Mental Models XXXVI

"Conscious apprehension seems to exist […] as happens in a mirror-image when the smooth and bright surface is peaceful." (Plotinus, "Enneads", cca. 270 AD)

“In the same way as regards the soul, when that kind of thing in us which mirrors the images of thought and intellect is undisturbed, we see them and know them in a way parallel to sense-perception, along with the prior knowledge that it is intellect and thought that are active. But when this is broken because the harmony of the body is upset, thought and intellect operate without an image, and then intellectual activity takes place without a mind-picture.” (Plotinus, “Enneads”, cca. 270 AD)

"The noetic act is without parts and has not, so to speak, come out into the open, but remains unobserved within, but the verbal expression unfolds its content and brings it out of the noetic act into the image making power, and so shows the noetic act as if in a mirror, and this is how there is conscious apprehension and persistence and memory of it." (Plotinus, "Enneads", cca. 270 AD)

"All the perceptions of the human mind resolve themselves into two distinct kinds, which I shall call impressions and ideas. The difference betwixt these consists in the degrees of force and liveliness, with which they strike upon the mind, and make their way into our thought or consciousness. Those perceptions, which enter with most force and violence, we may name impressions; and under this name I comprehend all our sensations, passions and emotions, as they make their first appearance in the soul. By ideas I mean the faint images of these in thinking and reasoning. I believe it will not be very necessary to employ many words in explaining this distinction." (David Hume, "A Treatise of Human Nature", 1738)

"You cannot crown the edifice by this abstraction. The scientific imagination, which is here authoritative, demands as the origin and cause of a series of ether waves a particle of vibrating matter quite as definite, though it may be excessively minute, as that which gives origin to a musical sound. Such a particle we name an atom or a molecule. I think the imagination when focused so as to give definition without penumbral haze, is sure to realise this image at last." (John Tyndall, "The Scientific Use of the Imagination", 1870) 

"The theory most prevalent among teachers is that mathematics affords the best training for the reasoning powers; […] The modem, and to my mind true, theory is that mathematics is the abstract form of the natural sciences; and that it is valuable as a training of the reasoning powers, not because it is abstract, but because it is a representation of actual things." (Truman H Safford, "Mathematical Teaching and Its Modern Methods", 1886)

"Images tell us nothing, either right or wrong, about the external world. […] It is just because forming images is a voluntary activity that it does not instruct us about the external world. […] When we form an image of something we are not observing. The coming and going of the pictures is not something that happens to us. We are not surprised by these pictures, saying ‘Look!’"  (Ludwig Wittgenstein, "Zettel", 1967)

“The power of images consists largely in the fact that they integrate different types of knowledge and experience.” (David Gooding, "Creative Rationality", 1996)

"When we talk of seeing an image, either in front of us or visualised with closed eyes, we invoke a range of metaphors and ideas which highlight the relationship between perception and imagery. For those of us with unimpaired vision, to see with ‘the mind’s eye’ conjures up a picture of perception where there is not a great deal of difference between an external or internal image." (Michael Forrester," Psychology of the Image", 2000) 

"[mental] images are sui generis, and should be added as a third great category of intentionality to the twin pillars of perception and cognition […] Neither is it at all obvious that images necessarily carry a [conceptual] thought component […] Images are not just minor variations on perception and thought, of negligible theoretical interest; they are a robust mental category in need of independent investigation." (Colin McGinn, Mindsight, 2009)

Plotinus - Collected Quotes

"All things are filled full of signs, and it is a wise man who can learn about one thing from another." (Plotinus, "Enneads", cca. 270 AD)

"But if anyone despises the arts because they produce their works by imitating nature, we must tell him, first, that natural things are imitations too. Then he must know that the arts do not simply imitate what they see, but they run back up to the forming principles from which nature derives" (Plotinus, "Enneads", cca. 270 AD)

"Conscious apprehension seems to exist […] as happens in a mirror-image when the smooth and bright surface is peaceful." (Plotinus, "Enneads", cca. 270 AD)

“In the same way as regards the soul, when that kind of thing in us which mirrors the images of thought and intellect is undisturbed, we see them and know them in a way parallel to sense-perception, along with the prior knowledge that it is intellect and thought that are active. But when this is broken because the harmony of the body is upset, thought and intellect operate without an image, and then intellectual activity takes place without a mind-picture.” (Plotinus, “Enneads”, cca. 270 AD)

"On the assumption that all happens by Cause, it is easy to discover the nearest determinants of any particular act or state to trace it plainly to them." (Plotinus, "Enneads", cca. 270 AD)

"[...] so external sensation is the image of this perception of the soul, which is in its essence truer and is a contemplation of forms alone without being affected. From these forms, from which the soul alone receives its lordship over the living being, come reasonings, and opinions and noetic acts; and this is precisely where ‘we’ are." (Plotinus, "Enneads", cca. 270 AD)

"The noetic act is without parts and has not, so to speak, come out into the open, but remains unobserved within, but the verbal expression unfolds its content and brings it out of the noetic act into the image making power, and so shows the noetic act as if in a mirror, and this is how there is conscious apprehension and persistence and memory of it." (Plotinus, "Enneads", cca. 270 AD)

"Beauty is rather a light that plays over the symmetry of things than that symmetry itself." (Plotinus)

"Knowledge has three degrees - opinion, science, illumination. The means or instrument of the first is sense; of the second, dialectic; of the third, intuition." (Plotinus)

"We must close our eyes and invoke a new manner of seeing […] a wakefulness that is the birthright of us all, though few put it to use." (Plotinus)

Nicholas of Cusa - Collected Quotes

"Wisdom is not to be found in the art of oratory, or in great books, but in a withdrawal from these sensible things and in a turning to the most simple and infinite forms. You will learn how to receive it into a temple purged from all vice, and by fervent love to cling to it until you may taste it and see how sweet That is which is all sweetness. Once this has been tasted, all things which you now consider as important will appear as vile, and you will be so humbled that no arrogance or other vice will remain in you. Once having tasted this wisdom, you will inseparably adhere to it with a chaste and pure heart. You will choose rather to forsake this world and all else that is not of this wisdom, and living with unspeakable happiness you will die." (Nicholas of Cusa [Nicolaus Cusanus], "De Docta Ignorantia" ["On Learned Ignorance"], 1440)

"Now discourse is necessarily limited by its point of departure and its point of arrival, and since these are in mutual opposition we speak of contradiction. For the discursive reason these terms are opposed and distinct. In the realm of the reason, therefore, there is a necessary disjunction between extremes, as, for example, in the rational definition of the circle where the lines from the center to the circumference are equal and where the center cannot coincide with the circumference." (Nicholas of Cusa, "Apologia Doctae ignorantiae" ["The Defense of Learned Ignorance"], 1449)

"Mind is the limit and measure of all things [...]" (Nicholas of Cusa [Nicolaus Cusanus], "Idiota de mente: The Layman: About Mind", 1450)

"You know how the divine Simplicity enfolds all things. Mind is the image of this enfolding Simplicity. If, then, you called this divine Simplicity infinite Mind, it will be the exemplar of our mind. If you called the divine mind the totality of the truth of things, you will call our mind the totality of the assimilation of things, so that it may be a totality of ideas. In the divine Mind conception is the production of things; in our mind conception is the knowledge of things. If the divine Mind is absolute Being, then its conception is the creation of beings; and conception in the human mind is the assimilation of beings." (Nicholas of Cusa [Nicolaus Cusanus], "Idiota de mente: The Layman: About Mind", 1450)

"[…] a great multitude cannot exist without much diversity […]" (Nicholas of Cusa[Nicolaus Cusanus], "De Pace Fidei" ["The Peace of Faith"], 1453)

"There can only be one wisdom. For if it were possible that there be several wisdoms, then these would have to be from one. Namely, unity is prior to all plurality." (Nicholas of Cusa [Nicolaus Cusanus], "De Pace Fidei" ["The Peace of Faith"], 1453)

"Within itself the soul sees all things more truly than as they exist in different things outside itself. And the more it goes out unto other things in order to know them, the more it enters into itself in order to know itself." (Nicholas of Cusa [Nicolaus Cusanus], "On Equality", 1459)

"If full knowledge about the very base of our existence could be described as a circle, the best we can do is to arrive at a polygon." (Nicholas of Cusa [Nicolaus Cusanus])

"The universe has no circumference, for if it had a center and a circumference there would be some and some thing beyond the world, suppositions which are wholly lacking in truth. Since, therefore, it is impossible that the universe should be enclosed within a corporeal center and corporeal boundary, it is not within our power to understand the universe, whose center and circumference are God. And though the universe." (Nicholas of Cusa[Nicolaus Cusanus])

"Time is to eternity as an image is to its exemplar, and those things which are temporal bear a resemblance to those things which are eternal." (Nicholas of Cusa[Nicolaus Cusanus])

Mental Models XXXV

"Here I am at the limit which God and nature has assigned to my individuality. I am compelled to depend upon word, language and image in the most precise sense, and am wholly unable to operate in any manner whatever with symbols and numbers which are easily intelligible to the most highly gifted minds." (Johann Wolfgang von Goethe, [Letter to Naumann] 1826)

"It may sound quite strange, but for me, as for other scientists on whom these kinds of imaginative images have a greater effect than other poems do, no science is at its very heart more closely related to poetry, perhaps, than is chemistry." (Just Liebig, 1854)

