from

Dialogues on Mathematics,

Alfred Renyi, Holden Day Publishers, San Francisco, 1967

A SOCRATIC DIALOGUE ON MATHEMATICS

SOCRATES Are you looking for somebody, my dear Hippocrates?

HIPPOCRATES No, Socrates, because I have already found him, namely you. I have been looking for you everywhere. Somebody told me at the agora that he saw you walking here along the River Ilissos; so I came after you.

SOCRATES Well then, tell me why you came, and then I want to ask you something about our discussion with Protagoras. Do you still remember it?

HIPPOCRATES How can you ask? Since that time not a single day has passed without my thinking about it. I came today to ask your advice because that discussion was on mind.

SOCRATES It seems, my dear Hippocrates, that you want to talk to me about the very question I wish to discuss with you; thus the two subjects are one and the same. It seems that the mathematicians are mistaken in saying that two is never equal to one.

HIPPOCRATES As a matter of fact, Socrates, mathematics is just the topic I want to talk to you about.

SOCRATES Hippocrates, you certainly know that I am not a mathematician. Why did you not take your questions to the celebrated Theodoros?

HIPPOCRATES You are amazing, Socrates, you answer my questions even before I tell you what they are. I came to ask your opinion about my becoming a pupil of Theodoros. When I came to you the last time, with the intention of becoming a pupil of Protagoras, we went to him together and you directed the discussion so that it became quite clear that he did not know the subject he taught. Thus I changed my mind and did not follow him. This discussion helped me to see what I should not do, but did not show me what I should do. I am still wondering about this. I visit banquets and the palaestra with young men of my age, I dare say I have a pleasant time, but this does not satisfy me. It disturbs me to feel myself ignorant. More precisely, I feel that the knowledge I have is rather uncertain. During the discussion with Protagoras, I realized that my knowledge about familiar notions like virtue, justice and courage was far from satisfactory. Nevertheless, I think it is great progress that I now see clearly my own ignorance.

SOCRATES I am glad, my dear Hippocrates, that you understand me so well. I always tell myself quite frankly that I know nothing. The difference between me and most other people is that I do not imagine I know what in reality I do not know.

HIPPOCRATES This clearly shows your wisdom, Socrates. But such knowledge is not enough for me. I have a strong desire to obtain some certain and solid knowledge, and I shall not be happy until I do. I am constantly pondering what kind of knowledge I should try to acquire. Recently, Theaitetos told me that certainty exists only in mathematics and suggested that I learn mathematics from his master, Theodoros, who is the leading expert on numbers and geometry in Athens. Now, I should not want to make the same mistake I made when I wanted to be a pupil of Protagoras. Therefore tell me, Socrates, shall I find the kind of sound knowledge I seek if I learn mathematics from Theodoros?

SOCRATES If you want to study mathematics, O son of Apollodoros, then you certainly cannot do better than go to my highly esteemed friend Theodoros. But you must decide for yourself whether or not you really do want to study mathematics. Nobody can know your needs better than you yourself.

HIPPOCRATES Why do you refuse to help me, Socrates? Perhaps I offended you without knowing it?

SOCRATES You misunderstand me, my young friend. I am not angry; but you ask the impossible of me. Everybody must decide for himself what he wants to do. I can do no more than assist as a midwife at the birth of your decision.

HIPPOCRATES Please, my dear Socrates, do not refuse to help me, and if you are free now, let us start immediately.

SOCRATES Well, if you want to. Let us lie down in the shadow of that plane-tree and begin. But first tell me, are you ready to conduct the discussion in the manner I prefer? I shall ask the questions and you shall answer them. By this method you will come to see more clearly what you already know, for it brings into blossom the seeds of knowledge already in your soul. I hope you will not behave like King Darius who killed the master of his mines because he brought only copper out of amine the king thought 'contained gold. I hope you do not forget that a miner can find in a mine only what it contains.

HIPPOCRATES I swear that I shall make no reproaches, but, by Zeus, let us begin mining at once.

SOCRATES All right. Then tell me, do you know what mathematics is? I suppose you can define it since you want to study it.

HIPPOCRATES I think every child could do so. Mathematics is one of the sciences, and one of the finest.

SOCRATES I did not ask you to praise mathematics, but to describe its nature. For instance, if I asked you about the art of physicians, you would answer that this art deals with health and illness, and has the aim of healing the sick and preserving health. Am I right?

HIPPOCRATES Certainly.

SOCRATES Then answer me this: does the art of the physicians deal with something that exists or with something that does not exist? If there were no physicians, would illness still exist?

HIPPOCRATES Certainly, and even more than now.

SOCRATES Let us have a look at another art, say that of astronomy. Do you agree with me that astronomers study the motion of the stars?

HIPPOCRATES To be sure.

SOCRATES And if I ask you whether astronomy deals with something that exists, what is your answer?

HIPPOCRATES My answer is yes.

SOCRATES Would stars exist if there were no astronomers in the world?

HIPPOCRATES Of course. And if Zeus in his anger extinguished all mankind, the stars would still shine in the sky at night. But why do we discuss astronomy instead of mathematics?

