AN INVERTED JACOB'S LADDER appears, and a Brand Corporation scientist starts climbing it to advocate confining and concealing nuclear explosions underground. CHORUS OF SPACE SCIENTISTS: Facilis descensus Averno Hoc opus hic labor est.* ULFILAS: That must mean that it is more difficult to conceal testing up in space than underground. If you are quoting Virgil, as you seem to be doing, you must know the end of the story.[6] I must say that my main worry is not war as a result of premeditated action, but accidents. If the probability in any one year is alpha, then— TOTILUS: Accidents, like mutations, are not always bad. You forget, Ulfilas, that there is perhaps a small probability beta-(catching himself) not so small, I should like to say that something good can come out of it. ULFILAS: It seems to me that we should all go more into space. This might be a tremendous distraction for all of us, for many reasons. I am in favor of hyperbolic orbits and space research. A lot of spare energy can go into it. It stimulates the economy as well, and the rivalry is perhaps less dangerous there. TOTILUS: I like parabolas myself. It is on these, you see, that you can show strength and deliver bombs. BENEFACIUS: I like ellipses. They are useful for communication satellites and weather prediction. TOTILUS (angry for not having said that himself, especially since elliptical orbits can also be used for ejecting clean bombs): I might agree to that. They might be important for the uncontaminated countries. SCIENTIST WITH LAPSED Q CLEARANCE (looking upon them with pity): I too have consulted for many years. But now I am studying the astronomy and the biology. Is better. Come, Ulfilas, let's go have drink. Finis It is very easy to descend to hell, but to retrace one's steps and reach the upper regions, that's the task, that's the toil.[5] [1] Livermore, of course. [2] Sherwood was the name of the research program at Los Alamos on controlled fusion. [3] A Roman proverb. [4] John Manley actually said that a toothbrush can not be improved forever. [5] From Book 6 of Virgil's Aeneid. [6] The significance of this reference to "the end of the story" is not clear. CONVERSATIONS The cultural affinities and intellectual differences between Stan and Rota were such that they could talk together for hours, though they were worlds apart mathematically and philosophically and never collaborated. I was fascinated by these spontaneous and informal discussions and recorded a number of those that took place in my presence, to transcribe and assemble loosely in a little collection. Los Alamos Science selected the following fragments to illustrate the way Stan and Rota interacted and, more generally, the way mathematicians converse about what occupies their minds. Françoise Ulam The Mathematical Dictionary ULAM: I think it is a very good idea to try to write a mathematical dictionary. First we must settle how many words to think about. Would you say two hundred or less? ROTA: Two hundred! No. Ten, maybe! ULAM: No, no. At least a hundred. They will have to be very diverse. It will be a long project. Logical words like but and even have a different character from words which have a topological or kinematical meaning like mix, find, search. Then there is another class of words like involve, intuitive, imaginary. There are many categories. I think we should have ULAM: And physical. Physics is almost ULAM: Already is difficult mathematical- ROTA: Logicians claim but is the same as and. ULAM: No! Its meaning is entirely different. How would you describe but logically? Something that leads us to a conclusion but does not? A disappointment in probability? A whole essay could be written about it. Someday there will be a tremendous theory devoted to its ramifications. It could be a germ like the word continuous. The study of topology is nothing else but the study of the word continuous. ROTA: When I was at Princeton, Alonzo Church gave a two-hour lecture on the meaning of but and and. It is now written up in his great Introduction to Mathematical Logic. ULAM: So you see! And what does he say? I never read it. I knew he was a logician but did not know he did things. like that. Now let's discuss things intelligently, professor. ROTA: O.K. Let us begin with the word but. Stan? ULAM: I would say that the word but suggests to me the following (we'll be more precise later): an element of an algebra whose elements are uttered sentences. I can imagine it as a point in a universe of points interpreted as sentences-physical facts. I see that it won't be easy to avoid circular definitions; we must not use the word but in developing a theory of but, right? The word but means that an element does not belong to given set of points that was defined before. But-I am just saying this on purpose now-but expresses that an element belongs to a set which is similar or slightly larger than the already given set. Of course, I did not really need to use the word but in my explanation. However-Oh! I just used the word however; you see how hard it is to avoid these words? By the way, this poses another interesting philosophical problem, the fact that we cannot explain a mathematical ... ROTA: Let's not digress. ULAM: I just want to see what is in my mind. I do not have a perfect definition right away. Do you agree that but is an element which does not belong to a set that was defined before? ROTA: Yes. Now let me try my definition. We have two sets, A and B, and a new relation between A and B which we will call contrast. The word but is the contrast between set A and set B. ULAM: The set B, in every example I know of, is usually given by the speaker. Set A is mentioned; set B is intended. It is not there at the beginning or maybe it is only in the mind of the second speaker. Would it be a good idea to consider it as a part of conversation? One person proposes something, and the other replies, "No, but ..."