"But I thoroughly believe myself, and hope to prove to you, that science is full of beautiful pictures, of real poetry, and of wonder-working fairies; and what is more […] though they themselves will always remain invisible, yet you will see their wonderful power at work everywhere around you. […] There is only one gift we must have before we can learn to know them - we must have imagination. I do not mean mere fancy, which creates unreal images and impossible monsters, but imagination, the power of making pictures or images in our mind, of that which is, though it is invisible to us." (Arabella Buckley, Fairyland, 1879)

"Ask your imagination if it will accept a vibrating multiple proportion - a numerical ratio in a state of oscillation? I do not think it will. You cannot crown the edifice with this abstraction. The scientific imagination, which is here authoritative, demands, as the origin and cause of a series of ether-waves, a particle of vibrating matter quite as definite, though it may be excessively minute, as that which gives origin to a musical sound. Such a particle we name an atom or a molecule. I think the intellect, when focused so as to give definition without penumbral haze, is sure to realize this image at the last." (John Tyndall, "Fragments of Science", 1892)

"The mathematical formula is the point through which all the light gained by science passes in order to be of use to practice; it is also the point in which all knowledge gained by practice, experiment, and observation must be concentrated before it can be scientifically grasped. The more distant and marked the point, the more concentrated will be the light coming from it, the more unmistakable the insight conveyed. All scientific thought, from the simple gravitation formula of Newton, through the more complicated formulae of physics and chemistry, the vaguer so called laws of organic and animated nature, down to the uncertain statements of psychology and the data of our social and historical knowledge, alike partakes of this characteristic, that it is an attempt to gather up the scattered rays of light, the different parts of knowledge, in a focus, from whence it can be again spread out and analyzed, according to the abstract processes of the thinking mind. But only when this can be done with a mathematical precision and accuracy is the image sharp and well-defined, and the deductions clear and unmistakable. As we descend from the mechanical, through the physical, chemical, and biological, to the mental, moral, and social sciences, the process of focalization becomes less and less perfect, - the sharp point, the focus, is replaced by a larger or smaller circle, the contours of the image become less and less distinct, and with the possible light which we gain there is mingled much darkness, the sources of many mistakes and errors. But the tendency of all scientific thought is toward clearer and clearer definition; it lies in the direction of a more and more extended use of mathematical measurements, of mathematical formulae." (John T Merz, "History of European Thought in the 19th Century" Vol. 1, 1904)

"Many people believe that reasoning, and therefore science, is a different activity from imagining. But this is a fallacy […] Reasoning is constructed with movable images just as certainly as poetry is." (Jacob Bronowski, "Visionary Eye", 1978)

"The thinking person goes over the same ground many times. He looks at it from varying points of view - his own, his arch-enemy’s, others’. He diagrams it, verbalizes it, formulates equations, constructs visual images of the whole problem, or of troublesome parts, or of what is clearly known. But he does not keep a detailed record of all this mental work, indeed could not. […] Deep understanding of a domain of knowledge requires knowing it in various ways. This multiplicity of perspectives grows slowly through hard work and sets the state for the re-cognition we experience as a new insight." (Howard E Gruber, "Darwin on Man", 1981)

"Mathematicians have always needed to ‘see’ the complex concepts they work with in order to reason with them effectively. In the past, they conjured up mental images as best they could, but the wonders of computer graphics provide them with far more detailed pictures to think with." (Richard Palais and Luc Bernard, "2006 Visualization Project", 2006)

"Time is to eternity as an image is to its exemplar, and those things which are temporal bear a resemblance to those things which are eternal." (Nicholas of Cusa)

"You know how the divine Simplicity enfolds all things. Mind is the image of this enfolding Simplicity. If, then, you called this divine Simplicity infinite Mind, it will be the exemplar of our mind. If you called the divine mind the totality of the truth of things, you will call our mind the totality of the assimilation of things, so that it may be a totality of ideas. In the divine Mind conception is the production of things; in our mind conception is the knowledge of things. If the divine Mind is absolute Being, then its conception is the creation of beings; and conception in the human mind is the assimilation of beings." (Nicholas of Cusa)

Hans Reichenbach - Collected Quotes

"It is characteristic of modern physics to represent all processes in terms of mathematical equations. But the close connection between the two sciences must not blur their essential difference." (Hans Reichenbach, "The Theory of Relativity and A Priori Knowledge", 1920)

"The mathematical object of knowledge is uniquely determined by the axioms and definitions of mathematics." (Hans Reichenbach, "The Theory of Relativity and A Priori Knowledge", 1920)

"The physical object cannot be determined by axioms and definitions. It is a thing of the real world, not an object of the logical world of mathematics. Offhand it looks as if the method of representing physical events by mathematical equations is the same as that of mathematics. Physics has developed the method of defining one magnitude in terms of others by relating them to more and more general magnitudes and by ultimately arriving at 'axioms', that is, the fundamental equations of physics. Yet what is obtained in this fashion is just a system of mathematical relations. What is lacking in such system is a statement regarding the significance of physics, the assertion that the system of equations is true for reality." (Hans Reichenbach, "The Theory of Relativity and A Priori Knowledge", 1920)

"If we wish to express our ideas in terms of the concepts synthetic and analytic, we would have to point out that these concepts are applicable only to sentences that can be either true of false, and not to definitions. The mathematical axioms are therefore neither synthetic nor analytic, but definitions. [....] Hence the question of whether axioms are a priori becomes pointless since they are arbitrary." (Hans Reichenbach, "The Philosophy of Space and Time", 1928) 

"Once a definition of congruence is given, the choice of geometry is no longer in our hands; rather, the geometry is now an empirical fact." (Hans Reichenbach, "The Philosophy of Space and Time", 1928)

"Whereas the conception of space and time as a four-dimensional manifold has been very fruitful for mathematical physicists, its effect in the field of epistemology has been only to confuse the issue. Calling time the fourth dimension gives it an air of mystery. One might think that time can now be conceived as a kind of space and try in vain to add visually a fourth dimension to the three dimensions of space. It is essential to guard against such a misunderstanding of mathematical concepts. If we add time to space as a fourth dimension it does not lose any of its peculiar character as time." (Hans Reichenbach, "The Philosophy of Space and Time", 1928)

"The urge to knowledge is so deeply rooted in man that it can scarcely be omitted from a list of life's important needs." (Hans Reichenbach, "Atom and Cosmos: The World of Modern Physics", 1933)

"The reliance on the concrete is the basis of both the charm and the power of physical research." (Hans Reichenbach, "Atom and Cosmos: The World of Modern Physics", 1933) 

"When science says that a law is valid, it means but one thing - that the law permits conclusions as to future observations." (Hans Reichenbach, "Atom and Cosmos: The World of Modern Physics", 1933)

“Geometrical truth is a product of reason; that makes it superior to empirical truth, which is found through generalization of a great number of instances.” (Hans Reichenbach, “The Rise of Scientific Philosophy”, 1951)

"If error is corrected whenever it is recognized as such, the path of error is the path of truth." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"Instead of asking for a cause of the universe, the scientist can ask only for the cause of the present state of the universe; and his task will consist in pushing farther and farther back the date from which he is able to account for the universe in terms of laws of nature." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"The belief in science has replaced in large measure, the belief in God. Even where religion was regarded as compatible with science, it was modified by the mentality of the believer in scientific truth." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"The essence of knowledge is generalization. That fire can be produced by rubbing wood in a certain way is a knowledge derived by generalization from individual experiences; the statement means that rubbing wood in this way will always produce fire. The art of discovery is therefore the art of correct generalization." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"The philosopher of science is not much interested in the thought processes which lead to scientific discoveries; he looks for a logical analysis of the completed theory, including the relationships establishing its validity. That is, he is not interested in the context of discovery, but in the context of justification" (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"The scientist who discovers a theory is usually guided to his discovery by guesses; he cannot name a method by means of which he found the theory and can only say that it appeared plausible to him, that he had the right hunch or that he saw intuitively which assumption would fit the facts." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"The study of inductive inference belongs to the theory of probability, since observational facts can make a theory only probable but will never make it absolutely certain." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"To say that observations of the past are certain, whereas predictions are merely probable, is not the ultimate answer to the question of induction; it is only a sort of intermediate answer, which is incomplete unless a theory of probability is developed that explains what we should mean by ‘probable’ and on what ground we can assert probabilities." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"There is no logical necessity for the existence of a unique direction of total time; whether there is only one time direction, or whether time directions alternate, depends on the shape of the entropy curve plotted by the universe." (Hans Reichenbach, "The Direction of Time", 1956)

"He who searches for truth must not appease his urge by giving himself up to the narcotic of belief." (Hans Reichenbach) 

On Abstraction (1900-1910)

"Our science, in contrast with others, is not founded on a single period of human history, but has accompanied the development of culture through all its stages. Mathematics is as much interwoven with Greek culture as with the most modern problems in Engineering. She not only lends a hand to the progressive natural sciences but participates at the same time in the abstract investigations of logicians and philosophers." (Felix Klein, "Klein und Riecke: Ueber angewandte Mathematik und Physik" 1900)

"The man of science deals with questions which commonly lie outside of the range of ordinary experience, which often have no immediately discernible relation to the affairs of everyday life, and which concentrate the mind upon apparent abstractions to an extraordinary degree." (Frank W Clarke, "The Man of Science in Practical Affairs", Appletons' Popular Science Monthly Vol. XLV, 1900)

