manner: what happened once will occur again. Its organization, (genetic and otherwise) is conservative and repeats only that which works. For this same reason living systems are historical systems; the relevance of a given conduct or mode of behavior is always determined in the past." (Humberto Maturana, "Biology of Cognition", 1970)

"The functional order maintained within living systems seems to defy the Second Law; nonequilibrium thermodynamics describes how such systems come to terms with entropy." (Ilya Prigogine, "Thermodynamics of Evolution", 1972)

"All nature is a continuum. The endless complexity of life is organized into patterns which repeat themselves - theme and variations - at each level of system. These similarities and differences are proper concerns for science. From the ceaseless streaming of protoplasm to the many-vectored activities of supranational systems, there are continuous flows through living systems as they maintain their highly organized steady states." (James G Miller, "Living Systems", 1978)

"Living systems are units of interactions; they exist in an ambience. From a purely biological point of view they cannot be understood independently of that part of the ambience with which they interact: the niche; nor can the niche be defined independently of the living system that specifies it." (Humberto Maturana, "Biology of Cognition", 1970)

"Information is carried by physical entities, such as books or sound waves or brains, but it is not itself material. Information in a living system is a feature of the order and arrangement of its parts, which arrangement provides the signs that constitute a ‘code’ or ‘language’." (John Z Young, "Programs of the Brain", 1978)

Living Systems Theory I

"A vital phenomenon can only be regarded as explained if it has been proven that it appears as the result of the material components of living organisms interacting according to the laws which those same components follow in their interactions outside of living systems." (Adolf E Fick, "Gesammelte Schriften" Vol. 3, 1904)

"Since the fundamental character of the living thing is its organization, the customary investigation of the single parts and processes cannot provide a complete explanation of the vital phenomena. This investigation gives us no information about the coordination of parts and processes. Thus, the chief task of biology must be to discover the laws of biological systems (at all levels of organization). We believe that the attempts to find a foundation for theoretical biology point at a fundamental change in the world picture. This view, considered as a method of investigation, we shall call ‘organismic biology’ and, as an attempt at an explanation, ‘the system theory of the organism’." (Ludwig von Bertalanffy, “Kritische Theorie der Formbildung”, 1928)

"[…] to the scientific mind the living and the non-living form one continuous series of systems of differing degrees of complexity […], while to the philosophic mind the whole universe, itself perhaps an organism, is composed of a vast number of interlacing organisms of all sizes." (James G Needham, "Developments in Philosophy of Biology", Quarterly Review of Biology Vol. 3 (1), 1928)

"General systems theory is a series of related definitions, assumptions, and postulates about all levels of systems from atomic particles through atoms, molecules, crystals, viruses, cells, organs, individuals, small groups, societies, planets, solar systems, and galaxies. General behavior systems theory is a subcategory of such theory, dealing with living systems, extending roughly from viruses through societies. A significant fact about living things is that they are open systems, with important inputs and outputs. Laws which apply to them differ from those applying to relatively closed systems." (James G Miller, "General behavior systems theory and summary", Journal of Counseling Psychology 3 (2), 1956)

"A system is primarily a living system, and the process which defines it is the maintenance of an organization which we know as life." (Ralph W Gerard, "Units and Concepts of Biology", 1958)

"System theory is basically concerned with problems of relationships, of structure, and of interdependence rather than with the constant attributes of objects. In general approach it resembles field theory except that its dynamics deal with temporal as well as spatial patterns. Older formulations of system constructs dealt with the closed systems of the physical sciences, in which relatively self-contained structures could be treated successfully as if they were independent of external forces. But living systems, whether biological organisms or social organizations, are acutely dependent on their external environment and so must be conceived of as open systems." (Daniel Katz, "The Social Psychology of Organizations", 1966)

"The homeostatic principle does not apply literally to the functioning of all complex living systems, in that in counteracting entropy they move toward growth and expansion." (Daniel Katz, "The Social Psychology of Organizations", 1966) 

"Conventional physics deals only with closed systems, i.e. systems which are considered to be isolated from their environment. [...] However, we find systems which by their very nature and definition are not closed systems. Every living organism is essentially an open system. It maintains itself in a continuous inflow and outflow, a building up and breaking down of components, never being, so long as it is alive, in a state of chemical and thermodynamic equilibrium but maintained in a so-called steady state which is distinct from the latter." (Ludwig von Bertalanffy, "General System Theory", 1968)

"My analysis of living systems uses concepts of thermodynamics, information theory, cybernetics, and systems engineering, as well as the classical concepts appropriate to each level. The purpose is to produce a description of living structure and process in terms of input and output, flows through systems, steady states, and feedbacks, which will clarify and unify the facts of life." (James G Miller, "Living Systems: Basic Concepts", 1969)

Chaos Theory I

"The chaos theory will require scientists in all fields to, develop sophisticated mathematical skills, so that they will be able to better recognize the meanings of results. Mathematics has expanded the field of fractals to help describe and explain the shapeless, asymmetrical find randomness of the natural environment." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)

"The term chaos is also used in a general sense to describe the body of chaos theory, the complete sequence of behaviours generated by feed-back rules, the properties of those rules and that behaviour." (Ralph D Stacey, "The Chaos Frontier: Creative Strategic Control for Business", 1991)

"Chaos theory reconciles our intuitive sense of free will with the deterministic laws of nature. However, it has an even deeper philosophical ramification. Not only do we have freedom to control our actions, but also the sensitivity to initial conditions implies that even our smallest act can drastically alter the course of history, for better or for worse. Like the butterfly flapping its wings, the results of our behavior are amplified with each day that passes, eventually producing a completely different world than would have existed in our absence!" (Julien C Sprott, "Strange Attractors: Creating Patterns in Chaos", 2000)