SOCRATES Do not be impatient, my good friend. Let us consider a few other arts in order to compare them with mathematics. How would you describe the man who knows about all the creatures living in the woods or in the depths of the sea?

HIPPOCRATES He is a scientist studying living nature.

SOCRATES And do you agree that such a man studies things which exist?

HIPPOCRATES I agree.

SOCRATES And if I say that every art deals with something that exists, would you agree?

HIPPOCRATES Completely.

SOCRATES Now tell me, my young friend, what is the object of mathematics? What things does a mathematician study ?

HIPPOCRATES I have asked Theaitetos the same question. He answered that a mathematician studies numbers and geometrical forms.

SOCRATES Well, the answer is right, but would you say that these things exist?

HIPPOCRATES Of course. How can we speak of them if they do not exist?

SOCRATES Then tell me, if there were no mathematicians, would there be prime numbers, and if so, where would they be?

HIPPOCRATES I really do not know what to answer. Clearly, if mathematicians think about prime numbers, then they exist in their consciousness; but if there were no mathematicians, the prime numbers would not be anywhere.

SOCRATES Do you mean that we have to say mathematicians study non-existing things?

HIPPOCRATES Yes, I think we have to admit that.

SOCRATES Let us look at the question from another point of view. Here, I wrote on this wax tablet the number 37. Do you see it?

HIPPOCRATES Yes, I do.

SOCRATES And can you touch it with your hand?

HIPPOCRATES Certainly.

SOCRATES Then perhaps numbers do exist?

HIPPOCRATES O Socrates, you are mocking me. Look here, I have drawn on the same tablet a dragon with seven heads. Does it follow that such a dragon exists? I have never met anybody who has seen a dragon, and I am convinced that dragons do not exist at all except in fairy tales. But suppose I am mistaken, suppose somewhere beyond the pillars of Heracles dragons really do exist, that still has nothing to do with my drawing.

SOCRATES You speak the truth, Hippocrates, and I agree with you completely. But does this mean that even though we can speak about them, and write them down, numbers nevertheless do not exist in reality?

HIPPOCRATES Certainly.

SOCRATES Do not draw hasty conclusions. Let us make another trial. Am I right in saying that we can count the sheep here in the meadow or the ships in the harbor of Pireus?

HIPPOCRATES Yes, we can.

SOCRATES And the sheep and the ships exist?

HIPPOCRATES Clearly.

SOCRATES But if the sheep exist, their number must be something that exists, too?

HIPPOCRATES You are making fun of me, Socrates. Mathematicians do not count sheep; that is the business of shepherds.

SOCRATES Do you mean, what mathematicians study is not the number of sheep or ships, or of other existing things, but the number itself? And thus they are concerned with something that exists only in their minds?

HIPPOCRATES Yes, this is what I mean.

SOCRATES You told me that according to Theaitetos, mathematicians study numbers and geometrical forms. How about forms? If I ask you whether they exist, what is your answer?

HIPPOCRATES Certainly they exist. We can see the form of a beautiful vessel, for example, and feel it with our hands, too.

SOCRATES Yet I still have one difficulty. If you look at a vessel what do you see, the vessel or its form?

HIPPOCRATES I see both.

SOCRATES Is that the same thing as looking at a lamb? Do you see the lamb and also its hair?

HIPPOCRATES I find the simile very well chosen.

SOCRATES Well, I think it limps like Hephaestus. You can cut the hair off the Iamb and then you see the Iamb without its hair, and the hair without the lamb. Can you separate in a similar way the form of a vessel from the vessel itself?

HIPPOCRATES Certainly not, and I dare say nobody can.

SOCRATES And nevertheless you still believe that you can see a geometric form?

HIPPOCRATES I am beginning to doubt it.

SOCRATES Besides this, if mathematicians study the forms of vessels, shouldn't we call them potters?

HIPPOCRATES Certainly.

SOCRATES Then if Theodoros is the best mathematician would he not be the best potter, too? I have heard many people praising him, but nobody has told me that he understands anything about pottery. I doubt whether he could make even the simplest pot. Or perhaps mathematicians deal with the form of statues or buildings?

HIPPOCRATES If they did, they would be sculptors and architects.

SOCRATES Well, my friend, we have come to the conclusion that mathematicians when studying geometry are not concerned with the forms of existing objects such as vessels, but with forms which exist only in their thoughts. Do you agree?

HIPPOCRATES I have to agree.

SOCRATES Having established that mathematicians are concerned with things that do not exist in reality, but only in their thoughts, let us examine the statement of Theaitetos, which you mentioned, that mathematics gives us more reliable and more trustworthy knowledge than does any other branch of science. Tell me, did Theaitetos give you some examples?

HIPPOCRATES Yes, he said for instance that one cannot know exactly how far Athens is from Sparta. Of course, the people who travel that way agree on the number of days one has to walk, but it is impossible to know exactly how many feet the distance is. On the other hand, one can tell, by means of the theorem of Pythagoras, what the length of the diagonal of a square is Theaitetos also -said that it is impossible to give the exact number of people living in Hellas. If somebody tried to count all of them, he would never get the exact figure, because during the counting some old people would die and children would be born; thus the total number could be only approximately correct. But if you ask a mathematician how many edges a regular dodecahedron has, he will tell you that the dodecahedron is bounded by 12 faces, each having 5 edges. This makes 60, but as each edge belongs to two faces and thus has been counted twice, the number of edges of the dodecahedron is equal to 30, and this figure is beyond every doubt.

