Plato on Mathematics

Plato wrote The Republic in around 375 BC, so about 75 years before Euclid wrote The Elements. In this work Plato sets out his ideas about education. For this, he believes, one must study the five mathematical disciplines, namely arithmetic, plane geometry, solid geometry, astronomy, and harmonics. After mastering mathematics, then one can proceed to the study of philosophy. First Plato gives an overview of his educational ideas, then goes on the talk about the five mathematical disciplines. Before we give a version of Plato's description as given in The Republic, let us note the style in which it is written. Plato, as narrator, describes a conversation he is having with his elder brother Glaucon, who addresses the narrator as Socrates.


Plato argues the merits of learning to calculate

'And to which class do unity and number belong?'

'I do not know,' he replied.

'Think a little,' I told him, 'and you will see that what has preceded will supply the answer; for if simple unity could be adequately perceived by the sight or by any other sense, then, there would be nothing to attract the mind towards reality any more than in the case of the finger we discussed. But when it is combined with the perception of its opposite, and seems to involve the conception of plurality as much as unity, then thought begins to be aroused within us, and the soul perplexed and wanting to arrive at a decision asks "What is absolute unity?" This is the way in which the study of the one has a power of drawing and converting the mind to the contemplation of reality.'

'And surely,' he said, 'this characteristic occurs in the case of one; for we see the same thing to be both one and infinite in multitude?'

'Yes,' I said, 'and this being true of one, it must be equally true of all number?'

'Certainly.'

'And all arithmetic and calculation have to do with number?'

'Yes.'

'And they both appear to lead the mind towards truth?'

'Yes, in a very remarkable manner.'

'Then this is knowledge of the kind for which we are seeking, having a double use, military and philosophical; for the soldier must learn the art of number or he will not know how to organise his army, and the philosopher also, because he has to rise out of the transient world and grasp reality, and therefore he must be able to calculate.'

'That is true.'

'And our guardians are both soldiers and philosophers?'

'Certainly.'

'Then this is a kind of knowledge which legislation must make a subject of study; and we must endeavour to persuade those who are in positions of authority in our State to go and learn arithmetic, not as amateurs, but they must carry on the study until they properly understand the nature of numbers; 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 mind itself; and because this will be the easiest way for it to pass from the world of becoming to that of truth and reality.'

'That is excellent,' he said.

'Yes,' I said, 'and now having spoken of it, I must add how charming the science of arithmetic is! and in how many ways it is a subtle and useful tool to achieve our purposes, if pursued in the spirit of a philosopher, and not of a shopkeeper!'

'How do you mean?', he asked.

'I mean, as I was saying, that arithmetic has a very great and elevating effect, compelling the mind 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 argue against and ridicule any one who attempts to divide absolute unity when he is calculating, and if you divide it, they multiply it, taking care that one shall never be shown to contain a multiplicity of parts.'

'That is very true.'

'Now, suppose a person were to say to them, Glaucon, "O my friends, what are these wonderful numbers about which you are reasoning, in which, as you say, there are constituent units, such as you demand, and each unit is equal to every other, invariable, and not divisible into parts," - what would they answer?'

'They would answer, as I should think, that they were speaking of those numbers which can only be realised in thought, and there is no other way of handling them.'

'Then you see,' I pointed out to him, 'that this knowledge may be truly called necessary, requiring the mind, as it clearly does, to use pure intelligence in the attainment of pure truth?'

'Yes; that is the effect of it,' he agreed.

'And here is another point, that those who have a natural talent for calculation are generally quick at every other kind of knowledge; and even the slow-witted if they have had an arithmetical training, although they may derive no other advantage from it, always become much quicker than they would otherwise have been.'

'Very true,' he said.

'And indeed, you will not easily find a more difficult study, which come harder to those who learn and practice it.'

'You will not.'

'And, for all these reasons, arithmetic is a kind of knowledge in which the brightest citizens should be trained, and which must not be given up.'

'I agree.'

Plato argues the merits of learning plane geometry

Plato's arguments here are similar in nature to those used to advocate the study of arithmetic. It must not be studied for its practical uses but for training the mind.

'Let arithmetic then be the first of our subjects of education. And next, shall we enquire whether the kindred science also concerns us?'

'Do you mean geometry?,' he asked.

'Exactly so.'

'Clearly,' he said, 'we are concerned with that part of geometry which relates to war; for in pitching a camp, or taking up a position, or closing or extending the lines of an army, or any other military manoeuvre, whether in actual battle or on a march, it will make all the difference whether a general does or does not know geometry.'

'Yes,' I said, 'but for that purpose a very little of either geometry or arithmetic will be sufficient; the question relates rather to the greater and more advanced part of geometry - whether that tends in any degree to make more easy the vision of the idea of good; and that, as I was saying, all things tend which compel the mind to turn its attention towards that place, where is the full perfection of being, which it ought, by all means, to see.'

'True,' he said.

'Then if geometry compels us to view reality, it concerns us; if the realm of change only, it does not concern us?'

'Yes, that is what we claim.'

