Jowett: number

  

Number, said to have been invented by Palamedes, Rep.   7. 522 D (cp. Laws 3. 677 D); number and being, Parm. 144 :—the number of the State, Rep. 8. 546;— number of the citizens, Laws 5. 737. 738; 6- 77i; 9- 877 D;— number of families, not to change, ib. 5. 740:—puzzles caused by numbers, Phaedo   96 E, 101 D : —the odd numbers sacred to the Gods above, the even to those below, Laws 4. 717 A.

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Socrates   : Is the false man, then, false about other things, but not about number, and would he not tell falsehoods when dealing with number ? LESSER HIPPIAS

Hippias : He is false about number also, by Zeus. LESSER HIPPIAS

Socrates : Shall we, then, assume this also, [367b] that there is such a person as a man who is false about calculation and number ? LESSER HIPPIAS

Socrates : Then each of us, if one, would be an odd number ; or do you not consider one an odd number ? GREATER HIPPIAS

Socrates : Then are we both an odd number, being two ? GREATER HIPPIAS

Socrates : But we are both an even number, are we not ? GREATER HIPPIAS

Soc. I believe that what you say may be true about numbers, which must be just what they are, or not be at all ; for example, the number ten at once becomes other than ten if a unit be added or subtracted, and so of any other number : but this does not apply to that which is qualitative or to anything which is represented under an image. I should say rather that the image, if expressing in every point the entire reality, would no longer be an image. Let us suppose the existence of two objects : one of them shall be Cratylus  , and the other the image of Cratylus ; and we will suppose, further, that some God makes not only a representation such as a painter would make of your outward form and colour, but also creates an inward organization like yours, having the same warmth and softness ; and into this infuses motion, and soul, and mind, such as you have, in a word copies all your qualities, and places them by you in another form ; would you say that this was Cratylus and the image of Cratylus, or that there were two Cratyluses ? CRATYLUS

Soc. This indication of my meaning may proceed from unlike as well as from like, for example in the l of sklerotes. But if this is true, then you have made a convention with yourself, and the correctness of a name turns out to be convention, since letters which are unlike are indicative equally with those which are like, if they are sanctioned by custom and convention. And even supposing that you distinguish custom from convention ever so much, still you must say that the signification of words is given by custom and not by likeness, for custom may indicate by the unlike as well as by the like. But as we are agreed thus far, Cratylus (for I shall assume that your silence gives consent), then custom and convention must be supposed to contribute to the indication of our thoughts ; for suppose we take the instance of number, how can you ever imagine, my good friend, that you will find names resembling every individual number, unless you allow that which you term convention and agreement to have authority in determining the correctness of names ? I quite agree with you that words should as far as possible resemble things ; but I fear that this dragging in of resemblance, as Hermogenes says, is a shabby thing, which has to be supplemented by the mechanical aid of convention with a view to correctness ; for I believe that if we could always, or almost always, use likenesses, which are perfectly appropriate, this would be the most perfect state of language ; as the opposite is the most imperfect. But let me ask you, what is the force of names, and what is the use of them ? CRATYLUS

Crat. Yes, Socrates, but observe ; the greater number express motion. CRATYLUS

Str. And all number is to be reckoned among things which are ? SOPHIST

Theaet. Yes, surely number, if anything, has a real existence. SOPHIST

Str. Then we must not attempt to attribute to not-being number either in the singular or plural ? SOPHIST

Str. But how can a man either express in words or even conceive in thought things which are not or a thing which is not without number ? SOPHIST

Str. Say no more of ourselves ; but until we find some one or other who can speak of not-being without number, we must acknowledge that the Sophist is a clever rogue who will not be got out of his hole. SOPHIST

Str. I think that Parmenides  , and all ever yet undertook to determine the number and nature of existences, talked to us in rather a light and easy strain. SOPHIST

Theaet. Surely we cannot admit that the number is less than it appeared to be just now. SOPHIST

Str. Then we may infer that being is not, in respect of as many other things as there are ; for not-being these it is itself one, and is : not the other things, which are infinite in number. SOPHIST

Str. And this appeared to be the case ; and therefore now, without hesitation, we shall number the different kinds as two. SOPHIST

