♾️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)

🅺Cassius J Keyser - Collected Quotes

"It [mathematics] is in the inner world of pure thought, where all entia dwell, where is every type of order and manner of correlation and variety of relationship, it is in this infinite ensemble of eternal verities whence, if there be one cosmos or many of them, each derives its character and mode of being, - it is there that the spirit of mathesis has its home and its life. [/] Is it a restricted home, a narrow life, static and cold and grey with logic, without artistic interest, devoid of emotion and mood and sentiment? That world, it is true, is not a world of solar light, not clad in the colours that liven and glorify the things of sense, but it is an illuminated world, and over it all and everywhere throughout are hues and tints transcending sense, painted there by radiant pencils of psychic light, the light in which it lies. It is a silent world, and, nevertheless, in respect to the highest principle of art - the interpenetration of content and form, the perfect fusion of mode and meaning - it even surpasses music. In a sense, it is a static world, but so, too, are the worlds of the sculptor and the architect. The figures, however, which reason constructs and the mathematic vision beholds, transcend the temple and the statue, alike in simplicity and in intricacy, in delicacy and in grace, in symmetry and in poise. Not only are this home and this life thus rich in aesthetic interests, really controlled and sustained by motives of a sublimed and supersensuous art, but the religious aspiration, too, finds there, especially in the beautiful doctrine of invariants, the most perfect symbols of what it seeks - the changeless in the midst of change, abiding things hi a world of flux, configurations that remain the same despite the swirl and stress of countless hosts of curious transformations." (Cassius J Keyser, "The Universe and Beyond", Hibbert Journal, 1904-1906)

"The apodictic quality of mathematical thought, the certainty and correctness of its conclusions, are due, not to a special mode of ratiocination, but to the character of the concepts with which it deals. What is that distinctive characteristic? I answer: precision, sharpness, completeness of definition. But how comes your mathematician by such completeness? There is no mysterious trick involved; some ideas admit of such precision, others do not; and the mathematician is one who deals with those that do." (Cassius J Keyser, "The Universe and Beyond", Hibbert Journal Vol. 3, 1904–1905)

"To think the thinkable - that is the mathematician's aim." (Cassius J Keyser, "The Universe and Beyond", Hibbert Journal Vol. 3, 1904-1905)

"For scientific theories are, each and all of them, and they will continue to be, built upon and about notions which, however sublimated, are nevertheless derived from common sense." (Cassius J Keyser, "Mathematics", 1907)

"It seems indeed as if the entire surface of the world of human consciousness were predestined to be covered over, in varying degrees of luxuriance, by the flora of mathematic science." (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1908)

"Mathematics is no more the art of reckoning and computation than architecture is the art of making bricks or hewing wood, no more than painting is the art of mixing colors on a palette, no more than the science of geology is the art of breaking rocks, or the science of anatomy the art of butchering." (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1908)

"Science does not seek emancipation in order to become a drudge, she consents to serve indeed but her service aims at freedom as an end." (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1908)

"Symbolic Logic is Mathematics, Mathematics is Symbolic Logic, the twain are one." (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1908)

"The great mathematician, like the great poet or great naturalist or great administrator, is born." (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1908)

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

"Curiosity is the aspect of the universe seeking to realise itself, and the fruit of such activity is new reality, stimulating to new research." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)

"Mathematics, like any other cardinal activity of the human spirit, has an individuality of its own." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)

"Knowledge - a kind of proliferating sphere, expanding along divergent lines by the outward-seeking of an inner life of wonder." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)
 
"Projective Geometry: a boundless domain of countless fields where reals and imaginaries, finites and infinites, enter on equal terms, where the spirit delights in the artistic balance and symmetric interplay of a kind of conceptual and logical counterpoint - an enchanted realm where thought is double and flows throughout in parallel streams." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)

"The domain of mathematics is the sole domain of certainty." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)

"The ideal of thought is rigor; mathematics is the name that usage employs to designate, not attainment of the ideal, for it cannot be attained, but its devoted pursuit and close approximation." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)

"The rigor of mathematics is not absolute - absolute rigor is an ideal, to be, like other ideals, aspired unto, forever approached, but never quite attained, for such attainment would mean that every possibility of error or indetermination, however slight, had been eliminated from idea, from symbol, and from argumentation." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)

