"Since a given system can never of its own accord go over into another equally probable state but into a more probable one, it is likewise impossible to construct a system of bodies that after traversing various states returns periodically to its original state, that is a perpetual motion machine." (Ludwig Boltzmann, "'The Second Law of Thermodynamics", [Address to a Formal meeting of the Imperial Academy of Science], 1886)
"[...] the task of the theory consists in constructing a picture of the external world that exists purely internally and must be our guiding star in all thought and experiment; that is in completing, as it were, the thinking process and carrying out globally what on a small scale occurs within us whenever we form an idea." (Ludwig E Boltzmann, "On the Significance of Theories", 1890)
"One is almost tempted to assert that quite apart from its intellectual mission, theory is the most practical thing conceivable, the quintessence of practice as it were, since the precision of its conclusions cannot be reached by any routine of estimating or trial and error; although given the hidden ways of theory, this will hold only for those who walk them with complete confidence." (Ludwig E Boltzmann, "On the Significance of Theories", 1890)
"Most surprising and far-reaching analogies revealed themselves between apparently quite disparate natural processes. It seemed that nature had built the most various things on exactly the same pattern; or, in the dry words of the analyst, the same differential equations hold for the most various phenomena. (Ludwig Boltzmann, "On the methods of theoretical physics", 1892)
"Every hypothesis must derive indubitable results from mechanically well-defined assumptions by mathematically correct methods." (Ludwig Boltzmann, "Certain Questions of the Theory of Gasses", Nature Vol. 51 (1322), 1895)
"Experience teaches that one will be led to new discoveries almost exclusively by means of special mechanical models." (Ludwig Boltzmann, "Lectures on Gas Theory", 1896)
"All our ideas and concepts are only internal pictures, or if spoken, combinations of sounds. The task of our thinking is so to use and combine them that by their means we always most readily hit upon the correct actions and guide others likewise. In this, metaphysics follows the most down-to-earth and practical point of view, so that extremes meet. The conceptual signs that we form thus exist only within us, we cannot measure external phenomena by the standard of our ideas. We can therefore pose such formal questions as whether only matter exists and force is a property of it, or whether force exists independently of matter or conversely whether matter is a product of force but none of these questions are significant since all these concepts are only mental pictures whose purpose is to represent phenomena correctly." (Ludwig Boltzmann, 1899)
"[…] no theory can be objective, actually coinciding with nature, but rather that each theory is only a mental picture of phenomena, related to them as sign is to designatum. From this it follows that it cannot be our task to find an absolutely correct theory but rather a picture that is, as simple as possible and that represents phenomena as accurately as possible. One might even conceive of two quite different theories both equally simple and equally congruent with phenomena, which therefore in spite of their difference are equally correct. (Ludwig Boltzmann, "On the development of the methods of theoretical physics", 1899)
"What, then, is meant by having perfectly correct understanding of a mechanism? Everybody knows that the practical criterion for this consists in being able to handle it correctly. However, I go further and assert that this is the only tenable definition of understanding a mechanism. (Ludwig Boltzmann, "On the principles of mechanics", 1902)
"I am of the opinion that the task of the theory consists in constructing a picture of the external world that exists purely internally and must be our guiding star in all thought and experiment." (Ludwig E Boltzmann)
"Mathematics and music! the most glaring possible opposites of human thought! and yet connected, mutually sustained! It is as if they would demonstrate the hidden consensus of all the actions of our mind, which in the revelations of genius makes us forefeel unconscious utterances of a mysteriously active intelligence." (Ludwig Boltzmann)
"Since in the differential equations of mechanics themselves there is absolutely nothing analogous to the second law of thermodynamics the latter can be mechanically represented only by means of assumptions regarding initial conditions." (Ludwig Boltzmann)
Quotes and Resources Related to Mathematics, (Mathematical) Sciences and Mathematicians
03 August 2019
William E Deming - Collected Quotes
“It is important to realize that it is not the one measurement, alone, but its relation to the rest of the sequence that is of interest.” (William E Deming, “Statistical Adjustment of Data”, 1938)
“The definition of random in terms of a physical operation is notoriously without effect on the mathematical operations of statistical theory because so far as these mathematical operations are concerned random is purely and simply an undefined term.” (Walter A Shewhart & William E Deming, “Statistical Method from the Viewpoint of Quality Control”, 1939)
"Scientific data are not taken for museum purposes; they are taken as a basis for doing something. If nothing is to be done with the data, then there is no use in collecting any. The ultimate purpose of taking data is to provide a basis for action or a recommendation for action. The step intermediate between the collection of data and the action is prediction." (William E Deming, "On a Classification of the Problems of Statistical Inference", Journal of the American Statistical Association Vol. 37 (218), 1942)
”Experience without theory teaches nothing.” (William E Deming, “Out of the Crisis”, 1986)
“It is important to realize that it is not the one measurement, alone, but its relation to the rest of the sequence that is of interest.” (William E Deming, “Statistical Adjustment of Data”, 1943)
“Sampling is the science and art of controlling and measuring the reliability of useful statistical information through the theory of probability.” (William E Deming, “Some Theory of Sampling”, 1950)
"We must know more about a plan than the probabilities of selection. We must know also the procedure by which to draw the sampling units, and the formula or procedure by which to calculate the estimate." (William E Deming, "Sample Design in Business Research", 1960)
“Experience by itself teaches nothing [...] Without theory, experience has no meaning. Without theory, one has no questions to ask. Hence without theory there is no learning.” (William E Deming, “The New Economics for Industry, Government, Education”, 1993)
“Knowledge is theory. We should be thankful if action of management is based on theory. Knowledge has temporal spread. Information is not knowledge. The world is drowning in information but is slow in acquisition of knowledge. There is no substitute for knowledge.” (William E Deming, “The New Economics for Industry, Government, Education”, 1993)
“What is a system? A system is a network of interdependent components that work together to try to accomplish the aim of the system. A system must have an aim. Without an aim, there is no system. The aim of the system must be clear to everyone in the system. The aim must include plans for the future. The aim is a value judgment.” (William E Deming, “The New Economics for Industry, Government, Education”, 1993)
"The only useful function of a statistician is to make predictions, and thus to provide a basis for action." (William E Deming)
On Worldviews (1700-1899)
"The largest views are not always the clearest, and that he who is short-sighted will be obliged to draw the object nearer, and may, perhaps, by a close and narrow survey, discern that which had escaped far better eyes." (George Berkeley, "A Treatise Concerning the Principles of Human Knowledge", 1710)
"It is not therefore the business of philosophy, in our present situation in the universe, to attempt to take in at once, in one view, the whole scheme of nature; but to extend, with great care and circumspection, our knowledge, by just steps, from sensible things, as far as our observations or reasonings from them will carry us, in our enquiries concerning either the greater motions and operations of nature, or her more subtile and hidden works." (Colin Maclaurin, "An Account of Sir Isaac Newton's Philosophical Discoveries, in Four Books", 1748)
“If the human mind is nonetheless to be able even to think the given infinite without contradiction, it must have within itself a power that is supersensible, whose idea of the noumenon cannot be intuited but can yet be regarded as the substrate underlying what is mere appearance, namely, our intuition of the world [worldview]." (Immanuel Kant, “Critique of Judgment”, 1790)
"It is not therefore the business of philosophy, in our present situation in the universe, to attempt to take in at once, in one view, the whole scheme of nature; but to extend, with great care and circumspection, our knowledge, by just steps, from sensible things, as far as our observations or reasonings from them will carry us, in our enquiries concerning either the greater motions and operations of nature, or her more subtile and hidden works." (Colin Maclaurin, "An Account of Sir Isaac Newton's Philosophical Discoveries, in Four Books", 1748)
“If the human mind is nonetheless to be able even to think the given infinite without contradiction, it must have within itself a power that is supersensible, whose idea of the noumenon cannot be intuited but can yet be regarded as the substrate underlying what is mere appearance, namely, our intuition of the world [worldview]." (Immanuel Kant, “Critique of Judgment”, 1790)
"The diversity of languages is not a diversity of signs and sounds but a diversity of views of the world." (Wilhelm von Humboldt, 1820)
"So it happens at times that a person believes that he has a world-view, but that there is yet one particular phenomenon that is of such a nature that it baffles the understanding, and that he explains differently and attempts to ignore in order not to harbor the thought that this phenomenon might overthrow the whole view, or that his reflection does not possess enough courage and resolution to penetrate the phenomenon with his world-view." (Søren Kierkegaard, 1844)
“The last change in our point of view gives the whole world a pictorial air.” (Ralph W Emerson, “Miscellanies, Embracing Nature, Addresses, and Lectures”, 1866)
“The mutual interdependence of thought and word illuminates clearly the truth that languages are not really means for representing already known truths, but are rather instruments for discovering previously unrecognised ones. The differences between languages are not those of sounds and signs but those of differing worldviews […] objective truth always rises from the entire energy of subjective individuality.” (Wilhelm von Humboldt)
"So it happens at times that a person believes that he has a world-view, but that there is yet one particular phenomenon that is of such a nature that it baffles the understanding, and that he explains differently and attempts to ignore in order not to harbor the thought that this phenomenon might overthrow the whole view, or that his reflection does not possess enough courage and resolution to penetrate the phenomenon with his world-view." (Søren Kierkegaard, 1844)
“The last change in our point of view gives the whole world a pictorial air.” (Ralph W Emerson, “Miscellanies, Embracing Nature, Addresses, and Lectures”, 1866)
