Howard W Eves - Collected Quotes

"A formal manipulator in mathematics often experiences the discomforting feeling that his pencil surpasses him in intelligence." (Howard W Eves, "An introduction to the history of mathematics", 1953)

"An expert problem solver must be endowed with two incompatible qualities, a restless imagination and a patient pertinacity." (Howard Eves, "In Mathematical Circles", 1969)

"Mathematics may be likened to a large rock whose interior composition we wish to examine. The older mathematicians appear as persevering stone cutters slowly attempting to demolish the rock from the outside with hammer and chisel. The later mathematicians resemble expert miners who seek vulnerable veins, drill into these strategic places, and then blast the rock apart with well-placed internal charges."     (Howard W Eves, "In Mathematical Circles", 1969)

"Older mathematics appears static while the newer appears dynamic, so that the older mathematics compares to the still-picture stage of photography while the newer mathematics compares to the moving-picture stage. Again, the older mathematics is to the newer much as anatomy is to physiology, wherein the former studies the dead body and the latter studies the living body. Once more, the older mathematics concerned itself with the fixed and the finite while the newer mathematics embraces the changing and the infinite." (Howard W Eves, "In Mathematical Circles", 1969)

"The development of mathematics over the ages may be viewed as a continent slowly rising from the sea. At first perhaps a single island appears, and, as it grows in size, other islands emerge at varying distances from one another. As the continent continues to rise, some of the islands become joined to others by isthmuses that widen until pairs of islands become single large islands. At length a point is reached where the shape of the continent is essentially defined, and there remain only a number of lakes and inland seas of various sizes. As the continent further rises, these lakes and seas shrink and vanish one by one. The older mathematics compares to the situation when the general shape of the rising continent is still undefined and the land area consists largely of islands of different sizes. The newer mathematics compares to the situation when the general shape of the rising continent has become essentially clear, with most of the former islands now joined by stretches of land." (Howard W Eves, "In Mathematical Circles", 1969)

"A good problem should be more than a mere exercise; it should be challenging and not too easily solved by the student, and it should require some ‘dreaming’ time." (Howard W Eves)

"It is impossible to overstate the importance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps. […] Every new discovery in mathematics, results from an attempt to solve some problem."   (Howard W Eves)

Howard Gardner - Collected Quotes

"An important symptom of an emerging understanding is the capacity to represent a problem in a number of different ways and to approach its solution from varied vantage points; a single, rigid representation is unlikely to suffice." (Howard Gardner, "The Unschooled Mind", 1991)

"[By understanding] I mean simply a sufficient grasp of concepts, principles, or skills so that one can bring them to bear on new problems and situations, deciding in which ways one’s present competencies can suffice and in which ways one may require new skills or knowledge." (Howard Gardner, "The Unschooled Mind", 1991)

"Indeed, knowledge that one will be judged on some criterion of ‘creativeness’ or ‘originality’ tends to narrow the scope of what one can produce (leading to products that are then judged as relatively conventional); in contrast, the absence of an evaluations seems to liberate creativity." (Howard Gardner,  "Creating Minds", 1993)

"An individual understands a concept, skill, theory, or domain of knowledge to the extent that he or she can apply it appropriately in a new situation." (Howard Gardner, "The Disciplined Mind", 1999)

“Education must ultimately justify itself in terms of enhancing human understanding.” (Howard Gardner, “Intelligence Reframed”, 1999)

"Anything that is worth teaching can be presented in many different ways. These multiple ways can make use of our multiple intelligences." (Howard Gardner)

"The biggest mistake of past centuries in teaching has been to treat all children as if they were variants of the same individual, and thus to feel justified in teaching them the same subjects in the same ways." (Howard Gardner)

Martin Gardner - Collected Quotes

"Chess combines the beauty of mathematical structure with the recreational delights of a competitive game." (Martin Gardner, "Mathematics, Magic, and Mystery", 1956)

"No branch of number theory is more saturated with mystery than the study of prime numbers: those exasperating, unruly integers that refuse to be divided evenly by any integers except themselves and 1. Some problems concerning primes are so simple that a child can understand them and yet so deep and far from solved that many mathematicians now suspect they have no solution. Perhaps they are 'undecideable'. Perhaps number theory, like quantum mechanics, has its own uncertainty principle that makes it necessary, in certain areas, to abandon exactness for probabilistic formulations." (Martin Gardner, "The remarkable lore of the prime numbers", Scientific American, 1964)

"In many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance." (Martin Gardner, "Mathematical Games", Scientific American, 1973)

"Surreal numbers are an astonishing feat of legerdemain. An empty hat rests on a table made of a few axioms of standard set theory. Conway waves two simple rules in the air, then reaches into almost nothing and pulls out an infinitely rich tapestry of numbers that form a real and closed field. Every real number is surrounded by a host of new numbers that lie closer to it than any other 'real' value does. The system is truly 'surreal.'" (Martin Gardner, "Mathematical Magic Show", 1977)

“All mathematical problems are solved by reasoning within a deductive system in which basic laws of logic are embedded.” (Martin Gardner, “Aha! Insight”, 1978)

"At the heart of mathematics is a constant search for simpler and simpler ways to prove theorems  and solve problems. [...] The sudden hunch, the creative leap of the mind that ‘sees’ in a flash how to solve a problem in a simple way, is something quite different from general intelligence." (Martin Gardner, "Aha! Insight", 1978)

“Combinatorial analysis, or combinatorics, is the study of how things can be arranged. In slightly less general terms, combinatorial analysis embodies the study of the ways in which elements can be grouped into sets subject to various specified rules, and the properties of those groupings. […] Combinatorial analysis often asks for the total number of different ways that certain things can be combined according to certain rules.” (Martin Gardner, "Aha! Insight", 1978)

"Every branch of mathematics has its combinatorial aspects […] There is combinatorial arithmetic, combinatorial topology, combinatorial logic, combinatorial set theory-even combinatorial linguistics, as we shall see in the section on word play. Combinatorics is particularly important in probability theory where it is essential to enumerate all possible combinations of things before a probability formula can be found." (Martin Gardner, "Aha! Insight", 1978)

"Every branch of geometry can be defined as the study of properties that are unaltered when a specified figure is given specified symmetry transformations. Euclidian plane geometry, for instance, concerns the study of properties that are 'invariant' when a figure is moved about on the plane, rotated, mirror reflected, or uniformly expanded and contracted. Affine geometry studies properties that are invariant when a figure is 'stretched' in a certain way. Projective geometry studies properties invariant under projection. Topology deals with properties that remain unchanged even when a figure is radically distorted in a manner similar to the deformation of a figure made of rubber." (Martin Gardner, "Aha! Insight", 1978)

“Geometry is the study of shapes. Although true, this definition is so broad that it is almost meaningless. The judge of a beauty contest is, in a sense, a geometrician because he is judging […] shapes, but this is not quite what we want the word to mean. It has been said that a curved line is the most beautiful distance between two points. Even though this statement is about curves, a proper element of geometry, the assertion seems more to be in the domain of aesthetics rather than mathematics.” (Martin Gardner, "Aha! Insight", 1978)

“Graph theory is the study of sets of points that are joined by lines.” (Martin Gardner, “Aha! Insight”, 1978)

“The great revolutions in science are almost always the result of unexpected intuitive leaps. After all, what is science if not the posing of difficult puzzles by the universe? Mother Nature does something interesting, and challenges the scientist to figure out how she does it. In many cases the solution is not found by exhaustive trial and error […] or even by a deduction based on the relevant knowledge.”  (Martin Gardner, "Aha! Insight", 1978)

“The word ‘induction’ has two essentially different meanings. Scientific induction is a process by which scientists make observations of particular cases, such as noticing that some crows are black, then leap to the universal conclusion that all crows are black. The conclusion is never certain. There is always the possibility that at least one unobserved crow is not black." (Martin Gardner, “Aha! Insight”, 1978)

"Mathematical induction […] is an entirely different procedure. Although it, too, leaps from the knowledge of particular cases to knowledge about an infinite sequence of cases, the leap is purely deductive. It is as certain as any proof in mathematics, and an indispensable tool in almost every branch of mathematics.” (Martin Gardner, “Aha! Insight”, 1978)

"The external world exists; the structure of the world is ordered; we know little about the nature of the order, nothing at all about why it should exist." (Martin Gardner, "Order and Surprise", 1983)

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

"Besides being essential in modern physics, the complex-number field provides pure mathematics with a multitude of brain-boggling theorems. It is worth keeping in mind that complex numbers, although they include the reals.as a subset, differ from real numbers in startling ways. One cannot, for example, speak of a complex number as being either positive or negative: those properties apply only to the reals and the pure imaginaries. It is equally meaningless to say that one complex number is larger or smaller than another." (Martin Gardner, "Fractal Music, Hypercards and More... Mathematical Recreations from Scientific American Magazine", 1992)

"The seemingly preposterous assumption that there is a square root of -1 was justified on pragmatic grounds: it simplified certain calculations and so could be used as long as 'real' values were obtained at the end. The parallel with the rules for using negative numbers is striking. If you are trying to determine how many cows there are in a field (that is, if you are working in the domain of positive integers), you may find negative numbers useful in the calculation, but of course the final answer must be in terms of positive numbers because there is no such thing as a negative cow." (Martin Gardner, "Fractal Music, Hypercards and More... Mathematical Recreations from Scientific American Magazine", 1992)

