Jowett: arithmetic

Soc. And yet surely, my dear friend Ion  , in a discussion about arithmetic, where many people are speaking, and one speaks better than the rest, there is somebody who can judge which of them is the good speaker ? ION

Soc. Yes, surely ; for if the subject of knowledge were the same, there would be no meaning in saying that the arts were different, — if they both gave the same knowledge. For example, I know that here are five fingers, and you know the same. And if I were to ask whether I and you became acquainted with this fact by the help of the same art of arithmetic, you would acknowledge that we did ? ION

Soc. And what sort of difference creates enmity and anger ? Suppose for example that you and I, my good friend, differ about a number ; do differences of this sort make us enemies and set us at variance with one another ? Do we not go at once to arithmetic, and put an end to them by a sum ? EUTHYPHRO  

Soc. But there are other arts which work wholly through the medium of language, and require either no action or very little, as, for example, the arts of arithmetic, of calculation, of geometry, and of playing draughts ; in some of these speech is pretty nearly co-extensive with action, but in most of them the verbal element is greater — they depend wholly on words for their efficacy and power : and I take your meaning to be that rhetoric is an art of this latter sort ? GORGIAS

Soc. And yet I do not believe that you really mean to call any of these arts rhetoric ; although the precise expression which you used was, that rhetoric is an art which works and takes effect only through the medium of discourse ; and an adversary who wished to be captious might say, "And so, Gorgias, you call arithmetic rhetoric." But I do not think that you really call arithmetic rhetoric any more than geometry would be so called by you. GORGIAS

Soc. Well, then, let me now have the rest of my answer : — seeing that rhetoric is one of those arts which works mainly by the use of words, and there are other arts which also use words, tell me what is that quality in words with which rhetoric is concerned : — Suppose that a person asks me about some of the arts which I was mentioning just now ; he might say, "Socrates  , what is arithmetic ?" and I should reply to him, as you replied to me, that arithmetic is one of those arts which take effect through words. And then he would proceed to ask : "Words about what ?" and I should reply, Words about and even numbers, and how many there are of each. And if he asked again : "What is the art of calculation ?" I should say, That also is one of the arts which is concerned wholly with words. And if he further said, "Concerned with what ?" I should say, like the clerks in the assembly, "as aforesaid" of arithmetic, but with a difference, the difference being that the art of calculation considers not only the quantities of odd and even numbers, but also their numerical relations to themselves and to one another. And suppose, again, I were to say that astronomy is only word — he would ask, "Words about what, Socrates ?" and I should answer, that astronomy tells us about the motions of the stars and sun and moon, and their relative swiftness. GORGIAS

Soc. Again, if we take the arts of which we were just now speaking : — do not arithmetic and the arithmeticians teach us the properties of number ? GORGIAS

Soc. Then arithmetic as well as rhetoric is an artificer of persuasion ? GORGIAS

Soc. At the Egyptian city of Naucratis, there was a famous old god, whose name was Theuth ; the bird which is called the Ibis is sacred to him, and he was the inventor of many arts, such as arithmetic and calculation and geometry and astronomy and draughts and dice, but his great discovery was the use of letters. Now in those days the god Thamus was the king of the whole country of Egypt ; and he dwelt in that great city of Upper Egypt which the Hellenes call Egyptian Thebes, and the god himself is called by them Ammon. To came Theuth and showed his inventions, desiring that the other Egyptians might be allowed to have the benefit of the he enumerated them, and Thamus enquired about their several uses, and praised some of them and censured others, as he approved or disapproved of them. It would take a long time to repeat all that Thamus said to Theuth in praise or blame of the various arts. But when they came to letters, This, said Theuth, will make the Egyptians wiser and give them better memories ; it is a specific both for the memory and for the wit. Thamus replied : O most ingenious Theuth, the parent or inventor of an art is not always the best judge of the utility or inutility of his own inventions to the users of them. And in this instance, you who are the father of letters, from a paternal love of your own children have been led to attribute to them a quality which they cannot have ; for this discovery of yours will create forgetfulness in the learners’ souls, because they will not use their memories ; they will trust to the external written characters and not remember of themselves. The specific which you have discovered is an aid not to memory, but to reminiscence, and you give your disciples not truth, but only the semblance of truth ; they will be hearers of many things and will have learned nothing ; they will appear to be omniscient and will generally know nothing ; they will be tiresome company, having the show of wisdom without the reality. PHAEDRUS  