"To humanize the teaching of mathematics means so to present the subject, so to interpret its ideas and doctrines, that they shall appeal, not merely to the computatory faculty or to the logical faculty but to all the great powers and interests of the human mind." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)

"Absolute certainty is a privilege of uneducated minds and fanatics. - It is, for scientific folk, an unattainable ideal." (Cassius J Keyser, "Mathematical Philosophy: A Study of Fate and Freedom", 1922)

"It can, you see, be said, with the same approximation to truth, that the whole of science, including mathematics, consists in the study of transformations or in the study of relations." (Cassius J Keyser. "Mathematical Philosophy: A Study of Fate and Freedom", 1922)

"Mathematics is, in many ways, the most precious response that the human spirit has made to the call of the infinite." (Cassius J. Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1925)

"Every major concern among the intellectual concerns of man is a concern of mathematics." (Cassius J Keyser, "Mole Philosophy and Other Essays", 1927)

"It is customary to speak of mathematics, of pure mathematics, and of applied mathematics, as if the first were a genus owning the other two as species. The custom is unfortunate because it is misleading." (Cassius J Keyser, "Mole Philosophy and Other Essay", 1927)

 "The validity of mathematical propositions is independent of the actual world -  the world of existing subject-matters - is logically prior to it, and would remain unaffected were it to vanish from being." (Cassius J Keyser, "The Pastures of Wonder: The Realm of Mathematics and the Realm of Science", 1929)

🆅Voltaire - Collected Quotes

"This method of subjecting the infinite to algebraic manipulations is called differential and integral calculus. It is the art of numbering and measuring with precision things the existence of which we cannot even conceive. Indeed, would you not think that you are being laughed at, when told that there are lines infinitely great which form infinitely small angles? Or that a line which is straight so long as it is finite would, by changing its direction infinitely little, become an infinite curve? Or that there are infinite squares, infinite cubes, and infinities of infinities, one greater than another, and that, as compared with the ultimate infinitude, those which precede it are as nought. All these things at first appear as excess of frenzy; yet, they bespeak the great scope and subtlety of the human spirit, for they have led to the discovery of truths hitherto undreamt of." (Voltaire, "Letters on the English", 1733)

"All certainty which does not consist in mathematical demonstration is nothing more than the highest probability; there is no other historical certainty." (Voltaire, "A Philosophical Dictionary", 1764)

"Chance is a world void of sense; nothing can exist without a cause." (Voltaire, A Philosophical Dictionary, 1764)

"History is the recital of facts represented as true. Fable, on the other hand, is the recital of facts represented as fiction." (Voltaire, "A Philosophical Dictionary", 1764)

"In geometry, as in most sciences, it is very rare that an isolated proposition is of immediate utility. But the theories most powerful in practice are formed of propositions which curiosity alone brought to light, and which long remained useless without its being able to divine in what way they should one day cease to be so. In this sense it may be said, that in real science, no theory, no research, is in effect useless." (Voltaire, "A Philosophical Dictionary", 1764) 

"Show all these fanatics a little geometry, and they learn it quite easily. But, strangely enough, their minds are not thereby rectified. They perceive the truths of geometry, but it does not teach them to weigh probabilities. Their minds have set hard. They will reason in a topsy-turvy wall all their lives, and I am sorry for it." (Voltaire, "False Minds" ["Esprits faux"], A Philosophical Dictionary],1764)

"We admit, in geometry, not only infinite magnitudes, that is to say, magnitudes greater than any assignable magnitude, but infinite magnitudes infinitely greater, the one than the other. This astonishes our dimension of brains, which is only about six inches long, five broad, and six in depth, in the largest heads." (Voltaire," Infinity" [in "A Philosophical Dictionary"], 1764)

"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." (Voltaire, "Oeuvres Complètes" ["Complete Works"] Vol. 35, 1888)

"The human brain is a complex organ with the wonderful power of enabling man to find reasons for continuing to believe whatever it is that he wants to believe." (Voltaire [attributed])

"The mathematics will always be a kind of mystery to the bulk of the nation, and consequently will always be an object of veneration." (Voltaire)

🆆William Whewell - Collected Quotes

"Discoveries are not generally made in the order of their scientific arrangement: their connexions and relations are made out gradually; and it is only when the fermentation of invention has subsided that the whole clears into simplicity and order. " (William Whewell, "An Elementary Treatise on Mechanics" Vol. I, 1819)

