Dear Miss Hamstrom:
I was glad to have your letter of April 27.
You say "I have had no formal work in point set theory other than that included in the Real Variable course I've been taking. This, and the little reading I have done, have appealed to me very strongly."
I wish you had never taken a course in Real Variable Theory and that you had read even less about point set theory than I imagine you have. You indicate that you would like to have some detailed information concerning courses to be offered next year and add "This is necessary so that I can plan my summer reading." Whatever else you read about this summer do not read any point set theory if you can help it.
Sometime ago an undergraduate student at Tulane was recommended to me quite highly but a statement was made to the effect that he had not yet taken a bachelor's degree. I indicated that, as far as I was concerned, that was no bar in his coming here and doing some parttime teaching. He came and enrolled in my graduate course P.M. 88 ("Foundations of Mathematics"). When he entered it his rank here was that of a junior. There were two others in this class, both graduates of a Northern University (though I believe they did not know each other there). After graduating at that institution one of them had studied at Harvard and at Colombia and had apparently read a good deal of point set theory, even including some of my work. The other one had studied at the Universities of Cincinnati and Chicago. He also had apparently studied some point set theory. For a while that young man from Tulane (his name was Gail S. Young) was, I think, quite discouraged, in that class with those students who had already had so much mathematics; but time passed, they both quit and he not only received a Ph.D. degree but has continued to produce and I believe will continue to do so. He told me of another student at Tulane and one day he brought him around to see me. There ensued a conversation a portion of which might be roughly indicated, if not accurately described, by saying: "He said he lacked only about two courses of having enough for his B.A. degree at Tulane and he wanted to continue there the following year and get that degree." I said "What courses will you take?" He told me the title of one course he had in mind. I said "it looks to me as if there would be some point set theory in that course. I would rather you wouldn't take it if you are coming here afterwards." He replied that he would really like to take it. I said, "All right, go ahead. But if you do, don't come here afterwards. Apply for a fellowship at Princeton or some other place, not here." Later I received a letter from him indicating that he was not taking that course and saying, I believe, something like this: "I am staying as ignorant as possible." He graduated at Tulane, came here and showed real ability. He took his Ph.D. degree last June. I hope he is glad he didn't take that course at Tulane. I know I am. His name is Moise. Dr. Kline knows of him.
As to my reasons for this attitude towards a person's taking courses in point set theory elsewhere or reading about it before coming here. I am not sure that I could explain it very satisfactorily in a letter without considerable difficulty. But after you have been here awhile I believe you will understand and I hope you will agree.
I expect to give, next year, P.M. 24 and P.M. 88. I am afraid that, in the course you are taking on Real Variable Theory, you have had a good deal about theory of measure of point sets and Riemann and Lebesgue integration. Please let me know whether or not this is the case. If it is, then I imagine, you had read or listened to statements and proofs (or approximations to proofs) of many theorems concerning these matters instead of working out proofs of them yourself and thus if you took 24 you would, I think, in many cases know in advance the answers to certain questions I would ask (and would have already seen approximations to proofs of many of them) while certain other members of the class (I have in mind, in particular a Mr. Pearson who is now taking a first course in calculus and showing unusual ability) would not know in advance what the answers are or, if they should have enough insight (as contrasted with information gathered from reading etc.) to have some idea of the answer, at least they would not have been deprived of the opportunity of working out proofs for themselves by having had someone tell them proofs either verbally or in books in a course previously taken. To read a proof of a theorem or to listen to some professor prove it is a very different thing from proving it yourself without any such assistance. Did it never occur to you that when you read or listen to a proof of a theorem that you had never heard of till someone stated it, (perhaps a few minutes before, without even giving you enough time to first realize just what it means) you are thereby acquiring information (of a sort) but depriving yourself of the opportunity to work it out for yourself and thereby, perhaps, to develop that much more power instead of just acquiring that much more information? What does information amount to compared to power? Suppose you and Mr. Pearson (and perhaps some others) should take 24 next year and I should define the word "countable" and say "See if you can prove next time we meet that the set of all points on the X-axis is countable" and Mr. Pearson will work hard trying to show that it is countable and you will say to yourself "Oh that question was settled long ago by Cantor. I can give you his argument right now. I don't need to wait till next time." Suppose Mr. Pearson however had never heard of this word countable before and starting out with no hint from anyone he would come in next time and say "I can settle that question" and would do so in a manner very different from any that you had ever seen before. And suppose at the next meeting I would say "prove that if a set of segments covers a segment some finite subset of it does so" and you would say to yourself "Oh, that is the Heine-borel Theorem (or is it?) and that was one of the theorems in Chapter I of that book we used last year, let's see how was it proved?" And suppose Mr. Pearson with no such previous knowledge would say to himself "That is an interesting question. I am going to get to work on that as soon as I can" and suppose at the next meeting he would come in with some wrong ideas and next time after that with some other wrong ones, but finally after a week or two would obtain a conclusive argument of his own. And suppose, you having told me that you had seen a proof of the Heine-Borel theorem long ago, I had said "well don't say anything to Mr. Pearson about it let him come to his own conclusions" and accordingly at every meeting for say about two weeks you watched Mr. Pearson try time after time, making mistake after mistake perhaps, and finally arrive at a correct conclusion and in doing so make some real progress towards developing power in the subject, would you or not be saying to yourself "I wish I didn't know so much and that I could be working these things out for myself like Mr. Pearson is doing instead of knowing, in advance, answers that have been told me or that I have read somewhere?"