Now God did not make the soul after the body, although we are speaking of them in this order ; for having brought them together he would never have allowed that the elder should be ruled by the younger ; but this is a random manner of speaking which we have, because somehow we ourselves too are very much under the dominion of chance. Whereas he made the soul in origin and excellence prior to and older than the body, to be the ruler and mistress, of whom the body was to be the subject. And he made her out of the following elements and on this wise : Out of the indivisible and unchangeable, and also out of that which is divisible and has to do with material bodies, he compounded a third and intermediate kind of essence, partaking of the nature of the same and of the other, and this compound he placed accordingly in a mean between the indivisible, and the divisible and material. He took the three elements of the same, the other, and the essence, and mingled them into one form, compressing by force the reluctant and unsociable nature of the other into the same. When he had mingled them with the essence and out of three made one, he again divided this whole into as many portions as was fitting, each portion being a compound of the same, the other, and the essence. And he proceeded to divide after this manner : — First of all, he took away one part of the whole [1], and then he separated a second part which was double the first [2], and then he took away a third part which was half as much again as the second and three times as much as the first [3], and then he took a fourth part which was twice as much as the second [4], and a fifth part which was three times the third [9], and a sixth part which was eight times the first [8], and a seventh part which was twenty-seven times the first [27]. After this he filled up the double intervals [i.e. between 1, 2, 4, 8] and the triple [i.e. between 1, 3, 9, 27] cutting off yet other portions from the mixture and placing them in the intervals, so that in each interval there were two kinds of means, the one exceeding and exceeded by equal parts of its extremes [as for example 1, 4/3, 2, in which the mean 4/3 is one-third of 1 more than 1, and one-third of 2 less than 2], the other being that kind of mean which exceeds and is exceeded by an equal number. Where there were intervals of 3/2 and of 4/3 and of 9/8, made by the connecting terms in the former intervals, he filled up all the intervals of 4/3 with the interval of 9/8, leaving a fraction over ; and the interval which this fraction expressed was in the ratio of 256 to 243. And thus the whole mixture out of which he cut these portions was all exhausted by him. This entire compound he divided lengthways into two parts, which he joined to one another at the centre like the letter X, and bent them into a circular form, connecting them with themselves and each other at the point opposite to their original meeting-point ; and, comprehending them in a uniform revolution upon the same axis, he made the one the outer and the other the inner circle. Now the motion of the outer circle he called the motion of the same, and the motion of the inner circle the motion of the other or diverse. The motion of the same he carried round by the side to the right, and the motion of the diverse diagonally to the left. And he gave dominion to the motion of the same and like, for that he left single and undivided ; but the inner motion he divided in six places and made seven unequal circles having their intervals in ratios of two and three, three of each, and bade the orbits proceed in a direction opposite to one another ; and three [Sun, Mercury, Venus] he made to move with equal swiftness, and the remaining four [Moon, Saturn, Mars, Jupiter] to move with unequal swiftness to the three and to one another, but in due proportion. TIMAEUS  

When the father creator saw the creature which he had made moving and living, the created image of the eternal gods, he rejoiced, and in his joy determined to make the copy still more like the original ; and as this was eternal, he sought to make the universe eternal, so far as might be. Now the nature of the ideal being was everlasting, but to bestow this attribute in its fulness upon a creature was impossible. Wherefore he resolved to have a moving image of eternity, and when he set in order the heaven, he made this image eternal but moving according to number, while eternity itself rests in unity ; and this image we call time. For there were no days and nights and months and years before the heaven was created, but when he constructed the heaven he created them also. They are all parts of time, and the past and future are created species of time, which we unconsciously but wrongly transfer to the eternal essence ; for we say that he "was," he "is," he "will be," but the truth is that "is" alone is properly attributed to him, and that "was" and "will be" only to be spoken of becoming in time, for they are motions, but that which is immovably the same cannot become older or younger by time, nor ever did or has become, or hereafter will be, older or younger, nor is subject at all to any of those states which affect moving and sensible things and of which generation is the cause. These are the forms of time, which imitates eternity and revolves according to a law of number. Moreover, when we say that what has become is become and what becomes is becoming, and that what will become is about to become and that the non-existent is non-existent — all these are inaccurate modes of expression. But perhaps this whole subject will be more suitably discussed on some other occasion. TIMAEUS