"It is not easy to anatomize the constitution and the operations of a mind which makes such an advance in knowledge. Yet we may observe that there must exist in it, in an eminent degree, the elements which compose the mathematical talent. It must possess distinctness of intuition, tenacity and facility in tracing logical connection, fertility of invention, and a strong tendency to generalization." (William Whewell, "History of the Inductive Sciences" Vol. 1, 1837)

"The peculiar character of mathematical truth is that it is necessarily and inevitably true; and one of the most important lessons which we learn from our mathematical studies is a knowledge that there are such truths." (William Whewell, "Principles of English University Education", 1838)

"Every stage of science has its train of practical applications and systematic inferences, arising both from the demands of convenience and curiosity, and from the pleasure which, as we have already said, ingenious and active-minded men feel in exercising the process of deduction." (William Whewell, "The Philosophy of the Inductive Sciences Founded Upon Their History", 1840)

"These sciences have no principles besides definitions and axioms, and no process of proof but deduction; this process, however, assuming a most remarkable character; and exhibiting a combination of simplicity and complexity, of rigor and generality, quite unparalleled in other subjects." (William Whewell, "The Philosophy of the Inductive Sciences Founded Upon Their History", 1840)

"Every detection of what is false directs us towards what is true: every trial exhausts some tempting form of error. Not only so; but scarcely any attempt is entirely a failure; scarcely any theory, the result of steady thought, is altogether false; no tempting form of error is without some latent charm derived from truth." (William Whewell, "Lectures on the History of Moral Philosophy in England", 1852)

"Scarcely any attempt is entirely a failure; scarcely any theory, the result of steady thought, is altogether false; no tempting form of Error is without some latent charm derived from Truth." (William Whewell, "Lectures on the History of Moral Philosophy in England", 1852)

"Geometry in every proposition speaks a language which experience never dares to utter; and indeed of which she but half comprehends the meaning. Experience sees that the assertions are true, but she sees not how profound and absolute is their truth. She unhesitatingly assents to the laws which geometry delivers, but she does not pretend to see the origin of their obligation. She is always ready to acknowledge the sway of pure scientific principles as a matter of fact, but she does not dream of offering her opinion on their authority as a matter of right; still less can she justly claim to herself the source of that authority." (William Whewell, "The Philosophy of the Inductive Sciences", 1858)

"The distinction of Fact and Theory is only relative. Events and phenomena, considered as particulars which may be colligated by Induction, are Facts; considered as generalities already obtained by colligation of other Facts, they are Theories." (William Whewell, "The Philosophy of the Inductive Sciences: Founded Upon Their History" Vol. 2, 1858)

“The ideas which these sciences, Geometry, Theoretical Arithmetic and Algebra involve extend to all objects and changes which we observe in the external world; and hence the consideration of mathematical relations forms a large portion of many of the sciences which treat of the phenomena and laws of external nature, as Astronomy, Optics, and Mechanics. Such sciences are hence often termed Mixed Mathematics, the relations of space and number being, in these branches of knowledge, combined with principles collected from special observation; while Geometry, Algebra, and the like subjects, which involve no result of experience, are called Pure Mathematics.” (Whewell, William, “The Philosophy of the Inductive Sciences”, 1858)

"This science, Geometry, is one of indispensable use and constant reference, for every student of the laws of nature; for the relations of space and number are the alphabet in which those laws are written. But besides the interest and importance of this kind which geometry possesses, it has a great and peculiar value for all who wish to understand the foundations of human knowledge, and the methods by which it is acquired. For the student of geometry acquires, with a degree of insight and clearness which the unmathematical reader can but feebly imagine, a conviction that there are necessary truths, many of them of a very complex and striking character; and that a few of the most simple and self-evident truths which it is possible for the mind of man to apprehend, may, by systematic deduction, lead to the most remote and unexpected results." (William Whewell, "The Philosophy of the Inductive Sciences", 1858)

🅹William James - Collected Quotes

"The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal." (William James, "Clifford's Lectures and Essays", 1897)

"Every definite image in the mind is steeped and dyed in the free water that flows around it. With it goes the sense of its relations, near and remote, the dying echo of whence it came to us, the dawning sense of whither it is to lead. The significance, the value, of the image is all in this halo or penumbra that surrounds and escorts it, - or rather that is fused into one with it and has become bone of its bone and flesh of its flesh; leaving it, it is true, an image of the same thing it was before, but making it an image of that thing newly taken and freshly understood." (William James, "The Principles of Psychology", 1890)

