Parapsychologists

"The structure of brain and nerves being similar, it is conceivable there may be present masses of such nerve coherers in the brain whose special function it may be to receive impulses brought from without through the connecting sequence of ether waves of appropriate order of magnitude. Röntgen has familiarized us with an order of vibrations of extreme minuteness compared with the smallest waves with which we have hitherto been acquainted, and of dimensions comparable with the distances between the centers of the atoms of which the material universe is built up; and there is no reason to suppose that we have here reached the limit of frequency. It is known that the action of thought is accompanied by certain molecular movements in the brain, and here we have physical vibrations capable, from their extreme minuteness, of acting direct on individual molecules, while their rapidity approaches that of the internal and external movements of the atoms themselves. Confirmation of telepathic phenomena is afforded by many converging experiments and by many spontaneous occurrences only thus intelligible. The most varied proof, perhaps, is drawn from analysis of the subconscious workings of the mind, when these, whether by accident or design, are brought into conscious survey. … This is not the place for details, but the vis medicatrix thus evoked, as it were, from the depths of the organism, is of good omen for the upward evolution of mankind."

"All the phenomena of the universe are presumably in some way continuous, and it is unscientific to call in the aid of mysterious agencies when, with every fresh advance in knowledge, it is shown that ether vibrations have powers and attributes abundantly equal to any demand — even to the transmission of thought."

"I am impelled to one further reflection, dealing with the conservation of energy. We say, with truth, that energy is transformed but not destroyed, and that whenever we can trace the transformation we find it quantitatively exact. So far as our very rough exactness goes, this is true for inorganic matter and for mechanical forces. But it is only inferentially true for organized matter and for vital forces. We can not express life in terms of heat or of motion. And thus it happens that just when the exact transformation of energy will be most interesting to watch, we can not really tell whether any fresh energy has been introduced into the system or not. Let us consider this a little more closely."

"I have raised a question which may be regarded as heretical. At the time when our modern conception of chemistry first dawned... the average chemist... accepted the elements as ultimate facts... absolutely simple, incapable of transmutation or decomposition, each a kind of barrier behind which we could not penetrate. ...[H]e said they were self-existent from all eternity ...But in our times... we cannot help asking what are the elements, whence do they come, what is their signification? ...These elements perplex us in our researches, baffle us in our speculations, and haunt us in our very dreams. They stretch like an unknown sea before us—mocking, mystifying and murmuring strange revelations and possibilities. If I venture to say that... elements are not simple and primordial... but have evolved from simpler matters—or... one sole kind of matter—I... give formal utterance to an idea... for some time "in the air" of science. Chemists, physicists, philosophers of the highest merit declare explicitly their belief that the seventy... elements of our text-books are not the which we must never hope to pass."

"The production of motion, molar or molecular, is governed by physical laws, which it is the business of the philosopher to find out and correlate. The law of the conservation of energy overrides all laws, and it is a preeminent canon of scientific belief that for every act done a corresponding expenditure of energy must be transformed. No work can be effected without using up a corresponding value in energy of another kind. But to us the other side of the problem is even of more importance. Granted the existence of a certain kind of molecular motion, what is it that determines its direction along one path rather than another?"

"Which was first, Matter or Force? If we think on this question, we shall find that we are unable to conceive of matter without force, or force without matter. When God created the elements of which the earth is composed, He created certain wondrous forces, which are set free and become evident when matter acts on matter."

"I will point out a curious, inveterate, and widespread illusion — the illusion that our earthly bodies are a kind of norm of humanity, so that ethereal bodies, if such there be, must correspond to them in shape and size. When we take a physical view of a human being in his highest form of development, he is seen to consist essentially of a thinking brain, the brain itself, among its manifold functions, being a transformer whereby intelligent will power is enabled to react on matter. To communicate with the external world, the brain requires organs by which it can be transported from place to place, and other organs by means of which energy is supplied to replace that expended in the exercise of its own special functions."

"The most helpful quality which has aided me in psychical problems and has made me lucky in physical discoveries (sometimes of rather unexpected kinds) has simply been my knowledge — my vital knowledge, if I may so term it — of my own ignorance. Most students of nature sooner or later pass through a process of writing off a large percentage of their supposed capital of knowledge as a merely illusory asset. As we trace more accurately certain familiar sequences of phenomena we begin to realize how closely these sequences, or laws, as we call them, are hemmed round by still other laws of which we can form no notion. With myself this writing off of illusory assets has gone rather far and the cobweb of supposed knowledge has been pinched (as some one has phrased) into a particularly small pill."