"A mathematical theorem and its demonstration are prose. But if the mathematician is overwhelmed with the grandeur and wondrous harmony of geometrical forms, of the importance and universal application of mathematical maxims, or, of the mysterious simplicity of its manifold laws which are so self-evident and plain and at the same time so complicated and profound, he is touched by the poetry of his science; and if he but understands how to give expression to his feelings, the mathematician turns poet, drawing inspiration from the most abstract domain of scientific thought." (Paul Carus, „Friedrich Schiller: A Sketch of His Life and an Appreciation of His Poetry", 1905)

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

[…] theory of numbers lies remote from those who are indifferent; they show little interest in its development, indeed they positively avoid it. [..] the pure theory of numbers is an extremely abstract thing, and one does not often find the gift of ability to understand with pleasure anything so abstract."  (Felix Klein, "Elementary Mathematics from an Advanced Standpoint", 1908)

On Abstraction (1940-1949)

"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)

"It is difficult, however, to learn all these things from situations such as occur in everyday life. What we need is a series of abstract and quite impersonal situations to argue about in which one side is surely right and the other surely wrong. The best source of such situations for our purposes is geometry. Consequently we shall study geometric situations in order to get practice in straight thinking and logical argument, and in order to see how it is possible to arrange all the ideas associated with a given subject in a coherent, logical system that is free from contradictions. That is, we shall regard the proof of each proposition of geometry as an example of correct method in argumentation, and shall come to regard geometry as our ideal of an abstract logical system. Later, when we have acquired some skill in abstract reasoning, we shall try to see how much of this skill we can apply to problems from real life." (George D Birkhoff & Ralph Beately, "Basic Geometry", 1940)

"[…] there is probably less difference between the positions of a mathematician and of a physicist than is generally supposed, [...] the mathematician is in much more direct contact with reality. This may seem a paradox, since it is the physicist who deals with the subject-matter usually described as 'real', but [...] [a physicist] is trying to correlate the incoherent body of crude fact confronting him with some definite and orderly scheme of abstract relations, the kind of scheme he can borrow only from mathematics." (Godfrey H Hardy, "A Mathematician's Apology", 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)

"We now come to a decisive step of mathematical abstraction: we forget about what the symbols stand for […] The mathematician] need not be idle; there are many operations which he may carry out with these symbols, without ever having to look at the things they stand for." (Hermann Weyl, "The Mathematical Way of Thinking", 1940)

"It is to be hoped that in the future more and more theoretical physicists will command a deep knowledge of mathematical principles; and also that mathematicians will no longer limit themselves so exclusively to the aesthetic development of mathematical abstractions." (George D Birkhoff, "Mathematical Nature of Physical Theories" American Scientific Vol. 31 (4), 1943)

"Mathematics being a very abstract science should be presented very concretely." (George Pólya, "How to Solve It", 1945)

"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)

"I think that it is a relatively good approximation to truth - which is much too complicated to allow anything but approximations - that mathematical ideas originate in empirics. But, once they are conceived, the subject begins to live a peculiar life of its own and is […] governed by almost entirely aesthetical motivations. In other words, at a great distance from its empirical source, or after much ‘abstract’ inbreeding, a mathematical subject is in danger of degeneration. Whenever this stage is reached the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas." (John von Neumann,  "The Mathematician", The Works of the Mind Vol. I (1), 1947)

On Abstraction (1910-1919)

"Poetry is a sort of inspired mathematics, which gives us equations, not for abstract figures, triangles, squares, and the like, but for the human emotions. If one has a mind which inclines to magic rather than science, one will prefer to speak of these equations as spells or incantations; it sounds more arcane, mysterious, recondite. " (Ezra Pound, "The Spirit of Romance", 1910)

"The ordinary mathematical treatment of any applied science substitutes exact axioms for the approximate results of experience, and deduces from these axioms the rigid mathematical conclusions. In applying this method it must not be forgotten that the mathematical developments transcending the limits of exactness of the science are of no practical value. It follows that a large portion of abstract mathematics remains without finding any practical application, the amount of mathematics that can be usefully employed in any science being in proportion to the degree of accuracy attained in the science. Thus, while the astronomer can put to use a wide range of mathematical theory, the chemist is only just beginning to apply the first derivative, i. e. the rate of change at which certain processes are going on; for second derivatives he does not seem to have found any use as yet." (Felix Klein, "Lectures on Mathematics", 1911)

"The belief that mathematics, because it is abstract, because it is static and cold and gray, is detached from life, is a mistaken belief. Mathematics, even in its purest and most abstract estate, is not detached from life. It is just the ideal handling of the problems of life, as sculpture may idealize a human figure or as poetry or painting may idealize a figure or a scene. Mathematics is precisely the ideal handling of the problems of life, and the central ideas of the science, the great concepts about which its stately doctrines have been built up, are precisely the chief ideas with which life must always deal and which, as it tumbles and rolls about them through time and space, give it its interests and problems, and its order and rationality. " (Cassius J Keyser, "The Humanization of the Teaching of Mathematics", 1912)

"Even the most refined statistics are nothing but abstractions." (Walter Lippmann, "Politics, The Golden Rule and After", 1913)

"[…] science deals with but a partial aspect of reality, and there is no faintest reason for supposing that everything science ignores is less real than what it accepts. [...] Why is it that science forms a closed system? Why is it that the elements of reality it ignores never come in to disturb it? The reason is that all the terms of physics are defined in terms of one another. The abstractions with which physics begins are all it ever has to do with." (John W N Sullivan, "The Limitations of Science", 1915)

"Abstract as it is, science is but an outgrowth of life. That is what the teacher must continually keep in mind. […] Let him explain […] science is not a dead system - the excretion of a monstrous pedantism - but really one of the most vigorous and exuberant phases of human life." (George A L Sarton, "The Teaching of the History of Science", The Scientific Monthly, 1918)

On Abstraction (1960-1969)

"It is of our very nature to see the universe as a place that we can talk about. In particular, you will remember, the brain tends to compute by organizing all of its input into certain general patterns. It is natural for us, therefore, to try to make these grand abstractions, to seek for one formula, one model, one God, around which we can organize all our communication and the whole business of living." (John Z Young, "Doubt and Certainty in Science: A Biologist’s Reflections on the Brain", 1960)

"Relativity is inherently convergent, though convergent toward a plurality of centers of abstract truths. Degrees of accuracy are only degrees of refinement and magnitude in no way affects the fundamental reliability, which refers, as directional or angular sense, toward centralized truths. Truth is a relationship." (R Buckminster Fuller, "The Designers and the Politicians", 1962)

"Scientists, it should already be clear, never learn concepts, laws, and theories in the abstract and by themselves. Instead, these intellectual tools are from the start encountered in a historically and pedagogically prior unit that displays them with and through their applications." (Thomas Kuhn, "The Structure of Scientific Revolutions", 1962)

"With even a superficial knowledge of mathematics, it is easy to recognize certain characteristic features: its abstractions, its precision, its logical rigor, the indisputable character of its conclusions, and finally, the exceptionally broad range for its applications." (Aleksandr D Aleksandrov, 1963)

"A quantity like time, or any other physical measurement, does not exist in a completely abstract way. We find no sense in talking about something unless we specify how we measure it. It is the definition by the method of measuring a quantity that is the one sure way of avoiding talking nonsense..." (Hermann Bondi. "Relativity and Common Sense", 1964)

"If you have a large number of unrelated ideas, you have to get quite a distance away from them to get a view of all of them, and this is the role of abstraction. If you look at each too closely you see too many details. If you get far away things may appear simpler because you can only see the large, broad outlines; you do not get lost in petty details." (John G Kemeny, "Random Essays on Mathematics, Education, and Computers", 1964)

"The interplay between generality and individuality, deduction and construction, logic and imagination - this is the profound essence of live mathematics. Anyone or another of these aspects of mathematics can be found at the center of a given achievement. In a far reaching development all of them will be involved. Generally speaking, such a development will start from the 'concrete', then discard ballast by abstraction and rise to the lofty layers of thin air where navigation and observation are easy: after this flight comes the crucial test for learning and reaching specific goals in the newly surveyed low plains of individual 'reality'. In brief, the flight into abstract generality must start from and return again to the concrete and specific." (Richard Courant, "Mathematics in the Modern World", Scientific American Vol. 211 (3), 1964) 

"A more problematic example is the parallel between the increasingly abstract and insubstantial picture of the physical universe which modern physics has given us and the popularity of abstract and non-representational forms of art and poetry. In each case the representation of reality is increasingly removed from the picture which is immediately presented to us by our senses." (Harvey Brooks, "Scientific Concepts and Cultural Change", 1965)

"Learning is any change in a system that produces a more or less permanent change in its capacity for adapting to its environment. Understanding systems, especially systems capable of understanding problems in new task domains, are learning systems." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"The more we are willing to abstract from the detail of a set of phenomena, the easier it becomes to simulate the phenomena. Moreover we do not have to know, or guess at, all the internal structure of the system but only that part of it that is crucial to the abstraction." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"We realize, however, that all scientific laws merely represent abstractions and idealizations expressing certain aspects of reality. Every science means a schematized picture of reality, in the sense that a certain conceptual construct is unequivocally related to certain features of order in reality […]" (Ludwig von Bertalanffy, "General System Theory", 1968)