? ROTA: No. I don't think it is a good idea to formalize conversation. It would get us too far from our purpose. If we are going to give definitions, they have to be objective. ULAM: Whatever is done, you always stick to tempus acti, and you do not want to do something unorthodox. Why reduce it to the existing formalism? It is good to try, but it is not necessary. ROTA: If possible, do it. Only when you have to, give up. ULAM: O.K. I agree. Continue. ROTA: So you have two sets and the contrast between two sets, and the word but is an expression of this contrast. And now I would say the word but is used when this contrast has to be brought out. ULAM: Very good. But is that really always true? ROTA: That is my story. ULAM: We should have examples, like in dictionaries. They always give quotes from Shakespeare. ROTA: Let me give an example: "We were going to go out touring today, but it is raining and we didn't go." Analysis: There are two situations, or sets, if you wish. One, going touring; two, assumption that the weather is fair. Then the weather turns out not to be fair, so there is contrast between fair and unfair and the word but arises. ULAM: I agree. Let me give another ex ample: "The snail is not an insect, but it is still an animal." ROTA: Here again you have a set. You presume the snail to belong to this set. The contrast arises because you see a further subdistinction inside this set. ULAM: Simply, these sets are not equal; one set contains the other. ROTA: Be that as it may, either not equal or partitioned, one contains the other. ULAM: This is part of it but perhaps not all. We will have to have detailed discussions like that about every word, as they do at l'Académie française! ROTA: Let me say that any definition is necessarily incomplete. It is a property of definitions to be incomplete. ULAM: Incomplete perhaps, but still it should try to be as broad as possible. ROTA: Then it will never end. There is a point where one says fine, adequate, even though it is not the whole story. ULAM: Yes, I agree. ROTA: Let's take another example. One says of a person: "He is good, but he is also careless." How would you analyze that? ULAM: A point belongs to two sets. If you say good, the presumption is that everything is good about him, so you add another set. ROTA: Suppose I replaced the word but by the word and. In your opinion how would the meaning of the sentence change? ULAM: It would be an entirely different meaning. ROTA: Why? ULAM: Because the set of carelessness is not a set which normally is associated with the set being good. ROTA: It is not a complete explanation. ROTA: Distinction is too weak. But requires contrast and unexpectedness. ULAM: Unexpectedness. Exactly. This is the essential thing to my mind. Namely the first set suggests something, and the second implies the suggestion does not hold. A set of properties implies a lot of others, but an exception is made. But suggests exception. ROTA: No exception is involved. For example, "I was going to go out but the phone rang." That is no exception. ULAM: That is yet a different meaning of but. It says that the normal pattern is being abruptly changed. ROTA: Let me say this. A lot of examples have the following structure. You have two sets, A and B, and you have an element c. You expect c to belong to A, but then it turns out to belong to B. That is the typical use of but. ULAM: Right. So it is not a relation of the contrast but of difference. ROTA: We have abstracted a set-theoretic relationship for the word but; namely we have two sets and an element. The element may belong to A but instead it belongs to B. ULAM: Very good! However the two sets are somehow close. They are not too different or one contains the other because you could not say, "The pencil is long but it is black." ROTA: Right. ULAM: Why? ROTA: There must be a similarity between the sets. ULAM: Ah! Now we have caught one essential point. ROTA: So there are these properties of sets which somehow are similar, and then there is the confusion of one element belonging to one instead of the other. ULAM: In general the two sets are in some relation of similarity, close in the sense of a Hausdorff distance or whatnot, and not completely separate. ROTA: Two or more sets are in turn subsets of a set of sets which is predetermined. They are members of the same family of sets. ULAM: What does it mean, the same family? ROTA: The family is the similarity class. ULAM: Right, that is what one could say. Very good. We are getting somewhere. ROTA: You see, I am becoming Ulamian. Set of sets! ULAM: My example about the pencil was crucial. It did not make sense. Let us take something else. For example, however. It is not quite the same as but. ROTA: Later. Let us finish with but. ULAM: We have to warn our readers and ourselves that there are words that mean almost the same, with subtle shades of difference. In French there is mais and cependant. We ought to analyze that. ROTA: What about nevertheless and yet? ULAM: Nevertheless has a greater degree of something. We should analyze all these. They are all coming together. ROTA: In spite of ... ULAM: I would very much like to define the word key or lock, because there is a sort of labyrinth, a maze. You have to enter a lock a certain way, which at random is difficult, and perform a sequence of operations. ROTA: Key is absolutely one of the best. ULAM: Key, lock, labyrinth- there is a whole topological, combinatorial meaning there. Logical too. Key also has an abstract meaning, a key to something. We are just beginning. This is a