'Yet anybody who has the least acquaintance with geometry will not deny that such a conception of the science is quite the opposite to the ordinary terms of those use it.'

'How so?'

'They have in view practice only, and are always talking in a narrow and ridiculous manner, of "squaring" and "extending" and "applying" and the like - they confuse the ways of geometry with those of daily life; whereas knowledge is the real object of the whole science.'

'Certainly that is very true,' he said.

'Then must we not make a further admission?'

'What admission?'

That the knowledge at which geometry aims is knowledge of the eternal, and not of anything transient which will decay.'

'That,' he replied, 'may be readily allowed, and it is certainly true that geometrical knowledge is eternal.'

'Then, my noble friend, geometry will draw the mind towards truth, and create the spirit of philosophy, and raise up that which is now sadly allowed to fall down.'

'Nothing will be more likely to have such an effect.'

'Then nothing should be more strongly required than that the inhabitants of your State should by all means learn geometry. Moreover the science has indirect advantages too, which are not small.'

'Of what kind?' he said.

'There are the military advantages of which you spoke,' I said, 'and in all departments of knowledge, as experience proves, any one who has studied geometry is infinitely quicker at learning other subjects than one who has not.'

'Yes indeed,' he said, 'there is an infinite difference between them.'

'Then shall we propose this as a second branch of knowledge which our youth must study?'

'Let us do so,' he replied.

Plato argues the merits of supporting solid geometry

It is clear that solid geometry was relatively undeveloped when Plato wrote The Republic so he argues strongly that its study should be encouraged.

'And suppose we make astronomy the third subject - what do you say?'

'I am strongly inclined to it,' he said, 'the observation of the seasons and of months and years is as essential to the general as it is to the farmer or sailor.'

'I am amused,' I said, 'at your fear of the disapproval of the public, which makes you guard against the appearance of insisting upon studies which appear useless; and I quite admit the difficulty of believing that in every man there is a faculty of the mind which, when it has been blinded and ruined by other pursuits, is by these purified and re-illumined; and is worth far more than ten thousand eyes, for by it alone the truth is seen. Now there are two types of persons: one type who will agree with you and will take your proposals with unqualified approval; another type to whom they will be complete nonsense, and who will naturally deem them to be idle tales, for they see no sort of profit to be obtained from them. And therefore you had better decide at once with which of the two types you are proposing to argue. You will very likely say with neither, and that your chief aim in carrying on the argument is for your own improvement; at the same time you do not grudge others any benefit which they may receive.

'I think,' he replied, 'that I should prefer to carry on the argument mainly for my own satisfaction.'

'Then take a step backward, for we have gone wrong in the order of the sciences coming after plane geometry.'

'What was the mistake?' he said.

'After plane geometry,' I said, 'we proceeded at once to solids in revolution, instead of taking solids in themselves; whereas after the second dimension the third, which is concerned with cubes and dimensions of depth, ought to have followed.'

'That is true, Socrates, but so little seems to be known as yet about this subject.'

'Why, yes,' I said, 'and for two reasons: - in the first place, no government places value on it; this leads to a lack of energy in the pursuit of it, and it is difficult. In the second place, students cannot learn it unless they have a teacher. But then a teacher can hardly be found, and even if he could, as matters now stand, the students, who are very conceited, would not listen to him. That, however, would be otherwise if the whole State became the director of these studies and gave value to them; then disciples would want to come forward, and there would be continuous and earnest investigations, and discoveries would be made; since even now, disregarded as they are by the world, and given inadequate treatment, and although neither the public nor students understand their real uses, still these studies make progress by their natural charm, and very likely, if they had assistance from the State, they would some day emerge into light.'

'Yes,' he said, 'there is a great attraction in them. But I do not clearly understand the change in the order. First you began with geometry of plane surfaces?'

'Yes,' I said.

'You first placed astronomy next, and then you went back on what you had said?'

Yes, the more hussy the less speed,' I said. 'In my hurry, the ludicrous state of solid geometry, which, in natural order, should have followed, made me pass over this branch and go on to astronomy, or the motion of solids.'

'True,' he said.

'Then assuming that the science now omitted would come into existence if encouraged by the State, let us go on to astronomy, which will be fourth subject.'

Plato argues the merits of astronomy

We note here that Plato has the aim of training the mind. He is creating a syllabus which would enable a student to thing for themselves. He is not advocating the study of the physical sciences in their own right in this context, so is against astronomical observations as such. He is, however, strongly arguing that the astronomer needs mathematical tools.

'The right order,' he replied. 'And now, Socrates, as you rebuked the vulgar manner in which I praised astronomy before, my praise shall now be given in your own spirit. For everyone, I think, must see that astronomy compels the mind to look upwards and leads us from the earth to the heavens.'

'Every one but myself,' I said, 'to every one else this may be clear, but it is not to me.'

'And what then would you say?'

'I should rather say that those who elevate astronomy into philosophy appear to me to make us look downwards and not upwards.'

'What do you mean?' he asked.