Now, when all the stars which were necessary to the creation of time had attained a motion suitable to them, — and had become living creatures having bodies fastened by vital chains, and learnt their appointed task, moving in the motion of the diverse, which is diagonal, and passes through and is governed by the motion of the same, they revolved, some in a larger and some in a lesser orbit — those which had the lesser orbit revolving faster, and those which had the larger more slowly. Now by reason of the motion of the same, those which revolved fastest appeared to be overtaken by those which moved slower although they really overtook them ; for the motion of the same made them all turn in a spiral, and, because some went one way and some another, that which receded most slowly from the sphere of the same, which was the swiftest, appeared to follow it most nearly. That there might be some visible measure of their relative swiftness and slowness as they proceeded in their eight courses, God lighted a fire, which we now call the sun, in the second from the earth of these orbits, that it might give light to the whole of heaven, and that the animals, as many as nature intended, might participate in number, learning arithmetic from the revolution of the same and the like. Thus then, and for this reason the night and the day were created, being the period of the one most intelligent revolution. And the month is accomplished when the moon has completed her orbit and overtaken the sun, and the year when the sun has completed his own orbit. Mankind, with hardly an exception, have not remarked the periods of the other stars, and they have no name for them, and do not measure them against one another by the help of number, and hence they can scarcely be said to know that their wanderings, being infinite in number and admirable for their variety, make up time. And yet there is no difficulty in seeing that the perfect number of time fulfils the perfect year when all the eight revolutions, having their relative degrees of swiftness, are accomplished together and attain their completion at the same time, measured by the rotation of the same and equally moving. After this manner, and for these reasons, came into being such of the stars as in their heavenly progress received reversals of motion, to the end that the created heaven might imitate the eternal nature, and be as like as possible to the perfect and intelligible animal. TIMAEUS

Thus far and until the birth of time the created universe was made in the likeness of the original, but inasmuch as all animals were not yet comprehended therein, it was still unlike. What remained, the creator then proceeded to fashion after the nature of the pattern. Now as in the ideal animal the mind perceives ideas or species of a certain nature and number, he thought that this created animal ought to have species of a like nature and number. There are four such ; one of them is the heavenly race of the gods ; another, the race of birds whose way is in the air ; the third, the watery species ; and the fourth, the pedestrian and land creatures. Of the heavenly and divine, he created the greater part out of fire, that they might be the brightest of all things and fairest to behold, and he fashioned them after the likeness of the universe in the figure of a circle, and made them follow the intelligent motion of the supreme, distributing them over the whole circumference of heaven, which was to be a true cosmos or glorious world spangled with them all over. And he gave to each of them two movements : the first, a movement on the same spot after the same manner, whereby they ever continue to think consistently the same thoughts about the same things ; the second, a forward movement, in which they are controlled by the revolution of the same and the like ; but by the other five motions they were unaffected, in order that each of them might attain the highest perfection. And for this reason the fixed stars were created, to be divine and eternal animals, ever-abiding and revolving after the same manner and on the same spot ; and the other stars which reverse their motion and are subject to deviations of this kind, were created in the manner already described. The earth, which is our nurse, clinging around the pole which is extended through the universe, he framed to be the guardian and artificer of night and day, first and eldest of gods that are in the interior of heaven. Vain would be the attempt to tell all the figures of them circling as in dance, and their juxtapositions, and the return of them in their revolutions upon themselves, and their approximations, and to say which of these deities in their conjunctions meet, and which of them are in opposition, and in what order they get behind and before one another, and when they are severally eclipsed to our sight and again reappear, sending terrors and intimations of the future to those who cannot calculate their movements — to attempt to tell of all this without a visible representation of the heavenly system would be labour in vain. Enough on this head ; and now let what we have said about the nature of the created and visible gods have an end. TIMAEUS

Thus he spake, and once more into the cup in which he had previously mingled the soul of the universe he poured the remains of the elements, and mingled them in much the same manner ; they were not, however, pure as before, but diluted to the second and third degree. And having made it he divided the whole mixture into souls equal in number to the stars, and assigned each soul to a star ; and having there placed them as in a chariot, he showed them the nature of the universe, and declared to them the laws of destiny, according to which their first birth would be one and the same for all, — no one should suffer a disadvantage at his hands ; they were to be sown in the instruments of time severally adapted to them, and to come forth the most religious of animals ; and as human nature was of two kinds, the superior race would here after be called man. Now, when they should be implanted in bodies by necessity, and be always gaining or losing some part of their bodily substance, then in the first place it would be necessary that they should all have in them one and the same faculty of sensation, arising out of irresistible impressions ; in the second place, they must have love, in which pleasure and pain mingle ; also fear and anger, and the feelings which are akin or opposite to them ; if they conquered these they would live righteously, and if they were conquered by them, unrighteously. He who lived well during his appointed time was to return and dwell in his native star, and there he would have a blessed and congenial existence. But if he failed in attaining this, at the second birth he would pass into a woman, and if, when in that state of being, he did not desist from evil, he would continually be changed into some brute who resembled him in the evil nature which he had acquired, and would not cease from his toils and transformations until he followed the revolution of the same and the like within him, and overcame by the help of reason the turbulent and irrational mob of later accretions, made up of fire and air and water and earth, and returned to the form of his first and better state. Having given all these laws to his creatures, that he might be guiltless of future evil in any of them, the creator sowed some of them in the earth, and some in the moon, and some in the other instruments of time ; and when he had sown them he committed to the younger gods the fashioning of their mortal bodies, and desired them to furnish what was still lacking to the human soul, and having made all the suitable additions, to rule over them, and to pilot the mortal animal in the best and wisest manner which they could, and avert from him all but self-inflicted evils. TIMAEUS

