Many years ago, at The University of Pennsylvania, I had a class of three students in a graduate course called Foundations of Mathematics.
I stated some axioms and theorems and proposed that they prove the theorems from the axioms. Sometime in December I stated Theorem 15. J. R. Kline presented a proof the following April. Another member of the class did not want to give up and listen to Kline's proof after he had tried for so long to get a proof of his own and he walked out of the classroom. Towards the end of the course he said he thought he could prove this theorem if he could only "get off the boundary" (as he expressed it). More than 12 years later when I was on a visit to a University where he had become a professor, I said "have you got Theorem 15 yet?" and he answered "No."
I am still giving a course bearing the same title but it is based on quite a different set of axioms and a theorem corresponding to the old Theorem 15 now comes much later in the treatment. The number of this course is 688. Axiom 0 states that every region is a point set and Axiom 1 now states that there exists a sequence satisfying certain conditions numbered 1, 2, 3 and 4. A large body of theorems can be derived from Axioms 0 and 1 without use of the fourth of these conditions and in 688 I usually do not state the fourth one until some months have gone by. In 1958-59, when the time finally arrived to state it, it appeared that there was one member of the class, a Mr. W., who did not want to be told what it was. So the others were told but he was not. One day about 2 years later, in my 690 class, he was at the board, near the door, explaining something and I started to make a remark. He seized the door knob, flung the door open and rushed out of the room. He thought I was about to state condition 4. I imagine some of you are thinking that it was ridiculous for him to be so determined to remain ignorant concerning this condition. If so I do not agree with you at all. In course of time he thought of quite a different condition such that if it is substituted in place of condition 4 in the statement of Axiom 1, the resulting axiom (Axiom 1w) is quite interesting and quite different from Axiom 1. If he had been told at the outset what condition 4 was he would, I believe, never have thought of Axiom 1w and never have discovered the interesting things he has proved to be true concerning it.
Often when I have raised some difficult questions in one of my graduate classes and days (or weeks or possibly even months) have gone by without its having been settled in that class and finally the day arrives when someone announces that he has a solution and he goes to the board to present it, some of the other members of the class (sometimes most of them) walk out and stay out until he is through. Is it a good thing for a graduate student to walk out under such circumstances? I think that often it is but sometimes it is not and in the case of some students it is very hard for me to decide whether or not to discourage it.
I do not believe I ever said anything to discourage it in the case of Mr. W. I don't believe there was a single instance where he walked out when it would have been better for him had he stayed in.
One year I had in my 688 class a student whom I will call Mr. X and whom I considered to be one of the very best, and probably the very youngest in the class to be walking out or even completely missing class too often and one day when he was present I made some remarks indicating that this sort of thing could be overdone. After class he came to my office and assured me that whenever he had stayed out when a theorem was being proved he had eventually obtained a proof of his own. After this he proceeded to stay away some more. One day when he was absent a member of the class gave a simple proof of Theorem 70, a theorem to the effect that if A, B, and O are three points of a connected point set M and no point separates O either from A or from B in M then no point other than O separates A from B in M. When he was through I said something like this "What would you say if someone would argue this way: since O is not separated either from A or from B, in M by any one point therefore if X is a point of M - (O + A + B), M - X contains a connected point set containing O and A and another one containing O and B and, since these two connected point sets have the point O in common, their sum is a connected subset of M - X containing both A and B and therefore A and B are not separated from each other by X"? Most of the class seemed to agree that that would be a good argument but a few seemed to be in doubt. I asked a student who I will call Mr G whether at the next meeting of the class attended by Mr X he would be willing to give this argument for Theorem 70 and see whether Mr X would offer any objections to it. He agreed to do so and when the proper time arrived I called on Mr G to prove Theorem 70. He went to the board and I noticed that Mr. X did not appear to be listening to him. I said "Are you listening to this argument? Listen to it. Listen carefully. When Mr G was through I asked X whether G's argument was all right. He said yes, I think so. You don't know that it is? Isn't it true that if every connected subset of M that contains O and A contains X then X separates O from A in M? He replied that it was. If you can't go from Austin to San Antonio without going through Buda then doesn't Buda separate Austin from San Antonio? He replied in the affirmative. I said "If you had staid out when Theorem 70 was being proved in class and you had subsequently thought of this argument then would you have felt that here was a case where you had stayed away when a theorem was being proved but you eventually proved it yourself? He said "yes." I asked whether there was anyone present who could show Mr. X that if he had thought so he would have been mistaken. Mr. P went to the board and drew a figure like this:
Here, for every positive integer n, Jn is a circle, of radius n, tangent to the line OA at the point X. If M denotes the line OA plus the sum of all the circles then neither X nor any other point separates O from A in M. But there is no connected subset of M - X containing O and A. I asked Mr X whether there may not have been cases where he thought he had proved that he had not proved. I indicated to him that by staying out a student may miss discussions on occasions when it would have been better for him to have stayed in and taken part in them.