'You,' I replied, 'have in your mind a truly sublime conception of our knowledge of the things above. And I dare say that if a person were to throw his head back and study the painted ceiling, you would still think that his mind was being used, and not his eyes. And you are very likely right, and I may be a simpleton: but, in my opinion, that knowledge only which is real and unseen can make the mind look upwards. And whether a man gapes at the heavens or blinks at the ground, seeking to learn some particular sense, I would deny that he can learn, for nothing of that sort is knowledge of science; his mind is looking downwards, not upwards, whether his way to knowledge is by water or by land, whether he floats, or only lies on his back.'

'I accept,' he said, that I should be scolded. Still, I should like to ascertain how astronomy ought to be learned in any manner more pertaining to that knowledge of which we are speaking?'

'I will tell you,' I said. 'The starry heaven which we behold is the finest and most perfect of visible things, but it must necessarily be deemed to be greatly inferior, just because they are visible, to the true motions of absolute swiftness and absolute slowness, which are relative to each other. The true realities of velocities are found in pure number and in every perfect figure. Now, these are to be apprehended by reason and intelligence, but not by sight. Do you agree?'

'Yes,' he replied.

'Well, then,' I continued, 'the visible splendours of the sky should be used as an illustration with a view to that higher knowledge; their beauty is like the beauty of figures or pictures excellently drawn by the hand of Daedalus, or some other great artist or draughtsman, which we may chance to behold. Anybody understanding geometry who saw them would appreciate the exquisiteness of their workmanship, but he would never dream of thinking that in them he could learn the truth about equality or about doubling, or the truth about any other proportion.'

'No,' he replied, 'such an idea would be absurd.'

'And will not a true astronomer have the same feeling when he looks at the movements of the stars?' I asked. 'Will he not think that the heavens and the heavenly bodies are put together by their Creator in the most perfect manner? But he will never imagine that the proportions of night and day, or of both to the month, or of the month to the year, or the periods of the stars to these and to one another, and any other things that are material and visible can also be eternal and subject to no deviation - that would be absurd; and it is equally absurd to take so much pains in investigating their exact truth.'

'I quite agree,' he replied, 'though I never thought of this before.'

'Then,' I said, 'in astronomy, as in geometry, we should set problems to be solved, and leave the visible heavens alone if we want to approach the subject in the right way and so to put the natural gift of reason to a real purpose.'

'That,' he said, 'is a work infinitely beyond our present astronomers.'

'Yes,' I said, 'and there are many other things which must also have a similar extension given to them, if our legislation is to be of any value. But can you tell me of any other suitable study?'

'No,' he said, 'not without giving it some thought.'

Plato argues the merits of harmonics

Here Plato gives arguments similar to those employed when he discussed astronomy.

'Motion,' I said, 'has many forms, and not just one. Two of them are obvious enough even to brains no better than ours; but there are others, as I imagine, which may be left to experts.'

'But what are the two?'

'There is a second,' I said, 'which is the counterpart of the one already named.'

'And what is that?'

'The second,' I said, 'would seem to relate to the ears in the same way that the first relates to the eyes. For I believe that as the eyes are designed to look up at the stars, so are the ears are designed to hear movements of harmony, and these are sister sciences - as the Pythagoreans say, and we, Glaucon, agree with them?'

'Yes,' he replied.

'But this,' I said, 'is a long and difficult study, and therefore we had better go and consult them on the subject and they will tell us whether there are any other applications of these sciences. At the same time, we must not lose sight of our own higher principles.'

'What is that?'

'There is a level which all knowledge ought to reach, and which our pupils ought also to attain, and not to fall short of, as I was saying that they did in astronomy. For in the science of harmony, as you probably know, the same thing happens. The teachers of harmony compare the sounds and consonances which are audible, and their labour, like that of the astronomers, is in vain.'

'Yes, by heaven!' he said, 'and its as good as a play to hear them talking about their condensed notes, as they call them. They put their ears close alongside of the strings like people trying to hear a sound through their neighbour's wall - some of them declaring that they can distinguish an intermediate note and have found the least interval which should be the unit of measurement, while the others insist that there is no difference between the two notes - both lots are putting their ears before their understanding.'

'You mean,' I said, 'those gentlemen who tease and torment the strings and twist them on the pegs of the instrument. I might continue the metaphor and speak after their manner of the blows which the plectrum gives, and make accusations against the strings, both of backwardness and forwardness to sound - but this would be tedious, and therefore I will only say that these are not the men, and that I am referring to the Pythagoreans, of whom I was just now proposing to enquire about harmony. For they too are in error in the same way as the astronomers. They investigate the numerical relationships between the harmonies which are heard, but they never get as far as formulating problems - that is to say, they never reach the natural harmonies of number, or reflect why some numbers are harmonious and others not.'

'That,' he said,' would be a fearsome job.'

'Nevertheless, a thing,' I replied, 'which I would rather call useful, that is, if investigated with a view to the beautiful and good. But if pursued in any other spirit, it is useless.'

'Very true,' he said.


JOC/EFR April 2007

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