SOCRATES Did he mention any other examples?

HIPPOCRATES Quite a few, but I do not remember all of them. He said that in reality you never find two things which are exactly the same. No two eggs are exactly the same, even the pillars of Poseidon's temple are slightly different from each other; but one may be sure that the two diagonals of a rectangle are exactly equal. He quoted Heraclitus who said that everything which exists. is constantly changing, and that sure knowledge is only possible about things which never change, for instance, the odd and the even, the straight line and the circle.

SOCRATES That will do. These examples convince me that in .mathematics we can get knowledge which is beyond doubt, while in other sciences or in everyday life it is impossible. Let us try to summarize the results of our inquiry into the nature of mathematics. Am I right in saying we came to the conclusion that mathematics studies non-existing things and is able to find out the full truth about them?

HIPPOCRATES Yes, that is what we established.

SOCRATES But tell me, for Zeus's sake, my dear Hippocrates, is it not mysterious that one can know more about things which do not exist than about things which do exist?

HIPPOCRATES If you put it like that, it certainly is a mystery. I am sure there is some mistake in our arguments.

SOCRATES No, we proceeded with the utmost care and we controlled every step of the argument. There cannot be any mistake in our reasoning. But listen, I remember something which may help us to solve the riddle.

HIPPOCRATES Tell me quickly, because I am quite bewildered.

SOCRATES This morning I was in the hall of the second archon, where the wife of a carpenter from the village Pitthos was accused of betraying and, with the aid of her lover, murdering her husband. The woman protested and swore to Artemis and Aphrodite that she was innocent, that she never loved anyone but her husband, and that her husband was killed by pirates. Many people were called as witnesses. Some said that the woman was guilty, others said that she was innocent. It was impossible to find out what really happened.

HIPPOCRATES Are you mocking me again? First you confused me completely, and now instead of helping me to find the truth you tell me such stories.

SOCRATES Do not be angry, my friend, I have serious reasons for speaking about this woman whose guilt it was impossible to ascertain. But one thing is sure. The woman exists. I saw her with my own eyes, and of anyone who' was there, many of whom have never lied in their lives, you can ask the same question and you will receive the same answer.

HIPPOCRATES Your testimony is sufficient for me, my dear Socrates. Let it be granted that the woman exists. But what has this fact to do with mathematics?

SOCRATES More than you imagine. But tell me first, do you know the story about Agamemnon and Clytemnestra?

HIPPOCRATES Everybody knows the story. I saw the trilogy of Aeschylus at the theatre last year.

SOCRATES Then tell me the story in a few words.

HIPPOCRATES While Agamemnon, the king of Mycenae, fought under the walls of Troy, his wife, Clytemnestra, committed adultery with Aegisthus, the cousin of her husband. After the fall of Troy, when Agamemnon returned home, his wife and her lover murdered him.

SOCRATES Tell, me Hippocrates, is it quite sure that Clytemnestra was guilty?

HIPPOCRATES I not understand why you ask me such questions. There can be no doubt about the story. According to Homer, when Odysseus visited the underworld he met Agamemnon, who told Odysseus his sad fate.

SOCRATES But are you sure that Clytemnestra and Agamemnon and all the other characters of the story really existed?

HIPPOCRATES Perhaps I would be ostracized if I said this in public, but my opinion is that it is impossible either to prove or disprove today, after so many centuries; whether the stories of Homer are true or not. But this is quite irrelevant. When I told you that Clytemnestra was guilty, I did not speak about the real Clytemnestra - if such a person ever lived - but about the Clytemnestra of our Homeric tradition, about the Clytemnestra in the trilogy of Aeschylus.

SOCRATES May I say that we know nothing about the real Clytemnestra? Even her existence is uncertain, but as regards the Clytemnestra who is a character in the trilogy of Aeschylus, we are sure that she was guilty and murdered Agamemnon because that is what Aeschylus tells us.

HIPPOCRATES Yes, of course. But why do you insist on all this?

SOCRATES You will see in a moment. Let me summarize what we found out It is impossible in the case of the flesh and blood woman was tried today in Athens to establish whether she is guilty, ~while there can be no doubt about the guilt of Clytemnestra who is a character in a play and who probably never existed. Do you agree?

HIPPOCRATES Now I am beginning to understand what you want to say. But it would be better if you draw the conclusions yourself.

SOCRATES The conclusion is this: we have much more certain knowledge about persons who exist only in our imagination, for example: about characters in a play, than about living persons. If we say that Clytemnestra was guilty, it means only that this is how Aeschylus imagined her and presented her in his play. The situation is exactly the same in mathematics. We may be sure that the diagonals of a rectangle are equal because this follows from the definition of a rectangle given by mathematicians.