“The mutual interdependence of thought and word illuminates clearly the truth that languages are not really means for representing already known truths, but are rather instruments for discovering previously unrecognised ones. The differences between languages are not those of sounds and signs but those of differing worldviews […] objective truth always rises from the entire energy of subjective individuality.” (Wilhelm von Humboldt)
Fritjof Capra - Collected Quotes
"If physics leads us today to a world view which is essentially mystical, it returns, in a way, to its beginning, 2,500 years ago. […] Eastern thought and, more generally, mystical thought provide a consistent and relevant philosophical background to the theories of contemporary science; a conception of the world in which scientific discoveries can be in perfect harmony with spiritual aims and religious beliefs. The two basic themes of this conception are the unity and interrelation of all phenomena and the intrinsically dynamic nature of the universe. The further we penetrate into the submicroscopic world, the more we shall realize how the modern physicist, like the Eastern mystic, has come to see the world as a system of inseparable, interacting and ever-moving components with the observer being an integral part of this system." (Fritjof Capra, "The Tao of Physics: An Exploration of the Parallels Between Modern Physics and Eastern Mysticism", 1975)
"The influence of modern physics goes beyond technology. It extends to the realm of thought and culture where it has led to a deep revision in man's conception of the universe and his relation to it." (Fritjof Capra, "The Tao of Physics: An Exploration of the Parallels Between Modern Physics and Eastern Mysticism", 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)
“[…] self-organization is the spontaneous emergence of new structures and new forms of behavior in open systems far from equilibrium, characterized by internal feedback loops and described mathematically by nonlinear equations.” (Fritjof Capra, “The web of life: a new scientific understanding of living systems”, 1996)
"The more complex the network is, the more complex its pattern of interconnections, the more resilient it will be." (Fritjof Capra, "The Web of Life: A New Scientific Understanding of Living Systems", 1996)
"The more we study the major problems of our time, the more we come to realise that they cannot be understood in isolation. They are systemic problems, which means that they are interconnected and interdependent." (Fritjof Capra, "The Web of Life: A New Scientific Understanding of Living Systems", 1996)
"The shift of paradigms requires an expansion not only of our perceptions and ways of thinking, but also of our values. […] scientific facts emerge out of an entire constellation of human perceptions, values, and actions-in one word, out of a paradigm-from which they cannot be separated. […] Today the paradigm shift in science, at its deepest level, implies a shift from physics to the life sciences." (Fritjof Capra, "The Web of Life: A New Scientific Understanding of Living Systems", 1996)
"Understanding ecological interdependence means understanding relationships. It requires the shifts of perception that are characteristic of systems thinking - from the parts to the whole, from objects to relationships, from contents to patterns. [...] Nourishing the community means nourishing those relationships." (Fritjof Capra, "The Web of Life: A New Scientific Understanding of Living Systems", 1996)
"What is sustained in a sustainable community is not economic growth, development, market share, or competitive advantage, but the entire web of life on which our long-term survival depends. In other words, a sustainable community is designed in such a way that its ways of life, businesses, economy, physical structures, and technologies do not interfere with nature’s inherent ability to sustain life." (Fritjof Capra, "Ecoliteracy: The Challenge for Education in the Next Century", 1999)
"One of the key insights of the systems approach has been the realization that the network is a pattern that is common to all life. Wherever we see life, we see networks." (Fritjof Capra, "The Hidden Connections: A Science for Sustainable Living", 2002)
"The phenomenon of emergence takes place at critical points of instability that arise from fluctuations in the environment, amplified by feedback loops. Emergence results in the creation of novelty, and this novelty is often qualitatively different from the phenomenon out of which it emerged." (Fritjof Capra, "The Hidden Connections: A Science for Sustainable Living", 2002)
"The influence of modern physics goes beyond technology. It extends to the realm of thought and culture where it has led to a deep revision in man's conception of the universe and his relation to it." (Fritjof Capra, "The Tao of Physics: An Exploration of the Parallels Between Modern Physics and Eastern Mysticism", 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)
"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)
"In microscopic systems, consisting of only a few molecules,
the second law is violated regularly, but in macroscopic systems, which consist
of vast numbers of molecules, the probability that the total entropy of the
system will increase becomes virtual certainty. Thus in any isolated system,
made up of a large number of molecules, the entropy - or disorder -will keep
increasing until, eventually, the system reaches a state of maximum entropy,
also known as 'heat death'; in this state all activity has ceased,
all material being evenly distributed and at the same temperature. According to
classical physics, the universe as a whole is going toward such a state of maximum
entropy; it is running down and will eventually grind to a halt."
"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)
"Systems theory looks at the world in terms of the interrelatedness and interdependence of all phenomena, and in this framework an integrated whole whose properties cannot be reduced to those of its parts is called a system. Living organisms, societies, and ecosystems are all systems." (Fritjof Capra, "The Turning Point: Science, Society, and the Turning Culture", 1982)
"The phenomena of the subatomic world are so complex that it
is by no means certain whether a complete, self-consistent theory will ever be
constructed, but one can envisage a series of partly successful models of
smaller scope. Each of them would be intended to cover only a part of the
observed phenomena and would contain some unexplained aspects, or parameters,
but the parameters of one model might be explained by another. Thus more and
more phenomena could gradually be covered with ever increasing accuracy by a
mosaic of interlocking models whose net number of unexplained parameters keeps decreasing."
"The phenomenon of self-organization is not limited to living
matter but occurs also in certain chemical systems […] [Ilya] Prigogine has
called these systems 'dissipative structures' to express the fact
that they maintain and develop structure by breaking down other structures in
the process of metabolism, thus creating entropy disorder - which is
subsequently dissipated in the form of degraded waste products. Dissipative
chemical structures display the dynamics of self-organization in its simplest
form, exhibiting most of the phenomena characteristic of life self-renewal,
adaptation, evolution, and even primitive forms of 'mental' processes."
“[…] self-organization is the spontaneous emergence of new structures and new forms of behavior in open systems far from equilibrium, characterized by internal feedback loops and described mathematically by nonlinear equations.” (Fritjof Capra, “The web of life: a new scientific understanding of living systems”, 1996)
"The more complex the network is, the more complex its pattern of interconnections, the more resilient it will be." (Fritjof Capra, "The Web of Life: A New Scientific Understanding of Living Systems", 1996)
"The more we study the major problems of our time, the more we come to realise that they cannot be understood in isolation. They are systemic problems, which means that they are interconnected and interdependent." (Fritjof Capra, "The Web of Life: A New Scientific Understanding of Living Systems", 1996)
"The shift of paradigms requires an expansion not only of our perceptions and ways of thinking, but also of our values. […] scientific facts emerge out of an entire constellation of human perceptions, values, and actions-in one word, out of a paradigm-from which they cannot be separated. […] Today the paradigm shift in science, at its deepest level, implies a shift from physics to the life sciences." (Fritjof Capra, "The Web of Life: A New Scientific Understanding of Living Systems", 1996)
"Understanding ecological interdependence means understanding relationships. It requires the shifts of perception that are characteristic of systems thinking - from the parts to the whole, from objects to relationships, from contents to patterns. [...] Nourishing the community means nourishing those relationships." (Fritjof Capra, "The Web of Life: A New Scientific Understanding of Living Systems", 1996)
"What is sustained in a sustainable community is not economic growth, development, market share, or competitive advantage, but the entire web of life on which our long-term survival depends. In other words, a sustainable community is designed in such a way that its ways of life, businesses, economy, physical structures, and technologies do not interfere with nature’s inherent ability to sustain life." (Fritjof Capra, "Ecoliteracy: The Challenge for Education in the Next Century", 1999)
"One of the key insights of the systems approach has been the realization that the network is a pattern that is common to all life. Wherever we see life, we see networks." (Fritjof Capra, "The Hidden Connections: A Science for Sustainable Living", 2002)
"The phenomenon of emergence takes place at critical points of instability that arise from fluctuations in the environment, amplified by feedback loops. Emergence results in the creation of novelty, and this novelty is often qualitatively different from the phenomenon out of which it emerged." (Fritjof Capra, "The Hidden Connections: A Science for Sustainable Living", 2002)
"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)
"All the variables we can observe in an ecosystem-population densities, availability of nutrients, weather patterns, and so forth-always fluctuate. This is how ecosystems maintain themselves in a flexible state, ready to adapt to changing conditions. The web of life is a flexible, ever-fluctuating network. The more variables are kept fluctuating, the more dynamic is the system; the greater is its flexibility; and the greater is its ability to adapt to changing conditions." (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)
"In linear systems, small changes produce small effects, and large effects are due either to large changes or to a sum of many small changes. In nonlinear systems, by contrast, small changes may have dramatic effects because they may be amplified repeatedly by self-reinforcing feedback. Such nonlinear feedback processes are the basis of the instabilities and the sudden emergence of new forms of order that are so characteristic of self-organization."
"In many nonlinear systems, however, small changes of
certain parameters may produce Dramatic changes in the basic characteristics of
the phase portrait. Attractors may disappear, or change into one another, or new
attractors may suddenly appear. Such systems are said to be structurally
unstable, and the critical points of instability are called 'bifurcation
points', because they are points in the system’s evolution where a fork
suddenly appears and the system branches off in a new direction.