Some years ago I had in 688 a student (Mr. S) of outstanding ability who, I think seldom, if ever, had any occasion to walk out. Seldom, if ever, did any other student present, in this class, a proof of a theorem that S had not already succeeded in proving. One of the other members of the class said that being in a class with S was like being in a lecture course (with S as a lecturer) and he did not like lecture courses so he did not want to take 689 (a sequel to 688) the next long session because S would be taking it then. He asked whether if he and another student should stay away from the University of Texas the whole of the next long session they could count on my giving 689 again in the long session immediately following the next one. I indicated that I expected to do so and they both got positions, stayed away from Texas one session and returned later according to plan. I think that under the circumstances their decision to stay away was a good one. They were both good students and went on to receive Ph.D. degrees.
I have sometimes said to a student "If you don't want to listen to someone else proving a theorem that you have not yet proved then how about trying to be the first one to prove it and if you fail to be then paying the penalty for not being by staying in and listening to someone else's proof?
On Tuesday March 10, 1964 I stated Theorem 55 to my class in 688. This theorem is as follows: If H is a countable collection of closed and compact point sets such that no nondegenerate component of an element of H intersects another element of H then the sum of all the elements of H is not a locally compact continuum. This class met on Tuesday, Thursday, and Saturday. I received no proof of Theorem 55 on March 12 or March 14 but after class on March 14 Mr. G indicated that he had what he had thought was a counterexample but discovered later that it was not one. Without knowing what his example was I suggested that he describe it at the next meeting of the class without disclosing that it was defective. On Tuesday, March 17 I asked whether anyone could prove Theorem 55. Mr. G said "I have a counter example." He went to the board and described a certain infinite collection H of mutually exclusive intervals lying on an interval M and indicated that M is the sum of all the intervals of the collection H, contrary to Theorem 55. Two members of this class of 15 had had 624 and were aquainted with an example resembling this one and two others had had work in point set theory elsewhere but I asked each of the remaining eleven except Mr. G whether this was (or served to be?) a counter example to Theorem 55 and they all replied in the affirmative. I said "Well you now have a choice. Either show there is nothing wrong with this example or prove Theorem 55."
On Thursday, March 19, I called on Mr. X and he indicated he thought he could prove that there is something wrong with this example and he could also prove Theorem 55. He went to the board and gave what I think was quite a good argument to show that the sum of the intervals of Mr. G's collection H is not a counter example to Theorem 55. But the problem of proving Theorem 55 remained. At the last meeting of the class before the examination I asked whether anyone could prove it. Mr. X raised his hand. Mr. Z went out. The others remained. Mr. X went to the board and his argument seemed to be proceeding nicely till he got to the point where, concerning a certain pair of elements of H he said let T denote an irreducible subcontinuum of M from one of them to the other one. The question was raised whether he was sure that there is such a subcontinuum of M. What if these two elements of H intersect? He seemed uncertain as to how to go on, the bell rang and we adjourned. Mr. Z has given to me what is supposed to be a written proof of Theorem 55 but I have not checked it. I am guessing that, at the first meeting of 689 next September, either Mr. X or Mr. Z will prove Theorem 55.
Years ago, a course in Advanced Calculus when given by me was usually so different from courses with the same title given by others that I decided to use a different title and a different number. About 1941 the title was changed to introduction to the Foundations of Analysis and the number was changed to 24 and later to 624.
I do not usually allow in 624 any student who has taken or is taking Advanced Calculus 321 or Differential Equations 322. A student who has had one of these courses is likely to know in advance the answers to so many of the questions raised in 624 that I would be inclined to call on him in that course either seldom or not at all. One of the questions that I ask, usually near the beginning, in 624 is whether or not there exists, on the X-axis, a closed and bounded point set M such that each point of M is a limit point of M but M contains no interval. On one occasion, sometime after I had raised the question, a student indicated that he could answer it and went to the board. I don't think he had written more than 2 or 3 sentences before I became suspicious, stopped him and asked him whether he had read anything on this subject. He replied that he had not but that he had talked to someone about it. I said "Well, that's enough! You have spoiled this question for this class" and he sat down. It was a long time before I called on him again about anything. In the hall after this class period another student said "He certainly did spoil this question. After he said what he did it was easy to see the answer to the question."
