UNITY AND NUMBER.
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CHAPTER I.
PRELIMINARY CONSIDERATIONS ON THE IDEA OF UNITY.
1. Before analyzing the idea of number, let us examine its simplest element, unity. Number is a connection of unities. We cannot know what number is, if we do not know what unity is.[25]
2. What is unity? When is a thing one? We all seem to know what unity is, since upon it we found the fabric of all our arithmetic cognitions. We all know when a thing is one, and we never equivocate on the meaning of the word. In this the learned and the unlearned stand on the same footing. The word one, in our language, has only one meaning for all who understand it. The same may be said of the word which in other languages expresses the same idea. When we meet the figure 1, which corresponds to this idea, and expresses it in a general manner, abstracting the difference of idioms, all men understand and apply it in the same manner.
3. The idea of unity is the same in all men; it is a common patrimony of the human race. It is not bound to this or that object, nor to this or that act of the mind; it extends to all in the same manner. Even composite and multiple things are called one only, inasmuch as they participate in a general idea. The indivisible point is one. The line composed of many points could not be one were there not a contiguous enchainment of these points, and did they not all unite to form one object, which gives us one impression, and is submitted to one act of our understanding.
4. The idea of unity is not a particular sensation, since it applies to all; neither is it sensation in general, since it pertains to what is not sensation. The sensation of color is one; so, also, the consciousness of the me is one, although this is not a sensation. The size of the rectangle which I see is one, and the relation of the equality of its angles is also one, but is not a sensation.
5. The idea of unity is a simple idea, and accompanies our mind from its first steps; we find it everywhere, and understand it well, but cannot explain it as we would, because it is simple, and cannot be decomposed and expressed by various words. We do not mean to say, however, that we must abjure all explanation of it; we only propose to warn the reader of the kind of explanation he may expect, which can be no other than the analysis of the fact, inasmuch as it is an object, and of the phenomenon as presented to our mind.
WHAT IS UNITY.
6. The scholastics were right in teaching that every being is one, and that whatever is one is being. Unity is a general attribute of every being, but is not distinct from it. However little we reflect, we cannot fail to perceive that unity and being are not distinguished: the unity of unity, by itself, offers us nothing real or even possible. What then would become of unity, if nothing but unity? This idea is involved in that of being; it is an aspect of it, a reason under which being is presented to the understanding.
7. But what is the conception of unity under which beings are presented to us? There is unity in the object when there is no distinction in the conception presenting it; and there is no distinction, when the perception of relative not-being is not combined in the object with that of being. We have unity whenever we perceive an object simply. Suppose that we perceive the object B. No matter what B is, it will to us be always one, unless we perceive it as composed of C, D, one of which is not the other. If we perceive in the object B, a distinction between C and D, unity disappears.
Evidently when we are aware of this composition we can abstract it and simply consider the result, the whole, B; and then unity appears anew.
8. We see by this that unity may be either real or fictitious. It is real and existing when there is no distinction in the thing either real or apparent; it is fictitious in those composites which of themselves include distinct things that may be offered to the understanding, inasmuch as they are subordinated to one unity of order, abstraction made of the real distinction contained in them.
9. The schoolmen sometimes defined what is one to be, "ens indivisum in se, et divisum ab aliis." The former part seems sufficiently exact if by indivisum is meant non-distinctum and not non-separatum; but the second part must be regarded at the best as superfluous. If there existed only one most simple and sole being, it would yet be one, although we could not say that it was divided from others, divisum ab aliis; for as there would be no others it could not be divided from them. This part of the definition is therefore superfluous.
10. It is no solution of the difficulty to say that this one being is divided from others, real or possible, and that in the supposition of one only being, others are possible although not real. The only being would be really one, and the division from others would be only possible; since there can be no real distinction between two terms when one of them is only possible. The division from others, divisis ab aliis, therefore is not a necessary element of unity, because unity is real, and this element is only possible.