"Great thinkers have vast premonitory glimpses of schemes of relations between terms, which hardly even as verbal images enter the mind, so rapid is the whole process. We all of us have this permanent consciousness of whither our thought is going." (William James, "The Principles of Psychology", 1890)

"Metaphysics means nothing but an unusually obstinate effort to think clearly." (William James, "The Principles of Psychology" Vol 1, 1890)

"The aim of science is always to reduce complexity to simplicity." (William James, "The Principles of Psychology", 1890)

"The aim of ‘science’ is to attain conceptions so adequate and exact that we shall never need to change them." (William James, "The Principles of Psychology", 1890)

"[…] as the sciences have developed further, the notion has gained ground that most, perhaps all, of our laws are only approximations." (William James, "Pragmatism: A New Name for Some Old Ways of Thinking", 1907)

"First [...] a new theory is attacked as absurd; then it is admitted to be true, but obvious and insignificant; finally it is seen to be so important that its adversaries claim they themselves discovered it." (William James, "Pragmatism: A New Name for Some Old Ways of Thinking", 1907)

"The truth of an idea is not a stagnant property inherent in it. Truth happens to an idea. It becomes true, is made true by events. Its verity is in fact an event, a process: the process namely of its verifying itself, its verification. Its validity is the process of its validation." (William James, "Pragmatism: A New Name for Some Old Ways of Thinking", 1907)

"The true, to put it very briefly, is only the expedient in the way of our thinking, just as the right is only the expedient in our way of behaving." (William James, "Pragmatism: A New Name for Some Old Ways of Thinking", 1907)

 "Theories thus become instruments, not answers to enigmas, in which we can rest. We don't lie back upon them, we move forward, and, on occasion, make nature over again by their aid." (William James, "Pragmatism: A New Name for Some Old Ways of Thinking", 1907)

"Reduced to their most pregnant difference, empiricism means the habit of explaining wholes by parts, and rationalism means the habit of explaining parts by wholes. Rationalism thus preserves affinities with monism, since wholeness goes with union, while empiricism inclines to pluralistic views. No philosophy can ever be anything but a summary sketch, a picture of the world in abridgment, a foreshortened bird's-eye view of the perspective of events. And the first thing to notice is this, that the only material we have at our disposal for making a picture of the whole world is supplied by the various portions of that world of which we have already had experience. We can invent no new forms of conception, applicable to the whole exclusively, and not suggested originally by the parts." (William James, "A Pluralistic Universe", 1908)

"An experience, perceptual or conceptual, must conform to reality in order to be true." (William James, "The Social Value of the College-Bred - Memories and Studies", 1911)

"A good hypothesis in science must have other properties than those of the phenomenon it is immediately invoked to explain, otherwise it is not prolific enough." (William James, "Selected Papers on Philosophy", 1918)

"Science herself consults her heart when she lays it down that the infinite ascertainment of fact and correction of false belief are the supreme goods for man." (William James, "Selected Papers on Philosophy", 1918)

🅽James R Newman - Collected Quotes

"Perhaps the greatest paradox of all is that there are paradoxes in mathematics […] because mathematics builds on the old but does not discard it, because its theorems are deduced from postulates by the methods of logic, in spite of its having undergone revolutionary changes we do not suspect it of being a discipline capable of engendering paradoxes." (James R Newman, "Mathematics and the Imagination", 1940) 

"The mathematician is still regarded as the hermit who knows little of the ways of life outside his cell, who spends his time compounding incredible and incomprehensible theorems in a strange, clipped, unintelligible jargon." (James R Newman, "Mathematics and the Imagination", 1940)

"A variety of natural phenomena exhibit what is called the minimum principle. The principle is displayed where the amount of energy expended in performing a given action is the least required for its execution, where the path of a particle or wave in moving from one point to another is the shortest possible, where a motion is completed in the shortest possible time, and so on." (James R Newman, "The World of Mathematics" Vol. II, 1956)