Mathematically, bifurcation points mark sudden changes in the system’s phase
portrait. Physically, they correspond to points of instability at which the
system changes abruptly and new forms of order suddenly appear."
"In other words, the web of life consists of networks within networks. At each scale, under closer scrutiny, the nodes of the network reveal themselves as smaller networks. We tend to arange these systems, all nesting within larger systems, in a hierarchical scheme by placing the larger systems above the smaller ones in pyramid fashion. But this is a human projection. In nature there is no 'above' or 'below', and there are no hierarchies. There are only networks nesting within other networks." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)
"In the nonlinear world – which includes most of the real
world, as we have discovered – simple deterministic equations may produce an
unsuspected richness and variety of behavior. On the other hand, complex and
seemingly chaotic behavior can give rise to ordered structures, to subtle and
beautiful patterns. In fact, in chaos theory the term 'chaos' has acquired a
new technical meaning. The behavior of chaotic systems only appears to be
random but in reality shows a deeper level of patterned order."
"Intersections of lines, for example, remain intersections, and the hole in a torus (doughnut) cannot be transformed away. Thus a doughnut may be transformed topologically into a coffee cup (the hole turning into a handle) but never into a pancake. Topology, then, is really a mathematics of relationships, of unchangeable, or 'invariant', patterns." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)
"It is evident that chaotic behavior, in the new
scientific sense of the term, is very different from random, erratic motion.
With the help of strange attractors a distinction can be made between mere
randomness, or 'noise', and chaos. Chaotic behavior is deterministic and patterned,
and strange attractors allow us to transform the seemingly random data into distinct
visible shapes."
"The first and most obvious property of any network is its nonlinearity – it goes in all directions. Thus the relationships in a network pattern are nonlinear relationships. In particular, an influence, or message, may travel along a cyclical path, which may become a feedback loop. In living networks, the concept of feedback is intimately connected with the network pattern." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)
"The impossibility of predicting which point in phase space the trajectory of the Lorenz attractor will pass through at a certain time, even though the system is governed by deterministic equations, is a common feature of all chaotic systems. However, this does not mean that chaos theory is not capable of any predictions. We can still make very accurate predictions, but they concern the qualitative features of the system’s behavior rather than the precise values of its variables at a particular time. The new mathematics thus represents the shift from quantity to quality that is characteristic of systems thinking in general. Whereas conventional mathematics deals with quantities and formulas, nonlinear dynamics deals with qualities and patterns." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)
"The study of pattern is crucial to the understanding of living systems because systemic properties […] arise from a configuration of ordered relationships. Systemic properties are properties of a pattern. What is destroyed when a living organism is dissected is its pattern. The components are still there, but the configuration of relationships between them – the pattern – is destroyed, and thus the organism dies." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)
"[…] the term 'information', as used in information theory, has nothing to do with meaning. It is a measure of the order, or nonrandomness, of a signal; and the main concern of information theory is the problem of how to get a message, coded as a signal, through a noisy channel." (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)
"To understand the phenomenon of self-organization, we first need to understand the importance of pattern. The idea of a pattern of organization – a configuration of relationships characteristic of a particular system – became the explicit focus of systems thinking in cybernetics and has been a crucial concept ever since. From the systems point of view, the understanding of life begins with the understanding of pattern." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)
"Topology is a geometry in which all lengths, angles, and areas can be distorted at will. Thus a triangle can be continuously transformed into a rectangle, the rectangle into a square, the square into a circle, and so on. Similarly, a cube can be transformed into a cylinder, the cylinder into a cone, the cone into a sphere. Because of these continuous transformations, topology is known popularly as 'rubber sheet geometry'. All figures that can be transformed into each other by continuous bending, stretching, and twisting are called 'topologically equivalent'." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)
"[…] topology is concerned precisely with those properties of geometric figures that do not change when the figures are transformed. Intersections of lines, for example, remain intersections, and the hole in a torus (doughnut) cannot be transformed away. Thus a doughnut may be transformed topologically into a coffee cup (the hole turning into a handle) but never into a pancake. Topology, then, is really a mathematics of relationships, of unchangeable, or 'invariant', patterns."
"Whenever we encounter living systems – organisms, parts of organisms, or communities of organisms – we can observe that their components are arranged in network fashion. Whenever we look at life, we look at networks." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)
Stanislaw M Ulam - Collected Quotes
“Mathematics is a self-contained microcosm, but it also has the potentiality of mirroring and modeling all the processes of thought and perhaps all of science. It has always had, and continues to an ever increasing degree to have, great usefulness. One could even go so far as to say that mathematics was necessary for man's conquest of nature and for the development of the human race through the shaping of its modes of thinking.” (Mark Kac & Stanislaw M Ulam, “Mathematics and Logic”, 1968)
"[…] the distinction between rigorous thinking and more vague ‘imaginings’; even in mathematics itself, all is not a question of rigor, but rather, at the start, of reasoned intuition and imagination, and, also, repeated guessing. After all, most thinking is a synthesis or juxtaposition of advances along a line of syllogisms - perhaps in a continuous and persistent ‘forward'’ movement, with searching, so to speak ‘sideways’, in directions which are not necessarily present from the very beginning and which I describe as ‘sending out exploratory patrols’ and trying alternative routes." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)
"Even the simplest calculation in the purest mathematics can have terrible consequences. Without the invention of the infinitesimal calculus most of our technology would have been impossible." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)
"In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug. Chess sometimes plays a similar role. In their unhappiness over the events of this world, some immerse themselves in a kind of self-sufficiency in mathematics." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)
"Mathematics may be a way of developing physically, that is anatomically, new connections in the brain." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)
"The esthetic side of mathematics has been of overwhelming importance throughout its growth. It is not so much whether a theorem is useful that matters, but how elegant it is." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)
"What exactly is mathematics? Many have tried but nobody has really succeeded in defining mathematics; it is always something else." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)
"Do not lose your faith. A mighty fortress is our mathematics. Mathematics will rise to the challenge, as it always has." (Stanislaw M Ulam)
"[…] the distinction between rigorous thinking and more vague ‘imaginings’; even in mathematics itself, all is not a question of rigor, but rather, at the start, of reasoned intuition and imagination, and, also, repeated guessing. After all, most thinking is a synthesis or juxtaposition of advances along a line of syllogisms - perhaps in a continuous and persistent ‘forward'’ movement, with searching, so to speak ‘sideways’, in directions which are not necessarily present from the very beginning and which I describe as ‘sending out exploratory patrols’ and trying alternative routes." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)
"Even the simplest calculation in the purest mathematics can have terrible consequences. Without the invention of the infinitesimal calculus most of our technology would have been impossible." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)
"In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug. Chess sometimes plays a similar role. In their unhappiness over the events of this world, some immerse themselves in a kind of self-sufficiency in mathematics." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)
"Mathematics may be a way of developing physically, that is anatomically, new connections in the brain." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)
"The esthetic side of mathematics has been of overwhelming importance throughout its growth. It is not so much whether a theorem is useful that matters, but how elegant it is." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)
"What exactly is mathematics? Many have tried but nobody has really succeeded in defining mathematics; it is always something else." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)
"Do not lose your faith. A mighty fortress is our mathematics. Mathematics will rise to the challenge, as it always has." (Stanislaw M Ulam)
George Berkeley - Collected Quotes
"A spirit is one simple, undivided, active being: as it perceives ideas, it is called the understanding, and as it produces or otherwise operates about them, it is called the will. Hence there can be no idea formed of a soul or spirit: for all ideas whatever, being passive and inert, vide, they cannot represent unto us, by way of image or likeness, that which acts." (George Berkeley, "Principles of Human Knowledge", 1710)
"Colour, Figure, Motion, Extension and the like, considered only so many Sensations in the Mind, are perfectly known, there being nothing in them which is not perceived. But if they are looked on as notes or Images, referred to Things or Archetypes existing without the Mind, then are we involved all in Scepticism."(George Berkeley, "Principles of Human Knowledge", 1710)
"In a large sense indeed, we may be said to have an idea, or rather a notion of spirit, that is, we understand the meaning of the word, otherwise we could not affirm or deny anything of it. Moreover, as we conceive the ideas that are in the minds of other spirits by means of our own, which we suppose to be resemblances of them: so we know other spirits by means of our own soul, which in that sense is the image or idea of them, it having a like respect to other spirits, that blueness or heat by me perceived hath to those ideas perceived by another." (George Berkeley, "Principles of Human Knowledge", 1710)
"Colour, Figure, Motion, Extension and the like, considered only so many Sensations in the Mind, are perfectly known, there being nothing in them which is not perceived. But if they are looked on as notes or Images, referred to Things or Archetypes existing without the Mind, then are we involved all in Scepticism."(George Berkeley, "Principles of Human Knowledge", 1710)