Soc. And thus, my friend, on every occasion, private as well as public, as I said at first, when he appears in a law-court, or in any place in which he has to speak of things which are at his feet and before his eyes, he is the jest, not only of Thracian handmaids but of the general herd, tumbling into wells and every sort of disaster through his inexperience. His awkwardness is fearful, and gives the impression of imbecility. When he is reviled, he has nothing personal to say in answer to the civilities of his adversaries, for he knows no scandals of any one, and they do not interest him ; and therefore he is laughed at for his sheepishness ; and when others are being praised and glorified, in the simplicity of his heart he cannot help going into fits of laughter, so that he seems to be a downright idiot. When he hears a tyrant or king eulogized, he fancies that he is listening to the praises of some keeper of cattle — a swineherd, or shepherd, or perhaps a cowherd, who is congratulated on the quantity of milk which he squeezes from them ; and he remarks that the creature whom they tend, and out of whom they squeeze the wealth, is of a less traitable and more insidious nature. Then, again, he observes that the great man is of necessity as ill-mannered and uneducated as any shepherd — for he has no leisure, and he is surrounded by a wall, which is his mountain-pen. Hearing of enormous landed proprietors of ten thousand acres and more, our philosopher deems this to be a trifle, because he has been accustomed to think of the whole earth ; and when they sing the, praises of family, and say that someone is a gentleman because he can show seven generations of wealthy ancestors, he thinks that their sentiments only betray a dull and narrow vision in those who utter them, and who are not educated enough to look at the whole, nor to consider that every man has had thousands and ten thousands of progenitors, and among them have been rich and poor, kings and slaves, Hellenes and barbarians, innumerable. And when people pride themselves on having a pedigree of twenty-five ancestors, which goes back to Heracles, the son of Amphitryon, he cannot understand their poverty of ideas. Why are they unable to calculate that Amphitryon had a twenty-fifth ancestor, who might have been anybody, and was such as fortune made him and he had a fiftieth, and so on ? He amuses himself with the notion that they cannot count, and thinks that a little arithmetic would have got rid of their senseless vanity. Now, in all these cases our philosopher is derided by the vulgar, partly because he is thought to despise them, and also because he is ignorant of what is before him, and always at a loss. THEAETETUS  

Soc. And further, when any one wishes to catch any of these knowledges or sciences, and having taken, to hold it, and again to let them go, how will he express himself ? — will he describe the "catching" of them and the original "possession" in the same words ? I will make my meaning clearer by an example : — You admit that there is an art of arithmetic ? THEAETETUS

Str. Well, and are not arithmetic and certain other kindred arts, merely abstract knowledge, wholly separated from action ? STATESMAN

Str. Are not all such sciences, no less than arithmetic and the like, subjects of pure knowledge ; and is not the difference between the two classes, that the one sort has the power of judging only, and the other of ruling as well ? STATESMAN

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  

Soc. I mean to say, that if arithmetic, mensuration, and weighing be taken away from any art, that which remains will not be much. PHILEBUS  

Pro. I see that you mean arithmetic, and the kindred arts of weighing and measuring. PHILEBUS

Soc. In the first place, arithmetic is of two kinds, one of which is popular, and the other philosophical. PHILEBUS

Pro. Undoubtedly there is, as you say, a great difference among the votaries of the science ; and there may be reasonably supposed to be two sorts of arithmetic. PHILEBUS

Pro. Certainly ; and let us say in reply, that those arts into which arithmetic and mensuration enter, far surpass all others ; and that of these the arts or sciences which are animated by the pure philosophic impulse are infinitely superior in accuracy and truth. PHILEBUS

Soc. That there are two arts of arithmetic, and two of mensuration ; and also several other arts which in like manner have this double nature, and yet only one name. PHILEBUS