"Nevertheless, there are three distinct types of paradoxes which do arise in mathematics. There are contradictory and absurd propositions, which arise from fallacious reasoning. There are theorems which seem strange and incredible, but which, because they are logically unassailable, must be accepted even though they transcend intuition and imagination. The third and most important class consists of those logical paradoxes which arise in connection with the theory of aggregates, and which have resulted in a re-examination of the foundations of mathematics." (James R Newman, "The World of Mathematics" Vol. III, 1956)

"Strictly speaking, mathematical propositions are neither true nor false; they are merely implied by the axioms and postulates which we assume. If we accept these premises and employ legitimate logical arguments, we obtain legitimate propositions. The postulates are not characterized by being true or false; we simply agree to abide by them. But we have used the word true without any of its philosophical implications to refer unambiguously to propositions logically deduced from commonly accepted axioms." (James R Newman, "The World of Mathematics" Vol. III, 1956)

"That mathematics is a handmaiden of science is a commonplace; but it is less well understood that experiments stimulate mathematical imagination, aid in the formulation of concepts and shape the direction and emphasis of mathematical studies. One of the most remarkable features of the relationship is the successful use of physical models and experiments to solve problems arising in mathematics. In some cases a physical experiment is the only means of determining whether a solution to a specific problem exists; once the existence of a solution has been demonstrated, it may then be possible to complete the mathematical analysis, even to move beyond the conclusions furnished by the model-a sort of boot-strap procedure. It is interesting to point out that what counts in this action and reaction is as much the 'physical way of thinking', the turning over in imagination of physical events, as the actual doing of the experiment." (James R Newman, "The World of Mathematics" Vol. II, 1956)

"The infinite in mathematics is always unruly unless it is properly treated." (James R Newman, "The World of Mathematics" Vol. III, 1956)

"The second law of thermodynamics provides a more modem (and a more discouraging) example of the maximum principle: the entropy (disorder) of the universe tends toward a maximum." (James R Newman, "The World of Mathematics" Vol. II, 1956)

"There are two ways to teach mathematics. One is to take real pains toward creating understanding - visual aids, that sort of thing. The other is the old British style of teaching until you’re blue in the face." (James R Newman, New York Times, 1956)

"Understanding mathematical logic, or the theory of relativity, is not an indispensable attribute of the cultured mind. But if one wishes to learn anything about these subjects, one must learn something. It is necessary to master the rudiments of the language, to practice a technique, to follow step by step a characteristic sequence of reasoning and to see a problem through from beginning to end." (James R Newman, "The World of Mathematics” Vol. I, 1956)

"When the mathematician says that such and such a proposition is true of one thing, it may be interesting, and it is surely safe. But when he tries to extend his proposition to everything, though it is much more interesting, it is also much more dangerous. In the transition from one to all, from the specific to the general, mathematics has made its greatest progress, and suffered its most serious setbacks, of which the logical paradoxes constitute the most important part. For, if mathematics is to advance securely and confidently it must first set its affairs in order at home." (James R Newman, "The World of Mathematics" Vol. III, 1956)

🅲Paul Carus - Collected Quotes

"The truth is that other systems of geometry are possible, yet after all, these other systems are not spaces but other methods of space measurements. There is one space only, though we may conceive of many different manifolds, which are contrivances or ideal constructions invented for the purpose of determining space." (Paul Carus, Science Vol. 18, 1903)

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

"Pythagoras says that number is the origin of all things, and certainly the law of number is the key that unlocks the secrets of the universe. But the law of number possesses an immanent order, which is at first sight mystifying, but on a more intimate acquaintance we easily understand it to be intrinsically necessary; and this law of number explains the wondrous consistency of the laws of nature." (Paul Carus, "Reflections on Magic Squares", Monist Vol. 16, 1906)

"There is no science which teaches the harmonies of nature more clearly than mathematics." (Paul Carus, "Andrews: Magic Squares and Cubes", 1908)

"Science is not the monopoly of the naturalist or the scholar, nor is it anything mysterious or esoteric. Science is the search for truth, and truth is the adequacy of a description of facts." (Paul Carus, "Philosophy as a Science", 1909)

"I do not say that the notion of infinity should be banished; I only call attention to its exceptional nature, and this so far as I can see, is due to the part which zero plays in it, and we must never forget that like the irrational it represents a function which possesses a definite character but can never be executed to the finish If we bear in mind the imaginary nature of these functions, their oddities will not disturb us, but if we misunderstand their origin and significance we are confronted by impossibilities." (Paul Carus, "The Nature of Logical and Mathematical Thought"; Monist Vol 20, 1910)