"Chaos theory, for example, uses the metaphor of the ‘butterfly effect’. At critical times in the formation of Earth’s weather, even the fluttering of the wings of a butterfly sends ripples that can tip the balance of forces and set off a powerful storm. Even the smallest inanimate objects sent back into the past will inevitably change the past in unpredictable ways, resulting in a time paradox." (Michio Kaku, "Parallel Worlds", 2004)

"This phenomenon, common to chaos theory, is also known as sensitive dependence on initial conditions. Just a small change in the initial conditions can drastically change the long-term behavior of a system. Such a small amount of difference in a measurement might be considered experimental noise, background noise, or an inaccuracy of the equipment." (Greg Rae, "Chaos Theory: A Brief Introduction", 2006)

"Yet, with the discovery of the butterfly effect in chaos theory, it is now understood that there is some emergent order over time even in weather occurrence, so that weather prediction is not next to being impossible as was once thought, although the science of meteorology is far from the state of perfection." (Peter Baofu, "The Future of Complexity: Conceiving a Better Way to Understand Order and Chaos", 2007)

"[chaos theory] presents a universe that is at once deterministic and obeys the fundamental physical laws, but is capable of disorder, complexity, and unpredictability. It shows that predictability is a rare phenomenon operating only within the constraints that science has filtered out from the rich diversity of our complex world." (Ziauddin Sardar & Iwona Abrams, "Introducing Chaos: A Graphic Guide", 2008)

"Complexity theory can be defined broadly as the study of how order, structure, pattern, and novelty arise from extremely complicated, apparently chaotic systems and conversely, how complex behavior and structure emerges from simple underlying rules. As such, it includes those other areas of study that are collectively known as chaos theory, and nonlinear dynamical theory." (Terry Cooke-Davies et al, "Exploring the Complexity of Projects", 2009)

"An ending is an artificial device; we like endings, they are satisfying, convenient, and a point has been made. But time does does not end, and stories march in step with time. Equally, chaos theory does not assume an ending; the ripple effect goes on, and on." (Penelope Lively, "How It All Began", 2011)

"The things that really change the world, according to Chaos theory, are the tiny things. A butterfly flaps its wings in the Amazonian jungle, and subsequently a storm ravages half of Europe." (Neil Gaiman, "Good Omens", 2011)

Complexity Theory

"Complexity theory began with an interest on how order spring from chaos. According to complexity theory, adaption is most effective in systems that are only partially connected. The argument is that too much structure creates gridlock, while too little structure creates chaos. […] Consequently, the key to effective change is to stay poised on this edge of chaos. Complexity theory focuses managerial thinking on the interrelationships among different parts of an organization and on the trade-off of less control for greater adaptation." (Shona Brown, "Competing on the Edge, 1998) 

"There is no over-arching theory of complexity that allows us to ignore the contingent aspects of complex systems. If something really is complex, it cannot by adequately described by means of a simple theory. Engaging with complexity entails engaging with specific complex systems. Despite this we can, at a very basic level, make general remarks concerning the conditions for complex behaviour and the dynamics of complex systems. Furthermore, I suggest that complex systems can be modelled." (Paul Cilliers, "Complexity and Postmodernism", 1998)

"Complexity theory is really a movement of the sciences. Standard sciences tend to see the world as mechanistic. That sort of science puts things under a finer and finer microscope. […] The movement that started complexity looks in the other direction. It’s asking, how do things assemble themselves? How do patterns emerge from these interacting elements? Complexity is looking at interacting elements and asking how they form patterns and how the patterns unfold. It’s important to point out that the patterns may never be finished. They’re open-ended. In standard science this hit some things that most scientists have a negative reaction to. Science doesn’t like perpetual novelty." (W Brian Arthur, "Coming from Your Inner Self", 1999)

"[…] most earlier attempts to construct a theory of complexity have overlooked the deep link between it and networks. In most systems, complexity starts where networks turn nontrivial. No matter how puzzled we are by the behavior of an electron or an atom, we rarely call it complex, as quantum mechanics offers us the tools to describe them with remarkable accuracy. The demystification of crystals-highly regular networks of atoms and molecules-is one of the major success stories of twentieth-century physics, resulting in the development of the transistor and the discovery of superconductivity. Yet, we continue to struggle with systems for which the interaction map between the components is less ordered and rigid, hoping to give self-organization a chance." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"[…] networks are the prerequisite for describing any complex system, indicating that complexity theory must inevitably stand on the shoulders of network theory. It is tempting to step in the footsteps of some of my predecessors and predict whether and when we will tame complexity. If nothing else, such a prediction could serve as a benchmark to be disproven. Looking back at the speed with which we disentangled the networks around us after the discovery of scale-free networks, one thing is sure: Once we stumble across the right vision of complexity, it will take little to bring it to fruition. When that will happen is one of the mysteries that keeps many of us going." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"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)

"Complexity Theory is concerned with the study of the intrinsic complexity of computational tasks. Its 'final' goals include the determination of the complexity of any well-defined task. Additional goals include obtaining an understanding of the relations between various computational phenomena (e.g., relating one fact regarding computational complexity to another). Indeed, we may say that the former type of goal is concerned with absolute answers regarding specific computational phenomena, whereas the latter type is concerned with questions regarding the relation between computational phenomena." (Oded Goldreich, "Computational Complexity: A Conceptual Perspective", 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)

"Complexity theory can be defined broadly as the study of how order, structure, pattern, and novelty arise from extremely complicated, apparently chaotic systems and conversely, how complex behavior and structure emerges from simple underlying rules. As such, it includes those other areas of study that are collectively known as chaos theory, and nonlinear dynamical theory." (Terry Cooke-Davies et al, "Exploring the Complexity of Projects", 2009)