"Pure mathematics are concerned only with abstract propositions, and have nothing to do with the realities of nature. There is no such thing in actual existence as a mathematical point, line or surface. There is no such thing as a circle or square. But that is of no consequence. We can define them in words, and reason about them. We can draw a diagram, and suppose that line to be straight which is not really straight, and that figure to be a circle which is not strictly a circle. It is conceived therefore by the generality of observers, that mathematics is the science of certainty." (William Godwin, "Thoughts on Man", 1969)

On Abstraction (1850-1899)

"Science gains from it [the pendulum] more than one can expect. With its huge dimensions, the apparatus presents qualities that one would try in vain to communicate by constructing it on a small [scale], no matter how carefully. Already the regularity of its motion promises the most conclusive results. One collects numbers that, compared with the predictions of theory, permit one to appreciate how far the true pendulum approximates or differs from the abstract system called 'the simple pendulum'." (Jean-Bernard-Léon Foucault, "Demonstration Experimentale du Movement de Rotation de la Terre", 1851)

"Beyond the little arithmetic required for the ordinary economies of life, the mass of college-bred men, unless engaged in the business of instruction or in pursuits which directly involve their application, from the time they leave their places of education, of whatever name, give up the Mathematics as a useless and hopeless abstraction." (Edward Everett, [address] 1857)

"The Mathematics, like language, (of which indeed they may be considered a species,) comprehending under that designation the whole science of number, space, form, time, and motion, as far as it can be expressed in abstract formulas, are evidently not only one of the most useful, but one of the grandest of studies." (Edward Everett, [address] 1857)

"However rapid and remote their flight of thought, it is a succession of images, not of abstractions. The details which give significance, and which by us are seen vaguely as through a vanishing mist, are by them seen in sharp outlines. The image which to us is a mere suggestion, is to them almost as vivid as the object. And it is because they see vividly that they can paint effectively." (George H Lewes, "The Principles of Success in Literature", 1865)

"Observe this: the abstraction of the philosopher is meant to keep the object itself, with its perturbing suggestions, out of sight, allowing only one quality to fill the field of vision; whereas the abstraction of the poet is meant to bring the object itself into more vivid relief, to make it visible by means of the selected qualities. In other words, the one aims at abstract symbols, the other at picturesque effects. The one can carry on his deductions by the aid of colourless signs, X or Y. The other appeals to the emotions through the symbols which will most vividly express the real objects in their relations to our sensibilities." (George H Lewes, "The Principles of Success in Literature", 1865)

"In abstract mathematical theorems the approximation to absolute truth is perfect, because we can treat of infinitesimals. In physical science, on the contrary, we treat of the least quantities which are perceptible." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"Purely mechanical phenomena do not exist […] are abstractions, made, either intentionally or from necessity, for facilitating our comprehension of things. The science of mechanics does not comprise the foundations, no, nor even a part of the world, but only an aspect of it." (Ernst Mach, "The Science of Mechanics", 1883)

"The theory most prevalent among teachers is that mathematics affords the best training for the reasoning powers; […] The modem, and to my mind true, theory is that mathematics is the abstract form of the natural sciences; and that it is valuable as a training of the reasoning powers, not because it is abstract, but because it is a representation of actual things." (Truman H Safford, "Mathematical Teaching and Its Modern Methods", 1886)

"[In mathematics] we behold the conscious logical activity of the human mind in its purest and most perfect form. Here we learn to realize the laborious nature of the process, the great care with which it must proceed, the accuracy which is necessary to determine the exact extent of the general propositions arrived at, the difficulty of forming and comprehending abstract concepts; but here we learn also to place confidence in the certainty, scope and fruitfulness of such intellectual activity." (Hermann Helmholtz, "Vorträge und Reden", 1896)

"In mathematics we see the conscious logical activity of our mind in its purest and most perfect form; here is made manifest to us all the labor and the great care with which it progresses, the precision which is necessary to determine exactly the source of the established general theorems, and the difficulty with which we form and comprehend abstract conceptions; but we also learn here to have confidence in the certainty, breadth, and fruitfulness of such intellectual labor." (Hermann von Helmholtz, "Vorträge und Reden", 1896)

"Mathematics is the most abstract of all the sciences. For it makes no external observations, nor asserts anything as a real fact. When the mathematician deals with facts, they become for him mere ‘hypotheses’; for with their truth he refuses to concern himself. The whole science of mathematics is a science of hypotheses; so that nothing could be more completely abstracted from concrete reality." (Charles S Peirce, "The Regenerated Logic", The Monist Vol. 7 (1), 1896)

"In order to comprehend and fully control arithmetical concepts and methods of proof, a high degree of abstraction is necessary, and this condition has at times been charged against arithmetic as a fault. I am of the opinion that all other fields of knowledge require at least an equally high degree of abstraction as mathematics, - provided, that in these fields the foundations are also everywhere examined with the rigour and completeness which is actually necessary." (David Hilbert, "Die Theorie der algebraischen Zahlkorper", 1897)

On Abstraction (1930-1939)

"Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field." (Paul A M Dirac, "The Principles of Quantum Mechanics", 1930)

"The steady progress of physics requires for its theoretical formulation a mathematics which get continually more advanced. […] it was expected that mathematics would get more and more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundation and gets more abstract. Non-Euclidean geometry and noncommutative algebra, which were at one time were considered to be purely fictions of the mind and pastimes of logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and the advance in physics is to be associated with continual modification and generalisation of the axioms at the base of mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation." (Paul A M Dirac, "Quantities singularities in the electromagnetic field", Proceedings of the Royal Society of London, 1931)

"The fundamental concepts of physical science, it is now understood, are abstractions, framed by our mind, so as to bring order to an apparent chaos of phenomena." (Sir William C Dampier, "A History of Science and its Relations with Philosophy & Religion", 1931)

"It is the function of notions in science to be useful, to be interesting, to be verifiable and to acquire value from anyone of these qualities. Scientific notions have little to gain as science from being forced into relation with that formidable abstraction, ‘general truth’." (Wilfred Trotter, [paper delivered before the Royal College of Surgeons of England] 1932)

"We love to discover in the cosmos the geometrical forms that exist in the depths of our consciousness. The exactitude of the proportions of our monuments and the precision of our machines express a fundamental character of our mind. Geometry does not exist in the earthly world. It has originated in ourselves. The methods of nature are never so precise as those of man. We do not find in the universe the clearness and accuracy of our thought. We attempt, therefore, to abstract from the complexity of phenomena some simple systems whose components bear to one another certain relations susceptible of being described mathematically." (Alexis Carrel, "Man the Unknown", 1935)

„[...] the abstract mathematical theory has an independent, if lonely existence of its own. But when a sufficient number of its terms are given physical definitions it becomes a part of a vital organism concerning itself at every instant with matters full of human significance. Every theorem can be given the form ‘if you do so and so, such and such will happen'." (Oswald Veblen, "Remarks on the Foundation of Geometry", Bulletin of the American Mathematical Society, Vol. 35, 1935)

"Given any domain of thought in which the fundamental objective is a knowledge that transcends mere induction or mere empiricism, it seems quite inevitable that its processes should be made to conform closely to the pattern of a system free of ambiguous terms, symbols, operations, deductions; a system whose implications and assumptions are unique and consistent; a system whose logic confounds not the necessary with the sufficient where these are distinct; a system whose materials are abstract elements interpretable as reality or unreality in any forms whatsoever provided only that these forms mirror a thought that is pure. To such a system is universally given the name Mathematics." (Samuel T. Sanders, "Mathematics", National Mathematics Magazine, 1937)

"Sooner or later the cold plunge into pure abstraction must be taken if one is to learn to swim in mathematics and to reason as rational, thinking human beings do." (Eric T Bell, "The Handmaiden of the Sciences", 1937)

"The longer mathematics lives the more abstract - and therefore, possibly also the more practical - it becomes." (Eric T Bell, "Men of Mathematics", 1937)

"Matter-of-fact is an abstraction, arrived at by confining thought to purely formal relations which then masquerade as the final reality. This is why science, in its perfection, relapses into the study of differential equations. The concrete world has slipped through the meshes of the scientific net." (Alfred N Whitehead, "Modes of Thought", 1938)

On Abstraction (1950-1959)

"The first thing to realize about physics […] is its extraordinary indirectness. […] For physics is not about the real world, it is about 'abstractions' from the real world, and this is what makes it so scientific. […] Theoretical physics runs merrily along with these unreal abstractions, but its conclusions are checked, at every possible point, by experiments." (Anthony Standen, "Science is a Sacred Cow", 1950)

"In mathematics […] we find two tendencies present. On the one hand, the tendency towards abstraction seeks to crystallise the logical relations inherent in the maze of materials [….] being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency towards intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations." (David Hilbert, "Geometry and the imagination", 1952)

"There is nothing mysterious, as some have tried to maintain, about the applicability of mathematics. What we get by abstraction from something can be returned. (Raymond L Wilder, Introduction to the Foundations of Mathematics, 1952)

"The theory of relativity is a fine example of the fundamental character of the modern development of theoretical science. The initial hypotheses become steadily more abstract and remote from experience. On the other hand, it gets nearer to the grand aim of all science, which is to cover the greatest possible number of empirical facts by logical deduction from the smallest possible number of hypotheses or axioms." (Albert Einstein, 1954)

"Beauty had been born, not, as we so often conceive it nowadays, as an ideal of humanity, but as measure, as the reduction of the chaos of appearances to the precision of linear symbols. Symmetry, balance, harmonic division, mated and mensurated intervals – such were its abstract characteristics." (Herbert Read, "Icon and Idea: The Function of Art in the Development of Human Consciousness", 1955)

