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I am going to write my master thesis in order to become a teacher of mathematics (with second subject business management). Supported by the ERASMUS program I have the opportunity to do this in France, and I have found already a very kind thesis advisor, who has proposed two alternative topics: Either analyzing the idea of infinity among school pupils and comparing it to that of adults (not mathematicians) or solving a long pending puzzle in connection with mathematical induction. The first topic would require a large number of interviews, the latter and as I confess much more attracting but also much more risky topic would require to find out how induction weakens when applied to normal life.

My thesis advisor gave me some hints on literature including MathOverflow where this problem (in connection with the advertized surprise-exam) has been discussed. But I have not been able to find it. Therefore my first question: Does anybody know where in MathOverflow I can find information on the failure of induction in the case of the surprize exam?

My second question concerns technical support about modelling the expectation. I am about to discern how induction, when transferred to real problems, vanishes step by step. For this sake I want to compare expected values with values from a random-number generator. If I look for the probability that the surprise exam is written on the first day of a period of one day, then I have clearly P(1) = 1. Further I know that P(n) = 0 for n tending to infinity (because we cannot reasonably expect a higher probability for one of infinitely many days). But if I investigate a finite periode, say the 5 days of a week, then it becomes somewhat ambiguous to estimate a probability for these days. By logical conclusion, the expectation P(5) for the last day has to be 0. But so have P(4) and all other days, unless the decay rate of induction is known already. But that is just what I want to find out. Further: Would it be meaningful to assume that the probability for the random number generator has to be chosen as 20 % for each day of a period of 5 days?

I have not yet decided what topic I will accept. It would be very nice to get some advice. (Of course support from here would be acknowledged in my thesis.)

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migrated from mathoverflow.net Jun 29 '13 at 16:22

This question came from our site for professional mathematicians.

This might be relevant: terrytao.wordpress.com/tag/unexpected-hanging-paradox –  Matteo Mio Jun 29 '13 at 15:44
Before worrying about the probability and induction aspects of the surprise-exam paradox, you might want to consider the simplified version where the professor says "There will be a surprise exam tomorrow. You have no way of knowing, before the exam, on which day it will occur." In other words, consider one day instead of the usual five. You might find, after you understand this simplified version, that you also understand the 5-day version without having to consider probability or induction. –  Andreas Blass Jun 29 '13 at 15:52
Andreas Blass, perhaps you have overlooked that I explicitly had written about the cases of 1 day and of 5 days? Do you wish to propose that the 100000 day case does not allow for a surprise? Explanation: The 1-day case is a logical contradiction while the 100000 day case is not. –  user84384 Jun 29 '13 at 16:40
In my opinion, the 100000-day case and the 5-day case and the 1-day case are all essentially the same paradox, not quite a logical contradiction but still leading to an unpleasant conclusion, namely that the students cannot know that what the teacher has told them is true. The only effect of increasing the number of days is to make it harder to understand the situation. That is why I suggested considering the 1-day case first; its solution and the associated unpleasantness apply equally well to longer weeks. –  Andreas Blass Jul 17 '13 at 21:39

2 Answers 2

"Does anybody know where in MathOverflow I can find information on the failure of induction in the case of the surprize exam?"

It was proposed for discussion here: http://mathoverflow.net/questions/134398/what-is-the-fade-away-rate-of-mathematical-induction-in-practical-applications --- but it was closed and deleted, so only highly reputed folks can see it. It also comes up in one of the answers at http://mathoverflow.net/questions/116445/positive-results-coming-from-paradoxes (with regard to unexpected hangings, rather than surprise exams). And it was discussed at m.se at Surprise exam paradox?

EDIT: Here is the deleted MO question:

The unexpected hanging paradox, hangman paradox, unexpected exam paradox, surprise test paradox or prediction paradox is a paradox about a person's expectations about the timing of a future event (e.g. a prisoner's hanging, or a school test) which he is told will occur at an unexpected time. http://en.wikipedia.org/wiki/Surprise_test_paradox

All these paradoxes and many others, like the blue-eyed islanders paradox, are mainly based on the unlimited validity of induction. But perhaps this assumption is incorrect with respect to application of mathematical induction to practical reality.

If the teacher announces "tomorrow we will write an unexpected exam", then this is clearly a self-contractory announcement, even in reality.