HIPPOCRATES Do you mean, Socrates, that our paradoxical result is really true and one can have a much more certain knowledge about non-existent things - for instance about the objects of mathematics - than about the real objects of nature? I think that now I also see the reason for this. The notions which we ourselves have created are by their very nature completely known to us, and we can find out the full truth about them because they have no other reality outside our imagination. However, the objects which exist in the real world are not identical with our picture of them, which is always incomplete and approximate; therefore our knowledge about these real things can never be complete or quite certain.

SOCRATES That is the truth, my young friend, and you stated it better than I could have.

HIPPOCRATES This is to your credit, Socrates, because you led me to understand these things. I see now not only that Theaitetos was quite right in telling me I must study mathematics if I want to obtain unfailing knowledge, but also why he was right. However, if you have guided with patience up to now, please do not abandon me yet because only of my questions, in fact the most important one, is still unanswered.

SOCRATES What is this question?

HIPPOCRATES Please remember Socrates that I came to ask your advice as to whether I should study mathematics. You helped me to realize that mathematics and only mathematics can give me the sort of sound knowledge I want. But what is the use of this knowledge? It is clear that if one obtains some knowledge about the existing world, even if this knowledge is incomplete and is not quite certain, it is nevertheless of value to the individual as well as to the state. Even if one gets some knowledge about things such as the stars, it may be useful, for instance in navigation at night. But what is the use of knowledge of non-existing things such as that which mathematics offers? Even if it is complete and beyond any doubt, what is the use of knowledge concerning things which do not exist in reality?

SOCRATES My dear friend, I am quite sure you know the answer, only you want to examine me.

HIPPOCRATES By Heracles, I do not know the answer. Please help me.

SOCRATES Well, let us try to find it. We have established that the notions of mathematics are created by the mathematician himself. Tell me, does this mean that the mathematician chooses his notions quite arbitrarily as it pleases him?

HIPPOCRATES As I told you, I do not yet know much about mathematics. But it seems to me that the mathematician is as free to choose the objects of his study as the poet is free to choose the characters of his play, and as the poet invests his characters with whatever traits please him, so can the mathematician endow his notions with such properties as he likes.

SOCRATES If this were so, there would be as many mathematical truths as there are mathematicians. How do you explain, then, that mathematicians study the same notions and problems? How do you explain that, as often happens, mathematicians living far from each other and having no contact independently discover the same truths? I never heard of two poets writing the same poem.

HIPPOCRATES Nor have I heard of such a thing. But I remember Theaitetos telling me about a very interesting theorem he discovered on incommensurable distances. He showed his results to his master, Theodoros, who produced a letter by Archytas in which the same theorem was contained almost word for word.

SOCRATES In poetry that would be impossible. Now you see that there is a problem. But let us continue. How do you explain that the mathematicians of different countries can usually agree about the truth, while about questions concerning the state, for example, the Persians and the Spartans have quite opposite views from ours in Athens, and, moreover, we here do not often agree with each other?

HIPPOCRATES I can answer that last question. In matters concerning the state everybody is personally interested, and these personal interests are often in contradiction. This is why it is difficult to come to an agreement. However, the mathematician is led purely by his desire to find the truth.

SOCRATES Do you mean to say that the mathematicians are trying to find a truth which is completely independent of their own person?

HIPPOCRATES Yes, I do.

SOCRATES But then we were mistaken in thinking that mathematicians choose the objects of their study at their own will. It seems that the object of their study has some sort of existence which is independent of their person. We have to solve this new riddle.

HIPPOCRATES I do not see how to start.

SOCRATES If you still have patience, let us try it together. Tell me, what is the difference between the sailor who finds an uninhabited island and the painter who finds a new color, one which no other painter has used before him?

HIPPOCRATES I think that the sailor may be called a discoverer, and the painter an inventor. The sailor discovers an island which existed before him, only it was unknown, while the painter invents a new color which before that did not exist at all.

SOCRATES Nobody could answer the question better. But tell me, the mathematician who finds a new truth, does he discover it or invent it? Is he a discoverer as the sailor or an inventor as the painter?

HIPPOCRATES It seems to me that the mathematician is more like a discoverer. He is a bold sailor who sails on the unknown sea of thought and explores its coasts, islands and whirlpools.

SOCRATES Well said, and I agree with you completely. I would add only that to a lesser extent the mathematician is an inventor too, especially when he invents new concepts. But every discoverer has to be, to a certain extent, an inventor too. For instance, if a sailor wants to get to places which other sailors before him were unable to reach, he has to build a ship that is better than the ships other sailors used. The new concepts invented by the mathematicians are like new ships which carry the discoverer farther on the great sea of thought.

HIPPOCRATES My dear Socrates, you helped me to find the answer to the question which seemed so difficult to me. The main aim of the mathematician is to explore the secrets and riddles of the sea of human thought. These exist independently of the person of the mathematician, though not from humanity as a whole. The mathematician has a certain freedom to invent new concepts as tools, and it seems that he could do this at his discretion. However, he is not quite free in doing this because the new concepts have to be useful for his work. The sailor also can build any sort of ship at his discretion, but, of course, he would be mad to build a ship which would be crushed to pieces by the first storm. Now I think that everything is clear.

SOCRATES If you see everything clearly, try again to answer the question: what is the object of mathematics?