"Infinity is the land of mathematical hocus pocus. There Zero the magician is king. When Zero divides any number he changes it without regard to its magnitude into the infinitely small [great?], and inversely, when divided by any number he begets the infinitely great [small?]. In this domain the circumference of the circle becomes a straight line, and then the circle can be squared. Here all ranks are abolished, for Zero reduces everything to the same level one way or another. Happy is the kingdom where Zero rules!" (Paul Carus, "The Nature of Logical and Mathematical Thought"; Monist Vol 20, 1910)

📒🆆Herbert G Wells - Collected Quotes

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

"At that I understood. At the risk of disappointing Richardson I stayed on, waiting for The Time Traveler; waiting for the second, perhaps still stranger story, and the specimens and photographs he would bring with him. But I am beginning now to fear that I must wait a lifetime. The Time Traveler vanished three years ago. And, as everybody knows now, he has never returned." (Herbert G Wells, "The Time Machine: An Invention", 1895)

"It is a law of nature we overlook, that intellectual versatility is the compensation for change, danger, and trouble. An animal perfectly in harmony with its environment is a perfect mechanism. Nature never appeals to intelligence until habit and instinct are useless. There is no intelligence where there is no change and no need of change. Only those animals partake of intelligence that have a huge variety of needs and dangers." (Herbert G Wells, "The Time Machine", 1895)

"Nature never appeals to intelligence until habit and instinct are useless. There is no intelligence where there is no need of change." (Herbert G Wells, "The Time Machine", 1895)

"'Upon that machine', said the Time Traveller, holding the lamp aloft, 'I intend to explore time'." (Herbert G Wells, "The Time Machine: An Invention", 1895)

"Until a scientific theory yields confident forecasts you know it is unsound and tentative; it is mere theorizing, as evanescent as art talk or the phantoms politicians talk about." (Herbert G Wells, [Annual Report of the Board of Regents of the Smithsonian Institution] 1902)

"Facts are the raw materials and not the substance of science." (Herbert G Wells, "The Discovery of the Future", 1902)

"The great body of physical science, a great deal of the essential fact of financial science, and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of one of the new great complex world-wide States that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and write." (Herbert G Wells, "Mankind in the Making", 1903)

"The new mathematics is a sort of supplement to language, affording a means of thought about form and quantity and a means of expression, more exact, compact, and ready than ordinary language." (Herbert G Wells, "Mankind in the Making", 1903)

"The forceps of our minds are clumsy forceps, and crush the truth a little in taking hold of it." (Herbert G Wells, "Scepticism of the Instrument: A Modern Utopia", 1905)

"When the mind grapples with a great and intricate problem, it makes its advances step by step, with but little realization of the gains it has made, until suddenly, with an effect of abrupt illumination, it realizes its victory." (Herbert G Wells, "A Short History of the World", 1922)

"Behind the adventurer, the speculator, comes that scavenger of adventurers, the statistician. […] The movement of the last hundred years is all in favor of the statistician." (Herbert G Wells, "The Work, Wealth and Happiness of Mankind", 1931)

"We live in reference to past experience and not to future events, however inevitable." (Herbert G Wells, "Mind at the End of Its Tether", 1946)

"Nothing destroys the powers of general observation quite so much as a life of experimental science." (Herbert G Wells)

"The popular idea of scientific investigation is a vehement, aimless collection of little facts, collected as a bower bird collects shells and pebbles, in methodical little rows, and out of this process, in some manner unknown to the popular mind, certain conjuring tricks - the celebrated 'wonders of science' - in a sort of accidental way emerge. ." (Herbert G Wells)

"You know of course that a mathematical line, a line of thickness nil, has no real existence. They taught you that? Neither has a mathematical plane. These things are mere abstractions." (Herbert G Wells)

🅲Gilbert K Chesterton - Collected Quotes

"Logic, then, is not necessarily an instrument for finding truth; on the contrary, truth is necessarily an instrument for using logic - for using it, that is, for the discovery of further truth and for the profit of humanity. Briefly, you can only find truth with logic if you have already found truth without it." (Gilbert K Chesterton, Daily News, 1905)

"There is no great harm in the theorist who makes up a new theory to fit a new event. But the theorist who starts with a false theory and then sees everything as making it come true is the most dangerous enemy of human reason." (Gilbert K Chesterton, "The Flying Inn", 1914)