"In a large sense indeed, we may be said to have an idea, or rather a notion of spirit, that is, we understand the meaning of the word, otherwise we could not affirm or deny anything of it. Moreover, as we conceive the ideas that are in the minds of other spirits by means of our own, which we suppose to be resemblances of them: so we know other spirits by means of our own soul, which in that sense is the image or idea of them, it having a like respect to other spirits, that blueness or heat by me perceived hath to those ideas perceived by another." (George Berkeley, "Principles of Human Knowledge", 1710)
"Some there are who make a distinction betwixt primary and secondary qualities: by the former, they mean extension, figure motion, rest, solidity or impenetrability and number: by the latter they denote all other sensible qualities, as colours, sounds, tastes, and so forth. The ideas we have of these they acknowledge not to be the resemblances of anything existing without the mind or unperceived; but they will have our ideas of the primary qualities to be patterns or images of things which exist without the mind, in an unthinking substance which they call matter. By matter therefore we are to under-stand an inert, senseless substance, in which extension, figure, and motion, do actually subsist." (George Berkeley, "Principles of Human Knowledge", 1710)
"That an idea which is inactive, and the existence whereof consists in being perceived, should be the image or likeness of an agent subsisting by itself, seems to need no other refutation, than barely attending to what is meant by those words. But perhaps you will say, that though an idea cannot resemble a spirit, in its thinking, acting, or subsisting by itself, yet it may in some other respects: and it is not necessary that an idea or image be in all respects like the original."(George Berkeley, "Principles of Human Knowledge", 1710)
"The ideas imprinted on the senses by the Author of Nature are called real things: and those excited in the imagination being less regular, vivid and constant, are more properly termed ideas, or images of things, which they copy and represent. But then our sensations, be they never so vivid and distinct, are nevertheless ideas, that is, they exist in the mind, or are perceived by it, as truly as the ideas of its own framing. The ideas of sense are allowed to have more reality in them, that is, to be more strong, orderly, and coherent than the creatures of the mind; but this is no argument that they exist without the mind. They are also less dependent on the spirit, or thinking substance which perceives them, in that they are excited by the will of another and more powerful spirit: yet still they are ideas, and certainly no idea, whether faint or strong, can exist otherwise than in a mind perceiving it." (George Berkeley, "Principles of Human Knowledge", 1710)
"The largest views are not always the clearest, and that he who is short-sighted will be obliged to draw the object nearer, and may, perhaps, by a close and narrow survey, discern that which had escaped far better eyes." (George Berkeley, "A Treatise Concerning the Principles of Human Knowledge", 1710)
"The method of Fluxions is the general key by help whereof the modern mathematicians unlock the secrets of Geometry, and consequently of Nature." (George Berkeley, "The Analyst: A Discourse Addressed to an Infidel Mathematician...", 1734)
"He that would make a real progress in knowledge must dedicate his age as well as first fruits - the latter growth as well as the first-fruits - at the altar of truth." (George Berkeley, "Siris", 1744)
"A ray of imagination or of wisdom may enlighten the universe, and glow into remotest centuries." (George Berkeley)
"The largest views are not always the clearest, and that he who is short-sighted will be obliged to draw the object nearer, and may, perhaps, by a close and narrow survey, discern that which had escaped far better eyes." (George Berkeley, "A Treatise Concerning the Principles of Human Knowledge", 1710)
"The only properties that we can conceive of as they are in themselves are ones that can be presented in, or constructed from features presented in, experience." (George Berkeley, "Principles of Human Knowledge", 1710)
"This perceiving, active being is what I call mind, spirit, soul, or myself. By which words I do not denote any one of my ideas, but a thing entirely distinct from them, wherein they exist, or, which is the same thing, whereby they are perceived; for the existence of an idea consists in being perceived." (George Berkeley, "Principles of Human Knowledge", 1710)
"But to pass over all this; if I understand you rightly, you say our ideas do not exist with-out the mind; but that they are copies, images, or representations of certain originals that do." (George Berkeley, "Three Dialogues", 1713)
"Doth the reality of sensible things consist in being perceived? or, is it something distinct from their being perceived, and that bears no relation to the mind?" (George Berkeley, "Three Dialogues", 1713)
"It is your opinion, the ideas we perceive by our senses are not real things, but images, or copies of them. Our knowledge therefore is no farther real, than as our ideas are the true representations of those originals. But as these supposed originals are in themselves unknown, it is impossible to know how far our ideas resemble them; or whether they resemble them at all. We cannot therefore be sure we have any real knowledge." (George Berkeley, "Three Dialogues", 1713)
"The same principles which at first view lead to scepticism, pursued to a certain point bring men back to common sense." (George Berkeley, "Three Dialogues", 1713)
"To speak the truth, Philonous, I think there are two kinds of objects, the one perceived immediately, which are likewise called ideas, the other are real things or external objects perceived by the mediation of ideas, which are their images and representations. Now I own, ideas do not exist without the mind; but the latter sort of objects do." (George Berkeley, "Three Dialogues", 1713)
"But to pass over all this; if I understand you rightly, you say our ideas do not exist with-out the mind; but that they are copies, images, or representations of certain originals that do." (George Berkeley, "Three Dialogues", 1713)
"Doth the reality of sensible things consist in being perceived? or, is it something distinct from their being perceived, and that bears no relation to the mind?" (George Berkeley, "Three Dialogues", 1713)
"It is your opinion, the ideas we perceive by our senses are not real things, but images, or copies of them. Our knowledge therefore is no farther real, than as our ideas are the true representations of those originals. But as these supposed originals are in themselves unknown, it is impossible to know how far our ideas resemble them; or whether they resemble them at all. We cannot therefore be sure we have any real knowledge." (George Berkeley, "Three Dialogues", 1713)
"The same principles which at first view lead to scepticism, pursued to a certain point bring men back to common sense." (George Berkeley, "Three Dialogues", 1713)
"To speak the truth, Philonous, I think there are two kinds of objects, the one perceived immediately, which are likewise called ideas, the other are real things or external objects perceived by the mediation of ideas, which are their images and representations. Now I own, ideas do not exist without the mind; but the latter sort of objects do." (George Berkeley, "Three Dialogues", 1713)
"As arithmetic and algebra are sciences of great clearness, certainty, and extent, which are immediately conversant about signs, upon the skillful use whereof they entirely depend, so a little attention to them may possibly help us to judge of the progress of the mind in other sciences, which, though differing in nature, design, and object, may yet agree in the general methods of proof and inquiry." (George Berkeley, "Alciphron: or the Minute Philosopher", 1732)
"The same principles which at first view lead to scepticism, pursued to a certain point bring men back to common sense." (George Berkeley, "Three Dialogues between Hylas and Philonous", 1734)
"The same principles which at first view lead to scepticism, pursued to a certain point bring men back to common sense." (George Berkeley, "Three Dialogues between Hylas and Philonous", 1734)
"And yet in the calculus differentialis, which method serves to all the same intents and ends with that of fluxions, our modern analysts are not content to consider only the differences of finite quantities: they also consider the differences of those differences, and the differences of the differences of the first differences. And so on ad infinitum. That is, they consider quantities infinitely less than the least discernible quantity; and others infinitely less than those infinitely small ones; and still others infinitely less than the preceding infinitesimals, and so on without end or limit. Insomuch that we are to admit an infinite succession of infinitesimals, each infinitely less than the foregoing, and infinitely greater than the following. As there are first, second, third, fourth, fifth, &c. fluxions, so there are differences, first, second, third, fourth, &c., in an infinite progression towards nothing, which you still approach and never arrive at. And (which is most strange) although you should take a million of millions of these infinitesimals, each whereof is supposed infinitely greater than some other real magnitude, and add them to the least given quantity, it shall never be the bigger. For this is one of the modest postulata of our modern mathematicians, and is a cornerstone or ground-work of their speculations." (George Berkeley, "The Analyst: A Discourse Addressed to an Infidel Mathematician...", 1734)
"And what are these fluxions? The velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities?" (George Berkeley, "The Analyst: A Discourse Addressed to an Infidel Mathematician...", 1734)
"He who can digest a second or third fluxion, a second or third difference, need not, we think, be squeamish about any point of divinity." (George Berkeley, "The Analyst: A Discourse Addressed to an Infidel Mathematician...", 1734)
"And what are these fluxions? The velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities?" (George Berkeley, "The Analyst: A Discourse Addressed to an Infidel Mathematician...", 1734)
"He who can digest a second or third fluxion, a second or third difference, need not, we think, be squeamish about any point of divinity." (George Berkeley, "The Analyst: A Discourse Addressed to an Infidel Mathematician...", 1734)
"It hath been an old remark, that Geometry is an excellent Logic. And it must be owned that when the definitions are clear; when the postulata cannot be refused, nor the axioms denied; when from the distinct contemplation and comparison of figures, their properties are derived, by a perpetual well-connected chain of consequences, the objects being still kept in view, and the attention ever fixed upon them; there is acquired a habit of reasoning, close and exact and methodical; which habit strengthens and sharpens the mind, and being transferred to other subjects is of general use in the inquiry after truth." (George Berkeley, "The Analyst: A Discourse Addressed to an Infidel Mathematician...", 1734)
"The foreign mathematicians are supposed by some, even of our own, to proceed in a manner less accurate, perhaps, and geometrical, yet more intelligible. Instead of flowing quantities and their fluxons, they consider the variable finite quantities as increasing or diminishing by the continual addition or subduction of infinitely small quantities. Instead of the velocities wherewith increments are generated, they consider the increments or decrements themselves, which they call differences, and which are supposed to be infinitely small. The difference of a line is an infinitely little line; of a plane an infinitely little plane.