HIPPOCRATES We came to the conclusion that besides the world in which we live, there exists another world, the world of human thought, and the mathematician is the fearless sailor who explores this world, not shrinking back from the troubles, dangers and adventures which await him.

SOCRATES My friend, your youthful vigor almost sweeps me off my feet, but I am afraid that in the ardor of your enthusiasm you overlook certain questions.

HIPPOCRATES What are these questions?

SOCRATES I do not want to disappoint you, but I feel that your main question has not yet been answered. We have not yet answered the question: what is the use of exploring the wonderful sea of human thought?

HIPPOCRATES You are right, my dear Socrates, as always. But won't you put aside your method this time and tell me the answer immediately?

SOCRATES No, my friend, even if I could, I would not do this, and it is for your sake. The knowledge somebody gets without work is almost worthless to him. We understand thoroughly only that which - perhaps with some outside help - we find out ourselves, just as a plant can use only the water which it sucks up from the soil through its own roots.

HIPPOCRATES All right, let us continue our search by the same method, but at least help me by a question.

SOCRATES Let us go back to the point where we established that the mathematician is not dealing with the number of sheep, ships or other existing things, but with the numbers themselves. Don't you think, however, that what the mathematicians discover to be true for pure numbers is true for the number of existing things too? For instance, the mathematician finds that 17 is a prime number. Therefore, is it not true that you cannot distribute 17 living sheep to a group of people, giving each the same number, unless there are 17 people?

HIPPOCRATES Of course, it is true.

SOCRATES Well, how about geometry? Can it not be applied in building houses, in making pots or in computing the amount of grain a ship can hold?

HIPPOCRATES Of course, it can be applied, though it seems to me that for these practical purposes of the craftsman not too much mathematics is needed. The simple rules known already by the clerks of the pharaohs in Egypt are sufficient for most such purposes, and the new discoveries about which Theaitetos spoke to me with overflowing fervor are neither used nor needed in practice.

SOCRATES Perhaps not at the moment, but they may be used in the future.

HIPPOCRATES I am interested in the present.

SOCRATES If you want to be a mathematician, you must realize you will be working mostly for the future. Now, let us return to the main question. We saw that knowledge about another world of thought, about things which do not exist in the usual sense of the word, can be used in everyday life to answer questions about the real world. Is this not surprising?

More than that, it is incomprehensible. It is really a miracle.

SOCRATES Perhaps it is not so mysterious at all, and if we open the shell of this question, we may find a real pearl.

HIPPOCRATES Please, my dear Socrates, do not speak in puzzles.

SOCRATES Tell me then, are you surprised when somebody who has traveled in distant countries, who has seen and experienced many things, returns to his city and uses his experience to give good advice to his fellow citizens?

HIPPOCRATES Not at all.

SOCRATES Even if the countries which the traveler has visited are very far away and are inhabited by quite a different sort of people, speaking another language, worshipping other gods?

HIPPOCRATES Not even in that case, because there is much that is common between different people.

SOCRATES Now tell me, if it turned out that the world of mathematics is, in spite of its peculiarities, in some sense similar to our the world in which we live, about the world of human thought, the question remained as to the use of this knowledge. Now we have found that the world of mathematics is nothing else but a reflection in our mind of the real world. This makes it clear that every discovery about the world of mathematics gives us some information about the real world. I am completely satisfied with this answer.

SOCRATES If I tell you the answer is not yet complete, I do so not because I want to confuse you, but because I am sure that sooner or later you will raise the question yourself and will reproach me for not having called your attention to it. You would say: "Tell me, Socrates, what is the sense of studying the reflected image if we can study the object itself?"

HIPPOCRATES You are perfectly right; it is an obvious question. You are a wizard, Socrates. You can totally confuse me by a few words, and you can knock down by an innocent looking question the whole edifice which we have built with so much trouble. I should, of course, answer that if we are able to have a look at the original thing, it makes no sense to look at the reflected image. But I am sure this shows only that our simile fails at this point. Certainly there is an answer, only I do not know how to find it.

SOCRATES Your guess is correct that the paradox arose because we kept too close to the simile of the reflected image. A simile is like a bow - if you stretch it too far, it snaps. Let us drop it and choose another one. You certainly know that travelers and sailors make good use of maps.

HIPPOCRATES I have experienced that myself. Do you mean that mathematics furnishes a map of the real world?

SOCRATES Yes. Can you now answer the question: what advantage would it be to look at the map instead of looking at the landscape?

HIPPOCRATES This is clear: using the map we can scan vast distances which could be covered only by traveling many weeks or months. The map shows us not every detail, but only the most important things. Therefore it is useful if we want to plan a long voyage.

SOCRATES Very well. But there is something else which occurred to me.

HIPPOCRATES What is it?

SOCRATES There is another reason why the study of the mathematical image of the world may be of use. If mathematicians discover some property of the circle, this at once gives us some information about any object of circular shape. Thus, the method of mathematics enables us to deal with different things at the same time.

HIPPOCRATES What about the following similes: If somebody looks at a city from the top of a nearby mountain, he gets a more comprehensive view than if he walks through its crooked streets; or if a general watches the movements of an enemy army from a hill, he gets a clearer picture of the situation than does the soldier in the front line who sees only those directly opposite him.