"I enjoy mathematics so much because it has a strange kind of unearthly beauty. There is a strong feeling of pleasure, hard to describe, in thinking through an elegant proof, and even greater pleasure in discovering a proof not previously known." (Martin Gardner, 2008)

"[…] if all sentient beings in the universe disappeared, there would remain a sense in which mathematical objects and theorems would continue to exist even though there would be no one around to write or talk about them. Huge prime numbers would continue to be prime, even if no one had proved them prime." (Martin Gardner, "When You Were a Tadpole and I Was a Fish", 2009)

"A surprising proportion of mathematicians are accomplished musicians. Is it because music and mathematics share patterns that are beautiful?" (Martin Gardner, The Dover Math and Science Newsletter, 2011)

"All mathematicians share […] a sense of amazement over the infinite depth and the mysterious beauty and usefulness of mathematics." (Martin Gardner)

"Mathematics is not only real, but it is the only reality. [The] entire universe is made of matter, obviously. And matter is made of particles. It's made of electrons and neutrons and protons. So the entire universe is made out of particles. Now what are the particles made out of? They're not made out of anything. The only thing you can say about the reality of an electron is to cite its mathematical properties. So there's a sense in which matter has completely dissolved and what is left is just a mathematical structure." (Martin Gardner)

"One would be hard put to find a set of whole numbers with a more fascinating history and more elegant properties surrounded by greater depths of mystery - and more totally useless - than the perfect numbers." (Martin Gardner)

"There are some traits all mathematicians share. An obvious one is a sense of amazement over the infinite depth and the mysterious beauty and usefulness of mathematics." (Martin Gardner)

Johann Wolfgang von Goethe - Collected Quotes

"One does not get to know that one exists until one rediscovers oneself in others." (Johann Wolfgang von Goethe, 1775)

"The study and appreciation of nature comes easier than that of art. The lowliest product of nature embodies the sphere of its perfection within itself, and to discover these relationships all I need is eyes to see. I am certain that within a small sphere a wholly true existence is confined. In a work of art, on the other hand, the principle of its perfection lies outside itself. There is most important of all the artists idea, rarely if ever matched by his execution. There are furthermore certain implicit laws which, though stemming from the nature of the craft, are not so easy to understand and decipher as the laws of living nature. In works of art there is always a large traditional factor, whereas the works of nature are like a word of God spoken this instant." (Johann Wolfgang von Goethe, 1786)

"Let the poet confine his use of individual models to what is necessary to make his subject alive and convincing. As for all the rest, let him rely on the living world as mirrored in his bosom." (Johann Wolfgang von Goethe, 1789)

"No hypothesis can lay claim to any value unless it assembles many phenomena under one concept." (Johann Wolfgang von Goethe, 1795)

"That the observation of nature leads to thinking; that its abundance makes us resort to a variety of methods in order to manipulate it even to some degree on this there seems to be general agreement. But only a limited few are equally aware of the fact that the contemplation of nature suggests ideas to which we ascribe the same degree of certainty as to nature itself a greater degree, in fact; and that we have a right to be guided by these ideas both in our search for data and in our attempts to arrange what we have found." (Johann Wolfgang von Goethe, 1801)

"The imagination lurks as the most powerful foe. It has an irresistible affinity for the absurd. Even cultured individuals are subject to this impulse to a high degree. It  is hostile to all civilized life and it confronts our decorous society with a reversion to the innate rudeness of the savage and his love of grimaces." (Johann Wolfgang von Goethe, "Annals", 1805)

"Former ages thought in terms of images of the imagination, whereas we moderns have concepts. Formerly the guiding ideas of life presented themselves in concrete visual form as divinities, whereas today they are conceptualized. The ancients excelled in creation; our own strength lies rather in destruction, in analysis." (Johann Wolfgang von Goethe, 1806)

"Each item of knowledge involves a second, a third step, and so on ad infinitum. If we pursue the life of the tree in its roots, or in its branches and twigs, one thing always follows from another. And the more vitally any concern of knowledge takes hold of us, the more we find ourselves driven to pursue it in its ramifications, both up and down." (Johann Wolfgang von Goethe, "Annals", 1807)

"Genuine works of art carry their own aesthetic theory implicit within them and suggest the standards according to which they are to be judged." (Johann Wolfgang von Goethe, 1808)

"Close observers of nature, however diverse their points of view, will agree that everything of a phenomenal nature must suggest either an original duality capable of being merged in unity, or an original unity capable of becoming a duality. Separating what is united and uniting what is separate is the life of nature. This is the eternal systole and diastole, the eternal synkrisis and diakrisis, the breathing in and out of the world in which we move and have our being." (Johann Wolfgang von Goethe, "On Theory of Color", 1810)

"In the sciences everything depends on what one calls an aperçu - the discovery of something that is at the bottom of phenomena. Such a discovery  is infinitely fruitful." (Johann Wolfgang von Goethe, "On Theory of Color", 1810)

"The highest gift we have received from God and nature is life, the rotating movement of the monad about itself, knowing neither pause nor rest. The impulse to nurture this life is ineradicably implanted in each individual, although its specific nature remains a mystery to ourselves and to Others." (Johann Wolfgang von Goethe, "On Theory of Color", 1810)

"The modern age has a false sense of superiority because of the great mass of data at its disposal. But the valid criterion of distinction is rather the extent to which man knows how to form and master the material at his command." (Johann Wolfgang von Goethe, "On Theory of Color", 1810)

"We might venture the statement that the history of science is science itself. We cannot really know what we possess until we have learned to know what others have possessed before us." (Johann Wolfgang von Goethe, "On Theory of Color", 1810)

"As soon as we proceed from the principle that knowledge and faith are not designed to cancel each other out but rather to supplement each other, we are on the right path pointing to just solutions." (Johann Wolfgang von Goethe, 1813)

"Faith is a sacred vessel in which each individual is prepared to sacrifice his feeling, his reason, his imagination, to the best of his ability. In the case of knowledge exactly the opposite holds true, I said. Not that one knows but what one knows, how well and how much one knows, is what counts. That is why knowledge is a subject for argument, inasmuch as it may be corrected, expanded, and concealed. Knowledge begins with isolated facts; it is endless and formless, and we can at most dream of grasping it as a totality. Hence it is diametrically opposed to faith." (Johann Wolfgang von Goethe, "The Autobiography", 1814)

"The touchstone of genuine poetry is that it has the ability, as a secular gospel, to liberate us from the weight of our earthly burden by an inner serenity and an outward sense of well-being." (Johann Wolfgang von Goethe, "The Autobiography", 1814)

"Idea and experience will never coincide in the center. Only art and action can effect a synthesis." (Johann Wolfgang von Goethe, 1816) 

"When all is said and done, nothing suits the theater except what also makes a symbolic appeal to the eyes a significant action suggesting an even more significant one." (Johann Wolfgang von Goethe, "Shakespeare and No End", 1816)

"That which is formed is straightway transformed again, and if we would to some degree arrive at a living intuition of Nature, we must on our part remain forever mobile and plastic, according to her own example." (Johann Wolfgang von Goethe, "Morphology", 1817)

"Everything that takes place is a symbol. In representing itself perfectly it suggests what lies beyond. In this reflection extreme modesty and extreme pretentiousness seem to me combined." (Johann Wolfgang von Goethe, 1818)

"As we contemplate the edifice of the universe, in its vastest extension, in its minutest divisibility, we cannot resist the notion that an idea underlies the whole, according to which God and Nature creatively interact forever and ever. Intuition, contemplation, reflection give us an approach to these mysteries, We are emboldened to venture upon ideas; in a more modest mood we fashion concepts that might bear some analogy to those primal beginnings." (Johann Wolfgang von Goethe, "Doubts and Resignation", 1820)

"The difficulty of bringing idea and experience into relation with one another makes itself very painfully felt in all investigation of nature. The idea is independent of space and time. Research is limited in space and time. Hence in the idea simultaneous and successive features are most intimately linked, whereas these are always separated in experience; and to think of a process of nature as simultaneous and successive at once, in accordance with the idea, makes our heads spin. The understanding is unable to conceive of those sense data as jointly present which experience transmitted to it one at a time. Thus, the contradiction between ideation and perception remains forever unresolved. " (Johann Wolfgang von Goethe, "Doubt and Resignation", 1820)

"The symbol. It is the thing without being the thing, and yet the thing: an image concentrated in the mirror of the mind and yet identical with the object. How inferior is allegory by comparison. Though it may have wit and subtle conceit, it is for the most part rhetorical and conventional. It always improves in proportion to its approach to what we call symbol." (Johann Wolfgang von Goethe, "Addenda on the Paintings of Philostratus", 1820) 

"Mathematics, like dialectics, is an instrument of the inner higher sense, while in practice it is an art like rhetoric. For both of these, nothing has value but form; content is immaterial. Whether mathematics is adding up pennies or guineas, whether rhetoric is defending truth or falsehood, makes no difference to either." (Johann Wolfgang  von Goethe, "Wilhelm Meisters Wanderjahre" ["Reflections in the Spirit of the Wanderers"], 1821)