"With whisper of her mellowing grain, With treble of brook and bud and tree, Earth joys for ever to sustain The bass eternal of the sea."

"Ah! what if some unshamed iconoclast Crumbling old fetish raiments of the past, Rises from dead cerements the Christ at last? What if men take to following where He leads, Weary of mumbling Athanasian creeds?"

"The following is exactly what we mean by a LIMIT. ...let the several values of x... bea1 a2 a3 a4. . . . &c.then if by passing from a1 to a2, from a2 to a3, &c., we continually approach to a certain quantity l [lower case L, for "limit"], so that each of the set differs from l by less than its predecessors; and if, in addition to this, the approach to l is of such a kind, that name any quantity we may, however small, namely z, we shall at last come to a series beginning, say with an, and continuing ad infinitum,an an+1 an+2. . . . &c.all the terms of which severally differ from l by less than z: then l is called the limit of x with respect to the supposition in question."

"Take a unit, halve it, halve the result, and so on continually. This gives—1 1⁄2 1⁄4 1⁄8 1⁄16 1⁄32 1⁄64 1⁄128 &c.Add these together, beginning from the first, namely, add the first two, the first three, the first four, &c... We see then a continual approach to 2, which is not reached, nor ever will be, for the deficit from 2 is always equal to the last term added. ...We say that—1, 1 + 1⁄2, 1 + 1⁄2 + 1⁄4, 1 + 1⁄2 + 1⁄4 + 1⁄8, &c. &c.is a series of quantities which continually approximate to the limit 2. Now the truth is, these several quantities are fixed, and do not approximate to 2. ...it is we ourselves who approximate to 2, by passing from one to another. Similarly when we say, "let x be a quantity which continually approximates to the limit 2," we mean, let us assign different values to x, each nearer to 2 than the preceding, and following such a law that we shall, by continuing our steps sufficiently far, actually find a value for x which shall be as near to 2 as we please."

"When... we have a series of values of a quantity which continually diminish, and in such a way, that name any quantity we may, however small, all the values, after a certain value, are severally less than that quantity, then the symbol by which the values are denoted is said to diminish without limit. And if the series of values increase in succession, so that name any quantity we may, however great, all after a certain point will be greater, then the series is said to increase without limit. It is also frequently said, when a quantity diminishes without limit, that it has nothing, zero or 0, for its limit: and that when it increases without limit it has infinity or ∞ or 1⁄0 for its limit."

"The student of the Differential Calculus may... be brought to think it possible that the terms and ideas which that science requires may exist in his own mind in the same rude form as that of a straight line in the conceptions of a beginner in geometry. ...he must be prepared to stop his course until he can form exact notions, acquire precise ideas, both of resemblance between those things which have appeared most distinct, and of distinction between those which have appeared most alike. To do this... formal definitions would be useless; for he cannot be supposed to have one single notion in that precise form which would make it worth while to attach it to a word. One reason of the great difficulty which is found in treatises on this subject... the tacit assumption that nothing is necessary previously to actually embodying the terms and rules of the science, as if mere statement of definitions could give instantaneous power of using terms rightly. We shall here attempt... a wider degree of verbal explanation than is usual with the view of enabling the student to come to the definitions in some state of previous preparation."

"The absolute requisites for the study of this work... are a knowledge of algebra to the binomial at least, plane and solid geometry, plane trigonometry, and the most simple part of the usual applications of algebra to geometry. ...A. De Morgan. London July 1, 1836"

"Find a fraction which, multiplied by itself, shall give 6, or... find the square root of 6. This can be shown to be an impossible problem; for it can be shown that no fraction whatsoever multiplied by itself, can give a whole number, unless it be itself a whole number disguised in a fractional form, such as 4⁄2 or 21⁄3. To this problem, then, there is but one answer, that it is self-contradictory. But if we propose the following problem,—to find a fraction which, multiplied by itself, shall give a product lying between 6 and 6 + a; we find that this problem admits of solution in every case. It therefore admits of solution however small a may be... as small as you please. ...there is such a thing as the square root of 6, and it is denoted by √6. But we do not say we actually find this, but that we approximate to it."

"In every age of the world there has been an established system, which has been opposed from time to time by isolated and dissentient reformers. The established system has sometimes fallen, slowly and gradually: it has either been upset by the rising influence of some one man, or it has been sapped by gradual change of opinion in the many."