project for several months. ROTA: We could get a grant! ULAM: From some cultural whatnot— there are such. Philosophers do not give grants, but we are rich old men, as Erdös says. If we could meet an hour a day, we could get somewhere in one month. FRANÇOISE: Next summer in Santa Fe. On Teaching And Learning ULAM: Most of what I've learned was subconscious, by osmosis. When I read, I am not aware that I am learning. I learn mainly from conversations, from people rather than from lectures, and I did not realize until a few years ago that I have a good memory. I could start teaching mathematics with courses for college freshmen and go to junior or senior courses without any preparation, because in mathematics one thing leads to another. Let us discuss whether teaching mathematics really makes any sense. Either the Gainesville January 1974 student is so good that he does not need a teacher, or else, if he needs help, he is not cut out to become a mathematician. At Harvard I had some good students with whom I could talk and feel that teaching was not merely an empty gesture. I think I influence people more than I teach them. I influence their taste or their choices. ROTA: I learned from you to argue in a short way, to give only ideas followed by simple examples. That is the Ulamian influence. ULAM: I don't mind teaching, but I don't like to do it regularly. When I have to do something at a fixed hour, even if it is a pleasant dinner or cocktail, I fret. I hate not feeling completely free. But of course, being completely free immediately brings on a feeling of restlessness, of not knowing what to do! Each of us has taught several thousand hours. If you think that a normal working year in America has about 2000 hoursan 8-hour day for about 250 days-that is quite a bit of your waking time, isn't it? But maybe it is not entirely waking time. One does it in a trance, partly asleep sometimes! I am told I teach calculus well. It is possible, for I believe one should concentrate on the essence. One should not teach everything at a uniform level either. One should stress some important as well as some unimportant details on purposein a sense to follow the way I think the memory works. When you remember a proof you remember a sequence of pleasant, unpleasant points, zeros and ones. Here comes a difficulty you try to remember, and you make an effort. Then you come to something that goes automatically and it is ULAM: I learn best from conversations. I love them, and that is how I learned physics in Los Alamos. Some people are different in this respect. They prefer to learn slowly and methodically. How about you? ROTA: I learn best when I am forced to do it. ULAM: Speaking of being forced to learn, in Poland it happened several times that I announced that I would speak on a certain subject at a meeting of the mathematical society before I had a proof. I felt absolutely confident that once I had agreed to speak, I would get a proof. It could have been an embarrassment otherwise. On the other hand, when I look at a paper of mine which has been published, I discard it after one glance, from fear that I will discover that it is wrong. There is also this tiny gnawing doubt about whether the result is new or not. Yet even in a field about which I know nothing, I can always tell whether a theorem or a point of view is good or not. This feeling comes somehow from the way the quanti fiers are arranged, from the tone or music of the piece. Do you remember what Galois wrote in his final letter before his fatal duel? He wrote that in their publications mathematicians really conceal the way they obtain their results because the process of discovery is different from what appears in print. It is important to repeat this again and again. John von Neumann ULAM: Hot! What is the temperature? ULAM: Pas possible! It must be the hottest day in thirty years. Which reminds me, once flying back to Los Alamos on Carco on a hot summer day, I opened the little window and my handkerchief flew out of the plane. Behind us there was a second plane carrying Johnny and others. What do you think the probability is that my handkerchief could have gotten enmeshed in the propeller of the other plane? ROTA: Von Neumann was older than you. ULAM: Six, seven years. ROTA: An older man! ULAM: Yes. You know how it is. In the beginning the percentage was twenty or so; later it went down to ten. ROTA: So you considered him a senior, and yet you made fun of him? ULAM: Oh always! Of Banach too. I was always impudent. ROTA: He did not treat you as someone younger? Gainesville February 1974 ULAM: No. I don't think he knew anybody more intimately and vice versa, despite our difference in age. For a man of his stature he was curiously insecure, but his understanding, intelligence, mathematical breadth, and appreciation of what mathematics is for, historically and in the future, was unsurpassed. His immense work stands at the crossroads of the deThe ravelopment of exact sciences. tionalization of the idea of infinity-the life blood of its history-with its mysterious power to encode succintly and generally the properties of numbers and the patterns of geometry, received some of its definite formulations from his work. His ideas also advanced immeasurably the attempts to formalize the new, strange world of physics in the philosophically strange work of quantum theory. Fundamental ideas of how to start and proceed with the formal modes of operations and the scope of computing machines owe an immense debt to his work, though they still today give hints that are only dimly. perceived about the workings of the nervous system and of the human brain itself. Other mathematicians strike me as virtuosi who play their own special instru |