SOCRATES Well, you surpass me in inventing new similes, but as I do not want to fall behind, let me also add one parable. Recently I looked at a painting by Aristophon, the son of Aglaophon, and the painter warned me, "If you go too near the picture, Socrates, you will see only colored spots, but you will not see the whole picture."

HIPPOCRATES Of course, he was right, and so were you, when you did not let us finish our discussion before we got to the heart of the question. But I think it is time for us to return to the city because the shadows of night are falling and I am hungry and thirsty. If you still have some patience, I would like to ask you something while we walk.

SOCRATES All right, let us start and you may ask your question.

HIPPOCRATES Our discourse convinced me fully that I should start studying mathematics and I am very grateful to you for this. But tell me, why are you yourself not doing mathematics? Judging from your deep understanding of the real nature and importance of mathematics, it is my guess that you would surpass all other mathematicians of Hellas, were you to concentrate on it. I would be glad to follow you as your pupil in mathematics, if you accepted me.

SOCRATES No, my dear Hippocrates, this is not my business. Theodoros knows much more about mathematics than I do and you cannot find a better master than him. As to your question of why I myself am not a mathematician, I shall give you the reasons. I do not conceal my high opinion about mathematics. I think that we Hellenes have in no other art made such important progress as in mathematics, and this is only the beginning. If we do not extinguish each other in mad wars, we shall obtain wonderful results as discoverers as well as inventors. You asked me why I do not join the ranks of those who develop this great science. As a matter of fact, I am some sort of a mathematician, only of a different kind. An inner voice, you may call it an oracle, to which I always listen carefully, asked me many years ago, "What is the source of the great advances which the mathematicians have made in their noble science?" I answered, "I think the source of the success of mathematicians lies in their methods, the high standards Of their logic, their striving without the least compromise to the full truth, their habit of starting always from first principles, of defining every notion used exactly and of avoiding self-contradictions." My inner voice answered, "Very well, but why do you think, Socrates, that this method of thinking and arguing can be used only for the study of numbers and geometric forms? Why do you not try to convince your fellow citizens to apply the same high logical standards in every other field, for instance in philosophy and politics, in discussing the problems of everyday private and public life?" From that time on, this has been my goal. I have demonstrated (you remember, for instance, our discussion with Protagoras) that those who are thought to be wise men are mostly ignorant fools. All their arguing lacks solid foundation, since they use - contrary to mathematicians - undefined and only half-understood notions. By this activity I have succeeded in making almost everybody my enemy. This is not surprising because for all people who are sluggish in thinking and idly content to use obscure terms, I am a living reproach. People do not like those who constantly remind them of the faults which they are unable or unwilling to correct. The day will come when these people will fall upon me and exterminate me. But until that day comes, I shall continue to follow my calling you, however, go to Theodoros.

AUTHOR’S POSTSCRIPT

An optimistic author does not write a preface to his book, because he is confident that it will speak for itself and he is convinced that the readers will understand what he wants to say without any additional explanation. While I am an optimist, I felt that in the case of this book, if not a preface at least a postscript was needed on the aims of the author and on the considerations which led him to choose the literary form of the dialogue. I add these remarks in the form of a postscript because I really want them to be read after the dialogues.

The interest in mathematics and its applications is increasing year by year in every country among an increasing number of people. I have been asked several times to give popular talks on mathematics; on such occasions I noticed that many people were primarily interested in finding out what mathematics really was, what its specific method consisted of, what its relation to the sciences and humanities was and what it could offer to those working in different fields. I found also that those who attended such lectures on mathematics or who were ready to read books on mathematics written for non-specialists usually wanted simply to broaden their outlook rather than to acquire specific mathematical methods. Even those who actually needed a knowledge of mathematics for their work, before deciding to study seriously a particular part of mathematics, wanted to find out what they could expect from it, especially since the study of mathematics is not easy for those unused to it.

While talking about mathematics to non-mathematicians I encountered quite a number of prejudices, misunderstandings and misconceptions, not only among people whose main interests and activities are quite far from mathematics but also among those who through their profession have a certain knowledge in some part of the field. This is really not surprising as those people who have some knowledge but do not have sufficiently broad vision or sufficiently deep insight, are most inclined to make false generalizations. I found also that the principles of mathematics and of its applications are often disputed even among mathematicians and many questions in this field are subject to controversy.

These circumstances convinced me that there exists a real need for a discussion of the basic questions of mathematics and its applications in a manner which while comprehensible to non-specialists, , yet presents these problems in their full complexity .I realized that it would not be an easy task to make such questions understandable to the general public, therefore I searched for a special method to bring abstract problems nearer to the layman. This search led me to experiment with the Socratic form of a dialogue. The Socratic dialogue presents thoughts while they are being created and dramatizes ideas. By so doing it keeps the attention awake and facilitates understanding.

I chose as the central theme of the first dialogue the question "What in fact is mathematics?" I consider the discussion of this question especially important because the teaching of mathematics in elementary and high schools is still far from giving a clear-cut, correct and up-to-date answer.