How then can we rightly order the distribution of the land ? In the first place, the number of the citizens has to be determined, and also the number and size of the divisions into which they will have to be formed ; and the land and the houses will then have to be apportioned by us as fairly as we can. The number of citizens can only be estimated satisfactorily in relation to the territory and the neighbouring states. The territory must be sufficient to maintain a certain number of inhabitants in a moderate way of life — more than this is not required ; and the number of citizens should be sufficient to defend themselves against the injustice of their neighbours, and also to give them the power of rendering efficient aid to their neighbours when they are wronged. After having taken a survey of theirs and their neighbours’ territory, we will determine the limits of them in fact as well as in theory. And now, let us proceed to legislate with a view to perfecting the form and outline of our state. The number of our citizens shall be 5040 — this will be a convenient number ; and these shall be owners of the land and protectors of the allotment. The houses and the land will be divided in the same way, so that every man may correspond to a lot. Let the whole number be first divided into two parts, and then into three ; and the number is further capable of being divided into four or five parts, or any number of parts up to ten. Every legislator ought to know so much arithmetic as to be able to tell what number is most likely to be useful to all cities ; and we are going to take that number which contains the greatest and most regular and unbroken series of divisions. The whole of number has every possible division, and the number 5040 can be divided by exactly fifty-nine divisors, and ten of these proceed without interval from one to ten : this will furnish numbers for war and peace, and for all contracts and dealings, including taxes and divisions of the land. These properties of number should be ascertained at leisure by those who are bound by law to know them ; for they are true, and should be proclaimed at the foundation of the city, with a view to use. Whether the legislator is establishing a new state or restoring an old and decayed one, in respect of Gods and temples — the temples which are to be built in each city, and the Gods or demi-gods after whom they are to be called — if he be a man of sense, he will make no change in anything which the oracle of Delphi, or Dodona, or the God Ammon, or any ancient tradition has sanctioned in whatever manner, whether by apparitions or reputed inspiration of Heaven, in obedience to which mankind have established sacrifices in connection with mystic rites, either originating on the spot, or derived from Tyrrhenia or Cyprus or some other place, and on the strength of which traditions they have consecrated oracles and images, and altars and temples, and portioned out a sacred domain for each of them. The least part of all these ought not to be disturbed by the legislator ; but he should assign to the several districts some God, or demi-god, or hero, and, in the distribution of the soil, should give to these first their chosen domain and all things fitting, that the inhabitants of the several districts may meet at fixed times, and that they may readily supply their various wants, and entertain one another with sacrifices, and become friends and acquaintances ; for there is no greater good in a state than that the citizens should be known to one another. When not light but darkness and ignorance of each other’s characters prevails among them, no one will receive the honour of which he is deserving, or the power or the justice to which he is fairly entitled : wherefore, in every state, above all things, every man should take heed that he have no deceit in him, but that he be always true and simple ; and that no deceitful person take any advantage of him. LAWS BOOK V