"I have throughout introduced the Integral Calculus in connexion with the Differential Calculus. ...Is it always proper to learn every branch of a direct subject before anything connected with the inverse relation is considered? If so why are not multiplication and involution in arithmetic made to follow addition and precede subtraction? The portion of the Integral Calculus, which properly belongs to any given portion of the Differential Calculus increases its power a hundred-fold..."

"I cannot see why it is necessary that every deduction from algebra should be bound to certain conventions incident to an earlier stage of mathematical learning, even supposing them to have been consistently used up to the point in question. I should not care if any one thought this treatise unalgebraical, but should only ask whether the premises were admissible and the conclusions logical."

"Experience has convinced me that the proper way of teaching is to bring together that which is simple from all quarters, and, if I may use such a phrase, to draw upon the surface of the subject a proper mean between the line of closest connexion and the line of easiest deduction. This was the method followed by Euclid, who, fortunately for us, never dreamed of a geometry of triangles, as distinguished from a geometry of circles, or a separate application of the arithmetics of addition and subtraction; but made one help out the other as he best could."

"A large quantity of examples is indispensable."

"It is not true, out of geometry, that the mathematical sciences are, in all their parts those models of finished accuracy which many suppose. The extreme boundaries of analysis have always been as imperfectly understood as the tract beyond the boundaries was absolutely unknown. But the way to enlarge the settled country has not been by keeping within it, but by making voyages of discovery, and I am perfectly convinced that the student should be exercised in this manner; that is, that he should be taught how to examine the boundary, as well as how to cultivate the interior. ...allowing all students whose capacity will let them read on the higher branches of applied mathematics, to have each his chance of being led to the cultivation of those parts of analysis on which rather depends its future progress than its present use in the sciences of matter."

"The following Treatise... has been endeavoured to make the theory of limits, or ultimate ratios... the sole foundation of the science, without any aid whatsoever from the theory of series, or algebraical expansions. I am not aware that any work exists in which this has been avowedly attempted, and I have been the more encouraged to make the trial from observing that the objections to the theory of limits have usually been founded either upon the difficulty of the notion itself, or its unalgebraical character, and seldom or never upon anything not to be defined or not to be received in the conception of a limit..."

"I am far from saying that this Treatise will be easy; the subject is a difficult one, as all know who have tried it."

"During the last two centuries and a half, physical knowledge has been gradually made to rest upon a basis which it had not before. It has become mathematical."

"My specific... object has been to contain, within the prescribed limits, the whole of the student's course, from the confines of elementary algebra and trigonometry, to the entrance of the highest works on mathematical physics. A learner who has a good knowledge of the subjects just named, and who can master the present treatise, taking up elementary works on conic sections, application of algebra to geometry, and the theory of equations, as he wants them, will, I am perfectly sure, find himself able to conquer the difficulties of anything he may meet with; and need not close any book of Laplace, Lagrange, Legendre, Poisson, Fourier, Cauchy, Gauss, Abel, Hindenburgh and his followers. or of any one of our English mathematicians, under the idea that it is too hard for him."

"The work now before the reader is the most extensive which our language contains on the subject."

"If much difficulty should be experienced in the elementary chapters, I know of no work which I can so confidently recommend to be used with the present one, as that of M. Duhamel."

"Although there is no study which presents so simple a beginning as that of geometry, there is none in which difficulties grow more rapidly as we proceed, and what may appear at first rather paradoxical, the more acute the student the more serious will the impediments in the way of his progress appear. This necessarily follows in a science which consists of reasoning from the very commencement, for it is evident that every student will feel a claim to have his objections answered, not by authority, but by argument, and that the intelligent student will perceive more readily than another the force of an objection and the obscurity arising from an unexplained difficulty, as the greater is the ordinary light the more will occasional darkness be felt. To remove some of these difficulties is the principal object of this Treatise."

"The lowest steps of the ladder are as useful as the highest."