For many years in 624 (and also in my section of calculus) I have defined a simple graph in the plane as a point set such that no vertical line contains two points of it and have pointed out that (1) no use of the notion function is involved in this definition but (2) the point set M is a simple graph if and only if it is the graph of some function.
The simple graph M is said to be continuous at the point A if and only if A belongs to M and for every two horizontal lines a and b with A between them there exist two vertical lines h and k with A between them such that every point of M between h and k is also between a and b.
C is said to be the slope of the simple graph M at the point A if and only if (1) C is a number and A belongs to M and for every two vertical lines with A between them there is a point of M distinct from A between them and (2) if l is a line of slope C containing A and a is an acute angle with vertex at A and some point of l in its interior then there exist two vertical lines h and k with A between them such that every point of M between h and k and distinct from A is in the interior either of a or of the angle vertical to a.
The line l is said to be tangent to the point set M at the point A if and only if (1) A belongs to M and every circle with center at A encloses a point of M distinct from A and (2) if a is an acute angle with vertex at A and some point of l in its interior then there exists a circle J with center at A such that every point of M in the interior of J distinct from A is in the interior of a or of the angle vertical to a.
Questions are asked concerning these properties and relationships between them. For example is it true that if a simple graph has a tangent at the point A then it has a slope at that point? If someone gives an example to show that is not true then can he think of some way to strengthen the hypothesis so that this conclusion will follow? Is it true that if the projection of the simple graph M onto the X-axis [is] closed and M is closed then M is continuous? If someone gives an example to show that that is not true then the conclusion would follow if the hypothesis were strengthened in what way? An example of what not to do would be to tell the class at the outset to prove that if M is a simple graph whose projection onto the X-axis is closed and bounded then M itself is closed and bounded if and only if it is continuous--to tell them that at the outset and thereby deprive them of the opportunity to both think of it and prove it.
I have often told a class to prove something that I know is not true for example to prove that if a point set is closed so is its projection onto the X-axis. Isn't this much better than to tell them to prove that the projection onto the X-axis of a closed and bounded point set is closed? Why should any teacher want to follow the latter procedure and therefore deprive a student of the opportunity to discover independently that one of these propositions is true and the other one is false?
But propositions, true or false, are not the only things to be considered. If it is granted that it is better not to prove or disprove a proposition for a student without at least giving him an opportunity to prove or disprove it for himself then what about concepts and definitions?
Suppose you would like to know whether there is anyone in a certain class who is capable of thinking of a certain concept, thinking of it without the help of any hint whatsoever. Suppose you know of a problem that you believe no one could solve without thinking of and using that concept. Then how about first carefully avoiding any reference to the concept or anything too closely related to it until you are ready to propose the problem and then proposing it.
This year, as I have done every year for a long time, I raised with my 624 class the question whether, in the plane, there exists a collection Q of closed and bounded point sets such that (1) each element of Q has at least three points, (2) if x and y are elements of Q and x is nondegenerate then y is the image of x under some continuous transformation and (3) if x belongs to Q and y is the image of x under some continuous transformation then y belongs to Q.
Wondering whether some member of the class might think of and introduce the concept connectedness in order to solve this problem, I had purposely refrained from saying anything whatsoever to this class about connectedness at any time before the day in the second semester when I stated the problem. On April 10, Miss S indicated that if Q is such a collection then no element of Q is the sum of two mutually exclusive straight line intervals -- For suppose an element of Q is the sum of two such intervals a and b. Let A and B denote two points. There is a continuous transformation throwing a into A and b into B. Therefore, by stipulation (3) concerning Q, A + B belongs to Q. Hence, by (2), a + b is the image of A + B under some continuous transformation. But this is impossible. I suggested that she try to find some other point sets that do not belong to Q. On April 13 she said "If there exists a distance between two subsets of M such that there is no point of M in that distance" then M does not belong to Q. She indicated that "there is no point of M in that distance" left something to be desired but she was trying to get words to describe a point set like -- which she wanted to call separated. I would much rather have a concept introduced this way by a student naturally in the course of an investigation even if (I am tempted to say especially if) it is not perfectly stated at first. I suggested that she continue to think about it. On April 15 she stated a definition of what she called a separated point set which is satisfied if and only if that set is not connected according to Lennes' definition.
Sometimes when a student gives a long argument to prove a theorem and I know of a much shorter one I do not tell him there is a shorter one.
[Here Moore left a space of several lines and wrote, "fill in."]
A mathematician from Europe once said to me Oh if you know a shorter proof you should tell him. If you don't you are not teaching.
I am tempted to paraphrase an often quoted saying about governments by saying "that student is taught the best who is told the least."