Imaginary number

“ But, while a series of mathematical points arranged in empty space expresses fairly well the process by which we form the idea of number, these mathematical points have a tendency to develop into lines in proportion as our attention is diverted from them, as if they were trying to reunite with one another. ”

“ The fingers become, in fact, when used thus, real numerical symbols, as much so as written ones, and the mind gradually becomes accustomed to thinking of the number of a set of objects as something which can be considered apart from the objects themselves, and which can be represented just as well by an equal number of other objects differing from the first set in all respects except that of number. Thus the abstract idea of number is formed. ”

“ In Number the quantum reaches its development and perfect mode. Like the One, the medium in which it exists, Number involves two qualitative factors or functions; Annumeration or Sum, which depends on the factor discreteness, and Unity, which depends on continuity. ”

“ No scientist or naturalist is so grossly stupid as to deny the infinite expansion of numbers? If this be so, it establishes the infinite of number, of which every finite number is a part, and thus we have a parallel in mathematics, the very cornerstone of the exact sciences, for a finite and an infinite mind. It is from the prototype of this infinite of number, namely, the infinite of intelligence, that spiritual dreams proceed. They are, therefore, the reflection of truth upon the dream mind and occur with less frequency than do dreams of the other two classes. ”

“ All the improved methods of teaching arithmetic to children proceed on a knowledge of this fact. All who wish to carry the child's mind along with them in learning arithmetic; all who wish to teach numbers, and not mere ciphers—now teach it through the evidence of the senses, in the manner we have described.
We may, if we please, call the proposition, "Three is two and one," a definition of the number three, and assert that arithmetic, as it has been asserted that geometry, is a science founded on definitions. ”

“ Innumerable as are the true propositions which can be formed concerning particular numbers, no adequate conception could be gained, from these alone, of the extent of the truths composing the science of number. Such propositions as we have spoken of are the least general of all numerical truths. It is true that even these are co-extensive with all nature; the properties of the number four are true of all objects that are divisible into four equal parts, and all objects are either actually or ideally so divisible. ”

“ The symbolism of numbers which greatly engaged the imaginative philosophy of earlier centuries has again acquired a fresh interest from the analytic investigations of Freud and his school. But in the material of number-dreams we no longer discover conscious puzzles of symbolic concatenations of numbers but the unconscious roots of the symbolism of numbers. ”

“ Is it a breath of God sent forth to organize the material universe, wherein nothing takes shape but as a result of divisibility which is an effect of number? The most minute as well as the most immense objects in creation are distinguished from each other by quantity, quality, dimension, and force,—are not these all conditions of number? That number is infinite is a fact proved to your intellect, but of which no material proof is obtainable. A mathematician will tell you that infinity of number is certain, but cannot be demonstrated. ”

“ But in fact a numerical equation of any order whatever has always a numerical root, and thus numbers (in the foregoing sense, number = quantity of the form α + βi) form (what real numbers do not) a universe complete in itself, such that starting in it we are never led out of it. ”

“ But mathematics uses these facts to deduce more and more complicated theorems, while philosophy seeks, by analysis, to go behind these facts to others, simpler, more fundamental, and inherently more fitted to form the premisses of the science of arithmetic. The question, “What is a number?” is the pre-eminent philosophic question in this subject, but it is one which the mathematician as such need not ask, provided he knows enough of the properties of numbers to enable him to deduce his theorems. We, since our object is philosophical, must grapple with the philosopher’s question. ”

“ To do justice to the appearance of measurable numerical relationships in nature, in whatever sphere, it is necessary to free ourselves from the abstract conception of number which governs modern scientific thought and to replace it by a more concrete one. ”

“ Arithmetic and its fundamental principles are thus independent of our experiences or of the order of the world. The matter of arithmetic is mental matter; its principles flow from the fact that the matter forms a series, which can be cut into by us wherever we like without the matter changing. The empiricist school has strangely tried to interpret the truths of number as results of coexistences among outward things. ”

“ There is the true science [pg 180] which deals with quantities really seen in objects and imagined in the mind, and an artificial science dealing with figures which have only a historical connection with real quantity. Of the latter, unfortunately, our arithmetical education chiefly consists. We are never taught to distinguish number and size in things by the 'eye,' that is, by reason. The symbolism that was originally intended to assist real arithmetical thought has ended by supplanting it. ”

“ But when at last they have been found, they are seen to be few in number, and to be the sole source of everything in pure mathematics. The discovery that all mathematics follows inevitably from a small collection of fundamental laws is one which immeasurably enhances the intellectual beauty of the whole ”

“ On the other hand, every abstract sum consists in reality of an infinite number of sense perceptions. Where do we find any indivisible unit outside of our abstract conceptions? Two halves, four fourths, eight eighths, or an infinite number of separate parts form the raw material out of which the mind fashions the mathematical unit. ”

“ Here we feel how nearly the secret of number is related to the secret of artistic creation. And so the born mathematician takes his place by the side of the great masters of the fugue, the chisel and the brush; he and they alike strive, and must strive, to actualize the grand order of all things by clothing it in symbol and so to communicate it to the plain fellow-man who hears that order within himself but cannot effectively possess it; the domain of number, like the domains of tone, line and colour, becomes an image of the world-form. ”