"Education is simply the soul of a society as it passes from one generation to another." (Gilbert K Chesterton)

"Imagination does not breed insanity. Exactly what does breed insanity is reason. Poets do not go mad […] mathematicians go mad." (Gilbert K Chesterton)

"Science boasts of the distance of its stars; of the terrific remoteness of the things of which it has to speak. But poetry and religion always insist upon the proximity, the almost menacing closeness of the things with which they are concerned." (Gilbert K Chesterton)

"The chief object of education is not to learn but to unlearn."  (Gilbert K Chesterton)

"The difference between the poet and the mathematician is that the poet tries to get his head into the heavens while the mathematician tries to get the heavens into his head." (Gilbert K Chesterton)

🅱Francis Bacon - Collected Quotes

"[…] no pleasure is comparable to the standing upon the vantage ground of truth […]" (Sir Francis Bacon, "Essays", 1597)

"For many parts of Nature can neither be invented with sufficient subtlety, nor demonstrated with sufficient perspicuity, nor accommodated to use with sufficient dexterity, without the aid and intervention of Mathematic: of which sort are Perspective, Music, Astronomy, cosmography, Architecture, Machinery, and some others." (Sir Francis Bacon, "De Augmentis", Bk. 3 [The Advancement of Learning], 1605)

"Physic […] is situate in a middle term or distance between natural history and metaphysic. For natural history describes the variety of things; physic the causes, but variable or respective causes; and metaphysic the fixed and constant causes." (Sir Francis Bacon, "The Advancement of Learning", 1605)

"Science is the labor and handicraft of the mind; poetry can only be considered its recreation." (Francis Bacon, "The Proficience and Advancement of Learning, Divine and Human", 1605)

"XI. As the present sciences are useless for the discovery of effects, so the present system of logic is useless for the discovery of the sciences. XII. The present system of logic rather assists in confirming and rendering inveterate the errors founded on vulgar notions than in searching after truth, and is therefore more hurtful than useful."  (Sir Francis Bacon, "Novum Organum", 1620)

"God forbid that we should give out a dream of our own imagination for a pattern of the world." (Francis Bacon, "The Great Instauration", 1620)

"Man prefers to believe what he prefers to be true." (Sir Francis Bacon, "Novum Organum", 1620)

"Man, as the minister and interpreter of nature, dies and understands as much as his observations on the order of nature, either with regard to things or the mind permit him, and neither knows or is capable of more." (Francis Bacon, "Novum Organum", 1620)

"No one has yet been found so firm of mind and purpose as resolutely to compel himself to sweep away all theories and common notions, and to apply the understanding, thus made fair and even, to a fresh examination of particulars. Thus it happens that human knowledge, as we have it, is a mere medley and ill-digested mass, made up of much credulity and much accident, and also of the childish notions which we at first imbibed." (Sir Francis Bacon, "Novum Organum" Book 2, 1620)

"The first and most ancient inquirers into truth were wont to throw their knowledge into aphorisms, or short, scattered, unmethodical sentences." (Sir Francis Bacon, "Novum Organum", 1620)

"The human mind is often so awkward and ill-regulated in the career of invention that is at first diffident, and then despises itself. For it appears at first incredible that any such discovery should be made, and when it has been made, it appears incredible that it should so long have escaped men’s research. All which affords good reason for the hope that a vast mass of inventions yet remains, which may be deduced not only from the investigation of new modes of operation, but also from transferring, comparing and applying those already known, by the methods of what we have termed literate experience." (Sir Francis Bacon, "Novum Organum", 1620)

"The human understanding resembles not a dry light, but admits a tincture of the will and passions, which generate their own system accordingly; for man always believes more readily that which he prefers." (Sir Francis Bacon, "Novum Organum", 1620)

"The end of our foundation is the knowledge of causes, and secret motions of things; and the enlarging of the bounds of human empire, to the effecting of all things possible." (Francis Bacon, "New Atlantis", 1627)

"If a man’s wit be wandering, let him study mathematics; for in demonstrations, if his wit be called away never so little, he must begin again." (Sir Francis Bacon)

"The human understanding is of its own nature prone to abstractions and gives us a substance and reality to thing which are fleeting. But to resolve nature into abstractions is less to our purpose than to dissect her into parts." (Sir Francis Bacon)