11. However, in confirmation of this doctrine, we may remark, that in common parlance, unity is opposed to distinction, and there is no unity where there is no distinction. If the only being be not conceived as multiple there can be no distinction; and this is so independently of its being compared with the rest. The words, others, and the rest, suppose single beings; the idea of unity precedes that of distinction; beings are not considered as distinct between themselves until after they are conceived as individually single.
12. It seems, therefore, that a single being ought to be defined as ens indivisum in se, or a being which includes no division. Unity, then, will depend upon non-distinction. If non-division denote non-distinction, there will be real unity; but if it denote non-separation or re-union, we shall only have a fictitious unity. The molecules without extension, of which many suppose matter to be composed, would be really one, because there is no distinction in them. Bodies are fictitiously one because their composite parts though united are really distinct.
13. A difficulty may be raised by asking whether a being, indivisible in itself, but not divided from others, would be really one, for in case it would not be one, it might be inferred that we had unjustly censured the definition of the schoolmen, since whatever wants the second property required by the definition would not be one. We reply, then, a being that includes no distinction in itself, and is not distinguished from others, would indeed be one, but in such a case there would be no others, since they cannot be when there is no distinction. In such an hypothesis, there would be only one unity, the unity of pantheism, the great all, the absolute in which all things would be identified.
14. We have already said that the unity which is confounded with being, is not the unity which originates number. We here in fact encounter two different conceptions of unity, the one marking only want of distinction, and the other expressing the property of engendering number. But we are not thence to infer that the one which is identified with being is distinct from that which engenders number. All beings, one in themselves, but distinct from each other, no matter what they may be, may be conceived under the idea of number. The number three enters into the august mystery of the Trinity, and we say with all truth that in God there are three persons.
15. It is not necessary that the unity which engenders number should be real; it suffices if it be fictitious. When we take a foot measure for unity, we employ a fictitious unity, since the foot is composed of parts, but the number which results therefrom is, nevertheless, a true number.
CHAPTER III.
UNITY AND SIMPLICITY.
16. Real unity and simplicity are identical. What is really one has no distinction in itself; nor is it composed of parts, of which it can be said, this is not that. Evidently simplicity requires nothing more; the simple is opposed to the composite, to what is formed of many beings whereof one is not the other.
17. We meet this simplicity in none of the objects of our intuitions, excepting the acts of our own mind; so that even when we know, by discursion, that there are substances really one or simple, we do not see them in themselves.
Extension consists essentially of parts; whence it happens that we never encounter real unity or simplicity in the corporeal world as object of our sensibility. But as the composite must be resolved into the simple, as it is hard to proceed ad infinitum, we infer that the corporeal universe itself is a union of substances which, whether called points without extension, or any thing else, cannot be decomposed into others; for which reason they are really one, or simple.
18. Hence we conclude that substances may be said to be in a certain manner simple; and that things called composite are unions of substances, which in their turn form a third substance by virtue of a law presiding over them and giving them that unity which we call factitious.
19. We cannot, then, do less than to remark that the transcendental analysis refutes those who deny simplicity to thinking beings, since we have seen that simplicity is prior to composition, which can neither be nor be conceived if it be not presupposed. Simplicity is a necessary law of every being: a composite being ought to be called a union of beings, rather than a being.
20. We have said that simple substances are not objects of our intuition, which has none worthy to be called simple excepting the acts of our mind. The reason of this is, that the principal medium of our intuition is sensibility, which is founded upon representations, themselves based upon extension. There can be no doubt that the acts of our mind, given us by intuition, in the inward sense are perfectly simple; for who can decompose a perception, a judgment, an act of the reason or of the will?
21. The perception of a certain object requires preparatory acts; and the same may be said of judgments and ratiocinations; yet these operations are in themselves exceedingly simple, and cannot be divided into various parts. Simplicity is met with alike in the acts of the will, whether of the pure, intellectual, or sensible will. How shall we divide such acts as these into parts: I desire, I do not desire, I love, I abhor, I suffer, I rejoice?