"Abstractions are wonderfully clever tools for taking things apart and for arranging things in patterns but they are very little use in putting things together and no use at all when it comes to determining what things are for." (Archibald MacLeish, "Why Do We Teach Poetry?", The Atlantic Monthly Vol. 197 (3), 1956)

"Behind these symbols lie the boldest, purest, coolest abstractions mankind has ever made. No schoolman speculating on essences and attributes ever approached anything like the abstractness of algebra." (Susanne K Langer, "Philosophy in a New Key", 1957)

"One great lesson that we can learn from its systematic absence in the work of the grand theorists is that every self-conscious thinker must at all times be aware of - and hence be able to control - the levels of abstraction on which he is working. The capacity to shuttle between levels of abstraction, with ease and with clarity, is a signal mark of the imaginative and systematic thinker." (C Wright Mills, "The Sociological Imagination", 1959)

"There is a logic of language and a logic of mathematics. The former is supple and lifelike, it follows our experience. The latter is abstract and rigid, more ideal. The latter is perfectly necessary, perfectly reliable: the former is only sometimes reliable and hardly ever systematic. But the logic of mathematics achieves necessity at the expense of living truth, it is less real than the other, although more certain. It achieves certainty by a flight from the concrete into abstraction." (Thomas Merton, "The Secular Journal of Thomas Merton", 1959)

On Abstraction (2000-2009)

"Abstraction is itself an abstract word and has no single meaning. […] Every word in our language is abstract, because it represents something else." (Eric Maisel, "The Creativity Book: A Year's Worth of Inspiration and Guidance", 2000)

"What cognitive capabilities underlie our fundamental human achievements? Although a complete answer remains elusive, one basic component is a special kind of symbolic activity - the ability to pick out patterns, to identify recurrences of these patterns despite variation in the elements that compose them, to form concepts that abstract and reify these patterns, and to express these concepts in language. Analogy, in its most general sense, is this ability to think about relational patterns." (Keith Holyoak et al, "Introduction: The Place of Analogy in Cognition", 2001)

"[…] we underestimate the share of randomness in about everything […]  The degree of resistance to randomness in one’s life is an abstract idea, part of its logic counterintuitive, and, to confuse matters, its realizations nonobservable." (Nassim N Taleb, "Fooled by Randomness", 2001)

"A model isolates one or a few causal connections, mechanisms, or processes, to the exclusion of other contributing or interfering factors - while in the actual world, those other factors make their effects felt in what actually happens. Models may seem true in the abstract, and are false in the concrete. The key issue is about whether there is a bridge between the two, the abstract and the concrete, such that a simple model can be relied on as a source of relevantly truthful information about the complex reality." (Uskali Mäki, "Fact and Fiction in Economics: Models, Realism and Social Construction", 2002)

"[Primes] are full of surprises and very mysterious […]. They are like things you can touch […] In mathematics most things are abstract, but I have some feeling that I can touch the primes, as if they are made of a really physical material. To me, the integers as a whole are like physical particles." (Yoichi Motohashi, "The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics", 2002)

"To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. Mathematics is the ultimate in technology transfer." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)

"Do not be afraid of the word 'theory'. Yes, it can sound dauntingly abstract at times, and in the hands of some writers can appear to have precious little to do with the actual, visual world around us. Good theory however, is an awesome thing. [...] But unless we actually use it, it borders on the metaphysical and might as well not be used at all." (Richard Howells,  Visual Culture, 2003)

"Group theory is a branch of mathematics that describes the properties of an abstract model of phenomena that depend on symmetry. Despite its abstract tone, group theory provides practical techniques for making quantitative and verifiable predictions about the behavior of atoms, molecules and solids." (Arthur M Lesk, "Introduction to Symmetry and Group Theory for Chemists", 2004)

"Mathematics is not about abstract entities alone but is about relation of abstract entities with real entities. […] Adequacy relations between abstract and real entities provide space or opportunity where mathematical and logical thought operates parsimoniously."  (Navjyoti Singh, "Classical Indian Mathematical Thought", 2005)

"That is, the physicist likes to learn from particular illustrations of a general abstract concept. The mathematician, on the other hand, often eschews the particular in pursuit of the most abstract and general formulation possible. Although the mathematician may think from, or through, particular concrete examples in coming to appreciate the likely truth of very general statements, he will hide all those intuitive steps when he comes to present the conclusions of his thinking to outsiders. It presents the results of research as a hierarchy of definitions, theorems and proofs after the manner of Euclid; this minimizes unnecessary words but very effectively disguises the natural train of thought that led to the original results." (John D Barrow, "New Theories of Everything", 2007)

"Abstraction is a mental process we use when trying to discern what is essential or relevant to a problem; it does not require a belief in abstract entities." (Tom G Palmer, Realizing Freedom: Libertarian Theory, History, and Practice, 2009)

"In order to deal with these phenomena, we abstract from details and attempt to concentrate on the larger picture - a particular set of features of the real world or the structure that underlies the processes that lead to the observed outcomes. Models are such abstractions of reality. Models force us to face the results of the structural and dynamic assumptions that we have made in our abstractions." (Bruce Hannon and Matthias Ruth, "Dynamic Modeling of Diseases and Pests", 2009)

"It is from this continuousness of thought and perception that the scientist, like the writer, receives the crucial flash of insight out of which a piece of work is conceived and executed. And the scientist (again like the writer) is grateful when the insight comes, because insight is the necessary catalyst through which the abstract is made concrete, intuition be given language, language provides specificity, and real work can go forward." (Vivian Gornick, "Women in Science: Then and Now", 2009)

On Abstraction (1990-1999)

"All of engineering involves some creativity to cover the parts not known, and almost all of science includes some practical engineering to translate the abstractions into practice." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)

"That is, the physicist likes to learn from particular illustrations of a general abstract concept. The mathematician, on the other hand, often eschews the particular in pursuit of the most abstract and general formulation possible. Although the mathematician may think from, or through, particular concrete examples in coming to appreciate the likely truth of very general statements, he will hide all those intuitive steps when he comes to present the conclusions of his thinking to outsiders. It presents the results of research as a hierarchy of definitions, theorems and proofs after the manner of Euclid; this minimizes unnecessary words but very effectively disguises the natural train of thought that led to the original results." (John D Barrow, "New Theories of Everything", 1991)

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

"The word theory, as used in the natural sciences, doesn’t mean an idea tentatively held for purposes of argument - that we call a hypothesis. Rather, a theory is a set of logically consistent abstract principles that explain a body of concrete facts. It is the logical connections among the principles and the facts that characterize a theory as truth. No one element of a theory [...] can be changed without creating a logical contradiction that invalidates the entire system. Thus, although it may not be possible to substantiate directly a particular principle in the theory, the principle is validated by the consistency of the entire logical structure." (Alan Cromer, "Uncommon Sense: The Heretical Nature of Science", 1993)

"A mental model is not normally based on formal definitions but rather on concrete properties that have been drawn from life experience. Mental models are typically analogs, and they comprise specific contents, but this does not necessarily restrict their power to deal with abstract concepts, as we will see. The important thing about mental models, especially in the context of mathematics, is the relations they represent. […]  The essence of understanding a concept is to have a mental representation or mental model that faithfully reflects the structure of that concept. (Lyn D. English & Graeme S. Halford, "Mathematics Education: Models and Processes", 1995)

"Music and math together satisfied a sort of abstract 'appetite', a desire that was partly intellectual, partly aesthetic, partly emotional, partly, even, physical." (Edward Rothstein, "Emblems of Mind: The Inner Life of Music and Mathematics", 1995)

"The larger, more detailed and complex the model - the less abstract the abstraction – the smaller the number of people capable of understanding it and the longer it takes for its weaknesses and limitations to be found out." (John Adams, "Risk", 1995)

"The representational nature of maps, however, is often ignored - what we see when looking at a map is not the word, but an abstract representation that we find convenient to use in place of the world. When we build these abstract representations we are not revealing knowledge as much as are creating it." (Alan M MacEachren, "How Maps Work: Representation, Visualization, and Design", 1995)

"Abstract concepts are largely metaphorical." (George Lakoff, "Philosophy in the Flesh: The Embodied Mind and Its Challenge to Western Thought", 1999)

"The abstractions of science are stereotypes, as two-dimensional and as potentially misleading as everyday stereotypes. And yet they are as necessary to the process of understanding as filtering is to the process of perception." (K C Cole, "First You Build a Cloud and Other Reflections on Physics as a Way of Life", 1999)

On Abstraction (1980-1989)

"Mathematical reality is in itself mysterious: how can it be highly abstract and yet applicable to the physical world? How can mathematical theorems be necessary truths about an unchanging realm of abstract entities and at the same time so useful in dealing with the contingent, variable and inexact happenings evident to the senses?" (Salomon Bochner, "The Role of Mathematics in the Rise of Science", 1981)

"Today abstraction is no longer that of the map, the double, the mirror, or the concept. Simulation is no longer that of a territory, a referential being or substance. It is the generation by models of a real without origin or reality: A hyperreal. The territory no longer precedes the map, nor does it survive it. It is nevertheless the map that precedes the territory - precession of simulacra - that engenders the territory." (Baudrillard Jean, "Simulacra and Simulation", 1981)