They suppose finite quantities to consist of parts infinitely little, and curves to be polygons, whereof the sides are infinitely little, which by the angles they make one with another determine the curvity of the line. Now to conceive a quantity infinitely small, that is, infinitely less than any sensible or imaginable quantity, or than any the least finite magnitude is, I confess, above my capacity. But to conceive a part of such infinitely small quantity that shall be still infinitely less than it, and consequently though multiplied infinitely shall never equal the minutest finite quantity, is, I suspect, an infinite difficulty to any man whatsoever; and will be allowed such by those who candidly say what they think; provided they really think and reflect, and do not take things upon trust." (George Berkeley, "The Analyst: A Discourse Addressed to an Infidel Mathematician...", 1734)
"The method of Fluxions is the general key by help whereof the modern mathematicians unlock the secrets of Geometry, and consequently of Nature." (George Berkeley, "The Analyst: A Discourse Addressed to an Infidel Mathematician...", 1734)
"He that would make a real progress in knowledge must dedicate his age as well as first fruits - the latter growth as well as the first-fruits - at the altar of truth." (George Berkeley, "Siris", 1744)
"A ray of imagination or of wisdom may enlighten the universe, and glow into remotest centuries." (George Berkeley)
"Make a point never so clear, it is great odds that a man whose habits and the bent of whose mind lie a contrary way, shall be unable to comprehend it. So weak a thing is reason in competition with inclination." (George Berkeley)
"That neither our thoughts, nor passions, nor ideas formed by the imagination, exist without the mind, is what every body will allow." (George Berkeley)
"Whatever is immediately perceived is an idea: and can any idea exist out of the mind?" (George Berkeley)
"That neither our thoughts, nor passions, nor ideas formed by the imagination, exist without the mind, is what every body will allow." (George Berkeley)
"Whatever is immediately perceived is an idea: and can any idea exist out of the mind?" (George Berkeley)
"Whenever I attempt to frame a simple idea of time, abstracted from the succession of ideas in my mind, which flows uniformly, and is participated by all beings, I am lost and embrangled in inextricable difficulties." (George Berkeley)
02 August 2019
Erwin Schrödinger - Collected Quotes
"In experimental science facts of the greatest importance are rarely discovered accidentally: more frequently new ideas point the way towards them." (Erwin Schrödinger, "Science and the Human Temperament", 1935)
"When two systems, of which we know the states by their respective representation, enter into a temporary physical interaction due to known forces between them and when after a time of mutual influence the systems separate again, then they can no longer be described as before, viz., by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics." (Erwin Schrödinger, 1935) [defining entanglement]
"[A living organism] feeds upon negative entropy […] Thus, the device by which an organism maintains itself stationary at a fairly high level of orderliness really consists in continually sucking orderliness from its environment." (Erwin Schrödinger, "What is Life? The Physical Aspect of the Living Cell", 1944)
"The scientist only imposes two things, namely truth and sincerity, imposes them upon himself and upon other scientists." (Erwin Schrödinger, "What is Life?", 1944)
"A theoretical science unaware that those of its constructs considered relevant and momentous are destined eventually to be framed in concepts and words that have a grip on the educated community and become part and parcel of the general world picture - a theoretical science, I say, where this is forgotten, and where the initiated continue musing to each other in terms that are, at best, understood by a small group of close fellow travelers, will necessarily be cut off from the rest of cultural mankind; in the long run it is bound to atrophy and ossify however virulently esoteric chat may continue within its joyfully isolated groups of experts." (Erwin Schrödinger, "Are There Quantum Jumps?", The British Journal for the Philosophy of Science Vol. 3, 1952)
"As our mental eye penetrates into smaller and smaller distances and shorter and shorter times, we find nature behaving so entirely differently from what we observe in visible and palpable bodies of our surroundings that no model shaped after our large-scale experiences can ever be ‘true’. A complete satisfactory model of this type is not only practically inaccessible, but not even thinkable. Or, to be precise, we can, of course, think of it, but however we think it, it is wrong; not perhaps quite as meaningless as a ‘triangular circle’, but more so than a ‘winged lion’." (Erwin Schrödinger, "Science and Humanism", 1952)
"'Mathematics' appears to be at the bottom of everything, since we find it unexpectedly where we have not put it in." (Erwin Schrödinger, "Nature and the Greeks", 1954)
"In particular, and most importantly, this is the reason why the scientific worldview contains of itself no ethical values, no esthetical values, not a word about our own ultimate scope or destination, and no God, if you please." (Erwin Schrödinger, "Nature and the Greeks", 1954)
"[…] the scientific picture of the real world around me is very deficient. It gives a lot of factual information, puts all our experience in a magnificently consistent order, but it is ghastly silent about all and sundry that is really near to our heart, that really matters to us. It cannot tell us a word about red and blue, bitter and sweet, physical pain and physical delight; it knows nothing of beautiful and ugly, good or bad, God and eternity. Science sometimes pretends to answer questions in these domains, but the answers are very often so silly that we are not inclined to take them seriously." (Erwin Schrödinger, "Nature and the Greeks", 1954)
"Speaking without metaphor we have to declare that we are here faced with one of these typical antinomies caused by the fact that we have not yet succeeded in elaborating a fairly understandable outlook on the world without retiring our own mind, the producer of the world picture, from it, so that mind has no place in it. The attempt to press it into it, after all, necessarily produces some absurdities." (Erwin Schrödinger, „Mind and Matter: the Tarner Lectures", 1956)
"While the stuff from which our world picture is built is yielded exclusively from the sense organs as organs of the mind, so that every man's world picture is and always remains a construct of his mind and cannot be proved to have any other existence, yet the conscious mind itself remains a stranger within that construct, it has no living space in it, you can spot it nowhere in space." (Erwin Schrödinger, „Mind and Matter: the Tarner Lectures", 1956)
"Primary scientific papers are not meant to be final statement of indisputable truths; each is merely a tiny tentative step forward, through the jungle of ignorance." (Erwin Schrödinger, "Information, Communication, Knowledge", Nature Vol. 224 (5217), 1969)
"A permanent state is reached, in which no observable events occur. The physicist calls this the state of thermodynamical equilibrium, or of ‘maximum entropy’. Practically, a state of this kind is usually reached very rapidly. Theoretically, it is very often not yet an absolute equilibrium, not yet the true maximum of entropy. But then the final approach to equilibrium is very slow. It could take anything between hours, years, centuries […]." (Erwin Schrödinger, "What is Life?", 1944)
"An isolated system or a system in a uniform environment (which for the present consideration we do best to include as a part of the system we contemplate) increases its entropy and more or less rapidly approaches the inert state of maximum entropy. We now recognize this fundamental law of physics to be just the natural tendency of things to approach the chaotic state (the same tendency that the books of a library or the piles of papers and manuscripts on a writing desk display) unless we obviate it. (The analogue of irregular heat motion, in this case, is our handling those objects now and again without troubling to put them back in their proper places.)" (Erwin Schrödinger, "What is Life?", 1944)
"Every process, event, happening - call it what you will; in a word, everything that is going on in Nature means an increase of the entropy of the part of the world where it is going on. Thus a living organism continually increases its entropy - or, as you may say, produces positive entropy – and thus tends to approach the dangerous state of maximum entropy, which is death. It can only keep aloof from it, i.e. alive, by continually drawing from its environment negative entropy – which is something very positive as we shall immediately see. What an organism feeds upon is negative entropy. Or, to put it less paradoxically, the essential thing in metabolism is that the organism succeeds in freeing itself from all the entropy it cannot help producing while alive." (Erwin Schrödinger, "What is Life?", 1944)
"Hence the awkward expression ‘negative entropy’ can be replaced by a better one: entropy, taken with the negative sign, is itself a measure of order. Thus the device by which an organism maintains itself stationary at a fairly high level of orderliness ( = fairly low level of entropy) really consists in continually sucking orderliness from its environment." (Erwin Schrödinger, "What is Life?", 1944)
"The great revelation of the quantum theory was that features of discreteness were discovered in the Book of Nature, in a context in which anything other than continuity seemed to be absurd according to the views held until then." (Erwin Schrödinger, "What is Life?", 1944)
"A theoretical science unaware that those of its constructs considered relevant and momentous are destined eventually to be framed in concepts and words that have a grip on the educated community and become part and parcel of the general world picture - a theoretical science, I say, where this is forgotten, and where the initiated continue musing to each other in terms that are, at best, understood by a small group of close fellow travelers, will necessarily be cut off from the rest of cultural mankind; in the long run it is bound to atrophy and ossify however virulently esoteric chat may continue within its joyfully isolated groups of experts." (Erwin Schrödinger, "Are There Quantum Jumps?", The British Journal for the Philosophy of Science Vol. 3, 1952)
"As our mental eye penetrates into smaller and smaller distances and shorter and shorter times, we find nature behaving so entirely differently from what we observe in visible and palpable bodies of our surroundings that no model shaped after our large-scale experiences can ever be ‘true’. A complete satisfactory model of this type is not only practically inaccessible, but not even thinkable. Or, to be precise, we can, of course, think of it, but however we think it, it is wrong; not perhaps quite as meaningless as a ‘triangular circle’, but more so than a ‘winged lion’." (Erwin Schrödinger, "Science and Humanism", 1952)
"'Mathematics' appears to be at the bottom of everything, since we find it unexpectedly where we have not put it in." (Erwin Schrödinger, "Nature and the Greeks", 1954)