22. We must take care not to confound the multiplicity of the acts with the acts themselves; there may be many acts, but in themselves they are simple. Thoughts, impressions and affections continually succeed one another in our mind; these phenomena are all distinct from each other, as is proved by their existing at different times, some at one time without the others, and by some being incompatible with others, because contradictory; but each individual phenomenon is by itself incapable of decomposition, and admits in itself no distinction into various parts; wherefore, it is simple.
23. True unity, therefore, is only found in simplicity; where there is no true simplicity, there may be factitious, but not real, unity; since even when there is no separation, there may be distinction between the various parts of which the composite is formed.
24. It may be inferred from this that indistinctum ought, perhaps, to take the place of indivisum in the definition of a one being; because distinction is opposed to unity of identity, and division to union. Absence of division is all that factitious unity requires; but real unity demands that there be no distinction. However closely united two things may be, if one is not the other they are distinct, and cannot, in strict metaphysical language, be called one.
25. The object of these observations is only to fix our ideas, not to modify our language. In common parlance, the idea of unity is used in a less rigorous sense, and, far from opposing this use, we readily accord it a reasonable foundation. There results from the union of two really distinct things, a conjunction, rightly called one so far as it also is subjected to a certain unity; and, were it not permitted to use this word in a sense less rigorous than that exacted by metaphysical analysis, we should be under the necessity of excluding unity from the great mass of objects. Simple substances, we have said, are not offered to us in immediate intuition, and we see compositions rather than their component elements. Could we apply unity only to simple elements, science would be greatly reduced, language would be impoverished, and literature and the fine arts would be despoiled of unity, one of their characteristic perfections.
CHAPTER IV.
ORIGIN OF THE TENDENCY OF OUR MIND TO UNITY.
26. Since we encounter multiplicity in all sensible objects, which are those chiefly demanding our attention, how does our mind acquire the idea of unity? In science, in literature, in the arts, and in every thing, we seek unity; and whence this irresistible tendency towards unity, which makes us seek a factitious when we cannot find a real unity, and this, too, notwithstanding the multiplicity presented by all the objects of our perception?
27. Two origins, if we mistake not, may be assigned to this tendency towards unity, the one objective, the other subjective. The former consists in the very character of unity in which the object of the understanding is mainly comprised; the other is the unity found in the intelligent being, and which it experiences in itself. We will explain these ideas more at length.
28. Unity is being; every being is one; and, properly speaking, being is not found without unity. Let us take a composite object: in it we discover two things; the simple component elements of it, and the union of them. The being, properly speaking, does not consist in the union, but in the united elements. The union is a mere relation, not even possible without the elements to be united. On the other hand, these elements in themselves, abstracted from their union, are true beings, existed before, and will exist after their union. What is an organized body? An aggregation of molecules united under a certain law, conformably to a principle presiding over their organization. The parts existed before their organization, and will continue to exist after its destruction. The being, therefore, properly consisted in the elements; and the organization was a relation of them among themselves.
29. Organization requires a principle to rule it, and subject its functions to determinate laws. Thus we see that even relation is subject to unity, to the unity of end and to the unity of a ruling and directing principle.
30. It is inconceivable how the union of distinct things can have any meaning, or lead to any result, if unity do not preside over it. In objects submitted to our experience, things are united in three ways: by juxtaposition in space; by co-existence in time; and by association in the exercise of their activity. The elements constitutive of extension are united in the first way; all objects belonging to the same time, in the second; and in the third all those which unite their forces and direct them to one and the same end.
31. The union consisting in the continuity of elements in space, has no value in the eyes of science, save inasmuch as there is an intelligent being who perceives the forms resulting from this continuity, by reducing them to unity under ideal types. Four lines of points, so disposed as to form a quadrilateral figure, have no scientific meaning until there comes an intelligence and perceives the form of a quadrilateral figure under the aspect of unity. We do not deny that the quadrilateral figure exists independently of intellectual perception: these lines will certainly exist, and be arranged in the same manner, although we prescind all intelligence; but this disposition in the quadrilateral form is a relation, not a being distinct from the aggregation of the elements disposed; and this relation, of itself alone, is no object of intelligence except inasmuch as presented to it under the unity of the quadrilateral form.