"A finished or even a competent reasoner is not the work of nature alone... education develops faculties which would otherwise never have manifested their existence. It is, therefore, as necessary to learn to reason before we can expect to be able to reason, as it is to learn to swim or fence, in order to attain either of those arts. Now, something must be reasoned upon, it matters not much what it is, provided that it can be reasoned upon with certainty. The properties of mind or matter, or the study of languages, mathematics, or natural history may be chosen for this purpose. Now, of all these, it is desirable to choose the one... in which we can find out by other means, such as measurement and ocular demonstration of all sorts, whether the results are true or not. ..Now the mathematics are peculiarly well adapted for this purpose, on the following grounds:— 1. Every term is distinctly explained, and has but one meaning, and it is rarely that two words are employed to mean the same thing. 2. The first principles are self-evident, and, though derived from observation, do not require more of it than has been made by children in general. 3. The demonstration is strictly logical, taking nothing for granted except the self-evident first principles, resting nothing upon probability, and entirely independent of authority and opinion. 4. When the conclusion is attained by reasoning, its truth or falsehood can be ascertained, in geometry by actual measurement, in algebra by common arithmetical calculation. This gives confidence, and is absolutely necessary, if... reason is not to be the instructor, but the pupil. 5. There are no words whose meanings are so much alike that the ideas which they stand for may be confounded. ...These are the principal grounds on which... the utility of mathematical studies may be shewn to rest, as a discipline for the reasoning powers. But the habits of mind which these studies have a tendency to form are valuable in the highest degree. The most important of all is the power of concentrating the ideas which a successful study of them increases where it did exist, and creates where it did not. A difficult position or a new method of passing from one proposition to another, arrests all the attention, and forces the united faculties to use their utmost exertions. The habit of mind thus formed soon extends itself to other pursuits, and is beneficially felt in all the business of life."

"...nor have I found occasion to depart from the plan... the rejection of the whole doctrine of series in the establishment of the fundamental parts both of the Differential and Integral Calculus. The method of Lagrange... had taken deep root in elementary works; it was the sacrifice of the clear and indubitable principle of limits to a phantom, the idea that an algebra without limits was purer than one in which that notion was introduced. But, independently of the idea of limits being absolutely necessary even to the proper conception of a convergent series, it must have been obvious enough to Lagrange himself, that all application of the science to concrete magnitude, even in his own system, required the theory of limits."

"The Object of this Treatise is—(1) To point out to the student of Mathematics, who has not the advantage of a tutor, the course of study which it is most advisable that he should follow, the extent to which he should pursue one part of the science before he commences another, and to direct him as to the sort of applications which he should make. (2) To treat fully of the various points which involve difficulties and which are apt to be misunderstood by beginners, and to describe at length the nature without going into the routine of the operations."

"The number of mathematical students, increased as it has been of late years, would be much augmented if those who hold the highest rank in science would condescend to give more effective assistance in clearing the elements of the difficulties which they present. If any one claiming that title should think my attempt obscure or enormus, he must share the blame with me, since it is through his neglect that I have been enabled to avail myself of an opportunity to perform a task which I would gladly have seen confided to more skilful hands."

"In order to see the difference which exists between... studies,—for instance, history and geometry, it will be useful to ask how we come by knowledge in each. Suppose, for example, we feel certain of a fact related in history... if we apply the notions of evidence which every-day experience justifies us in entertaining, we feel that the improbability of the contrary compels us to take refuge in the belief of the fact; and, if we allow that there is still a possibility of its falsehood, it is because this supposition does not involve absolute absurdity, but only extreme improbability. In mathematics the case is wholly different... and the difference consists in this—that, instead of showing the contrary of the proposition asserted to be only improbable, it proves it at once to be absurd and impossible. This is done by showing that the contrary of the proposition which is asserted is in direct contradiction to some extremely evident fact, of the truth of which our eyes and hands convince us. In geometry, of the principles alluded to, those which are most commonly used are— I. If a magnitude is divided into parts, the whole is greater than either of those parts. II. Two straight lines cannot inclose a space. III. Through one point only one straight line can be drawn, which never meets another straight line, or which is parallel to it. It is on such principles as these that the whole of geometry is founded, and the demonstration of every proposition consists in proving the contrary of it to be inconsistent with one of these."

"I did not hear what you said, but I absolutely disagree with you."

"There never has been, and till we see it we never shall believe that there can be, a system of geometry worthy of the name, which has any material departures (we do not speak of corrections or extensions or developments) from the plan laid down by Euclid."

"The two races which have founded the mathematics, those of the Sanscrit and Greek languages, have been the two which have independently formed systems of logic."