"All hypotheses get in the way of the anatheorismos the urge to look again, to contemplate the objects, the phenomena in question, from all angles." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1822)

"Caught up in the limitless maze, the fragmentation and complication of modern natural science, and yearning for the recapture of simplicity, we must forever ask ourselves: Supposing he had known nature in its present state of complexity, a basic unity withal, how would Plato have coped with it?" (Johann Wolfgang von Goethe, "Maxims and Reflections", 1822)

"Even in the sciences, mere knowing is of no avail. It is always a matter of doing." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1822)

"Every idea appears at first as a strange visitor, and when it begins to be realized, it is hardly distinguishable from fantasy." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1822)

"General concepts and great conceit are always poised to make a terrible mess of things." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1822)

"It were the height of insight to realize that everything factual as such is, in a sense, theory. The blue of the sky exhibits the basic law of chromatics. There is no sense in looking for something behind phenomena. They are theory." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1822)

"Science helps us above all in facilitating that faculty of marveling to which we are destined. Its other function is to provide life in its ceaseless evolution with new techniques for warding off what is harmful and promoting what is useful." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1822)

"The highest gift we have received from God and nature is life, the rotating movement of the monad about itself, knowing neither pause nor rest. The impulse to nurture this life is ineradicably implanted in each individual, although its specific nature remains a mystery to ourselves and to Others. (Johann Wolfgang von Goethe, "Maxims and Reflections", 1822)

"The most foolish of all errors is for young people  to believe that they lose their originality by accepting the truths which have already been accepted by their predecessors." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1822)

"Theories are as a rule impulsive reactions of an overhasty understanding which would like to have done with phenomena and therefore substitute images, concepts, or often even just words, in their place. One has an inkling, sometimes even a clear realization, of the fact that a theory is only a dodge. But are not passion and partisanship always on the lookout for dodges? And rightly so, since they are so much in need of them." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1822)

"Theory as such is of no use except in so far as it makes us believe in the coherence of phenomena." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1822)

"'Natural system' - a contradiction in terms. Nature has no system; she has, she is life and its progress from an unknown center toward an unknowable goal. Scientific research is therefore endless, whether one proceed analytically into minutiae or follow the trail as a whole, in all its breadth and height." (Johann Wolfgang von Goethe, 1823)

"There are some problems in natural science which cannot properly be discussed without recourse to metaphysics not in the sense of scholastic verbiage, but as that which was, is, and shall be before, with, and after physics." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1823)

"There can and will be new inventions, but there can be no inventing of anything new as regards the moral nature of man. Everything has already been thought and said; the most we can do is to give it new forms and new phrasing." (Johann Wolfgang von Goethe, 1823)

"When Nature begins to reveal her manifest mystery to a man, he feels an irresistible longing for her worthiest interpreter art." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1823)

"Truth is like a torch, but of gigantic proportions. It is all we can do to grope our way with dazzled eyes, in fear even of getting scorched." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1824)

"Man was not born to solve the problems of the universe, but rather to seek to lay bare the heart of the problem and then confine himself within the limits of what is amenable to understanding." (Johann Wolfgang von Goethe, 1825)

"The true, which is identical with the divine, transcends our grasp as such. We perceive it only as reflection, parable, symbol, in specific and related manifestations.  We become aware of it as life that defies comprehension, and for all that we cannot renounce the wish to comprehend. " (Johann Wolfgang von Goethe, "Essay on Meteorology", 1825)

"Error finds ceaseless repetition in deed, for which reason one must never tire of repeating the truth in words." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1826)

"True symbolism is present where the specific represents the more general, not as a dream and shadow, but as a living momentary revelation of the inscrutable." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1826) 

"There is in nature what is within reach and what is beyond reach. Ponder this well and with respect. A great deal is already gained if we impress this general fact upon our mind, even though it always remains difficult to see where the one ends and the other begins. He who is unaware of the distinction may waste himself in lifelong toil trying to get at the inaccessible without ever getting close to truth. But he who knows it and is wise will stick to what is accessible; and in exploring this region in all directions and confirming his gains he will even push back the confines of the inaccessible. Even so he will have to admit in the end that some things can be mastered only to a certain degree and that nature always retain a problematic aspect too deep for human faculties to fathom." (Johann Wolfgang von Goethe, 1827)

"Knowing is possible only when one knows little. As one comes to experience more, one gets gradually assailed by doubts. [...] No phenomenon can be explained, taken merely by itself. Only many, surveyed in their connection, and methodically arranged, finally yield something that can pass for theory." (Johann Wolfgang von Goethe, 1828)

"The greatest art, both in teaching and in life itself, consists in transforming the problem into a postulate." (Johann Wolfgang von Goethe, 1928)

"The vintner's occupation [...] Nature, from whatever angle you approach her, has a glorious way of becoming ever truer, ever more manifest, unfolding ever more, ever deeper, although she remains herself, always the same." (Johann Wolfgang von Goethe, 1828)

"For truth is simple and without fuss, whereas error affords opportunity for dissipating time and energy." (Johann Wolfgang von Goethe, 1829)

"In music the dignity of art seems to find supreme expression. There is no subject matter to be discounted. It is all form and significant content. It elevates and ennobles whatever it expresses." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1829)

 "Man must cling to the belief that the incomprehensible is comprehensible. Else he would give up investigating. " (Johann Wolfgang von Goethe, "Maxims and Reflections", 1829)  

"Music in the best sense does not require the appeal of novelty. As a matter of fact, the older and the more familiar it is, the more it affects us." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1829)

"To concentrate on a craft is the best procedure. For the person of inferior gifts it will always remain a craft. The more gifted person  will raise it to an art. And as for the man of highest endowment, in doing one thing he does all things; or, to put it less paradoxically, in the one thing that he does properly, he sees a symbol of all things that are done right. " (Johann Wolfgang von Goethe, 1829)

"'Nature does nothing for nothing' is an old Philistine saying. She is eternally alive, prodigal and extravagant in her workings, to keep the infinite ever present, because nothing can endure without change." (Johann Wolfgang von Goethe, 1831)

"Although we gladly acknowledge Nature's mysterious encheiresis, her faculty of creating and furthering life, and, without being mystics, admit the existence of ultimate limits to our explorations, we are nevertheless convinced that man, if he is serious about it, cannot desist from the attempt to keep encroaching upon the region of the unexplorable. In the end, of course, he has to give up and willingly concede his defeat." (Johann Wolfgang von Goethe, 1832)

"Hypotheses are scaffoldings erected in front of a building and then dismantled when the building is finished. They are indispensable for the workman; but you mustn't mistake the scaffolding for the building." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1833)

"The desire to explain what is simple by what is complex, what is easy by what is difficult, is a calamity affecting the whole body of science, known, it is true, to men of insight, but not generally admitted." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1833)

"Theories are usually the over-hasty efforts of an impatient understanding that would gladly be rid of phenomena, and so puts in their place pictures, notions, nay, often mere words." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1833)

"With the growth of knowledge our ideas must from time to time be organized afresh. The change takes place usually in accordance with new maxims as they arise, but it always remains provisional. (Johann Wolfgang von Goethe, "Maxims and Reflections", 1833)

"What is exact about mathematics but exactness? And is not this a consequence of the inner sense of truth?" (Johann Wolfgang von Goethe, "Sprüche in Prosa", 1840)

"Nothing hurts a new truth more than an old error." (Johann Wolfgang von Goethe, "Sprüche in Prosa", 1840)

"All truly wise thoughts have been thought already thousands of times; but to make them truly ours, we must think them over again honestly, till they take root in our personal experience." (Johann Wolfgang von Goethe) 

"By the word symmetry […] one thinks of an external relationship between pleasing parts of a whole; mostly the word is used to refer to parts arranged regularly against one another around a centre. We have […] observed [these parts] one after the other, not always like following like, but rather a raising up from below, a strength out of weakness, a beauty out of ordinariness." (Johann Wolfgang von Goethe)

"Content without method leads to fantasy; method without content to empty sophistry; matter without form to unwieldy crudition, form without matter to hollow speculation." (Johann Wolfgang von Goethe, "Scientfic Studies", Colleted Works Vol. 12)

"Everything is simpler than one can imagine, at the same time more involved than can be comprehended." (Johann Wolfgang von Goethe)

"In nature we never see anything isolated, but everything in connection with something else which is before it, beside it, under it and over it." (Johann Wolfgang von Goethe)

"In science everything depends on what one calls an apercu - the discovery of something that is at the bottom of phenomena. Such a discovery is infinitely fruitful." (Johann Wolfgang von Goethe)

"In tearing down a position, all false arguments carry weight; not so in building up. Only the truth is constructive." (Johann Wolfgang von Goethe, "Maxims and Reflections" [posthumous])

"It requires a much higher organ to seize upon truth than it does to defend error." (Johann Wolfgang von Goethe, "Maxims and Reflections" [posthumous])

"Man is not born to solve the problems of the universe, but to find out where the problems begin, and then to take his stand within the limits of the intelligible." (Johann Wolfgang von Goethe)

"Mathematics must subdue the flights of our reason; they are the staff of the blind; no one can take a step without them; and to them and experience is due all that is certain in physics." (Johann Wolfgang von Goethe)

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

"No phenomenon can be explained in and of itself; only many comprehended together, methodically arranged, in the end yield something that could be regarded as theory." (Johann Wolfgang von Goethe)