The intelligence in search of a true being, can find none, save in elements; and if it wishes to perceive their relation, it must recur to the unity of form.
32. Co-existence in time, is a relation, which, of itself alone, neither gives any thing to, nor takes any thing from objects. These exist independently of this relation; for they must, of necessity, exist, in order to co-exist. This relation denotes something perceptible to the understanding, only as it is presented to it under unity, which, in this case, is unity of time, as in the former it was unity of space.
33. Neither has the association of activities any meaning, except when it expresses the convergence of forces towards one and the same object. If unity be wanting to the point of their direction, their union will express nothing, and the intelligence will have for its object only scattered and unrelated activities.
34. We have then shown that unity is a law of our understanding, founded upon the very nature of things. Absolute being is never found in the composite, but only in the simple, and relative being is not even conceivable, if it be not submitted to unity.
35. We discover in the very nature of our mind, the second origin of its tendency to unity. It in itself is one, is simple, and therefore disposed to assimilate every thing to itself under this same unity and simplicity. It feels that it is one in the midst of multiplicity, permanent even in succession, and under all the immense variety of sensible phenomena, intellectual and moral, which it unceasingly experiences. The inward sense attests with irresistible certainty the identity of the me. This unity, this identity, is as certain, as evident to the child who begins to feel pleasure or pain, and is sure that he is one and the same that experiences both impressions, as they are to the philosopher who has spent long years in profoundly investigating the idea of the me and the unity of consciousness.
The unity and simplicity which we experience in ourselves force us to reduce the composite to the simple, the multiple to the one. The perception of things the most composite refers to a consciousness essentially one: even were we to perceive the whole complicated universe by a single act, this act would be most simple, since otherwise the me could not say, I perceive.
36. Two reasons, then, exist why our mind in all things seeks unity. Objects are unintelligible, except so far as subjected to a certain perceptible unity, to a form, under which the multiple is made one, and the composite simple. The object of the understanding is being, and being consists in the simple. The composite involves an aggregation of simple elements with the relation called union; but unless this be presented under a certain unity, it does not constitute a perceptible object.
Without the indivisible unity of consciousness, no intelligent subject is conceivable. Every intelligent being requires this link to unite the variety of phenomena of which it is the subject. If this unity fail, the phenomena become an informal aggregation, unrelated among themselves: intellectual acts without an intelligent being.
The tendency to unity originates in the perfection of our mind, and is itself a perfection; but it needs to be carefully watched, lest it go astray, and seek real unity there, where only a factitious unity can be found. This exaggeration is the cause of pantheism, the fatal error of our day. Our mind is one, so also is the infinite essence, cause of all finite beings; but the aggregation of these beings is not one, for even when united by many ties, they cease not to be distinct. There is in the world unity of order, of harmony, of origin, and of end; but there is no absolute unity. Number also enters into unity of harmony, but it is incompatible with absolute unity, as reason and experience both show.
GENERATION OF THE IDEA OF NUMBER.
37. Unity is the first element of number, but does not of itself alone constitute number, which is not unity, but the collection of unities.
38. Two is a number. What is our idea of the number two? Evidently it is not confounded with its sign, for signs are many and very different, but it is one and always the same.
39. It would seem at first sight that the idea of two is independent of the mode of its generation, and that, being one, it may be formed by addition or subtraction, by adding one to one, or taking one from three: 1 + 1 = 2; 3 - 1 = 2. But if we reflect upon these two expressions, we shall see that the latter is impossible without the former. We should not know that 3 -1 = 2 if we did not previously know that two entered into the composition of three, and how it entered. We could know nothing of this had we not already the idea of two, and this idea is nothing else than the perception of this sum.
40. The idea of two is no sensation, for it extends alike to the sensible and the non-sensible, to the simultaneous and the successive. In itself it is simple, its object is composite.
41. Since the collection of objects is small in two, the imagination can easily figure to itself what the understanding perceives; and the idea seems clearer to us because made sensible by a representation. The idea of addition made, in facto, that is, the idea of the sum, enters into that of two, but not of addition in fieri. Our idea of this number is perfectly clear, and yet we do not continually think of one plus one.