"There is a mistake into which several have fallen, and have deceived others, and perhaps themselves, by clothing some false reasoning in what they called a mathematical dress, imagining that by the application of mathematical symbols to their subject, they secured mathematical argument. This could not have happened if they had possessed a knowledge of the bounds within which the empire of mathematics is contained. That empire is sufficiently wide, and might have been better known, had the time which has been wasted in aggressions upon the domains of others, been spent in exploring the immense tracts which are yet untrodden."

"I... subjoin references to those parts of the work for which I have not been indebted to my knowledge of what has been written before me: much of what is cited is probably not new, indeed it is dangerous for any one at the present day to claim anything as belonging to himself; several things which I once thought to have entered in this list have been since found (either by myself, or by a friend to whom I referred it) in preceding writers."

"A great many individuals ever since the rise of the mathematical method, have, each for himself, attacked its direct and indirect consequences. ...I shall call each of these persons a paradoxer, and his system a paradox. I use the word in the old sense: ...something which is apart from general opinion, either in subject-matter, method, or conclusion. ...Thus in the sixteenth century many spoke of the earth's motion as the paradox of Copernicus, who held the ingenuity of that theory in very high esteem, and some, I think, who even inclined towards it. In the seventeenth century, the depravation of meaning took place... Phillips says paradox is "a thing which seemeth strange"—here is the old meaning...—"and absurd, and is contrary to common opinion," which is an addition due to his own time."

"Dr. George Boole, author of The Laws of Thought had introduced himself in the year 1842 to Mr. De Morgan by a letter on the Differential and Integral Calculus then recently published. His character and pursuits were in many points like those of the author who found great pleasure in his correspondence and friendship. ...In 1847, his attention having been drawn to the subject by the publication of Mr. De Morgan's Formal Logic, he published the Mathematical Analysis of Logic and in the following year communicated... a paper on the Calculus of Logic. His great work, An Investigation into the Laws of Thought... was a development of the principle laid down in the Calculus..."

"A very interesting detailed account of the peculiarities of the circle squarer, and of the futility of the attempts on the part of the Mathematicians to convince him of his errors, will be found in Augustus De Morgan's Budget of Paradoxes."

"‘European science could never have reached its present height had it not been fertilised by successive wafts from the […] knowledge stored up in the East.’ ‘Think what must have been the effect of the intense Hinduizing of three such men as Babbage, De Morgan and George Boole on the mathematical atmosphere of 1830–1865.’ ‘I do as George Boole and De Morgan did: I bow my head inreverent thankfulness to that mysterious East, whence come to us wafts of some transcendent power the nature of which we ourselves can hardly state in words.’"

"The fact is known that having very thoroughly worked at the generalisations of Mathematics in theory and practice, Mr. De Morgan was enabled to establish with perfect precision the most highly generalised conception of Logic, perhaps, which it is possible to entertain. It is no new doctrine that Logic deals with the necessary laws of action of thought, and that Mathematics apply these laws to necessary matter of thought; but by showing that these laws can and must be applied with equal precision and equal necessity to all kinds of relations, and not only to those which the Aristotelian theory takes account of, he so enlarged the scope and intensified the power of Logic as an instrument, that we may hope for coming generations, as he must have hoped... another instalment of the kind... Mathematics are, meanwhile, and perhaps will always remain, the completest and most accurate example of the generalised Logic. At any rate, in the mind of the author, Logic and Mathematics as 'the two great branches of exact science, the study of the necessary laws of thought, the study of the necessary matter of thought, were always viewed in connection and antithesis."

"All existing things upon this earth, which have knowledge of their own existence, possess, some in one degree and some in another, the power of thought, accompanied by perception, which is the awakening of thought by the effects of external objects upon the senses."

"The moving power of mathematical invention is not reasoning, but imagination."

"We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of exact science are mathematics and logic: the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it can see better with one eye than with two."

"They [the English] forget that at this moment there still exists [in India] a body of literature and science which might well be the nucleus of a new civilisation, though every trace of Christian and Mohammedan civilization were blotted out of existence."

"All the men who are now called discoverers, in every matter ruled by thought, have been men versed in the minds of their predecessors, and learned in what had been before them. There is not one exception. I do not say that every man has made direct acquantance with the whole of his mental ancestry... But... it is remarkable how many of the greatest names in all departments of knowledge have been real antiquaries in their several subjects. I may cite among those... in science, Aristotle, Plato, Ptolemy, Euclid, Archimedes, Roger Bacon, Copernicus, Francis Bacon, Ramus, Tycho Brahe, Galileo, Napier, Descartes, Leibnitz, Newton, Locke."