"In particular, and most importantly, this is the reason why the scientific worldview contains of itself no ethical values, no esthetical values, not a word about our own ultimate scope or destination, and no God, if you please." (Erwin Schrödinger, "Nature and the Greeks", 1954)
"[…] the scientific picture of the real world around me is very deficient. It gives a lot of factual information, puts all our experience in a magnificently consistent order, but it is ghastly silent about all and sundry that is really near to our heart, that really matters to us. It cannot tell us a word about red and blue, bitter and sweet, physical pain and physical delight; it knows nothing of beautiful and ugly, good or bad, God and eternity. Science sometimes pretends to answer questions in these domains, but the answers are very often so silly that we are not inclined to take them seriously." (Erwin Schrödinger, "Nature and the Greeks", 1954)
"Speaking without metaphor we have to declare that we are here faced with one of these typical antinomies caused by the fact that we have not yet succeeded in elaborating a fairly understandable outlook on the world without retiring our own mind, the producer of the world picture, from it, so that mind has no place in it. The attempt to press it into it, after all, necessarily produces some absurdities." (Erwin Schrödinger, „Mind and Matter: the Tarner Lectures", 1956)
"While the stuff from which our world picture is built is yielded exclusively from the sense organs as organs of the mind, so that every man's world picture is and always remains a construct of his mind and cannot be proved to have any other existence, yet the conscious mind itself remains a stranger within that construct, it has no living space in it, you can spot it nowhere in space." (Erwin Schrödinger, „Mind and Matter: the Tarner Lectures", 1956)
"Primary scientific papers are not meant to be final statement of indisputable truths; each is merely a tiny tentative step forward, through the jungle of ignorance." (Erwin Schrödinger, "Information, Communication, Knowledge", Nature Vol. 224 (5217), 1969)
01 August 2019
Eric T Bell - Collected Quotes
"Roughly it amounts to this: mathematical analysis as it works today must make use of irrational numbers (such as the square root of two); the sense if any in which such numbers exist is hazy. Their reputed mathematical existence implies the disputed theories of the infi nite. The paradoxes remain. Without a satisfactory theory of irrational numbers, among other things, Achilles does not catch up with the tortoise, and the earth cannot turn on its axis. But as Galileo remarked, it does. It would seem to follow that something is wrong with our attempts to compass the infinite." (Eric T Bell, "Debunking Science", 1930)
"The theory of numbers is the last great uncivilized continent of mathematics. It is split up into innumerable countries, fertile enough in themselves, but all the more or less indifferent to one another’s welfare and without a vestige of a central, intelligent government. If any young Alexander is weeping for a new world to conquer, it lies before him." (Eric T Bell, "The Queen of the Sciences", 1931)
"Pick your assumptions to pieces till the stuff they are made of is exposed to plain view." (Eric T Bell, 1935)
"All measurements in science are statistical in character." (Eric T Bell, "The Handmaiden of the Sciences", 1937)
"A circle no doubt has a certain appealing simplicity at the first glance, but one look at a healthy ellipse should have convinced even the most mystical of astronomers that that the perfect simplicity of the circle is akin to the vacant smile of complete idiocy. Compared to what an ellipse can tell us, a circle has nothing to say." (Eric T Bell, "The Handmaiden of the Sciences", 1937)
"The algebraic numbers are spotted over the plane like stars against a black sky; the dense blackness is the firmament of the transcendentals." (Eric T Bell, "Men of Mathematics", 1937)
"The longer mathematics lives the more abstract - and therefore, possibly also the more practical - it becomes." (Eric T Bell, "Men of Mathematics", 1937)
"The pursuit of pretty formulas and neat theorems can no doubt quickly degenerate into a silly vice, but so can the quest for austere generalities which are so very general indeed that they are incapable of application to any particular." (Eric T Bell, "Men of Mathematics", 1937)
"Abstractness, sometimes hurled as a reproach at mathematics, is its chief glory and its surest title to practical usefulness. It is also the source of such beauty as may spring from mathematics." (Eric T Bell, "The Development of Mathematics", 1940)
"A circle no doubt has a certain appealing simplicity at the first glance, but one look at a healthy ellipse should have convinced even the most mystical of astronomers that that the perfect simplicity of the circle is akin to the vacant smile of complete idiocy. Compared to what an ellipse can tell us, a circle has nothing to say." (Eric T Bell, "The Handmaiden of the Sciences", 1937)
"The algebraic numbers are spotted over the plane like stars against a black sky; the dense blackness is the firmament of the transcendentals." (Eric T Bell, "Men of Mathematics", 1937)
"The longer mathematics lives the more abstract - and therefore, possibly also the more practical - it becomes." (Eric T Bell, "Men of Mathematics", 1937)
"The pursuit of pretty formulas and neat theorems can no doubt quickly degenerate into a silly vice, but so can the quest for austere generalities which are so very general indeed that they are incapable of application to any particular." (Eric T Bell, "Men of Mathematics", 1937)
"Abstractness, sometimes hurled as a reproach at mathematics, is its chief glory and its surest title to practical usefulness. It is also the source of such beauty as may spring from mathematics." (Eric T Bell, "The Development of Mathematics", 1940)
"Nearly always it is the recondite and complicated which is elaborated first; and it is only when some relatively unsophisticated mind attacks a problem that its deep simplicity is revealed." (Eric T Bell, "The Development of Mathematics", 1940)
"There are no absolutes [...] in mathematics or in its history." (Eric Temple Bell, "The Development of Mathematics", 1940)
"The mistakes and unresolved difficulties of the past in mathematics have always been the opportunities of its future; and should analysis ever appear to be without or blemish, its perfection might only be that of death." (Eric T Bell, "The Development of Mathematics", 1940)
"The only royal road to elementary geometry is ingenuity." (Eric T Bell, "The Development of Mathematics", 1940)
"This abstracting of common experience is one of the principal sources of the utility of mathematics and the secret of its scientific power. The world that impinges on the senses of all but introverted solipsists is too intricate for any exact description yet imagined by human beings. By abstracting and simplifying the evidence of the senses, mathematics brings the worlds of science and daily life into focus with our myopic comprehension, and makes possible a rational description of our experiences which accords remarkably well with observation." (Eric T Bell, "The Development of Mathematics", 1940)
"Without the strictest deductive proof from admitted assumptions, explicitly stated as such, mathematics does not exist." (Eric T Bell, "The Development of Mathematics", 1940)
"The straight line of the geometers does not exist in the material universe. It is a pure abstraction, an invention of the imagination or, if one prefers, an idea of the Eternal Mind." (Eric T Bell, "The Magic of Numbers", 1946)
"Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions." (Eric T Bell, Mathematics Magazine, 1949)
"’Obvious’ is the most dangerous word in mathematics." (Eric T Bell, "Mathematics: Queen and Servant of Science", 1951)
"Science makes no pretension to eternal truth or absolute truth; some of its rivals do." (Eric T Bell, "Mathematics: Queen and Servant of Science", 1951)
"Without the strictest deductive proof from admitted assumptions, explicitly stated as such, mathematics does not exist." (Eric T Bell, "The Development of Mathematics", 1940)
"The straight line of the geometers does not exist in the material universe. It is a pure abstraction, an invention of the imagination or, if one prefers, an idea of the Eternal Mind." (Eric T Bell, "The Magic of Numbers", 1946)
"Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions." (Eric T Bell, Mathematics Magazine, 1949)
"’Obvious’ is the most dangerous word in mathematics." (Eric T Bell, "Mathematics: Queen and Servant of Science", 1951)
"Science makes no pretension to eternal truth or absolute truth; some of its rivals do." (Eric T Bell, "Mathematics: Queen and Servant of Science", 1951)
"Something more than impeccable logic is required in mathematics. An expert logician will not necessarily be a passable mathematician for all his skill in logic, any more than a scholarly prosodist will be a respectable poet for all his mastery of meter." (Eric T Bell, "Mathematics, Queen and Servant of Science", 1951)
"The technical analysis of any large collection of data is a task for a highly trained and expensive man who knows the mathematical theory of statistics inside and out. Otherwise the outcome is likely to be a collection of drawings - quartered pies, cute little battleships, and tapering rows of sturdy soldiers in diversified uniforms - interesting enough in the colored Sunday supplement, but hardly the sort of thing from which to draw reliable inferences." (Eric T Bell, "Mathematics: Queen and Servant of Science", 1951)
"Wherever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos." (Eric T Bell, "Mathematics, Queen and Servant of Science", 1951)
"Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness." (Eric T Bell)"Mathematics has a light and wisdom of its own, above any possible applications to science, and it will richly reward any intelligent human being to catch a glimpse of what mathematics means to itself. This is not the old doctrine of art for art's sake; it is art for humanity's sake." (Eric T Bell)
"To appreciate the living spirit rather than the dry bones of mathematics, it is necessary to inspect the work of a master at first hand.Textbooks and treatises are an unavoidable evil [...] The very crudities of the first attack on a significant problem by a master are more illuminating than all the pretty elegance of the standard texts which has been won at the cost of perhaps centuries of finicky polishing." (Eric T Bell, "Mathematics: Queen and Servant of Science", 1951)
Plato - Collected Quotes
"[Arithmetic] has a very great and elevating effect, compelling the soul to reason about abstract numbers, and rebelling against the introduction of visible or tangible objects into the argument." (Plato)
"As being is to become, so is truth to belief" (Plato, "Timaeus", cca. 360 BC)
"Astronomy compels the soul to look up upward and leads us from this world to another." (Plato, cca. 400 BC)
"Do not train a child to learn by force or harshness; but direct them to it by what amuses their minds, so that you may be better able to discover with accuracy the peculiar bent of the genius of each." (Plato)
"Education is teaching our children to desire the right things." (Plato)