In this dialogue I tried to follow as closely as possible the method and even the language of the original Socratic dialogues. Socrates himself is the main actor and the discussion takes place in the period when mathematics, in the sense that it has been understood ever since, was born; thus mathematics is presented to the reader "in statu nascendi." In the dialogue Socrates applies his peculiar method of discussion: by the phrasing of his questions he leads his partner to understand the issue. Thus a Socratic dialogue is not the clash of two points of view; rather the participants try to find out the truth together. By a logical analysis of the concepts involved they arrive at an answer to the questions step-by-step. During the discussion the participants often make statements - sometimes in a quite categorical form - which they later realize to be false. Thus a Socratic dialogue is an organic whole and its real meaning can be understood only if one reads it from beginning to end, if possible without interruption. All these features make a Socratic dialogue lively and vivid, and so I found this form particularly suitable to my aims.

I had still another reason for choosing this form: it is my firm belief that the Socratic method is basically cognate with the mathematical method. In this belief I was very much strengthened by the recent fundamental research work of Arpad Szab, which has thrown quite a new light on the origin of Ancient Greek mathematics.

The first dialogue was published in Hungarian1 in 1962. In 1963 a French translation appeared in Les Cahiers Rationalistes.2 In 1963 I presented this dialogue as an after-dinner talk to the meeting of American Physicists in Edmonton, and an English version was published both in the Canadian Mathematical Bulletin3 and in Physics Today4 and was reprinted by the journal Simon Stevin5 too. Since then it has also appeared both in German6 and Portuguese7 translations.

The favorable reception of the first dialogue both among mathematicians and among non-mathematicians encouraged me to continue experimenting with this genre. A second dialogue was first presented at the University of Toronto in 1964 and appeared in English in the Ontario Mathematics Gazette8 and later in Simon Stevin9.

Since in the first dialogue I had discussed the relation of mathematics to reality only in a general philosophical sense, in the second I wanted to make central a more detailed discussion of the applications of mathematics. It was logical to choose Archimedes as the chief character of such a dialogue as his name even in ancient times was inseparably connected with such applications. The historical frame of the second dialogue, however, did not allow me to say all that I wanted about this controversial topic.

Thus I felt I had to write a third dialogue, the chief character of which was Galileo, the first thinker in modem times who fully realized the central importance of the mathematical method in discovering the laws of nature, and who propagated his conviction with great force. The second and third dialogues thus complement each other, and also the first. They are, however, essentially different from the first in form and style. Archimedes and Galileo do not, of course, use the method of Socrates: instead of guiding their partner to guess their thoughts, they express them themselves. Thus I had to dispense with the main source of inner tension which the Socratic dialogue provides. I tried to compensate for this loss by putting these dialogues in extremely decisive historical situations, the dynamics of which were inseparably connected with the issues of the dialogues and would thus amplify their tension.

Featuring Archimedes and Galileo made it possible to touch in these dialogues on much more specialized mathematical topics than were discussed in the first one, especially on such ideas which originated with Archimedes and Galileo themselves; I tried to incorporate in some form or other most of their famous achievements.

In this connection I must say a few words about how I dealt with historical facts. In all three dialogues I tried hard to avoid every sort of anachronism. I was careful not to attribute to my characters any such knowledge of mathematics as well as of other things) which they could not possibly possess at that time. However as both Archimedes and Galileo were pioneers whose ideas and way of thinking were not only far ahead of their time but also are modern even when measured by present day standards, I was not prevented from including in these dialogues everything I deemed important to say. Of course, in order to avoid anachronism I had to restrict myself mainly to examples from elementary mathematics; I could thus go into infinitesimal mathematics but only as far as Archimedes and Galileo did themselves. This restriction, however, had certain advantages because it forced me to avoid examples which would have been too difficult for the non-mathematician.

I did not, however, interpret the requirement of historical faithfulness so rigidly as to attribute to my characters only such views and ideas which they certainly possessed; I felt free to attribute to them views and ideas at which they may have arrived, particularly if these were logical developments of such ideas with which they were definitely familiar. In cases, however, where it is known they had erroneous beliefs, I felt compelled not to hide the fact. Thus, for instance, it is known that Galileo thought that the planets move in circles around the Sun and he did not understand the role of gravitation; so Galileo speaks about these questions accordingly. On the other hand, I thought it admissible to make such bold conjectures as, for instance, that Archimedes arrived at certain ideas which are nowadays classified under cybernetics and that he planned a machine for sieving primes.10 I cannot support such conjectures by any document, and of course do not consider them as well founded; the only thing I claim is that it is not unthinkable that these conjectures are true and, furthermore, that the facts at our disposal are as insufficient to disprove these conjectures as to prove them. I thought that "poetic license" entitled me to use such hypotheses as these.

As for the historical background of the second and third dialogues, I kept to the facts in every essential point. The only exception, where I departed consciously from the facts, is in the second dialogue where King Hieron is directing the defense of Syracuse in the siege of the year 212 B.C., while in reality he died three years earlier. However, both dialogues contain the description of hypothetical events about which we have no definite knowledge, but which are not contradicted by known facts. This is the case, for instance, with the plan of helping Galileo to escape: we do not know whether Torricelli and his friends really had such a plan, but it is not at all impossible.