All these are to be reckoned among the second and co-operative causes which God, carrying into execution the idea of the best as far as possible, uses as his ministers. They are thought by most men not to be the second, but the prime causes of all things, because they freeze and heat, and contract and dilate, and the like. But they are not so, for they are incapable of reason or intellect ; the only being which can properly have mind is the invisible soul, whereas fire and water, and earth and air, are all of them visible bodies. The lover of intellect and knowledge ought to explore causes of intelligent nature first of all, and, secondly, of those things which, being moved by others, are compelled to move others. And this is what we too must do. Both kinds of causes should be acknowledged by us, but a distinction should be made between those which are endowed with mind and are the workers of things fair and good, and those which are deprived of intelligence and always produce chance effects without order or design. Of the second or co-operative causes of sight, which help to give to the eyes the power which they now possess, enough has been said. I will therefore now proceed to speak of the higher use and purpose for which God has given them to us. The sight in my opinion is the source of the greatest benefit to us, for had we never seen the stars, and the sun, and the heaven, none of the words which we have spoken about the universe would ever have been uttered. But now the sight of day and night, and the months and the revolutions of the years, have created number, and have given us a conception of time, and the power of enquiring about the nature of the universe ; and from this source we have derived philosophy, than which no greater good ever was or will be given by the gods to mortal man. This is the greatest boon of sight : and of the lesser benefits why should I speak ? even the ordinary man if he were deprived of them would bewail his loss, but in vain. Thus much let me say however : God invented and gave us sight to the end that we might behold the courses of intelligence in the heaven, and apply them to the courses of our own intelligence which are akin to them, the unperturbed to the perturbed ; and that we, learning them and partaking of the natural truth of reason, might imitate the absolutely unerring courses of God and regulate our own vagaries. The same may be affirmed of speech and hearing : they have been given by the gods to the same end and for a like reason. For this is the principal end of speech, whereto it most contributes. Moreover, so much of music as is adapted to the sound of the voice and to the sense of hearing is granted to us for the sake of harmony ; and harmony, which has motions akin to the revolutions of our souls, is not regarded by the intelligent votary of the Muses as given by them with a view to irrational pleasure, which is deemed to be the purpose of it in our day, but as meant to correct any discord which may have arisen in the courses of the soul, and to be our ally in bringing her into harmony and agreement with herself ; and rhythm too was given by them for the same reason, on account of the irregular and graceless ways which prevail among mankind generally, and to help us against them. TIMAEUS

Thus have I concisely given the result of my thoughts ; and my verdict is that being and space and generation, these three, existed in their three ways before the heaven ; and that the nurse of generation, moistened by water and inflamed by fire, and receiving the forms of earth and air, and experiencing all the affections which accompany these, presented a strange variety of appearances ; and being full of powers which were neither similar nor equally balanced, was never in any part in a state of equipoise, but swaying unevenly hither and thither, was shaken by them, and by its motion again shook them ; and the elements when moved were separated and carried continually, some one way, some another ; as, when rain is shaken and winnowed by fans and other instruments used in the threshing of corn, the close and heavy particles are borne away and settle in one direction, and the loose and light particles in another. In this manner, the four kinds or elements were then shaken by the receiving vessel, which, moving like a winnowing machine, scattered far away from one another the elements most unlike, and forced the most similar elements into dose contact. Wherefore also the various elements had different places before they were arranged so as to form the universe. At first, they were all without reason and measure. But when the world began to get into order, fire and water and earth and air had only certain faint traces of themselves, and were altogether such as everything might be expected to be in the absence of God ; this, I say, was their nature at that time, and God fashioned them by form and number. Let it be consistently maintained by us in all that we say that God made them as far as possible the fairest and best, out of things which were not fair and good. And now I will endeavour to show you the disposition and generation of them by an unaccustomed argument, which am compelled to use ; but I believe that you will be able to follow me, for your education has made you familiar with the methods of science. TIMAEUS