Do you see why I want to know something about what you have had in the Real Variable Theory course?
Suppose you have had so much Real Variable Theory that it would be inadvisable in certain ways for you to take 24, that your previous knowledge of the subject would make you too much of a spectator in that course, an onlooker observing others working to get answers that you already know. Then what? Perhaps take 88? I think that in 88 next year I will probably have four or five students who are now taking 24 and who have been developing power working on question that I have raised, somehow trying to prove something is true that is really false and finally discovering that it is really false, sometimes coming in with arguments and having it pointed out that they are wrong, trying again, again getting something wrong and finally either settling the question correctly or having the experience of listening to some other member of the class finally straighten it out, etc. I remember that in one class I had a student who for a while, made mistake after mistake. For I don't know how long I was inclined to think, if he said he had a theorem proved, "he thinks he does, but he probably doesn't." But after a certain number of months had passed if he said "I have proved this theorem" I was inclined to think "If he says he does he probably does." Do you think such a change could have taken place in him if he had been given no opportunity to work things out for himself, no opportunity to make mistake after mistake and have them corrected instead of listening to (or reading) other people's arguments?
While I feel that I would have preferred that you come without a previous course in Real Variable Theory and start here in 24, I am inclined to think that, with the situation as it now is, it might be preferable for you to start in 88 in spite of the fact that if you do, you will probably be competing with some students who have had 24 and who have thus been accustomed to working things out for themselves for quite a while. Would it not be better to do that than to take 24 and be, for a very considerable part, if not most, of the time, an onlooker watching others work on theorems which are new and interesting to them but which you have already heard or read about?
As to the question of a minor subject for the Ph.D. degree, in the past it has for sometime been possible for a student, with Pure Mathematics as a major subject, to absolve the requiring of a course in a minor subject by taking a course in differential equations given by a member of our department but counting as Applied Mathematics 22 instead of Pure Mathematics 22 if the student desires to count it that way. I think that Mr. Anderson, a candidate for a Ph.D. degree next month, will do this.
If you are going on to a Ph.D. degree and are going to become a research worker in mathematics and it is not necessary that you first get an M.A. degree for financial reasons then I don't think you should divert, to the fulfillment of the additional requirements for an M.A. degree, time and energy that might otherwise be devoted to the development of power to do mathematical research. Of the last three people who took Ph.D. degrees here in Mathematics, only one had a masters degree and Mr. Anderson has none.
I do not think you should take more than three courses (each meeting three times a week) next long session unless you plan to take an M.A. degree. I think it might be a good idea to take Dr. Ettlinger's 22 (Differential Equations and Applications), and, for reasons indicated above, register for it as A.M. 22 instead of P.M. 22. According to you transcript you had an apparently one semester course in Differential Equations at Pennsylvania but I do not think that would be a good reason for not taking 22. In fact in this case I think it might be a good thing that you have had a preliminary course and indeed if you have access to Lester R. Ford's Differential Equations, which Dr. Ettlinger has been using as a textbook in this course, I should think it might be a good idea to study it this summer in preparation for taking 22.
I think Dr. Lubben will probably give a course called Topics in Modern Algebra (P.M. 37). I believe the last time he gave it he used Birkhoff and MacLane's "A Survey of Modern Algebra" as a textbook. I don't know (and I don't think he does) whether or not he will use it again next year. He has apparently also been thinking of Albert's Modern Higher Algebra. I talked to him today and he seemed uncertain as to just what topics he will take up. The subject of Matrices is one thing he has been thinking about. If you should decide to take this course I should think it might not be a bad idea to some reading along these lines this summer if you have time and feel so inclined. To what extent, if any, the apparently 3 semester hour courses Mathematics 631 (Modern Algebra) which I understand you are taking now may overlap with our 6 semester hour 37 I don't know. I have an idea that Dr. Lubben would not confine himself entirely to any text book that he might use for this course.
I am not sure yet as to what are all the other courses that will be given here next year but I think that among them will probably be 45 (Probability), 46 (Mathematical Statistics), 47 (Actuarial Mathematics), and the one semester courses 336f (Elementary Number Theory), 380 (Finite Groups) and 381s (Algebraic Solvability).
Don't hesitate to write me if there is anything I have written which you would like to have explained in more detail or there is anything else you would like to ask about.
Please remember me to Miss Mullikin.