The Twentieth Century saw many advances in mathematics, including the one by Gödel that says that any mathematical system sufficiently sophisticated to contain number theory must contain propositions that are not provable in that theory. I.e., there is no end to mathematics. Among the devices that Gödel used were sentences that invoke themselves and cause contradictions, such as "This sentence is false." or "This proposition cannot be proved." Research into these is at the cutting edge of mathematics. However, there are much more mundane problems that carry us into this same realm, and furthermore, invoke the human quality of trust, which is critical to the functioning of human society. One of these is posed by the mathematics professor himself. Actually, any professor can pull this thing on his students, but since analysis of it involves mathematical reasoning, let's choose a math professor.
Our mathematics professor is giving an introductory calculus course that meets for one hour Monday through Friday. Near the end of a Friday class, he announces a pop quiz for the following week. He says that the students should be prepared for it, since he is going to give it next week; furthermore, the students are not going to know which day it is going to be on until he actually hands the quiz papers out at the beginning of a class next week. The students leave the class somewhat worried, and they began talking amongst themselves:
Art: Wouldn't you know he would have to give a quiz next week! I have two term papers to prepare and another quiz to prepare for!
Betty: I haven't begun to learn how to get all these epsilons and deltas in order. If he gives it Monday, I'm sunk.
Charlie: What if he gives it Friday?
Betty: That would be better. I would have time to study for it.
Doris: Hey! He can't give it Friday!
Charlie: Why not?
Doris: Did you hear him say that we won't know what date it will be until he arrives in class on quiz morning?
Art: Oh, I see. If Thursday night arrives and he has not given it, we can conclude that it will be held on Friday! We will know in advance that is when he gives it, and he said we will not know in advance.
Art: So he has to give it on Monday through Thursday. But aha. If he hasn't given it by Wednesday, then we will know for sure that it will be on Thursday.
Betty: Well maybe I can get it studied by then.
Charlie: No, wait a minute. It can't be Thursday, for we will know in advance if it is Thursday, because it can't be Friday. So it can't be Thursday, either.
Betty: Please, guys! I need to study for this. You seem to be making it earlier and earlier.
Art: Well, yeah. At least we know it will be Monday, Tuesday, or Wednesday.
Doris: Well, then it can't be Wednesday, because we will know. Right?
Betty: You keep talking like this and I'll have to ask you to help me study.
Charlie: Sure we will help you study. We will have to do it soon, since it will be Monday or Tuesday.
Art: Well, then it can't be Tuesday.
Charlie: So it has to be Monday.
Betty: (feeling really scared) Really?
Charlie: Yes, I know for certain it has to be Monday.
Art:: That means it's certain it won't be Monday! For we know it will be Monday.
Doris: So he can't give it at all!
With that, the four students threw their books in the air and leaped for joy. No math quiz to study for. The professor can't give it any of the days of the week, for you can eliminate each one in succession. They partied all night on Friday and Saturday, and played Frisbee all day on Sunday. Then they went back to classes.
On Tuesday morning, the professor said when he entered the classroom with a handful of papers. "Put your books under the desk. Pop quiz is today."
Betty complained, "Hey you can't give it today."
The professor said, "Yes I can. You didn't know it would be today?"
Astonished, Betty answered, "Yes." Further, the other students were taken aback as well. Indeed the professor did exactly as he said he would. He said there would be a calculus quiz, and that the students would not know what day it would be, and indeed that is exactly what happened.
Maybe he did, but there seems to be something amiss here. The professor can't give the exam on Friday, so he can't give it on Thursday, and so forth, for then the students will know in advance. But he gave it on Tuesday, and indeed the students did not know in advance, because they concluded he can't give it at all. But he said he would, so it is really a contradiction instead. The professor won't give the exam and he will. How about if he gave it on Thursday? Wouldn't the students claim that they predicted the date?