"Sciences destroy themselves in two ways: by the breadth they reach and by the depth they plumb." (Johann Wolfgang von Goethe)

"Science has been seriously retarded by the study of what is not worth knowing, and of what is not knowable." (Johann Wolfgang von Goethe)

"Science helps us before all things in this, that it somewhat lightens the feeling of wonder with which Nature fills us; then, however, as life becomes more and more complex, it creates new facilities for the avoidance of what would do us harm and the promotion of what will do us good." (Johann Wolfgang von Goethe)

"Symbolism transforms the phenomenon into the idea, and the idea into an image in such a fashion that in the image the idea remains infinitely active and incommensurable, and if all languages were used to express it, it would still remain inexpressible." (Johann Wolfgang von Goethe, "Maxims and Reflections", [posthumous])

"That is the way of youth and life generally, that we usually come to understand the strategy only after the campaign is over. " (Johann Wolfgang von Goethe, "The Autobiography",  [posthumous])

"The highest happiness of man as a thinking being is to have probed what is knowable and quietly to revere what is unknowable. " (Johann Wolfgang von Goethe, "Maxims and Reflections" [posthumous])

"The mathematician is perfect only in so far as he is a perfect being, in so far as he perceives the beauty of truth; only then will his work be thorough, transparent, comprehensive, pure, clear, attractive, and even elegant." (Johann Wolfgang von Goethe)

"The orbits of certainties touch one another; but in the interstices there is room enough for error to go forth and prevail." (Johann Wolfgang von Goethe)

"The tissue of the world is built from necessities and randomness; the intellect of men places itself between both and can control them; it considers the necessity and the reason of its existence; it knows how randomness can be managed, controlled, and used." (Johann Wolfgang von Goethe)

"Thinking by analogy is not to be despised. Analogy has this merit, that it does not settle things - does not pretend to be conclusive. On the other hand, that induction is pernicious which, with a preconceived end in view, and working right forward for only that, drags in its train a number of unshifted observations, both false and true." (Johann Wolfgang von Goethe)

"Thinking is more interesting than knowing, but not more so than beholding." (Johann Wolfgang von Goethe, "Maxims and Reflections", [posthumous])

"To the mathematician everything appears tangible, comprehensible, and mechanical, and he comes under suspicion of being secretly an atheist, inasmuch as he fancies himself as comprehending in his scheme the most immeasurable essence which we call God, and thereby seems to renounce his specific or pre-eminent existence. " (Johann Wolfgang von Goethe, "Maxims and Reflections", [posthumous])

Baruch Spinoza - Collected Quotes

"Measure, time and number are nothing but modes of thought or rather of imagination." (Baruch Spinoza, [Letter to Ludvicus Meyer] 1663)

"In practical life we are compelled to follow what is most probable; in speculative thought we are compelled to follow truth. […] we must take care not to admit as true anything, which is only probable. For when one falsity has been let in, infinite others follow." (Baruch Spinoza, [letter to Hugo Boxel], 1674)

"The highest endeavor of the mind, and the highest virtue, it to understand things by intuition." (Baruch Spinoza, "The Road to Inner Freedom: The Ethics", 1667)

"For the Mind feels those things that it conceives in understanding no less than those it has in the memory. For the eyes of the mind, by which it sees and observes things, are demonstrations [descriptions] themselves." (Baruch Spinoza, "Ethics, Demonstrated in Geometrical Order", 1677)

"From a given determined cause an effect follows of necessity, and on the other hand, if no determined cause is granted, it is impossible that an effect should follow." (Baruch Spinoza, "Ethics", 1677)

"I understand that to be CAUSE OF ITSELF (causa sui) whose essence involves existence and whose nature cannot be conceived unless existing." (Baruch Spinoza, "Ethics", 1677)

"[...] in ordering our thoughts and images, we must always attend to those things which arc good in each thing so that in this way we are always determined to acting from an affect of joy." (Baruch Spinoza, "Ethics", 1677)

"Many errors, of a truth, consist merely in the application of the wrong names of things. For if a man says that the lines which are drawn from the centre of the circle to the circumference are not equal, he understands by the circle, at all events for the time, something else than mathematicians understand by it." (Baruch Spinoza, "Ethics", Book I, 1677)

"Men judge things according to the disposition of their minds, and had rather imagine things than understand them." (Baruch Spinoza, "Ethics", Book I, 1677)

"Neither can the body determine the mind to think, nor the mind the body to move or to rest nor to anything else, if such there be." (Baruch Spinoza, "Ethics", 1677)

"Nothing in Nature is random. […] A thing appears random only through the incompleteness of our knowledge." (Baruch Spinoza, "Ethics", Book I, 1677)

"Nothing in the universe is contingent, but all things are conditioned to exist and operate in a particular manner by the necessity of the divine nature." (Baruch Spinoza, "Ethics", Book I, 1677)

"The idea of any mode in which the human body is affected by external bodies must involve the nature of the human body and at the same rime the nature of the external body." (Baruch Spinoza, "Ethics", 1677)

"[...] that all men are born ignorant of the causes of things, and that all have a desire of acquiring what is useful; [...]" (Baruch Spinoza, "Ethics", 1677)

"The images of things are affections of the human body whose ideas represent external bodies as present to us. […] the affections of the human body whose ideas present external bodies as present to us, we shall call things, though they do not reproduce [external] figures of things. And when the mind regards bodies in this way, we shall say that it imagines." (Baruch Spinoza, "Ethics", 1677)

"Truly, as light manifests itself and darkness, thus truth is the standard of itself and of error." (Baruch Spinoza, "Ethics", 1677)

William R Hamilton - Collected Quotes

“There seems to me to be something analogous to polarized intensity in the pure imaginary part; and to unpolarized energy (indifferent to direction) in the real part of a quaternion: and thus we have some slight glimpse of a future Calculus of Polarities. This is certainly very vague […]” (Sir William R Hamilton, “On Quaternions; or on a new System of Imaginaries in Algebra”, 1844)

“The difficulties which so many have felt in the doctrine of Negative and Imaginary Quantities in Algebra forced themselves long ago on my attention […] And while agreeing with those who had contended that negatives and imaginaries were not properly quantities at all, I still felt dissatisfied with any view which should not give to them, from the outset, a clear interpretation and meaning [...] It early appeared to me that these ends might be attained by our consenting to regard Algebra as being no mere Art, nor Language, nor primarily a Science of Quantity; but rather as the Science of Order in Progression.” (William R Hamilton, “Lectures on Quaternions: Containing a Systematic Statement of a New Mathematical Method… “, 1853)

“Each mathematician for himself, and not anyone for any other, not even all for one, must tread that more than royal road which leads to the palace and sanctuary of mathematical truth.” (Sir William R Hamilton, “Report of the Fifth Meeting of the British Association for the Advancement of Science”, [Address] 1835)

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

”Mathematical language, precise and adequate, nay, absolutely convertible with mathematical thought, can afford us no example of those fallacies which so easily arise from the ambiguities of ordinary language; its study cannot, therefore, it is evident, supply us with any means of obviating those illusions from which it is itself exempt. The contrast of mathematics and philosophy, in this respect, is an interesting object of speculation; tut, as imitation is impossible, one of no practical result.” (Sir William R Hamilton)

"Metaphysics, in whatever latitude the term be taken, is a science or complement of sciences exclusively occupied with mind." (Sir William R Hamilton)

"Time is said to have only one dimension, and space to have three dimensions [...] The mathematical quaternion partakes of both these elements; in technical language it may be said to be ‘time plus space’, or ‘space plus time’: and in this sense it has, or at least involves a reference to, four dimensions. 
And how the One of Time, of Space the Three, 
Might in the Chain of Symbols girdled be." (Sir William R Hamilton)

“We are naturally disposed to refer everything we do not know to principles with which we are familiar.” (Sir William R Hamilton)

Friedrich W Nietzsche - Collected Quotes

"Thus the man who is responsive to artistic stimuli reacts to the reality of dreams as does the philosopher to the reality of existence; he observes closely, and he enjoys his observation: for it is out of these images that he interprets life, out of these processes that he trains himself for life." (Friedrich Nietzsche," The Birth of Tragedy", 1872) 

"Everything which distinguishes man from the animals depends upon this ability to volatilize perceptual metaphors in a schema, and thus to dissolve an image into a concept. For something is possible in the realm of these schemata which could never be achieved with the vivid first impressions: the construction of a pyramidal order according to castes and degrees, the creation of a new world of laws, privileges, subordinations, and clearly marked boundaries - a new world, one which now confronts that other vivid world of first impressions as more solid, more universal, better known, and more human than the immediately perceived world, and thus as the regulative and imperative world." (Friedrich Nietzsche, "On Truth and Lie in an Extra-Moral Sense", 1873)

"That immense framework and planking of concepts to which the needy man clings his whole life long in order to preserve himself is nothing but a scaffolding and toy for the most audacious feats of the liberated intellect. And when it smashes this framework to pieces, throws it into confusion, and puts it back together in an ironic fashion, pairing the most alien things and separating the closest, it is demonstrating that it has no need of these makeshifts of indigence and that it will now be guided by intuitions rather than by concepts. There is no regular path which leads from these intuitions into the land of ghostly schemata, the land of abstractions. There exists no word for these intuitions; when man sees them he grows dumb, or else he speaks only in forbidden metaphors and in unheard - of combinations of concepts. He does this so that by shattering and mocking the old conceptual barriers he may at least correspond creatively to the impression of the powerful present intuition." (Friedrich Nietzsche, "On Truth and Lie in an Extra-Moral Sense", 1873)