"[…] a mathematician's ultimate concern is that his or her inventions be logical, not realistic. This is not to say, however, that mathematical inventions do not correspond to real things. They do, in most, and possibly all, cases. The coincidence between mathematical ideas and natural reality is so extensive and well documented, in fact, that it requires an explanation. Keep in mind that the coincidence is not the outcome of mathematicians trying to be realistic - quite to the contrary, their ideas are often very abstract and do not initially appear to have any correspondence to the real world. Typically, however, mathematical ideas are eventually successfully applied to describe real phenomena […]"(Michael Guillen,"Bridges to Infinity: The Human Side of Mathematics", 1983)

"Language is the most formless means of expression. Its capacity to describe concepts without physical or visual references carries us into an advanced state of abstraction." (Ian Wilson, "Conceptual Art", 1984)

"Theoretical scientists, inching away from the safe and known, skirting the point of no return, confront nature with a free invention of the intellect. They strip the discovery down and wire it into place in the form of mathematical models or other abstractions that define the perceived relation exactly. The now-naked idea is scrutinized with as much coldness and outward lack of pity as the naturally warm human heart can muster. They try to put it to use, devising experiments or field observations to test its claims. By the rules of scientific procedure it is then either discarded or temporarily sustained. Either way, the central theory encompassing it grows. If the abstractions survive they generate new knowledge from which further exploratory trips of the mind can be planned. Through the repeated alternation between flights of the imagination and the accretion of hard data, a mutual agreement on the workings of the world is written, in the form of natural law." (Edward O Wilson, "Biophilia", 1984)

"A central problem in teaching mathematics is to communicate a reasonable sense of taste - meaning often when to, or not to, generalize, abstract, or extend something you have just done." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"There is no agreed upon definition of mathematics, but there is widespread agreement that the essence of mathematics is extension, generalization, and abstraction [… which] often bring increased confidence in the results of a specific application, as well as new viewpoints."  (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"In mathematics itself abstract algebra plays a dual role: that of a unifying link between disparate parts of mathematics  and that of a research subject with a highly active life of its own." (Israel N Herstein, "Abstract Algebra", 1986)

"A mental model is a data structure, in a computational system, that represents a part of the real world or of a fictitious world. It is assumed that there can be mental models of abstract realms, such as that of mathematics, but little more will be said about them. A model-theoretic semanticist is free to think of the entities in his model as actual items in the world.[...] Mental model is an appropriate term for the mental representations that underlie everyday reasoning about the world. To understand the everyday world is to have a theory of how it works." (Alan Granham, "Mental Models as Representations of Discourse and Text", 1987)

"Metaphor [is] a pervasive mode of understanding by which we project patterns from one domain of experience in order to structure another domain of a different kind. So conceived metaphor is not merely a linguistic mode of expression; rather, it is one of the chief cognitive structures by which we are able to have coherent, ordered experiences that we can reason about and make sense of. Through metaphor, we make use of patterns that obtain in our physical experience to organise our more abstract understanding. " (Mark Johnson, "The Body in the Mind", 1987)

"The essence of modeling, as we see it, is that one begins with a nontrivial word problem about the world around us. We then grapple with the not always obvious problem of how it can be posed as a mathematical question. Emphasis is on the evolution of a roughly conceived idea into a more abstract but manageable form in which inessentials have been eliminated. One of the lessons learned is that there is no best model, only better ones."  (Edward Beltrami, "Mathematics for Dynamic Modeling", 1987)

"Probabilities are summaries of knowledge that is left behind when information is transferred to a higher level of abstraction." (Judea Pearl, Probabilistic Reasoning in Intelligent Systems: Network of Plausible, Inference, 1988)

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

"[…] a model is the picture of the real - a short form of the whole. Hence, a model is an abstraction or simplification of a system. It is a technique by which aspects of reality can be 'artificially' represented or 'simulated' and at the same time simplified to facilitate comprehension." (Laxmi K Patnaik, "Model Building in Political Science", The Indian Journal of Political Science, Vol. 50, No. 2, 1989)

"As a practical matter, mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, mathematics relies on logic rather than observation as its standard of truth, yet employs observation, simulation, and even experimentation as a means of discovering truth. "(National Research Council, "Everybody Counts", 1989)

"Modeling in its broadest sense is the cost-effective use of something in place of something else for some [cognitive] purpose. It allows us to use something that is simpler, safer, or cheaper than reality instead of reality for some purpose. A model represents reality for the given purpose; the model is an abstraction of reality in the sense that it cannot represent all aspects of reality. This allows us to deal with the world in a simplified manner, avoiding the complexity, danger and irreversibility of reality." (Jeff Rothenberg, "The Nature of Modeling. In: Artificial Intelligence, Simulation, and Modeling", 1989)

On Simplicity X

"One should not be deceived by philosophical works that pretend to be mathematical, but are merely dubious and murky metaphysics. Just because a philosopher can recite the words lemma, theorem and corollary doesn't mean that his work has the certainty of mathematics. That certainty does not derive from big words, or even from the method used by geometers, but rather from the utter simplicity of the objects considered by mathematics." (Pierre L Maupertuis, "Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique", 1746)

"The model of an object, process or phenomenon is some other object, process or phenomenon having certain features in common with the original. It is ordinarily assumed that the model is a simplified version of the object of study. However, it is not always easy to give precise meaning to the concept 'simpler than the original', for the simple reason that in reality all entities or phenomena are infinitely complicated and their study can be carried out with differing and constantly increasing degrees of accuracy." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"Certainty, simplicity, vividness originate in popular knowledge. That is where the expert obtains his faith in this triad as the ideal of knowledge. Therein lies the general epistemological significance of popular science." (Ludwik Fleck, "Genesis and Development of a Scientific Fact", 1979)

"Nature is capable of building complex structures by processes of self-organization; simplicity begets complexity." (Victor J Stenger, "God: The Failed Hypothesis", 2010)
 
"Simplicity in a system tends to increase that system's efficiency. Because less can go wrong with fewer parts, less will. Complexity in a system tends to increase that system's inefficiency; the greater the number of variables, the greater the probability of those variables clashing, and in turn, the greater the potential for conflict and disarray. Because more can go wrong, more will. That is why centralized systems are inclined to break down quickly and become enmeshed in greater unintended consequences." (Lawrence K Samuels, "Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)

"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)
 
"[…] if one really understood the central point and its necessity in the construction of the world, one ought to be able to state it in one clear, simple sentence. Until we see the quantum principle with this simplicity we can well believe that we do not know the first thing about the universe, about ourselves, and about our place in the universe." (John A Wheeler)

Music and Mathematics VI

"Mathematics is the instrument by which the engineer tunnels out mountains, bridges our rivers, constructs our aqueducts, erects out factories and makes them musical by the busy hum of spindles. Take away the results of the reasoning of mathematics, and there would go with it nearly all the material achievements which give convenience and glory to modern civilization." (Edward Brooks, "Mental Science and Culture", 1891)

"The best proofs in mathematics are short and crisp like epigrams, and the longest have swings and rhythms that are like music." (Scott Buchanan, Poetry and Mathematics, 1929)

"Mathematics, like music and poetry, is a creation of the mind; [...] the primary task of the mathematician, like that of any other artist, is to extend man's mental horizon by representation and interpretation." (Graham Sutton, "Mathematics in Action", 1954)

"The question ‘What is mathematics?’ cannot be answered meaningfully by philosophical generalities, semantic definitions or journalistic circumlocutions. Such characterizations also fail to do justice to music or painting. No one can form an appreciation of these arts without some experience with rhythm, harmony and structure, or with form, color and composition. For the appreciation of mathematics actual contact with its substance is even more necessary." (Richard Courant, "Mathematics in the Modern World", Scientific American Vol. 211 (3), 1964)

"The structures with which mathematics deals are more like lace, the leaves of trees, and the play of light and shadow on a human face, than they are like buildings and machines, the least of their representatives. The best proofs in mathematics are short and crisp like epigrams, and the longest have swings and rhythms that are like music. The structures of mathematics and the propositions about them are ways for the imagination to travel and the wings, or legs, or vehicles to take you where you want to go." (Scott Buchanan, "Poetry and Mathematics", 1975)

"There are three reasons for the study of inequalities: practical, theoretical and aesthetic. On the aesthetic aspects, as has been pointed out, beauty is in the eyes of the beholder. However, it is generally agreed that certain pieces of music, art, or mathematics are beautiful. There is an elegance to inequalities that makes them very attractive." (Richard E Bellman, 1978)

"The way the mathematics game is played, most variations lie outside the rules, while music can insist on perfect canon or tolerate a casual accompaniment." (Marvin Minsky, "Music, Mind, and Meaning", 1981)

"Music is mathematics, the mathematics of listening, mathematics for the ears." (Karlheinz Stockhausen, "Conversations with Stockhausen", 1987)

"Mathematical notation is for the scientist what musical notation is for the composer." (John Holland, "Emergence: From Chaos to Order", 1998)

"Music is a science which should have definite rules; these rules should be drawn from an evident principle; and this principle cannot really be known to us without the aid of mathematics." (Jean-Philippe Rameau, Treatise on Harmony, 2012)

On Observation (1970-1979)

"Science consists simply of the formulation and testing of hypotheses based on observational evidence; experiments are important where applicable, but their function is merely to simplify observation by imposing controlled conditions." (Henry L Batten, "Evolution of the Earth", 1971)