In fact, lets limit the number of days to two. Suppose the professor gives the announcement just before end of class on Wednesday, and that he is going to give the pop quiz on Thursday or Friday and the students won't know which it is until he actually gives the exam. In this case, if Thursday night arrives without an exam, the students will know that it has to be Friday. So can we conclude the exam will be on Thursday?
I would certainly think the professor should give the exam on Thursday. If he gave it on Friday, the students would rightfully claim that they predicted the date before the exam, contrary to what the professor said he was going to do. That would cause the professor to lose trust among the students.
But can the students prove that the date of the exam will be Thursday? No. You can't prove what anyone does. I can't prove you will go to work tomorrow, even though you are expected to. I can't prove that students will show up for class, or that the professor will show up to give it. There have been many instances to the contrary. The professor meant "prove" when he said "deduce", so indeed the professor can give the exam on Thursday and the students can't deduce that he will give it on Thursday. They can argue that almost certainly he won't give it on Friday but they can't prove mathematically or beyond a doubt that he won't.
So by giving the exam on Thursday the professor will fulfill his end of the bargain.
This whole argument brings up the issue of trust, which is needed for our society to function and for it to provide for us. We trust that electricity will remain on in our homes and businesses, that the grocery store will be stocked with items, that our devices will work when we want them to work, that accountants won't embezzle the company's money, and so forth. In the case of this math professor, the professor needs the trust of his students. They will complain hotly if he calls for homework to be turned in on Thursday if he said it was due Friday. In the case of the pop quiz, then, has the professor earned the trust of the students? If we persue this argument, we get a vicious circle.
Assume the professor can be trusted. Then the students will conclude that the exam will not be on Friday. Since Thursday is the last day he can give it and if it is that day they can trust that it will be Thursday, then they conclude that it can't be Thursday. And so forth, through the week, until we conclude that if the professor can be trusted, we get a contradiction: the professor will give the exam if and only if he doesn't give the exam.
This implies that our hypothesis is false; therefore, the professor can't be trusted. He utters statements that can't possibly be true. If he can't be trusted, his word is worthless. This means he has essentially told nothing about when the exam will be next week. So when he steps into the classroom with a handful of blank quiz papers on Tuesday, the students will have had no way of predicting that in advance. This means the word of the professor was met: he gave the exam, and the students were unable to predict it. Therefore, he can be trusted.
So therefore if the professor can be trusted, he can't, and if he can't be trusted, then he can. It is not a case of a professor that can't be trusted; it is a case of a contradiction no matter what you say about trusting the professor. This sounds much like the statements "This sentence is false." and "This sentence can't be proved." Statements like these strike at the heart of mathematical foundations, and nowadays, mathematicians say you can't talk about a logical language in the same language. You can't talk about English in English, and in particular, sentences can't talk about themselves without causing contradictions like "This sentence is false. Now Gödel did use a statement that referred to itself by the techniques of quining and Gödel numbering, which I won't get into here. But he did so within the theory itself. His sentence was stated in number theory, in the theory he was talking about. Timothy Y. Chow (American Mathematical Monthly, 1998 January) did show what application of Gödel's techniques would do to the mathematics professor problem, and showed that a contradiction is derived.
This means the resolution of this paradox can be made not by any fancy mathematical means but by arguing on the basis of trust. The weak part of the argument above is when I said that the professor can be trusted because he met his end of the bargain in this problem. True, he can be trusted here, but the professor asserted also that in general he is not trustworthy. He said in effect, "I can't be trusted.". If that sentence is false, then he can be trusted, so his word is as good as gold, so the statement is true, so he can't be trusted, contradiction. Therefore the sentence is true, so the professor can't in general be trusted, even though he told the truth just this one time.
This is a hotly debated problem, so this is probably not the final answer on it. But it is my take on it. In the meantime, whenever a professor tells you that you need to study for a quiz, you better be prepared for it.
Dr. Jim Blowers
2002 June 15