The essential content (though usually not the wording) of some sentences in the dialogues is either directly attributable to my characters or attributed to them by their contemporaries. This is the case, for example, when Socrates talks about himself11, Archimedes about his method12 and Galileo about the language of the book of nature.13 Such sentences (and only these) are printed in italics.

I have tried to present the personalities of my characters as faithfully as possible. In the case of the third dialogue, the drama of L. Nemeth influenced me greatly: I took from it, among other things, the idea of presenting Torricelli and Signora Niccolini.

For those who want to study the historical background of these dialogues, a selected bibliography is added which does not aim at completeness; it contains only such books I found particularly useful in the collection of my material.

I hope this postscript makes clear what my aims were in writing these dialogues. It is up to the reader to judge how far I was able to realize my intentions.

ALFRED RENYI

1 Dialogus a matematikarol, Valosag, 3, 1-19, 1962.

2 On dialogue, Les Cahiers Rationalistes, 33, NO.208-209. Janvier- Fevrier, 1963.

3 A Socratic dialogue on mathematics, Canadian Mathematical Bulletin, 7, 441-462, 1964.

4 A Socratic dialogue on mathematics, Physics Today, December, 1964, pp. 1-36.

5 A Socratic dialogue on mathematics, Simon Stevin, 38, 125-144, 1964-1965.

6 Sokratischer Dialog, Neue Sammlung, 6, 284-304, 1966.

7 A matematica – Un Dialogo Socratico, Gazeta de Matemtitica, 7.6, No. 100, Julho-Dezembro 1965, pp. 59-71.

8 A dialogue on the applications of mathematics, Ontario Mathematics Gazette, 3, No.2, 28-40, 1964.

9 A dialogue on the applications of mathematics, Simon Stevin, 39, 3-17, 1965.

10 Such an apparatus was first described by D. H. Lehmer (A photoelectric number sieve, American Mathematical Monthly, 4°, 401-406, 1933).

11 See for example "The Apology of Socrates" (Great Dialogues of Plato, translation by W. H. D. Rouse, edited by Eric H. Warmington and Philip G. Rouse, Mentor Books, 8th printing, New York, 423-446, 1962).

12 See the letter of Archimedes to Eratosthenes (The works of Archimedes, with the method of Archimedes, edited by T. L. Heath, Dover, New York, 1960 ) .See particularly the following sentences on page 13:

"Certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards, because their investigation by the said method did not furnish an actual demonstration. But it is of course easier when we have previously acquired by the method, some knowledge of the question, to supply the proof than it is to find it without any previ- ous knowledge."

13 See particularly in the letter of Galileo called "The Assayer" (Discoveries and opinions of Galileo, translated with an introduction and notes by Stillman Drake, Doubleday Anchor Books, New York, 237-238, 1957), the following sentences:

"Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics.”

SELECTED BIBLIOGRAPHY

A Socratic Dialogue on Mathematics

Rouse, W. H. D. (trans.), Great Dialogues of Plato (Mentor Books, New York, 1962).

Szabo, A., Wie ist die Mathematik zu einer deduktiven Wissenschaft geworden, Acta Antiqua Acad. Sci. Hung., 4, 109-152, 1956.

• Die Grundlagen der fri.ihgriechischen Mathematik, Studi Italiani di Filologia Classica, 3°, 1-51, 1958.

• The transformation of mathematics into deductive science and the beginnings of its foundation on definition and axioms, Scripta Mathematica, 27, 27-48, 1960.

• Anfange des Euklidischen Axiomensystems, Archive for History Of Exact Sciences, 1, 37-106, 1960.

• Der alteste Versuch einer definitorisch-axiomatischen Grundlegung der Mathematik, Osiris, 14, 308-369, 1962.

A Dialogue on the Applications of Mathematics

Clagett, M., Greek science in antiquity (Collier, New York, 1955). Heath, T. L., The works Of Archimedes with the method Of Archimedes (Dover, New York, 1960).

• A manual of Greek mathematics (Dover, New York, 1963).

A Dialogue on the Language of the Book of Nature

Annitage, A., The world of Copernicus (Signet Science Library, New York, 1947).

Drake, S., Discoveries and opinions of Galileo (Doubleday, New York, 1957).

Fenni, L. and G. Bernardini, Galileo and the scientific revolution (Fawcett World Library, New York, 1965).

Galilei, G., Dialogues concerning two new sciences (Dover, New York, 1914).

• Dialogue concerning the two chief systems-Ptolemaic and Copernican (translated by Stillman Drake, foreword by Albert Einstein. University of California Press, Berkeley and Los Angeles, 1962).

Geymonat, L., Galileo Galilei (Einaudi, Rome, 1957).

Santillana, G. de, The crime of Galileo (Mercury, London, 1961).

A note from an article on the net:

Alfred Renyi was a famous probabilist and number theorist, co-creator with Paul Erdos of the subject random graphs, and for many years director of the Institute of Mathematics in Budapest. This is a most inviting, charming and thought-provoking tour de force and jeu d’esprit. It poses the basic problem, and answers it in a way that invites further questioning and deeper development. Apparently Prof. Renyi used to give live performances of this work, assisted by his daughter Zsuzsanna, to whom he dedicated the book Dialogues on Mathematics (1967).

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