If the teacher announces that the unexpected exam will be written next week or in any specified interval of days, then many mathematicians tend to conclude that this is also self-contradictory in reality because induction shows that the last day of the interval cannot apply, therefore also the day before the last one cannot apply, and so on.

But for a really long interval induction fails as can be proved. Consider that the teacher announces one or even 100 surprise tests during the next 3000 days, then induction won't help at all to determine the dates. In order to prove that, guess 100 dates and compare with a set of 100 random numbers of that interval.

That suggests: in these cases the reasoning based upon induction does not remain valid for large intervals in reality. (Compare the blue-eyed islanders paradox with $10^{10}$ islanders.) The validity of inductive reasoning is certainly absolute for $n = 1$, but near to zero for $n = 10^{10}$ and has limit 0 for an infinite interval.

The question is of course, whether this problem belongs to mathematics or to reality only. But I would plead the case that also problems with importance for reality should be scrutinized by mathematicians.

Therefore my question: What is the fade-away-rate? Can a "function of validity" $f(n)$ be defined concerning the state of knowledge at time zero? Has this already been done somewhere? And, if so, how has this fuction be calculated?

A statistical approximation could be this: Take intervals of $n$ days, guess the day of exam and compare your guess with a random number $mod n$.

Disclaimer: I am well aware that there are different interesting approaches to analyse and explain the paradox. Despite significant academic interest, there is no consensus on its precise nature and consequently a final 'correct' resolution has not yet been established. http://en.wikipedia.org/wiki/Surprise_test_paradox. (Further information is given in the references to this Wikipedia article.) But my question does not ask for different approaches. It concerns only the possible existence of $f(n)$.

And now I'll try to add the comments, but the formatting may go pear-shaped:

The question you ask is not a mathematical one. While the question might be research-worthy, the research in question is not mathematical. Voting to close as off-topic. – Boris Bukh Jun 21 at 13:41 1

You may be right. But my question is just whether this problem has been considered in mathematical literature. I guess this is possible. Why should mathematicians refuse to help in practical questions? Perhaps mathematical methods will be useful to find such a function? – user35078 Jun 21 at 14:12 2

Mathematicians might well love to help in a practical question. They just might not have a clue how to do so. – Lee Mosher Jun 21 at 14:15 2

Suppose you only knew that the exam would be held and would be a surprise with certain probabilities, perhaps very high. Then one might hope to propagate the inductive reasoning through this probability, perhaps ultimately giving a probability distribution for when the exam would occur? Can someone give an answer along these lines? – Joel David Hamkins Jun 21 at 14:50

@Joel, the Wikipedia entry links to a Math Monthly article by Timothy Chow from 1998, which includes a discussion of an idea, attributed by Chow to Karl Narveson, along the lines you're asking about. – Barry Cipra Jun 21 at 20:27

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Thank you, Gerry Myerson, for that information. Meanwhile I have found a link on Google to this paper: What is the fade-away-rate of mathematical ... - MathOverflowmathoverflow.net/.../what-is-the-fade-away-rate-of-...‎21.06.2013 – The unexpected hanging paradox, hangman paradox, unexpected exam paradox, surprise test paradox or prediction paradox is a paradox ... It is probably the same that you mention. What can I do can I have a look at it? –  user84453 Jun 30 '13 at 10:14

If the problem that your adviser is suggesting is the hangman's paradox, then abandon hope. And I say this for your sake, your adviser is ill-informed.

The hangman's paradox is a bit like Fermat's Last Theorem. Everyone, even those without much background, can understand the basic idea of the problem. However, unlike Fermat's Last Theorem, everyone and their mother seriously thinks that they have a solution.

This problem is hard. Just look at the talk page on Wikipedia on the article. It's fifteen pages of everyone insisting that they have the solution. In order to be able to actually resolve this paradox you need to be very strong in arguments of logic, and 2 people who assume 2 different logic systems may very well reach 2 different equally valid resolutions. And it is very likely that neither will care about the other's solution.

It's hard to come up with a worse idea for a thesis for someone interested in teaching mathematics.

If I was to suggest a thesis for someone interested in teaching mathematics, it would be a research paper on how the history of mathematical knowledge was passed from person to person. We didn't always use equations. In the past most mathematical results involved a lot of prose and geometrical descriptions, and in the future who knows where we will go? It could be stable like what we do now, or it could move towards more lisp-like notations, or even somewhere else.

Knowing the history of the different way people understood math helps explain these things to students.

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