"The drive toward the formation of metaphors is the fundamental human drive, which one cannot for a single instant dispense with in thought, for one would thereby dispense with man himself." (Friedrich Nietzsche, "On Truth and Lies in a Nonmoral Sense", 1873)

"The man who is guided by concepts and abstractions only succeeds by such means in warding off misfortune, without ever gaining any happiness for himself from these abstractions. And while he aims for the greatest possible freedom from pain, the intuitive man, standing in the midst of a culture, already reaps from his intuition a harvest of continually inflowing illumination, cheer, and redemption - in addition to obtaining a defense against misfortune. To be sure, he suffers more intensely, when he suffers; he even suffers more frequently, since he does not understand how to learn from experience and keeps falling over and over again into the same ditch." (Friedrich Nietzsche,"On Truth and Lie in an Nonmoral Sense", 1873) 

"What then is truth? A movable host of metaphors, metonymies, and anthropomorphisms: in short, a sum of human relations which have been poetically and rhetorically intensified, transferred, and embellished, and which, after long usage, seem to a people to be fixed, canonical, and binding." (Friedrich W Nietzsche, "On Truth and Lies in a Nonmoral Sense", 1873)

"Words are but symbols for the relations of things to one another and to us; nowhere do they touch upon absolute truth." (Friedrich Nietzsche, "Philosophy in the Tragic Age of the Greeks", 1873)

"Convictions are more dangerous enemies of truth than lies." (Friedrich W Nietzsche, "Human, All Too Human: A book for Free Spirits", 1878) 

"It is true, there could be a metaphysical world; the absolute possibility of it is hardly to be disputed. We behold all things through the human head and cannot cut off this head; while the question nonetheless remains what of the world would still be there if one had cut it off." (Friedrich W Nietzsche, "Human, All Too Human: A Book for Free Spirits", 1878)

"All sciences are now under the obligation to prepare the ground for the future task of the philosopher, which is to solve the problem of value, to determine the true hierarchy of values." (Friedrich W Nietzsche, "The Genealogy of Morals", 1887)

"We say it is ‘explanation’ but it is only in ‘description’ that we are in advance of the older stages of knowledge and science. We describe better we explain just as little as our predecessors." (Friedrich W Nietzsche, "The Joyful Wisdom", 1887)

"In mathematics there is no understanding. In mathematics there are only necessities, laws of existence, invariant relationships. Thus any mathematico-mechanistic outlook must, in the last analysis, waive all understanding. For, we only understand when we know the motives; where there are no motives, all understanding ceases." (Friedrich W Nietzsche) [attributed]

"Science is about finding ever better approximations rather than pretending you have already found ultimate truth." (Friedrich W Nietzsche) [attributed]

Galileo Galilei - Collected Quotes

“Two truths cannot contradict one another.” (Galileo Galilei, [letter to Madame Christina of Lorraine] 1615)

"Nature's great book is written in mathematical symbols." (Galileo Galilei, “The Assayer”, 1623)

"Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth." (Galileo Galilei, “The Assayer”, 1623)

"The great book of nature can be read only by those who know the language in which it was written. And this language is mathematics." (Galileo Galilei, “The Assayer”, 1623)

"[…] nature does not multiply things unnecessarily; that she makes use of the easiest and simplest means for producing her effects; that she does nothing in vain, and the like" (Galileo Galilei, "Dialogue Concerning the Two Chief World Systems", 1632)

"Well, since paradoxes are at hand, let us see how it might be demonstrated that in a finite continuous extension it is not impossible for infinitely many voids to be found." (Galileo Galilei, "Dialogue Concerning the Two Chief World Systems", 1632)

"[Paradoxes of the infinite arise] only when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited; but this […] is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another.” (Galileo Galilei, "Two New Sciences", 1638)

“Infinities and indivisibles transcend our finite understanding, the former on account of their magnitude, the latter because of their smallness; Imagine what they are when combined. In spite of this men cannot refrain from discussing them.” (Galileo Galilei, "Two New Sciences", 1638)

“The length of strings is not the direct and immediate reason behind the forms [ratios] of musical intervals, nor is their tension, nor their thickness, but rather, the ratios of the numbers of vibrations and impacts of air waves that go to strike our eardrum.” (Galileo Galilei, "Two New Sciences", 1638)

”[…] it is astonishing and incredible to us, but not to Nature; for she performs with utmost ease and simplicity things which are even infinitely puzzling to our minds, and what is very difficult for us to comprehend is quite easy for her to perform.” (Galileo Galilei)

“All truths are easy to understand once they are discovered; the point is to discover them.” (Galileo Galilei)

“By denying scientific principles, one may maintain any paradox.” (Galileo Galilei)

“Mathematics is the key and door to the sciences.” (Galileo Galilei)

“The laws of Nature are written in the language of mathematics […]” (Galileo Galilei)

“These are among the marvels that surpass the bounds of our imagination, and that must warn us how gravely one errs in trying to reason about infinites by using the same attributes that we apply to finites.” (Galileo Galilei)

"We cannot teach people anything; we can only help them discover it within themselves." (Galileo Galilei)

On Complex Numbers IIX

"There seems to me to be something analogous to polarized intensity in the pure imaginary part; and to unpolarized energy (indifferent to direction) in the real part of a quaternion: and thus we have some slight glimpse of a future Calculus of Polarities. This is certainly very vague […]" (Sir William R Hamilton, "On Quaternions; or on a new System of Imaginaries in Algebra", 1844) 

"Those who can, in common algebra, find a square root of -1, will be at no loss to find a fourth dimension in space in which ABC may become ABCD: or, if they cannot find it, they have but to imagine it, and call it an impossible dimension, subject to all the laws of the three we find possible. And just as √-1 in common algebra, gives all its significant combinations true, so would it be with any number of dimensions of space which the speculator might choose to call into impossible existence." (Augustus De Morgan, "Trigonometry and Double Algebra", 1849)

“The conception of the inconceivable [imaginary], this measurement of what not only does not, but cannot exist, is one of the finest achievements of the human intellect. No one can deny that such imaginings are indeed imaginary. But they lead to results grander than any which flow from the imagination of the poet. The imaginary calculus is one of the master keys to physical science. These realms of the inconceivable afford in many places our only mode of passage to the domains of positive knowledge. Light itself lay in darkness until this imaginary calculus threw light upon light. And in all modern researches into electricity, magnetism, and heat, and other subtile physical inquiries, these are the most powerful instruments.” (Thomas Hill, “The Imagination in Mathematics”, North American Review Vol. 85, 1857)

“When we consider that the whole of geometry rests ultimately on axioms which derive their validity from the nature of our intuitive faculty, we seem well justified in questioning the sense of imaginary forms, since we attribute to them properties which not infrequently contradict all our intuitions.” (Gottlob Frege, “On a Geometrical Representation of Imaginary forms in the Plane”, 1873) 

“[…] with few exceptions all the operations and concepts that occur in the case of real numbers can indeed be carried over unchanged to complex ones. However, the concept of being greater cannot very well be applied to complex numbers. In the case of integration, too, there appear differences which rest on the multplicity of possible paths of integration when we are dealing with complex variables. Nevertheless, the large extent to which imaginary forms conform to the same laws as real ones justifies the introduction of imaginary forms into geometry.” (Gottlob Frege, “On a Geometrical Representation of Imaginary forms in the Plane”, 1873) 

“When we consider complex numbers and their geometrical representation, we leave the field of the original concept of quantity, as contained especially in the quantities of Euclidean geometry: its lines, surfaces and volumes. According to the old conception, length appears as something material which fills the straight line between its end points and at the same time prevents another thing from penetrating into its space by its rigidity. In adding quantities, we are therefore forced to place one quantity against another. Something similar holds for surfaces and solid contents. The introduction of negative quantities made a dent in this conception, and imaginary quantities made it completely impossible. Now all that matters is the point of origin and the end point; whether there is a continuous line between them, and if so which, appears to make no difference whatsoever; the idea of filling space has been completely lost. All that has remained is certain general properties of addition, which now emerge as the essential characteristic marks of quantity. The concept has thus gradually freed itself from intuition and made itself independent. This is quite unobjectionable, especially since its earlier intuitive character was at bottom mere appearance. Bounded straight lines and planes enclosed by curves can certainly be intuited, but what is quantitative about them, what is common to lengths and surfaces, escapes our intuition.” (Gottlob Frege, “Methods of Calculation based on an Extension of the Concept of Quantity”, 1874)

"The discovery of Minkowski […] is to be found […] in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space. In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude, √-1*ct, proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same role as the three space-coordinates. Formally, these four co-ordinates correspond exactly to the three space co-ordinates in Euclidean geometry." (Albert Einstein,"Relativity: The Special and General Theory", 1920) 