“ It is this view of number as generated by counting which has been the chief psychological obstacle to the understanding of infinite numbers. Counting, because it is familiar, is erroneously supposed to be simple, whereas it is in fact a highly complex process, which has no meaning unless the numbers reached in counting have some significance independent of the process by which they are reached. And infinite numbers cannot be reached at all in this way. ”

“ Three pebbles in two separate parcels, and three pebbles in one parcel, do not make the same impression on our senses; and the assertion that the very same pebbles may by an alteration of place and arrangement be made to produce either the one set of sensations or the other, though a very familiar proposition, is not an identical one. It is a truth known to us by early and constant experience: an inductive truth; and such truths are the foundation of the science of Number. The fundamental truths of that science all rest on the evidence of sense ”

“ The origin of numbers resembles that of the myth. Primitive man elevates indefinable nature-impressions (the “alien,” in our terminology) into deities, numina, at the same time capturing and impounding them by a name which limits them. So also numbers are something that marks off and captures nature-impressions, and it is by means of names and numbers that the human understanding obtains power over the world. In the last analysis, the number-language of a mathematic and the grammar of a tongue are structurally alike. Logic is always a kind of mathematic and vice versa. ”

“ And, indeed, it is the distinctive peculiarity of the calculus in general, that it proceeds on the hypothesis of that absolute equality of its abstract units, which no real units possess; and that the exactness of its results holds only in virtue of this hypothesis. The first ideas of number must necessarily then have been derived from like or equal magnitudes as seen chiefly in organic objects; and as the like magnitudes most frequently observed were magnitudes of extension, it follows that geometry and arithmetic had a simultaneous origin. ”

“ Another and much more common oddity is the tendency to visualize numerals in a peculiar manner, which characterizes, as I have roughly reckoned, about one woman in every fifteen, and one man in every thirty. Those who do so are never able to dissociate any single number from its own particular place in the field of their mental view, so that when they think of a series of numbers they always visualize them in a certain form. ”

“ From these remarks it appears, I think, that even number one is no exception to the Pythagorean law, which declares that the elements of all number are the monad and the duad, and that these are not themselves numbers. Thus, by considering numbers, we obtain light as to the constitution of the universe. Everything in the universe has some point in which it resembles everything else, and it has some point or points in which it differs from everything else; just as every number has some point in which it resembles all other numbers, and some in which it differs from all other numbers. ”

“ The mathematician comes down into his own consciousness, and finds certain conscious affirmations pertaining to number and quantity. He puts them down as truths not to be disregarded, and calls them self-evident truths or axioms. They are such affirmations of the consciousness as everybody must, per force, admit to be true; and when he has obtained the affirmations of his consciousness pertaining to number and quantity, he puts them down as truths not to be disregarded. They are always true everywhere, and under all circumstances, where number and quantity are to be investigated. ”

“ The concepts of number and form have only come to us by the way of the real world. The ten fingers on which men count and thereby performed the first arithmetical calculations are anything but a free creation of the mind. ”

“ Number is a phase of the kosmogonic movement, a measurer of the intellect and the establisher of the geometrism of space, answering tentatively to the numericity of pure being. In fact, being actually expresses number and number itself is an evolution and not a thing posited once for all as a pure, invariable form in the universe. ”

“ Or if an observer were stationed at a certain point, and 10,000 persons were to pass him in single file without his counting them as they passed, what sort of an estimate would he make of their number? The truth seems to be that our mental conception of number is much more limited than is commonly thought, and that we unconsciously adopt some new unit as a standard of comparison when we wish to render intelligible to our minds any number of considerable magnitude. ”

“ And of this nature are all the theorems of the science of number. They assert the identity of the result of different modes of formation. They affirm that some mode of formation from x, and some mode of formation from a certain function of x, produce the same number.
Such, as above described, is the aim and end of the calculus. As for its processes, every one knows that they are simply deductive. In demonstrating an algebraical theorem, or in resolving an equation, we travel from the datum to the quæsitum by pure ratiocination ”

“ The whole difficulty of the subject lies in the necessity of thinking in an unfamiliar way, and in realising that many properties which we have thought inherent in number are in fact peculiar to finite numbers. If this is remembered, the positive theory of infinity, which will occupy the next lecture, will not be found so difficult as it is to those who cling obstinately to the prejudices instilled by the arithmetic which is learnt in childhood. ”

“ I have said how much the intuition of pure number, whence comes rigorous mathematical induction, differs from sensible intuition to which the imagination, properly so called, is the principal contributor. ”

“ Computation and theoretical music have a closer affinity than others with arithmetic; for this department, being one also of quantity and ratio, approaches it in number and proportion. [17] Optics and geodesy, again, are more in affinity with geometry. And mechanics and astrology are in general affinity with both.
As to mathematics having its principles [18] in hypothesis and about hypothesis. Now, the term hypothesis is used in three ways, or indeed in many ways. For according to one usage of the term we have the dramatic revolution ”

“ You know neither where number begins, where it pauses, nor where it ends. Now you call it time, anon you call it space; by number only does anything exist; but for number all substance would be one and the same; it alone differentiates and modifies matter. Number is to your mind what it is to matter, an intangible agent. ”