"I can show you that the art of computation has to do with odd and even numbers in their numerical relations to themselves and to each other." (Plato, "Charmides", cca. 5 century BC)
"I know too well that these arguments from probabilities are imposters, and unless great caution is observed in the use of them, they are apt to be deceptive." (Plato," Phaedo" [On the Soul], 4th century BC)
"If in a discussion of many matters […] we are not able to give perfectly exact and self-consistent accounts, do not be surprised: rather we would be content if we provide accounts that are second to none in probability." (Plato, "Timaeus", cca. 360 BC)
"Number is the bond of the eternal continuance of things." (Plato)
"The knowledge of which geometry aims is the knowledge of the eternal." (Plato)
"The object of education is to teach us to love what is beautiful." (Plato, "The Republic")
"The qualities of number appear to lead to the apprehension of truth." (Plato)
"They say that the greatest and fairest things are the work of nature and of chance, the lesser of art, which, receiving from nature the greater and primeval creations, molds and fashions all those lesser works which are generally termed artificial." (Plato, "Nomoi" ["Laws"], cca. 360 BC)
"We must make a threefold distinction and think of that which becomes, that in which it becomes, and the model which it resembles" (Plato, "Timaeus")
"As being is to become, so is truth to belief" (Plato, "Timaeus", cca. 360 BC)
"Astronomy compels the soul to look up upward and leads us from this world to another." (Plato, cca. 400 BC)
"Do not train a child to learn by force or harshness; but direct them to it by what amuses their minds, so that you may be better able to discover with accuracy the peculiar bent of the genius of each." (Plato)
"Education is teaching our children to desire the right things." (Plato)
"I can show you that the art of computation has to do with odd and even numbers in their numerical relations to themselves and to each other." (Plato, "Charmides", cca. 5 century BC)
"I know too well that these arguments from probabilities are imposters, and unless great caution is observed in the use of them, they are apt to be deceptive." (Plato," Phaedo" [On the Soul], 4th century BC)
"If in a discussion of many matters […] we are not able to give perfectly exact and self-consistent accounts, do not be surprised: rather we would be content if we provide accounts that are second to none in probability." (Plato, "Timaeus", cca. 360 BC)
"Number is the bond of the eternal continuance of things." (Plato)
"The knowledge of which geometry aims is the knowledge of the eternal." (Plato)
"The object of education is to teach us to love what is beautiful." (Plato, "The Republic")
"The qualities of number appear to lead to the apprehension of truth." (Plato)
"They say that the greatest and fairest things are the work of nature and of chance, the lesser of art, which, receiving from nature the greater and primeval creations, molds and fashions all those lesser works which are generally termed artificial." (Plato, "Nomoi" ["Laws"], cca. 360 BC)
"[...] this knowledge of the kind [Le., mathematics] for which we are seeking, has a double use, military and philosophical; for the man of war must learn the art of number or he will not know how to array his troops, and the philosopher also, because he has to rise out of the sea of change and lay hold of true being, and therefore he must be an arithmetician." (Plato, "Republic" Book VII)
"We must make a threefold distinction and think of that which becomes, that in which it becomes, and the model which it resembles" (Plato, "Timaeus")
Aristotle - Collected Quotes
"A deduction is a demonstration when the premisses from which the deduction starts are true and primitive, or are such that our knowledge of them has originally come through premisses which are primitive and true." (Aristotle, "Topics", cca. 350 BC)
"A quantity is infinite if it is such that we can always take a part outside what has been already taken." (Aristotle, "Physics")
"All human actions have one or more of these seven causes: chance, nature, compulsions, habit, reason, passion, desire." (Aristotle)
"But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end." (Aristotle, "Generation of Animals")
"Educating the mind without educating the heart is no education at all." (Aristotle)
"A problem that is suitable for argumentative reasoning is one for which there are numerous and good arguments." (Aristotle, "Topics", cca. 350 BC)
"A quantity is infinite if it is such that we can always take a part outside what has been already taken." (Aristotle, "Physics")
"All human actions have one or more of these seven causes: chance, nature, compulsions, habit, reason, passion, desire." (Aristotle)
"But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end." (Aristotle, "Generation of Animals")
"Educating the mind without educating the heart is no education at all." (Aristotle)
"Education and morals will be found almost the whole that goes to make a good man." (Aristotle)
"Education is the best provision for old age." (Aristotle)
"Education is the best provision for old age." (Aristotle)
"For all who argue per impossibile infer by syllogism a false conclusion, and prove the original conclusion hypothetically when something impossible follows from a contradictory assumption, as, for example, that the diagonal [of a square] is incommensurable [with the side] because odd numbers are equal to even if it is assumed to be commensurate. It is inferred by syllogism that odd numbers are equal to even, and proved hypothetically that the diagonal is incommensurate, since a false conclusion follows from the contradictory assumption." (Aristotle, "Laws")
"For generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite […]." (Aristotle, "Physics")
"For that which is probable is that which generally happens." (Aristotle, "The Art of Rhetoric", 4th century BC)
"For the things we have to learn before we can do, we learn by doing." (Aristotle, "Nicomachean Ethics", Book II, 349 BC)
"[Imagination is] that in virtue of which we say that an image occurs to us and not as we speak of it metaphorically." (Aristotle, "De Anima" III, cca. 350 BC)
"In all disciplines in which there is systematic knowledge of things with principles, causes, or elements, it arises from a grasp of those: we think we have knowledge of a thing when we have found its primary causes and principles, and followed it back to its elements." (Aristotle, "Physics", cca. 350 BC)
"Intuition is the source of scientific knowledge." (Aristotle)
"Metaphysics is universal and is exclusively concerned with primary substance. […] And here we will have the science to study that which is, both in its essence and in the properties which, just as a thing that is, it has. […] That among entities there must be some cause which moves and combines things. […] There must then be a principle of such a kind that its substance is activity." (Aristotle, "Metaphysics", cca. 335-323 BC)
"Of the figures, the first is especially scientific. The mathematical sciences carry out their demonstrations through it – e.g. arithmetic and geometry and optics – and so do almost all those sciences which inquire into the reason why. For deductions giving the reason why are carried out, either in general or for the most part and in most cases, through this figure. For this reason, then, it is especially scientific; for study of the reason why has most importance for knowledge." (Aristotle, "Posterior Analytics", cca. 350 BC)
"Our account does not rob mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase, in the sense of the untraceable. In point of fact they do not need the infinite and do not use it. They postulate any that the finite straight line may be produced as far as they wish." (Aristotle, "Physics")
"For generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite […]." (Aristotle, "Physics")
"For that which is probable is that which generally happens." (Aristotle, "The Art of Rhetoric", 4th century BC)
"For the things we have to learn before we can do, we learn by doing." (Aristotle, "Nicomachean Ethics", Book II, 349 BC)
"[Imagination is] that in virtue of which we say that an image occurs to us and not as we speak of it metaphorically." (Aristotle, "De Anima" III, cca. 350 BC)
"If 'bounded by a surface' is the definition of body there cannot be an infinite body either intelligible or sensible." (Aristotle, "De Caelo" ["On the Heavens"], cca. 350 BC)
"In all disciplines in which there is systematic knowledge of things with principles, causes, or elements, it arises from a grasp of those: we think we have knowledge of a thing when we have found its primary causes and principles, and followed it back to its elements." (Aristotle, "Physics", cca. 350 BC)
"In all things which have a plurality of parts, and which are not a total aggregate but a whole of some sort distinct from the parts, there is some cause." (Aristotle, "Metaphysics", cca. 335-323 BC)
"Intuition is the source of scientific knowledge." (Aristotle)
"It is impossible that the existence of something [one thing] should necessitate both the existence and the non-existence of something else [i.e., one (same) other thing]." (Aristotle, "Posterior Analytics", cca. 350 BC)
"It is obvious then, that memory belongs to that part of the soul to which imagination belongs. […] Just as the picture painted on the panel is at once a picture and a portrait, and though one and the same, is both, yet the essence of the two is not the same, and it is possible to think of it both as a picture and as a portrait, so in the same way we must regard the mental picture within us both as an object of contemplation in itself and as a mental picture of something else […]. Insofar as we consider it in relation to something else, e.g. as a likeness, it is also an aid to memory." (Aristotle, "De Memoria et Reminiscentia" [On Memory and Recollection], 4th century BC)
"It is the mark of an educated mind to rest satisfied with the degree of precision which the nature of the subject admits and not to seek exactness where only an approximation is possible." (Aristotle, "Nicomachean Ethics", Book II, 349 BC)
"Knowledge, then, is a state of capacity to demonstrate, and has the other limiting characteristics which we specify in the Analytics; for it is when one believes in a certain way and the principles are known to him that he has knowledge, since if they are not better known to him than the conclusion, he will have his knowledge only on the basis of some concomitant." (Aristotle," Nicomachean Ethics", cca. 340 BC)
"It is obvious then, that memory belongs to that part of the soul to which imagination belongs. […] Just as the picture painted on the panel is at once a picture and a portrait, and though one and the same, is both, yet the essence of the two is not the same, and it is possible to think of it both as a picture and as a portrait, so in the same way we must regard the mental picture within us both as an object of contemplation in itself and as a mental picture of something else […]. Insofar as we consider it in relation to something else, e.g. as a likeness, it is also an aid to memory." (Aristotle, "De Memoria et Reminiscentia" [On Memory and Recollection], 4th century BC)