42. The idea of two refers to the simultaneous as well as to the successive; but our mind does not discover it until after it has the idea of succession. The object of this perception is the relation of united things; the understanding perceives them as such, and then only has it the idea of two.
43. Neither the successive nor simultaneous perception of two objects unaccompanied by relation is the idea of two. Hence the saying: a man and a horse do not make two, but only one and one; and the reason of this is that the man and the horse are represented to the understanding by their difference, not by their resemblance; and things must be presented to the mind under a common idea in order to give number. Thus, if we abstract their difference, and consider them only as animals, or corporeal beings, or beings simply, or things, they will make two.
44. In objects, then, totally unlike, or not comprehended under some common idea, there can be no number. Abstract number is number by excellence; because it eliminates all that distinguishes the things numbered, and considers them only as beings, consequently as similar, as contained in the general idea of being. Concrete numbers are only numbers so far as they participate in this property. Two is applicable to one horse and another horse, but not to a horse and a man, unless we identify them under the idea of animal, and abstract rationality and irrationality. Concrete number requires a common denomination; otherwise it is not number.
45. The idea of distinction, that is, that the one is not the other, enters into the idea of two, so that this idea necessarily involves an affirmation and a negation. The affirmation is of the real, possible, or imaginary existence of the things counted; the negation is of the one with respect to the other. Affirmation without distinction or negation involves identity. The idea of two, as well as that of every other number, includes the ideas of identity and distinction. The identity is of each extreme with itself; the distinction is of the extremes among themselves. Identity in the thing is the thing itself: identity in the idea is the simple perception of the thing. Distinction in the thing is the negation of it with respect to others: distinction in the idea is the perception of negation. We always perceive a thing as identical, and consequently every perception includes the idea of unity. But we do not always, when we perceive a thing, observe its negation with respect to others, and consequently do not always perceive number. The idea of number originates in comparison, when we see an object which is not another.
46. The ideas of being, distinction, and similarity enter into that of two. The idea of being, because nothing cannot be counted: that of distinction, or negation of the one being the other, because the identical does not constitute number: that of similarity, because things are only numbered when abstraction is made of their difference. Being is the basis of perception; distinction, of comparison; and similarity, of union. Perception begins with unity, proceeds with distinction, and ends with similarity, which is a kind of unity. The perception of this similarity unites what is distinct; but the union need not always be of the things, but may be in the idea comprising them. There are two poles of the world, but they are not united. The perception of the number two requires something more than the simple perception of objects; they must be susceptible of comparison, and consequently united in a common idea. This perception, therefore, demands comparison and abstraction, and this is why animals cannot numerate; they can neither compare nor generalize.
47. The analysis of the idea of two is the analysis of all numbers; the difference is not of nature, but of more and less; in the repetition of the same perception.
48. If any one now ask whether number be in the things, or in the mind alone, we reply that it is in things as in its foundation, because both distinction and similarity are in the things; that is, the one is not the other, and both have something in common; but it is the mind that sees all this.
49. After having perceived the distinction and union of two objects, we can also perceive another object, which will be neither the one nor the other of them, and will yet be comprehended in one general idea with them. This is the perception or idea of the number three. No matter how many numbers be imagined, nothing will ever be discovered in any of them except a simultaneous perception of objects, distinction of objects, and similarity of objects. If these be determinate, we shall have concrete number; if they be comprised in the general idea of being, of thing, we shall have abstract number.
50. The limits of our mind prevent it from comparing many objects at one time, and from easily recollecting the comparisons it has already made. To assist the memory, and the perception of these relations, we make use of signs. When we pass beyond three or four, our power of simultaneous perception fails, and we divide the object into groups which serve us as new units, and are expressed by signs. Ten is clearly the general group in the decimal system; but before we reach the number ten we have already formed other subalternate groups; since to count ten, we do not say one and one and one, etc., but one and one, two; two and one, three; three and one, four, etc. Each unit added forms a new group, which, in its turn, serves to form another. With two, we form three; with three, four, and so on. This affords an idea of the relation of numbers with their signs; but, as this matter is too important to be here dismissed, we will further develop it in the following chapters.