Having determined that there is to be a distribution into twelve parts, let us now see in what way this may be accomplished. There is no difficulty in perceiving that the twelve parts admit of the greatest number of divisions of that which they include, or in seeing the other numbers which are consequent upon them, and are produced out of them up to 5040 ; wherefore the law ought to order phratries and demes and villages, and also military ranks and movements, as well as coins and measures, dry and liquid, and weights, so as to be commensurable and agreeable to one another. Nor should we fear the appearance of minuteness, if the law commands that all the vessels which a man possesses should have a common measure, when we consider generally that the divisions and variations of numbers have a use in respect of all the variations of which they are susceptible, both in themselves and as measures of height and depth, and in all sounds, and in motions, as well those which proceed in a straight direction, upwards or downwards, as in those which go round and round. The legislator is to consider all these things and to bid the citizens, as far as possible, not to lose sight of numerical order ; for no single instrument of youthful education has such mighty power, both as regards domestic economy and politics, and in the arts, as the study of arithmetic. Above all, arithmetic stirs up him who is by nature sleepy and dull, and makes him quick to learn, retentive, shrewd, and aided by art divine he makes progress quite beyond his natural powers. All such things, if only the legislator, by other laws and institutions, can banish meanness and covetousness from the souls of men, so that they can use them properly and to their own good, will be excellent and suitable instruments of education. But if he cannot, he will unintentionally create in them, instead of wisdom, the habit of craft, which evil tendency may be observed in the Egyptians and Phoenicians, and many other races, through the general vulgarity of their pursuits and acquisitions, whether some unworthy legislator theirs has been the cause, or some impediment of chance or nature. For we must not fail to observe, O Megillus and Cleinias, that there is a difference in places, and that some beget better men and others worse ; and we must legislate accordingly. Some places are subject to strange and fatal influences by reason of diverse winds and violent heats, some by reason of waters ; or, again, from the character of the food given by the earth, which not only affects the bodies of men for good or evil, but produces similar results in their souls. And in all such qualities those spots excel in which there is a divine inspiration, and in which the demi-gods have their appointed lots, and are propitious, not adverse, to the settlers in them. To all these matters the legislator, if he have any sense in him, will attend as far as man can, and frame his laws accordingly. And this is what you, Cleinias, must do, and to matters of this kind you must turn your mind since you are going to colonize a new country. LAWS BOOK V

You argue like an informer, Socrates. Do you mean, for example, that he who is mistaken about the sick is a physician in that he is mistaken ? or that he who errs in arithmetic or grammar is an arithmetician or grammarian at the time when he is making the mistake, in respect of the mistake ? True, we say that the physician or arithmetician or grammarian has made a mistake, but this is only a way of speaking ; for the fact is that neither the grammarian nor any other person of skill ever makes a mistake in so far as he is what his name implies ; they none of them err unless their skill fails them, and then they cease to be skilled artists. No artist or sage or ruler errs at the time when he is what his name implies ; though he is commonly said to err, and I adopted the common mode of speaking. But to be perfectly accurate, since you are such a lover of accuracy, we should say that the ruler, in so far as he is a ruler, is unerring, and, being unerring, always commands that which is for his own interest ; and the subject is required to execute his commands ; and therefore, as I said at first and now repeat, justice is the interest of the stronger. THE REPUBLIC   BOOK I

Then I will try again ; you will understand me better when I have made some preliminary remarks. You are aware that students of geometry, arithmetic, and the kindred sciences assume the odd, and the even, and the figures, and three kinds of angles, and the like, in their several branches of science ; these are their hypotheses, which they and everybody are supposed to know, and therefore they do not deign to give any account of them either to themselves or others ; but they begin with them, and go on until they arrive at last, and in a consistent manner, at their conclusion ? THE REPUBLIC BOOK VI

Can we deny that a warrior should have a knowledge of arithmetic ? THE REPUBLIC BOOK VII

I will try, I said ; and I wish you would share the inquiry with me, and say "yes" or "no" when I attempt to distinguish in my own mind what branches of knowledge have this attracting power, in order that we may have clearer proof that arithmetic is, as I suspect, one of them. THE REPUBLIC BOOK VII

And all arithmetic and calculation have to do with number ? THE REPUBLIC BOOK VII

Then this is a kind of knowledge which legislation may fitly prescribe ; and we must endeavor to persuade those who are to be the principal men of our State to go and learn arithmetic, not as amateurs, but they must carry on the study until they see the nature of numbers with the mind only ; nor again, like merchants or retail-traders, with a view to buying or selling, but for the sake of their military use, and of the soul herself ; and because this will be the easiest way for her to pass from becoming to truth and being. THE REPUBLIC BOOK VII

I mean, as I was saying, that arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible or tangible objects into the argument. You know how steadily the masters of the art repel and ridicule anyone who attempts to divide absolute unity when he is calculating, and if you divide, they multiply, taking care that one shall continue one and not become lost in fractions. THE REPUBLIC BOOK VII

And, for all these reasons, arithmetic is a kind of knowledge in which the best natures should be trained, and which must not be given up. THE REPUBLIC BOOK VII