"It is the mark of an educated mind to rest satisfied with the degree of precision which the nature of the subject admits and not to seek exactness where only an approximation is possible." (Aristotle, "Nicomachean Ethics", Book II, 349 BC)
"Knowledge, then, is a state of capacity to demonstrate, and has the other limiting characteristics which we specify in the Analytics; for it is when one believes in a certain way and the principles are known to him that he has knowledge, since if they are not better known to him than the conclusion, he will have his knowledge only on the basis of some concomitant." (Aristotle," Nicomachean Ethics", cca. 340 BC)
"Metaphors do somehow make known what they mean by the comparison they involve. Whenever we use them, we transfer meaning according to some similarity. But statements such as the aforesaid do not make anything known. There is no inherent resemblance to justify calling law ‘a measure’ or ‘an image’ , nor is it customary to refer to it as such. If one says that law is literally ‘a measure’ or ‘an
image’, he is either deceiving or being deceived. An image is something fashioned in the likeness of something else, but such is not an inherent characteristic of law. If, on the other hand, the statement is not made in a literal sense, it is evident that it is obscure, and worse than any metaphorical expression." (Aristotle)
"Metaphysics is universal and is exclusively concerned with primary substance. […] And here we will have the science to study that which is, both in its essence and in the properties which, just as a thing that is, it has. […] That among entities there must be some cause which moves and combines things. […] There must then be a principle of such a kind that its substance is activity." (Aristotle, "Metaphysics", cca. 335-323 BC)
"Of magnitude that which (extends) one way is a line, that which (extends) two ways a
plane, and that which (extends) three ways a body. And there is no magnitude besides
these, because the dimensions are all that there are, and thrice extended means extended
all ways. For, as the Pythagoreans say, the All and all things in it are determined by three
things; end, middle and beginning give the number of the All, and these give the number
of the Triad." (Aristotle, "De Caelo" ["On the Heavens"], cca. 350 BC)
"Of the figures, the first is especially scientific. The mathematical sciences carry out their demonstrations through it – e.g. arithmetic and geometry and optics – and so do almost all those sciences which inquire into the reason why. For deductions giving the reason why are carried out, either in general or for the most part and in most cases, through this figure. For this reason, then, it is especially scientific; for study of the reason why has most importance for knowledge." (Aristotle, "Posterior Analytics", cca. 350 BC)
"Our account does not rob mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase, in the sense of the untraceable. In point of fact they do not need the infinite and do not use it. They postulate any that the finite straight line may be produced as far as they wish." (Aristotle, "Physics")
"Sense perception is a prerequisite for memory; the memory of frequently repeated sense perceptions results in experimental proof; experimental proofs provide the materials for a science or an art.” (Aristotle)
"So poetry is something more philosophical and more worthy of serious attention than history, for while poetry is concerned with universal truth, history treats of particular facts [...]" (Aristotle, "Poetics", cca. 350 BC)
"The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree. And since these (e.g. order and definiteness) are obviously causes of many things, evidently these sciences must treat this sort of causative principle also (i.e. the beautiful) as in some sense a cause." (Aristotle, "Metaphysica", cca. 350 BC)
"The infinite is imperfect, unfinished and therefore, unthinkable; it is formless and confused." (Aristotle)
"The intellectual capacity thinks the forms in the phantasmata (mental images) […] And for the following reason, as without having perceptual awareness no one could either learn or understand anything, so when one engages in intellectual activity one must at that time do so by means of a phantasma. For, phantasmata are just as perceptual states (aisthemata) are [in actual external perception] but without matter." (Aristotle, "De Anima" III, cca. 350 BC)
"The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful." (Aristotle, "Metaphysics", cca. 335-323 BC)
"The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree. And since these (e.g. order and definiteness) are obviously causes of many things, evidently these sciences must treat this sort of causative principle also (i.e. the beautiful) as in some sense a cause." (Aristotle, "Metaphysica", cca. 350 BC)
"The infinite is imperfect, unfinished and therefore, unthinkable; it is formless and confused." (Aristotle)
"The intellectual capacity thinks the forms in the phantasmata (mental images) […] And for the following reason, as without having perceptual awareness no one could either learn or understand anything, so when one engages in intellectual activity one must at that time do so by means of a phantasma. For, phantasmata are just as perceptual states (aisthemata) are [in actual external perception] but without matter." (Aristotle, "De Anima" III, cca. 350 BC)
"The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful." (Aristotle, "Metaphysics", cca. 335-323 BC)
"The only possible way to conceive universal is by induction, since we come to know abstractions by induction. But unless we have sense experience, we cannot make inductions. Even though sense perception relates to particular things, scientific knowledge concerning such can only be constructed by the successive steps of sense perception, induction, and formulation of universals." (Aristotle, "Posterior Analytics", cca. 350 BC)
"The point is prior to, and in itself more evident than the line. The same may be said of the line relative to the plane surface, and of the plane surface with reference to the solid. It is likewise true of unity in relation to plurality, for which unity is the principle. This also holds in regard to the letter relative to the syllable. [...] The reverse, however, sometimes occurs in the case of our knowledge. Generally the average mind more readily perceives what is posterior, whereas the comprehension of what is prior is reserved to the more profound and learned intellect." (Aristotle, "Topics", cca. 350 BC)
"The whole is more than the sum of its parts." (Aristotle, "Metaphysics", cca. 335-323 BC)
"Those who assert that the mathematical sciences make no affirmation about what is fair or good make a false assertion; for they do speak of these and frame demonstrations of them in the most eminent sense of the word. […] Of what is fair, however, the most important species are order and symmetry, and that which is definite, which the mathematical sciences make manifest in a most eminent degree." (Aristotle, "Metaphysics", cca. 335-323 BC)
"Time and space are divided into the same and equal divisions. Wherefore also, Zeno’s argument, that it is impossible to go through an infinite collection or to touch an infinite collection one by one in a finite time, is fallacious. For there are two senses in which the term ‘infinte’ is applied both to length and to time and in fact to all continuous things: either in regard to divisibility or in regard to number. Now it is not possible to touch things infinite as to number in a finite time, but it is possible to touch things infinite in regard to divisibility; for time itself is also infinite in this sense." (Aristotle)
"What we know is not capable of being otherwise; of things capable of being otherwise we do not know, when they have passed outsideour observation, whether they exist or not. Therefore the object of knowledge is of necessity. Therefore it is eternal; for things that are of necessity in the unqualified sense are all eternal; and things that are eternal are ungenerated and imperishable. " (Aristotle, "Nicomachean Ethics", cca. 340 BC)
"When the consequences of either assumption are the same, we should always assume that things are finite rather than infinite in number, since in things constituted by nature that which is infinite and that which is better ought, if possible, to be present rather than the reverse […]" (Aristotle)
"While those whom devotion to abstract discussions has rendered unobservant of the facts are too ready to dogmatize on the basis of a few observations." (Aristotle, "De Caelo" ["On the Heavens"], cca. 350 BC)
"The same ideas, one must believe, recur in men’s minds not once or twice but again and again." (Aristotle, "De Caelo" ["On the Heavens"], cca. 350 BC)
"The soul never thinks without a picture." (Aristotle)"The whole is more than the sum of its parts." (Aristotle, "Metaphysics", cca. 335-323 BC)
"Those who assert that the mathematical sciences make no affirmation about what is fair or good make a false assertion; for they do speak of these and frame demonstrations of them in the most eminent sense of the word. […] Of what is fair, however, the most important species are order and symmetry, and that which is definite, which the mathematical sciences make manifest in a most eminent degree." (Aristotle, "Metaphysics", cca. 335-323 BC)
"Time and space are divided into the same and equal divisions. Wherefore also, Zeno’s argument, that it is impossible to go through an infinite collection or to touch an infinite collection one by one in a finite time, is fallacious. For there are two senses in which the term ‘infinte’ is applied both to length and to time and in fact to all continuous things: either in regard to divisibility or in regard to number. Now it is not possible to touch things infinite as to number in a finite time, but it is possible to touch things infinite in regard to divisibility; for time itself is also infinite in this sense." (Aristotle)
"What is one is indivisible whatever it may be, e.g. a man is one man, not many. Number on the other hand is a plurality of 'ones' and a certain quantity of them. Hence number must stop at the indivisible: for 'two' and 'three' are merely derivative terms, and so with each of the other numbers." (Aristotle, "Physics", cca. 4th-century BC)
"What we know is not capable of being otherwise; of things capable of being otherwise we do not know, when they have passed outsideour observation, whether they exist or not. Therefore the object of knowledge is of necessity. Therefore it is eternal; for things that are of necessity in the unqualified sense are all eternal; and things that are eternal are ungenerated and imperishable. " (Aristotle, "Nicomachean Ethics", cca. 340 BC)
"When the consequences of either assumption are the same, we should always assume that things are finite rather than infinite in number, since in things constituted by nature that which is infinite and that which is better ought, if possible, to be present rather than the reverse […]" (Aristotle)
"While those whom devotion to abstract discussions has rendered unobservant of the facts are too ready to dogmatize on the basis of a few observations." (Aristotle, "De Caelo" ["On the Heavens"], cca. 350 BC)
Subscribe to:
Posts (Atom)
🌌On Cosmology: On Big Bang (2000-)
"A more extreme form of exponential growth was probably responsible for the start of the universe. Astronomer and physicists now genera...