"As an operation, multiplication by i x i has the same effect as multiplication by -1; multiplication by i has the same effect as a rotation by a right angle, and these interpretations […] are consistent. […] Although the interpretation by means of rotations proves nothing, it may suggest that there is no occasion for anyone to muddle himself into a state of mystic wonderment over nothing about the grossly misnamed ‘imaginaries’." (Eric T Bell, "Gauss, the Prince of Mathematicians", 1956)

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

 "All these questions he [the master?] does not pose. So we have to ask them: is the ‘Ring I’ a trap to catch the master or is the ‘Ring I’ a vessel of understanding? In quantum theory it specifies a formula which includes the irrational in a symbol of totality, in a holistic ‘cosmogramm’. But the formula has a catch. If one squares i = √-1, although a negative, one obtains a rationally understandable negative number -1. So one can make the irrational disappear through a slight of hand. This formula does not correspond to reality because the irrational that we call the collective unconscious or the objective psyche can never be rational. It remains always creatively spontaneous, not predictable, not manipulatable. Each holistic formula is in that sense also a trap, because it brings about the illusion that one has understood the whole." (Marie Louise von Franz, "Reflexionen zum ‘Ring I’")

Nassim N Taleb - Collected Quotes

"[…] 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 mistake is not something to be determined after the fact, but in the light of the information until that point." (Nassim N Taleb, "Fooled by Randomness", 2001)

"Probability is not about the odds, but about the belief in the existence of an alternative outcome, cause, or motive." (Nassim N Taleb, "Fooled by Randomness", 2001)

"A Black Swan is a highly improbable event with three principal characteristics: It is unpredictable; it carries a massive impact; and, after the fact, we concoct an explanation that makes it appear less random, and more predictable, than it was. […] The Black Swan idea is based on the structure of randomness in empirical reality. [...] the Black Swan is what we leave out of simplification." (Nassim N Taleb, “The Black Swan”, 2007)

"A theory is like medicine (or government): often useless, sometimes necessary, always self-serving, and on occasion lethal. So, it needs to be used with care, moderation and close adult supervision." (Nassim N Taleb, "The Black Swan: The Impact of the Highly Improbable", 2007)

"Prediction, not narration, is the real test of our understanding of the world." (Nassim N Taleb, “The Black Swan”, 2007)

"Probability is a liberal art; it is a child of skepticism, not a tool for people with calculators on their belts to satisfy their desire to produce fancy calculations and certainties." (Nassim N Taleb, “The Black Swan”, 2007)

"The inability to predict outliers implies the inability to predict the course of history." (Nassim N Taleb, "The Black Swan" , 2007)

"While in theory randomness is an intrinsic property, in practice, randomness is incomplete information." (Nassim N Taleb, "The Black Swan", 2007)

"Antifragility is beyond resilience or robustness. The resilient resists shocks and stays the same; the antifragile gets better." (Nassim N Taleb, "Antifragile: Things that gain from disorder", 2012)

"Black Swans (capitalized) are large-scale unpredictable and irregular events of massive consequence - unpredicted by a certain observer, and such un - predictor is generally called the 'turkey' when he is both surprised and harmed by these events. [...] Black Swans hijack our brains, making us feel we 'sort of' or 'almost' predicted them, because they are retrospectively explainable. We don’t realize the role of these Swans in life because of this illusion of predictability. […] An annoying aspect of the Black Swan problem - in fact the central, and largely missed, point - is that the odds of rare events are simply not computable." (Nassim N Taleb, "Antifragile: Things that gain from disorder", 2012)

"Complex systems are full of interdependencies - hard to detect - and nonlinear responses. […] Man-made complex systems tend to develop cascades and runaway chains of reactions that decrease, even eliminate, predictability and cause outsized events. So the modern world may be increasing in technological knowledge, but, paradoxically, it is making things a lot more unpredictable." (Nassim N Taleb, "Antifragile: Things that gain from disorder", 2012)

"Heuristics are simplified rules of thumb that make things simple and easy to implement. But their main advantage is that the user knows that they are not perfect, just expedient, and is therefore less fooled by their powers. They become dangerous when we forget that." (Nassim N Taleb, "Antifragile: Things that gain from disorder", 2012)

"It is all about redundancy. Nature likes to overinsure itself." (Nassim N Taleb, "Antifragile: Things That Gain from Disorder", 2012)

"In fact, the most interesting aspect of evolution is that it only works because of its antifragility; it is in love with stressors, randomness, uncertainty, and disorder - while individual organisms are relatively fragile, the gene pool takes ad - vantage of shocks to enhance its fitness. […] So evolution benefits from randomness by two different routes: randomness in the mutations, and randomness in the environment - both act in a similar way to cause changes in the traits of the surviving next generations." (Nassim N Taleb, "Antifragile: Things that gain from disorder", 2012)

"Social scientists use the term 'equilibrium' to describe balance between opposing forces, say, supply and demand, so small disturbances or deviations in one direction, like those of a pendulum, would be countered with an adjustment in the opposite direction that would bring things back to stability." (Nassim N Taleb, "Antifragile: Things that gain from disorder", 2012)

"Some things benefit from shocks; they thrive and grow when exposed to volatility, randomness, disorder, and stressors and love adventure, risk, and uncertainty. Yet, in spite of the ubiquity of the phenomenon, there is no word for the exact opposite of fragile. Let us call it antifragile." (Nassim N Taleb, "Antifragile: Things that gain from disorder", 2012)

"Systems subjected to randomness - and unpredictability - build a mechanism beyond the robust to opportunistically reinvent themselves each generation, with a continuous change of population and species." (Nassim N Taleb, "Antifragile: Things that gain from disorder", 2012)

"Technology is the result of antifragility, exploited by risk-takers in the form of tinkering and trial and error, with nerd-driven design confined to the backstage." (Nassim N Taleb, "Antifragile: Things that gain from disorder", 2012)

"This is the central illusion in life: that randomness is risky, that it is a bad thing - and that eliminating randomness is done by eliminating randomness. Randomness is distributed rather than concentrated." (Nassim N Taleb, "Antifragile: Things that gain from disorder", 2012)

"We can simplify the relationships between fragility, errors, and antifragility as follows. When you are fragile, you depend on things following the exact planned course, with as little deviation as possible - for deviations are more harmful than helpful. This is why the fragile needs to be very predictive in its approach, and, conversely, predictive systems cause fragility. When you want deviations, and you don’t care about the possible dispersion of outcomes that the future can bring, since most will be helpful, you are antifragile. Further, the random element in trial and error is not quite random, if it is carried out rationally, using error as a source of information. If every trial provides you with information about what does not work, you start zooming in on a solution - so every attempt becomes more valuable, more like an expense than an error. And of course you make discoveries along the way." (Nassim N Taleb, "Antifragile: Things that gain from disorder", 2012)

"When some systems are stuck in a dangerous impasse, randomness and only randomness can unlock them and set them free. You can see here that absence of randomness equals guaranteed death. The idea of injecting random noise into a system to improve its functioning has been applied across fields. By a mechanism called stochastic resonance, adding random noise to the background makes you hear the sounds (say, music) with more accuracy." (Nassim N Taleb, "Antifragile: Things that gain from disorder", 2012)

"The higher the dimension, in other words, the higher the number of possible interactions, and the more disproportionally difficult it is to understand the macro from the micro, the general from the simple units. This disproportionate increase of computational demands is called the curse of dimensionality." (Nassim N Taleb, "Skin in the Game: Hidden Asymmetries in Daily Life", 2018)

"Behavioral finance so far makes conclusions from statics not dynamics, hence misses the picture. It applies trade-offs out of context and develops the consensus that people irrationally overestimate tail risk (hence need to be 'nudged' into taking more of these exposures). But the catastrophic event is an absorbing barrier. No risky exposure can be analyzed in isolation: risks accumulate. If we ride a motorcycle, smoke, fly our own propeller plane, and join the mafia, these risks add up to a near-certain premature death. Tail risks are not a renewable resource." (Nassim N Taleb, "Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications" 2nd Ed., 2022)

"But note that any heavy tailed process, even a power law, can be described in sample (that is finite number of observations necessarily discretized) by a simple Gaussian process with changing variance, a regime switching process, or a combination of Gaussian plus a series of variable jumps (though not one where jumps are of equal size […])." (Nassim N Taleb, "Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications" 2nd Ed., 2022)

"[…] it is not merely that events in the tails of the distributions matter, happen, play a large role, etc. The point is that these events play the major role and their probabilities are not (easily) computable, not reliable for any effective use. The implication is that Black Swans do not necessarily come from fat tails; the problem can result from an incomplete assessment of tail events." (Nassim N Taleb, "Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications" 2nd Ed., 2022)

"Once we know something is fat-tailed, we can use heuristics to see how an exposure there reacts to random events: how much is a given unit harmed by them. It is vastly more effective to focus on being insulated from the harm of random events than try to figure them out in the required details (as we saw the inferential errors under thick tails are huge). So it is more solid, much wiser, more ethical, and more effective to focus on detection heuristics and policies rather than fabricate statistical properties." (Nassim N Taleb, "Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications" 2nd Ed., 2022)

"[…] whenever people make decisions after being supplied with the standard deviation number, they act as if it were the expected mean deviation." (Nassim N Taleb, "Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications" 2nd Ed., 2022)