CONNECTION OF THE IDEAS OF NUMBER WITH THEIR SIGNS.
51. The connection of ideas and impressions, in a sign, is a most wonderful intellectual phenomenon, and at the same time of the greatest help to our mind. Were it not for this connection, we could scarcely reflect at all upon objects somewhat complex, and above all our memory would be exceedingly limited.[26]
52. Condillac made some excellent remarks upon this matter: in his opinion, we cannot, unaided by signs, count more than three or four. If, indeed, we had no sign but that of unity, we could readily count two, saying one and one. Having only two ideas, we could easily satisfy ourselves that we had twice repeated one. But it is not so easy to be certain of the exactness of our repetition when we have to count three, by saying one and one and one; still, this is not difficult. It is more so to count four, and next to impossible to go as far as ten. If we undertake to abstract the signs, we shall find that it is impossible to form an idea of ten by repeating one; and that it will be alike impossible, if we employ no sign, to make sure that we have repeated one exactly ten times.
53. Suppose the sign two, and one half of the difficulty is obviated; thus it will be much easier to say two and one, than one and one and one. In this supposition four will be no more difficult than was two, since, just as we before said, one and one, two; we now say, two and two, four. The attention before divided four times by the repetition of one, is now only divided twice. Six was before a hard number to count, but, in the present supposition, it is as easy as three was before; for, if we repeat two and two and two, we shall have six. The attention before distracted by six signs, is now distracted only by three. Evidently, if we continue to form the numbers three, four, and so on, expressive of distinct collections, we shall gradually facilitate numeration, until we attain the decimal simplicity now in use.
54. It may here be asked if the actual system be the most perfect possible? And if facility depend upon the distribution of collections in signs, can there be any thing more perfect than this distribution? Either there is question of new signs to denote new collections, or of the combination of signs. There can be no number which we cannot express with our present system, and consequently there is no need of inventing any thing to denote new collections. New signs might perhaps be invented for these collections, and these collections might possibly be distributed in a simpler and more convenient manner. In this case we admit an amelioration to be possible, though very difficult; but none in the former. In a word, the only possible progress would be in expressing better, not in expressing more.
55. The sign connects many ideas which, without it, would be isolated; hence its necessity in many cases, its utility in all cases. With the word hundred, or its numerical representative, 100, we know that we have one repeated a hundred times. Were this help to fail, we could not speak of a hundred, base calculations upon it, or even form it. It is, however, well said that we do not succeed in forming it except by tens, by repeating the calculation ten ten times.
56. Let it not, therefore, be thought that the idea of the number is the idea of the sign; for evidently the same idea of ten corresponds to the word ten, whether written, spoken, or numerically represented by the figures 10, although these three signs are very different. Every language has a word of its own to express ten, and all people have the same idea of it.
57. This last remark creates a difficulty as to what the idea of ten consists in. We cannot say that it is the recollection of the repetition of one ten times; first, because we do not think of this recollection when thinking of ten; and second, because, according to what has already been said, a clear recollection of this repetition is impossible. Neither is it the idea of the sign, for the idea signified existed before the sign was invented, otherwise the invention would have had no object, and would even have been impossible. There can be no sign where there is nothing to signify.
The idea of number includes more difficulties than Condillac ever imagined; who, if he had, after his close analysis of what facilitates numeration, profoundly meditated upon the idea itself, would not so readily have censured St. Augustine, Malebranche, and the whole Platonic school, for having said that numbers perceived by the pure understanding are something superior to those perceived by the senses.
CHAPTER VII.
ANALYSIS OF THE IDEA OF NUMBER IN ITSELF AND IN ITS RELATIONS WITH SIGNS.
58. In order clearly to conceive the idea of number, and the way it is engendered in our mind, let us study its formation in a deaf and dumb person.