"A hypothesis is empirical or scientific only if it can be tested by experience. […] A hypothesis or theory which cannot be, at least in principle, falsified by empirical observations and experiments does not belong to the realm of science." (Francisco J Ayala, "Biological Evolution: Natural Selection or Random Walk", American Scientist, 1974)

"Science is systematic organisation of knowledge about the universe on the basis of explanatory hypotheses which are genuinely testable. Science advances by developing gradually more comprehensive theories; that is, by formulating theories of greater generality which can account for observational statements and hypotheses which appear as prima facie unrelated." (Francisco J Ayala, "Studies in the Philosophy of Biology: Reduction and Related Problems", 1974)

"All perceiving is also thinking, all reasoning is also intuition, all observation is also invention." (Rudolf Arnheim, "Entropy and Art: An Essay on Disorder and Order", 1974)

"The essential function of a hypothesis consists in the guidance it affords to new observations and experiments, by which our conjecture is either confirmed or refuted." (Ernst Mach, "Knowledge and Error: Sketches on the Psychology of Enquiry", 1976)

"[…] a body of practices widely regarded by outsiders as well organized, logical, and coherent, in fact consists of a disordered array of observations with which scientists struggle to produce order." (Bruno Latour & S Woolgar, Laboratory Life: The Social Construction of Scientific Facts, 1979)

Claude Bernard - Collected Quotes

"A theory is merely a scientific idea controlled by experiment." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"An anticipative idea or an hypothesis is, then, the necessary starting point for all experimental reasoning. Without it, we could not make any investigation at all nor learn anything; we could only pile up sterile observations. If we experiment without a preconceived idea, we should move at random […]" (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Considered by itself, the experimental method is nothing but reasoning by whose help we methodically submit our ideas to experience, - the experience of facts." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"It seems, indeed, a necessary weakness of our mind to be able to reach truth only across a multitude of errors and obstacles." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"If the facts used as a basis of reasoning are ill-established or erroneous, everything will crumble or be falsified; and it is thus that errors in scientific theories most often originate in errors of fact." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Men who have excessive faith in their theories or ideas are not only ill prepared for making discoveries; they also make very poor observations." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Observation is a passive science, experimentation an active science." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Observation, then, is what shows facts.; experiment is what teaches about facts and gives experience in relation to anything." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Only when a phenomenon includes conditions as yet undefined, can we compile; we must learn, therefore, that we compile statistics only when we cannot possibly help it." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Only within very narrow boundaries can man observe the phenomena which surround him; most of them naturally escape his senses, and mere observation is not enough." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Pile up facts or observations as we may, we shall be none the wiser. To learn, we must necessarily reason about what we have observed, compare the facts and judge them by other facts used as controls."  (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Speaking concretely, when we say ‘making experiments or making observations’, we mean that we devote ourselves to investigation and to research, that we make attempts and trials in order to gain facts from which the mind, through reasoning, may draw knowledge or instruction.
Speaking in the abstract, when we say, ‘relying on observation and gaining experience’, we mean that observation is the mind’s support in reasoning, and experience the mind’s support in deciding, or still better, the fruit of exact reasoning applied to the interpretation of facts.
Observation, then, is what shows facts; experiment is what teaches about facts and gives experience in relation to anything." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"When we meet a fact which contradicts a prevailing theory, we must accept the fact and abandon the theory, even when the theory is supported by great names and generally accepted." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

On Observation VII

Physics builds from observations. No physical theory can succeed if it is not confirmed by observations, and a theory strongly supported by observations cannot be denied. (William N Cropper, Great Physicists, 2001)

"[…] because observations are all we have, we take them seriously. We choose hard data and the framework of mathematics as our guides, not unrestrained imagination or unrelenting skepticism, and seek the simplest yet most wide-reaching theories capable of explaining and predicting the outcome of today’s and future experiments." (Brian Greene, "The Fabric of the Cosmos", 2004)

"We have to be aware that probabilities are relative to a level of observation, and that what is most probable at one level is not necessarily so at another. Moreover, a state is defined by an observer, being the conjunction of the values for all the variables or attributes that the observer considers relevant for the phenomenon being modeled. Therefore, we can have different degrees of order or ‘entropies’ for different models or levels of observation of the same entity."(Carlos Gershenson, "Design and Control of Self-organizing Systems", 2007)

"The reasoning of the mathematician and that of the scientist are similar to a point. Both make conjectures often prompted by particular observations. Both advance tentative generalizations and look for supporting evidence of their validity. Both consider specific implications of their generalizations and put those implications to the test. Both attempt to understand their generalizations in the sense of finding explanations for them in terms of concepts with which they are already familiar. Both notice fragmentary regularities and - through a process that may include false starts and blind alleys - attempt to put the scattered details together into what appears to be a meaningful whole. At some point, however, the mathematician’s quest and that of the scientist diverge. For scientists, observation is the highest authority, whereas what mathematicians seek ultimately for their conjectures is deductive proof." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures and Proofs", 2009)

"A mathematician possesses a mental model of the mathematical entity she works on. This internal mental model is accessible to her direct observation and manipulation. At the same time, it is socially and culturally controlled, to conform to the mathematics community's collective model of the entity in question. The mathematician observes a property of her own internal model of that mathematical entity. Then she must find a recipe, a set of instructions, that enables other competent, qualified mathematicians to observe the corresponding property of their corresponding mental model. That recipe is the proof. It establishes that property of the mathematical entity." (Reuben Hersh," Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)

On Observation (1980-1989)

"After all of this it is a miracle that our models describe anything at all successfully. In fact, they describe many things well: we observe what they have predicted, and we understand what we observe. However, this last act of observation and understanding always eludes physical description." (Yuri I Manin, "Mathematics and Physics", 1981)

"In natural science we are concerned ultimately, not with convenient arrangements of observational data which can be generalized into universal explanatory form, but with movements of thought, at once theoretical and empirical, which penetrate into the intrinsic structure of the universe in such a way that there becomes disclosed to us its basic design and we find ourselves at grips with reality.… We cannot pursue natural science scientifically without engaging at the same time in meta-scientific operations." (Thomas F Torrance, "Divine and Contingent Order", 1981)

"The vision of the Universe that is so vivid in our minds is framed by a few iron posts of true observation - themselves resting on theory for their meaning - but most of all the walls and towers in the vision are of papier-mâché, plastered in between those posts by an immense labor of imagination and theory." (John A Wheeler & Wojciech H Zurek, "Quantum Theory and Measurement", 1983)

"Science is a process. It is a way of thinking, a manner of approaching and of possibly resolving problems, a route by which one can produce order and sense out of disorganized and chaotic observations. Through it we achieve useful conclusions and results that are compelling and upon which there is a tendency to agree." (Isaac Asimov, "‘X’ Stands for Unknown", 1984)

"Science is defined as a set of observations and theories about observations." (F Albert Matsen, "The Role of Theory in Chemistry", Journal of Chemical Education Vol. 62 (5), 1985)

"The only touchstone for empirical truth is experiment and observation." (Heinz Pagels, "Perfect Symmetry: The Search for the Beginning of Time", 1985) 

"Although science literally means ‘knowledge’, the scientific attitude is concerned much more with rational perception through the mind and with testing such perceptions against actual fact, in the form of experiments and observations." (David Bohm & F David Peat, "Science, Order, and Creativity", 1987)

"The model is only a suggestive metaphor, a fiction about the messy and unwieldy observations of the real world. In order for it to be persuasive, to convey a sense of credibility, it is important that it not be too complicated and that the assumptions that are made be clearly in evidence. In short, the model must be simple, transparent, and verifiable." (Edward Beltrami, "Mathematics for Dynamic Modeling", 1987)

"As a practical matter, mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, mathematics relies on logic rather than observation as its standard of truth, yet employs observation, simulation, and even experimentation as a means of discovering truth. "(National Research Council, "Everybody Counts", 1989)

Music and Mathematics VI

"But mathematics, certainly, does not play the smallest part in the charm and movement of the mind produced by music. Rather is it only the indispensable condition (conditio sine qua non) of that proportion of the combining as well as changing impressions which makes it possible to grasp them all in one and prevent them from destroying one another, and to let them, rather, conspire towards the production of a continuous movement and quickening of the mind by affections that are in unison with it, and thus towards a serene self-enjoyment." (Immanuel Kant, "The Critique of Judgment", 1790)

"May not Music be described as the Mathematic of sense, Mathematic as the Music of the reason? the soul of each the same! Thus the musician feels Mathematic, the mathematician thinks Music - Music the dream, Mathematic the working life - each to receive its consummation from the other when the human intelligence, elevated to its perfect type […]" (James J Sylvester, "On Newton’s Rule for the Discovery of Imaginary Roots", 1865)

"Mathematics and music! the most glaring possible opposites of human thought! and yet connected, mutually sustained!" (Hermann von Helmholtz, "Popular Lectures on Scientific Subjects", 1885)

"Mathematics has a triple end. It is to furnish an instrument for the study of nature. But that is not all. It has a philosophic end, and I dare say it, an esthetic end. […] Those skilled in mathematics find in it pleasure akin to those which painting and music give. They admire the delicate harmony of numbers and of forms; they marvel when a new discovery opens an unexpected perspective; and is this pleasure not esthetic, even though the senses have no part in it?" (Henri Poincaré, "Sur les rapports de l’analyse pur et de la physique mathématique", [Report to the Zurich International Congress of Mathathematics], 1897)