Thomas Aquinas - Collected Quotes

"[…] the least initial deviation from the truth is multiplied later a thousand-fold. Admit, for instance, the existence of a minimum magnitude, and you will find that the minimum which you have introduced, small as it is, causes the greatest truths of mathematics to totter. The reason is that a principle is great rather in power than in extent; hence that which was small at the start turns out a giant at the end." (St. Thomas Aquinas, "De Ente et Essentia", cca. 1252)

"If a thing can be done adequately by means of one, it is superfluous to do it by means of several; for we observe that nature does not employ two instruments where one suffices." (Thomas Aquinas, "Summa Theologica", cca. 1266-1273)

"It is superfluous to suppose that what can be accounted for by a few principles has been produced by many." (Thomas Aquinas, "Summa Theologica", cca. 1266-1273)

"Reason may be employed in two ways to establish a point: first for the purpose of furnishing sufficient proof of some principle, as in natural science, where sufficient proof can be brought to show that the movement of the heavens is always of uniform velocity. Reason is employed in another way, not as furnishing a sufficient proof of a principle, but as confirming an already established principle, by showing the congruity of its results […]" (St. Thomas Aquinas, "Summa Theologica", cca. 1266-1273)

"The existence of an actual infinite multitude is impossible. For any set of things one considers must be a specific set. And sets of things are specified by the number of things in them. Now no number is infinite, for number results from counting through a set of units. So no set of things can actually be inherently unlimited, nor can it happen to be unlimited." (St. Thomas Aquinas, "Summa Theologica", cca. 1266-1273)

"The universal cause is one thing, a particular cause another. An effect can be haphazard with respect to the plan of the second, but not of the first. For an effect is not taken out of the scope of one particular cause save by another particular cause which prevents it, as when wood dowsed with water, will not catch fire. The first cause, however, cannot have a random effect in its own order, since all particular causes are comprehended in its causality. When an effect does escape from a system of particular causality, we speak of it as fortuitous or a chance happening […]" (Thomas Aquinas, "Summa Theologica", cca. 1266-1273)

"We see that things which lack intelligence, such as natural bodies, act for an end, and this is evident from their acting always, or nearly always, in the same way, so as to obtain the best result. Hence it is plain that not fortuitously, but designedly, do they achieve their end. Now whatever lacks intelligence cannot move towards an end, unless it be directed by some being endowed with knowledge and intelligence; as the arrow is shot to its mark by the archer. Therefore some intelligent being exists by whom all natural things are directed to their end; and this being we call God." (St. Thomas Aquinas, "Summa Theologia", cca. 1266-1273)

"Nor is it enough to say that the intelligible notions formed by the active intellect subsist somehow in the phantasmata (mental image), which are certainly intrinsic to us; for as we have already observed in treating the passive intellect, objects only become actually intelligible when abstracted from phantasmata; so that merely by way of the phantasmata, we cannot attribute the work of the active intellect to ourselves" (St. Thomas Aquinas, "De Anima" III, cca. 1268) [On Aristotle's phantasmata]

"Practical sciences proceed by building up; theoretical sciences by resolving into components." (St. Thomas Aquinas, "Sententia libri Ethicorum" [Commentary on the Nicomachean Ethics], 1271)

"For it is necessary in every practical science to proceed in a composite (i.e. deductive) manner. On the contrary in speculative science, it is necessary to proceed in an analytical manner by breaking down the complex into elementary principles." (St. Thomas Aquinas, "Sententia libri Ethicorum" [Commentary on the Nicomachean Ethics], 1271)

"A scrap of knowledge about sublime things is worth more than any amount about trivialities." (St. Thomas Aquinas)

"Because philosophy arises from awe, a philosopher is bound in his way to be a lover of myths and poetic fables. Poets and philosophers are alike in being big with wonder." (St. Thomas Aquinas)

"Distinctions drawn by the mind are not necessarily equivalent to distinctions in reality." (St. Thomas Aquinas)

"Eternity is called whole, not because it has parts, but because it is lacking in nothing." (St. Thomas Aquinas)

"Mistakes are made on two counts: an argument is either based on error or incorrectly developed." (St. Thomas Aquinas)

Auguste Comte - Collected Quotes

"Geometry is a true natural science: - only more simple, and therefore more perfect than any other. We must not suppose that, because it admits the application of mathematical analysis, it is therefore a purely logical science, independent of observation. Everybody studied by geometers presents some primitive phenomena which, not being discoverable by reasoning, must be due to observation alone." (Auguste Comte, “Course of Positive Philosophy”, 1830)  

"In mathematics we find the primitive source of rationality; and to mathematics must the biologists resort for means to carry on their researches."  (Auguste Comte, “Course of Positive Philosophy”, 1830) 

“[…] in order to observe, our mind has need of some theory or other. If in contemplating phenomena we did not immediately connect them with principles, not only would it be impossible for us to combine these isolated observations, and therefore to derive profit from them, but we should even be entirely incapable of remembering facts, which would for the most remain unnoted by us.” (Auguste Comte, “Course of Positive Philosophy”, 1830)

"It must ever be remembered that the true positive spirit first came forth from the pure sources of mathematical science; and it is only the mind that has imbibed it there, and which has been face to face with the lucid truths of geometry and mechanics, that can bring into full action its natural positivity, and apply it in bringing the most complex studies into the reality of demonstration. No other discipline can fitly prepare the intellectual organ." (Auguste Comte, “Course of Positive Philosophy”, 1830) 

“The business of concrete mathematics is to discover the equations which express the mathematical laws of the phenomenon under consideration; and these equations are the starting-point of the calculus, which must obtain from them certain quantities by means of others.” (Auguste Comte, “Course of Positive Philosophy”, 1830) 

"The domain of physics is no proper field for mathematical pastimes. The best security would be in giving a geometrical training to physicists, who need not then have recourse to mathematicians, whose tendency is to despise experimental science. By this method will that union between the abstract and the concrete be effected which will perfect the uses of mathematical, while extending the positive value of physical science. Meantime, the uses of analysis in physics is clear enough. Without it we should have no precision, and no co-ordination; and what account could we give of our study of heat, weight, light, etc.? We should have merely series of unconnected facts, in which we could foresee nothing but by constant recourse to experiment; whereas, they now have a character of rationality which fits them for purposes of prevision." (Auguste Comte, “The Positive Philosophy”, 1830)

 “The limitations of Mathematical science are not, then, in its nature. The limitations are in our intelligence: and by these we find the domain of the science remarkably restricted, in proportion as phenomena, in becoming special, become complex.” (Auguste Comte, “The Positive Philosophy”, 1830)

"There is no inquiry which is not finally reducible to a question of Numbers; for there is none which may not be conceived of as consisting in the determination of quantities by each other, according to certain relations." (Auguste Comte, “The Positive Philosophy”, 1830)

“To understand a science it is necessary to know its history.” (Auguste Comte, “The Positive Philosophy”, 1830)

"We may therefore define Astronomy as the science by which we discover the laws of the geometrical and mechanical phenomena presented by the heavenly bodies." (Auguste Comte, “The Positive Philosophy”, 1830)

“[Algebra] has for its object the resolution of equations; taking this expression in its full logical meaning, which signifies the transformation of implicit functions into equivalent explicit ones. In the same way arithmetic may be defined as destined to the determination of the values of functions. […] We will briefly say that Algebra is the Calculus of functions, and Arithmetic is the Calculus of Values.” (Auguste Comte, “Philosophy of Mathematics”, 1851)

“Every science consists in the coordination of facts; if the different observations were entirely isolated, there would be no science.” (Auguste Comte, “Philosophy of Mathematics”, 1851)

"Mathematical Analysis is […] the true rational basis of the whole system of our positive knowledge." (Auguste Comte, "System of Positive Polity", 1851)

 "No science can be really understood apart from its special history, which again cannot be separated from the general history of Humanity." (Auguste Comte, "System of Positive Polity", 1851)

"The formation of the differential equations proper to the phenomena is, independent of their integration, a very important acquisition, on account of the approximations which mathematical analysis allows between questions, otherwise heterogeneous [...]" (Auguste Comte, "System of Positive Polity", 1851)

“[…] it is only through Mathematics that we can thoroughly understand what true science is. Here alone can we find in the highest degree simplicity and severity of scientific law, and such abstraction as the human mind can attain.” (Auguste Comte)

Heinrich Hertz - Collected Quotes

"For of an image one requires some kind of sameness with the pictured object, of a statue sameness of form, of a delineation sameness of perspective projection in the visual field, of a painting also sameness of color."  (Heinrich Hertz, "The Facts in Perception", 1878) 

"All physicists agree that the problem of physics consists in tracing the phenomena of nature back to the simple laws of mechanics.” (Heinrich Hertz, "The Principles of Mechanics Presented in a New Form", 1894)

"Experience is the collecting of what is similar in different particular perceptions." (Heinrich Hertz, "The Principles of Mechanics Presented in a New Form", 1894)

"For our purpose it is not necessary that they [images] should be in conformity with the things in any other respect whatever. As a matter of fact, we do not know, nor have we any means of knowing, whether our conceptions of things are in conformity with them in any other than this one fundamental respect."  (Heinrich Hertz, "The Principles of Mechanics Presented in a New Form", 1894)