We have no better way of giving such a one an idea of unity than by presenting an object to him. Now, if we would convey to him the idea of two, we show him two fingers, then two oranges, then two books, and in each of these operations make a sign which must be always the same. If we repeat this operation a number of times, the deaf and dumb person will associate the idea of two with that of the sign, and one will suggest the others; and he will endeavor to show us that he has seen two objects of some kind, by uniting the expression of the object with the sign of two. The same will take place with three, or four. When we reach higher numbers, the sign becomes more indispensable; since the less easily the idea of number is represented, the more necessary is the sign to secure it. But what we do to convey an idea of number to the deaf and dumb person, what he himself must do to express the number which he conceives, we must all do if we would obtain the idea.
59. Numeration is a repetition of operations; and the art of facilitating it consists in instituting signs which recall to our memory what we have done. It is an exceedingly complicated labyrinth, and we cannot trust ourselves to its windings with any expectation of finding our way out again, if we do not take care to mark the path we have followed.
It is to the admirable simplicity of the decimal system, united to its inexhaustible variety, that the facility and fecundity of our arithmetic are due. Algebra, going a step beyond, expresses without determining numbers, and presents the results of its operations without effacing its footsteps on the road travelled, is far superior to arithmetic, and has made the human mind take gigantic strides. But how? Solely by aiding the memory. Thus, the very principle that enables the child to say four and one, five, instead of adding unity five times to unity, the dumb man to express five by a hand, a hundred by a grain, enables the algebraist to express the result of his longest operations by a formula easy of retention by the memory. Both attain their object simply by aiding the memory. A grain of wheat denotes to the dumb man the idea of hundred, and this he applies to all similar collections; a few letters combined in a simple manner designate to the mathematician a property of certain quantities, and this he applies to all which are found in the same case.
60. Numeration is only an aggregation of formulas; and the more easy these are of mutual transformation with a slight modification, the more perfect will be the numeration. The better one knows the relations of these formulas and the manner of transforming them, the better will he know how to count. The greater a person's intellectual power of fixing simultaneously the attention upon many formulas, and of composing them, the more perfect arithmetician will he be, because the simultaneous comparison of many, leads to the perception of new relations.
61. What is our idea of hundred? The union of the units composing it, a union which we have made more or less frequently when learning to count. But how do we know that it is the same union? Because we have a formula called a hundred, expressed by a sign 100. This formula is so easily recollected that we have no difficulty in recollecting the idea of hundred and all the properties connected with it. We may be asked if a hundred is more than ninety. Were we under the necessity counting one and one and one, we should be bewildered, and never succeed in distinguishing the greater; but knowing as we do that to reach the formula hundred, we must pass by another formula ninety, and that this was in ascending, we know, once for all, that hundred expresses ninety and something more, that is, a hundred is more than ninety. And if it be further inquired what is the excess, we shall not undertake to ascertain this by adding units, but by the two formulas ninety and ten which compose the formula hundred.
62. By generalization we unite many similar things in one idea. The general idea is a kind of formula. Numeration unites in one sign many things contained in a general idea, but this sign has, at the same time, its own distinctive character. Thus the general idea belongs as a predicate to each of its particular objects; number belongs to no one in particular, but to all joined. We perceive in abstraction a common property, and lay aside all the particular objects which it presents; in numeration, we perceive similarity, but always with distinction. Abstraction is the result of comparison, but not comparison. Numeration implies a permanent comparison, or the recollection of it.
63. The idea of number is not conventional; a hundred is always a hundred with all its properties and relations, and this, too, prior to all convention and even to all human perception. The sign, and the sign only, is conventional. Were there no intellectual creature, and a hundred beings distinct among themselves were to exist, there would really be this number. The number three exists in the august mystery of the Trinity, from all eternity, and of absolute necessity. Number requires only the existence of distinct things; since, however unlike they may be, they always have something in common being, which may be included in a general idea, and consequently they fulfil the two conditions necessary to number.
64. The perception of being and of distinction, that is, of substantive being and of relative not-being, is the perception of number. The science of the relations of every collection, with its measure, which is unity, is the science of numbers.
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[Pg 199]