"[…] mathematics, accessible in its full depth only to the very few, holds a quite peculiar position amongst the creation of the mind. It is a science of the most rigorous kind, like logic but more comprehensive and very much fuller; it is a true art, along with sculpture and music, as needing the guidance of inspiration and as developing under great conventions of form […]" (Oswald Spengler, "The Decline of the West" Vol. 1, 1926)

"If all the arts aspire to the condition of music, all the sciences aspire to the condition of mathematics." (George Santayana, Some Turns of Thought in Modern Philosophy: Five Essays, 1933)

"[…] mathematics is like music, freely exploring the possibilities of form. And yet, notoriously, mathematics holds true of things; hugs and permeates them far more closely than does confused and inconstant human perception; so that the dream of many exasperated critics of human error has been to assimilate all science to mathematics, so as to make knowledge safe by making it, as Locke wished, direct perception of the relations between ideas […]" (George Santayana, "The Realm of Truth: Book Third of Realms of Being", 1937)

"Mathematizing may well be a creative activity of man, like language or music, of primary originality, whose historical decisions defy complete objective rationalizations." (Hermann Weyl, “Obituary for David Hilbert”, Royal Society Biographies Vol. 4, 1944)

"The fact is that there are few more ‘popular’ subjects than mathematics. Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune; and there are probably more people really interested in mathematics than in music. Appearances may suggest the contrary, but there are easy explanations. Music can be used to stimulate mass emotion, while mathematics cannot; and musical incapacity is recognized (no doubt rightly) as mildly discreditable [...]" (Godfrey H Hardy, "A Mathematician’s Apology", 1967)

"I see some parallels between the shifts of fashion in mathematics and in music. In music, the popular new styles of jazz and rock became fashionable a little earlier than the new mathematical styles of chaos and complexity theory. Jazz and rock were long despised by classical musicians, but have emerged as art-forms more accessible than classical music to a wide section of the public. Jazz and rock are no longer to be despised as passing fads. Neither are chaos and complexity theory. But still, classical music and classical mathematics are not dead. Mozart lives, and so does Euler. When the wheel of fashion turns once more, quantum mechanics and hard analysis will once again be in style." (Freeman J Dyson, "Book Review of ‘Nature’s Numbers’", The American Mathematical Monthly, Vol. 103 (7), 1996)

Avicenna Latinus - Collected Quotes

"The faculty which grasps such concepts acquires intelligible forms from sense-perception by force of an inborn disposition, so that forms, which are in the form-bearing faculty and the memorizing faculty, are made present to [the rational soul] with the assistance of the imaginative and estimative [faculties]." (Avicenna Latinus [Ibn Sina], "A Compendium on the Soul", cca. 996-997)

"When knowledge reaches the specific natures and what is accidental to them, inquiry stops and is not followed by the fleeting knowledge of individuals to which our souls are not at all inclined." (Avicenna Latinus [Ibn Sina], "The Physics of The Healing", cca. 1014)

"a [thing’s] likeness often appears and seems to those who perceive it as if the image itself were speaking, as if they heard the words that they held and read." (Avicenna Latinus [Ibn Sina], "Liber de Anima", cca. 1014-1027)

"It seems that all perception is but the grasping of the form of the perceived object in some manner. If, then, it is a perception of some material object, it consists in an apprehension of its form by abstracting it from matter in some way. But the kinds of abstraction are different and their degrees various. This is because, owing to matter, the material form is subject to certain states and conditions which do not belong to [the form] by itself insofar as it is this form. So sometimes the abstraction from matter is effected with all or some of these attachments, and sometimes it is complete in that the concept is abstracted from matter and from the accidents it possesses on account of the matter."(Avicenna Latinus [Ibn Sina], "Liber De anima", cca. 1014-1027)

"The animal faculties assist the rational soul in various ways, one of them being that sense-perception brings to it particulars, from which four things result in [the rational soul]: One of them is that the mind extracts single universals from the particulars, by abstracting their concepts from matter and the appendages of matter and its accidents, by considering what is common in it and what different, and what in its existence is essential and what accidental. From this the principles of conceptualization come about [in] the soul: and this with the help of its employing imagination and estimation." (Avicenna Latinus [Ibn Sina], "Liber De Anima", cca. 1014-1027)

"When the intellectual faculty considers the particulars which are [stored] in imagination and the light of the above-mentioned active intellect shines upon them in us, then the [particulars] are transformed into something abstracted from matter and from the [material] attachments and get imprinted in the rational soul, but not in the sense that the particulars themselves are transferred from imagination to our intellect, nor in the sense that the concept buried in [material] attachments - which in itself and with regard to its essence is abstract - produces a copy of itself, but in the sense that looking at the particulars disposes the soul for something abstracted to flow upon it from the active intellect. (Avicenna Latinus [Ibn Sina], "Liber De Anima", cca. 1014-1027)

"Sometimes a thing is perceived [via sense-perception] when it is observed; then it is imagined, when it is absent [in reality] through the representation of its form inside, Sense-perception grasps [the concept] insofar as it is buried in these accidents that cling to it because of the matter out of which it is made without abstracting it from [matter], and it grasps it only by means of a connection through position [ that exists] between its perception and its matter. It is for this reason that the form of [the thing] is not represented in the external sense when [sensation] ceases. As to the internal [faculty of] imagination, it imagines [the concept] together with these accidents, without being able to entirely abstract it from them. Still, [imagination] abstracts it from the afore-mentioned connection [through position] on which sense-perception depends, so that [imagination] represents the form [of the thing] despite the absence of the form's [outside] carrier." (Avicenna Latinus [Ibn Sina], "Pointer and Reminders", cca. 1030)

"The multiplicity of the soul's occupations with sense-perceptible imaginable forms and connotational images, which are in the form-bearing and the remembering [faculties respectively], with the help of the estimative and cogitative faculty, makes the soul obtain a disposition for the reception of abstractions of them [i.e., of the imaginable forms and images] from the separate substance through some kind of relationship between the two. Observation and inspection of the issue verify this. These occupations [with imaginable forms and images] are those which give [the soul] a perfect disposition that is specific for [the reception of] each individual form, though an intellectual concept may [also] provide this specific [disposition] for [the reception of] another intellectual concept." (Avicenna Latinus [Ibn Sina], "Pointer and Reminders", cca. 1030)

"Absence of understanding does not warrant absence of existence." (Avicenna Latinus [Ibn Sina]) 

"Anyone who denies the law of non-contradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned." (Avicenna Latinus [Ibn Sina])

"Cultivate the self with learning in order to progress and leave all else; for knowledge is an abode of all things. 

The self is like glass, the knowledge, like a lamp, and the wisdom of God, like oil. 

When your self is illuminated - you are alive, and when there is darkness - you are dead." (Avicenna Latinus [Ibn Sina])

"Friendship is always a sweet responsibility, never an opportunity." (Avicenna Latinus [Ibn Sina])

"Further, this power which conceives these ideas does at times gain from sense forms mental, imaginative, and innate in (instinctive to) itself; and in such a case it does this in that it lays before itself the forms that are in the conceiving power and in the remembering (preserving) power, by employing the imaginative and the conjecturing power, and then contemplates them, and finds them to have participated in some forms and to have differed in some other forms; and finds some amongst the forms that are in these powers to be essential and others to be accidental." (Avicenna Latinus [Ibn Sina])

"It is evident that everything which does not exist at first and then exists, is determined by something other than itself." (Avicenna Latinus [Ibn Sina])

"Medicine deals with the states of health and disease in the human body. It is a truism of philosophy that a complete knowledge of a thing can only be obtained by elucidating its causes and antecedents, provided, of course, such causes exist. In medicine it is, therefore, necessary that causes of both health and disease should be determined." (Avicenna Latinus [Ibn Sina]) 

"Now it is established in the sciences that no knowledge is acquired save through the study of its causes and beginnings, if it has had causes and beginnings; nor completed except by knowledge of its accidents and accompanying essentials." (Avicenna Latinus [Ibn Sina]) 

"Pain is a sensation produced by something contrary to the course of nature and this sensation is set up by one of two circumstances: either a very sudden change of the temperament (or the bad effect of a contrary temperament) or a solution of continuity." (Avicenna Latinus [Ibn Sina]) 

"The different sorts of madness are innumerable." (Avicenna Latinus [Ibn Sina]) 

 

"The knowledge of anything, since all things have causes, is not acquired or complete unless it is known by its causes." (Avicenna Latinus [Ibn Sina]) 

"The mind (Understanding, Reason) is in fact and deed wholly and solely nothing else than the forms of mentally-grasped things, if these be arrayed in the very mind potentially, and through it they are brought out to effective action; and hence it is said that the mind is in fact and deed at once both understanding and understood. Amongst the properties of the understanding power is this, that it unifies the many and multiplies the one through analysis and synthesis. As to multiplication, it is such as the analysis of one man into essence, body, nourishment-getting, animal, speaking (rational). As to unification of the many, it is such as the composition (synthesis) of this one man out of essence, body, animal, speaking (rational) into one notion which is mankind (human being)."(Avicenna Latinus [Ibn Sina])

"The theory of medicine, therefore, presents what is useful in thought, but does not indicate how it is to be applied in practice - the mode of operation of these principles. The theory, when mastered, gives us a certain kind of knowledge. Thus we say, for example, there are three forms of fevers and nine constitutions. The practice of medicine is not the work which the physician carries out, but is that branch of medical knowledge which, when acquired, enables one to form an opinion upon which to base the proper plan of treatment." (Avicenna Latinus [Ibn Sina])