"In this sense the fundamental ideas of mechanics, together with the principles connecting them, represent the simplest image which physics can produce of things in the sensible world and the processes which occur in it. By varying the choice of the propositions which we take as fundamental, we can give various representations of the principles of mechanics. Hence we can thus obtain various images of things; and these images we can test and compare with each other in respect of permissibility, correctness, and appropriateness." (Heinrich Hertz, "The Principles of Mechanics Presented in a New Form", 1894)

"It is a common and necessary feature of human intelligence that we can neither conceive of things nor define them conceptually without adding attributes to them that simply do not exist. This applies not only to every thought and imagination of ordinary life, even the sciences do not proceed otherwise. Only philosophy seeks and finds the difference between things that exist and things that we perceive, and also sees the necessity of this difference. […] What we add are therefore not incorrect conceptions but the conditions for such conceptions in general. We cannot simply remove them and replace them with better ones; either we must add them, or we must abstain from all conceptions of this kind." (Heinrich Hertz, "The Principles of Mechanics Presented in a New Form", 1894)

"Mature knowledge regards logical clearness as of prime importance: only logically clear images does it test as to correctness; only correct images does it compare as to appropriateness. By pressure of circumstances the process is often reversed. Images are found to be suitable for a certain purpose; are next tested as to their correctness ; and only in the last place purged of implied contradictions." (Heinrich Hertz, "The Principles of Mechanics Presented in a New Form", 1894)

"The images which we may form of things are not determined without ambiguity by the requirement that the consequents of the images must be the images of the consequents. Various images of the same objects are possible, and these images may differ in various respects. We should at once denote as inadmissible all images which implicitly contradict the laws of our thought. Hence we postulate in the first place that all our images shall be logically permissible or, briefly, that they shall be permissible. We shall denote as incorrect any permissible images, if their essential relations contradict the relations of external things, i.e. if they do not satisfy our first fundamental requirement. Hence we postulate in the second place that our images shall be correct. But two permissible and correct images of the same external objects may yet differ in respect of appropriateness. Of two images of the same object that is the more appropriate which pictures more of the essential relations of the object, the one which we may call the more distinct. Of two images of equal distinctness the more appropriate is the one which contains, in addition to the essential characteristics, the smaller number of superfluous or empty relations, the simpler of the two. Empty relations cannot be altogether avoided: they enter into the images because they are simply images, images produced by our mind and necessarily affected by the characteristics of its mode of portrayal." (Heinrich Hertz, "The Principles of Mechanics Presented in a New Form”, 1894)

"The most direct, and in a sense the most important, problem which our conscious knowledge of nature should enable us to solve is the anticipation of future events, so that we may arrange our present affairs in accordance with such anticipation. As a basis for the solution of this problem we always make use of our knowledge of events which have already occurred, obtained by chance observation or by prearranged experiment. In endeavouring thus to draw inferences as to the future from the past, we always adopt the following process. We form for ourselves images or symbols of external objects; and the form which we give them is such that the necessary consequents of the images in thought are always the images of the necessary consequents in nature of the things pictured. In order that this requirement may be satisfied, there must be a certain conformity between nature and our thought." (Heinrich Hertz, "The Principles of Mechanics Presented in a New Form", 1894) 

"[…] we cannot a priori demand from nature simplicity, nor can we judge what in her opinion is simple. But with regard to images of our own creation we can lay down requirements. We are justified in deciding that if our images are well adapted to the things, the actual relations of the things must be represented by simple relations between the images. And if the actual relations between the things can only be represented by complicated relations, which are not even intelligible to an unprepared mind, we decide that those images are not sufficiently well adapted to the things. Hence our requirement of simplicity does not apply to nature, but to the images thereof which we fashion ; and our repugnance to a complicated statement as a fundamental law only expresses the conviction that, if the contents of the statement are correct and comprehensive, it can be stated in a simpler form by a more suitable choice of the fundamental conceptions." (Heinrich Hertz, "The Principles of Mechanics Presented in a New Form", 1894)

"One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them." (Heinrich Hertz) 

"The rigor of science requires that we distinguish well the undraped figure of nature itself from the gay-coloured vesture with which we clothe it at our pleasure." (Heinrich Hertz) 

"This is often the way it is in physics - our mistake is not that we take our theories too seriously, but that we do not take them seriously enough. It is always hard to realize that these numbers and equations we play with at our desks have something to do with the real world." (Heinrich Hertz) 

On Complex Numbers VII

“The remark which you make concerning roots that can not be extracted, and containing imaginary quantities which when added together give none the less a real quantity, is surprising and entirely new. One would never have believed that √(1 + √-3) + √(1 - √- 3) would make √6, and there is something hidden in this which is incomprehensible.” (Christaan Huygens, [letter to Gottfried W Leibniz] cca. 1670)

"But if now a simple, that is, a linear equation, is multiplied by a quadratic, a cubic equation will result, which will have  real roots if the quadratic is possible, or two imaginary roots and only one real one if the quadratic is impossible. […] How can it be, that a real quantity, a root of the proposed equation, is expressed by the intervention of an imaginary? For this is the remarkable thing, that, as calculation shows, such an imaginary quantity is only observed to enter those cubic equations that have no imaginary root, all their roots being real or possible, as has been shown by trisection of an angle, by Albert Girard and others. […] This difficulty has been too much for all writers on algebra up to the present, and they have all said they that in this case Cardano’s rules fail." (Gottfried W Leibniz, cca. 1675)

"For this evil I have found a remedy and obtained a method, by which without experimentation the roots of such binomials can be extracted, imaginaries being no hindrance, and not only in the case of cubics but also in higher equations. This invention rests upon a certain peculiarity which I will explain later. Now I will add certain rules derived from the consideration of irrationals (although no mention is made of irrationals), by which a rational root can easily be extracted from them." (Gottfried W Leibniz, cca. 1675)

“Infinities and infinitely small quantities could be taken as fictions, similar to imaginary roots, except that it would make our calculations wrong, these fictions being useful and based in reality.” (Gottfried W Leibniz, [letter to Johann Bernoulli] 1689)

“For it ought to be considered that both –b   and –c  , as they stand alone, are, in some Sense, as much impossible Quantities as √(-b)  and √(-c) ; since the Sign –, according to the established Rules of Notation, shews the Quantity, to which it is prefixed, is to be subtracted, but to subtract something from nothing is impossible, and the Notion or Supposition of a Quantity actually less than Nothing, absurd and shocking to the Imagination.” (Thomas Simpson, “A Treatise of Algebra”, 1745) 

“After exponential quantities the circular functions, sine and cosine, should be considered because they arise when imaginary quantities are involved in the exponential."  (Leonhard Euler, ”Introductio in analysin infinitorum”, 1748)

“Moreover, the whole method has the essential disadvantage that it occupies the mind with the distinction of a great number of cases that can be recognized only by inner intuition, and thus neutralizes an important part of that which algebra is supposed to accomplish, which is relieving the power of inner intuition. Finally, in such a treatment algebra loses a great part of the generality that it can obtain by the mutual connection of different problems, which becomes evident so easily when one uses isolated negative quantities. [...] Since imaginary quantities have to occur, science would certainly not win that much by avoiding negative quantities than it would lose in terms of clarity and generality.” (Johann P W Stein,  “Die Elemente der Algebra: Erster Cursus”, 1828) 

"Originally assuming the concept of the absolute integers, it extended its domain step by step; integers were supplemented by fractions, rational numbers by irrational numbers, positive numbers by negative numbers, and real numbers by imaginary numbers. This advance, however, occurred initially with a fearfully hesitant step. The first algebraists preferred to call negative roots of equations false roots, and it is precisely these where the problem to which they refer was always termed in such a way as to ensure that the nature of the quantity sought did not admit any opposite.” (Carl F Gauss, “Theoria residuorum biquadraticum. Commentatio secunda. [Selbstanzeige]”, Göttingische gelehrte Anzeigen 23 (4), 1831)

“[T]he notion of a negative magnitude has become quite a familiar one […] But it is far otherwise with the notion which is really the fundamental one (and I cannot too strongly emphasize the assertion) underlying and pervading the whole of modern analysis and geometry, that of imaginary magnitude in analysis and of imaginary space (or space as a locus in quo of imaginary points and figures) in geometry: I use in each case the word imaginary as including real. This has not been, so far as I am aware, a subject of philosophical discussion or inquiry. […] considering the prominent position which the notion occupies-say even that the conclusion were that the notion belongs to mere technical mathematics, or has reference to nonentities in regard to which no science is possible, still it seems to me that (as a subject of philosophical discussion) the notion ought not to be thus ignored; it should at least be shown that there is a right to ignore it.” (Arthur Cayley, [address before the meeting of the British Association at Southport] 1870) 

“A satisfactory theory of the imaginary quantities of ordinary algebra, which is essentially a simple case of multiple algebra, with difficulty obtained recognition in the first third of this century. We must observe that this double algebra, as it has been called, was not sought for or invented; - it forced itself, unbidden, upon the attention of mathematicians, and with its rules already formed.
But the idea of double algebra, once received, although as it were unwillingly, must have suggested to many minds more or less distinctly the possibility of other multiple algebras, of higher orders, possessing interesting or useful properties.” (Josiah W Gibbs, “On multiple Algebra”, Proceedings of the American Association for the Advancement of Science Vol. 35, 1886)