Thursday, November 02, 2006

A Thesis to Forget

It is rumored that a graduate student once wrote a whole thesis on the set of functions mapping reals to reals such that for some fixed c and α>1,
|f(x)-f(y)| ≤ c|x-y|α
for all reals x and y.

A committee member then asked what happens when you compute the derivative of f.

[From The Way of Analysis by Robert Strichartz]


  1. The minatory lesson (which is ?) of this seeming urban legend is just as interesting IMHO as its mathematical humor!

    Over the years I have heard this same story in many variants; very likely it communicates an important social message. Which is?

  2. Can someone explain the story to me? I really didn't get it.

  3. α is meant to be smaller than 1 i suppose?

  4. I vaguely remember another legend, of a research in Sociology that has given evidence to what turned out to be a special case of Ramsey's theorem. Can anyone with more details enlighten me?

  5. Hint for anonymous: let y approach x and use the definition of the derivative.

  6. Eldar, it is a legend because no computer scientist believes it is possible to have more than one friend. : )

  7. I heard this story in an undergraduate math class. The lesson there was that concrete examples are important (in addition to good abstract theory).

  8. Do you have a page number for where he mentions this "rumor"?

  9. How many different urban legends can you
    make out of this:

    There was a PhD defense in Math in
    the topic of CHOOSE(Set Theory,
    Logic, Category Theory, Analysis, Geometry)
    at the school CHOOSE(Harvard, Yale, Standford,
    Berkeley, MIT, OTHERS) where the student
    was just finishing when it was pointed out
    that the object he was studying
    CHOOSE(didn't exist, was the empty set,
    was N, was a contradictory theory,
    was the set of all constant functions).

  10. Is this an urban legend? OMG. A professor mentioned this in an undergrad class back in 1989! He alluded that it really happened at the same university.

  11. "He alluded that it really happened"

    They always do.

    And the lady who microwaved her cat
    is my great aunt. Really! :-)

    bill gasarch

  12. Well, considering that the greatest of logicians spent a significant part of his life studying a contradictory theory, I don't think the joke is on the student, after all.


  13. I witnessed first hand a similar exchange at a conference. Person A describes many neat theorems about objects of type X, audience member asks "can you give *one* example of an object of type X", researcher can't produce an example.

  14. Somebody was telling stories about expander graphs some decades ago.

  15. By the way, Lance, you ought to mention your story in the New Scientist on the blog. Congrats on the ink?

  16. you mean there is only one solution

  17. Personally, I love urban legends & I hope Lance posts more examples of mathematical ones. "The truth is precious: let us economize" (Twain).

  18. For Anonymous 8, the "rumor" is on page 164, exercise 2. I first heard it when I took this course from Strichartz back in the 80's.

  19. To anonymous 16: Set y = x + h and divide both sides by |x - y| to get |f(x+h)-f(x)|/h <= c*h^{\alpha}, \alpha, h > 0. Since it holds for any h > 0, take the limit as h-->0 to find that limit of the left is 0 (i.e. the derivative is 0). So yes, f is constant.

  20. I have heard this rumor so many times, in some cases with inconsistent variations, that I should ask Strichartz about its actual foundation. It could indeed be an urban legend.

    You know, the famous quip from Wolfgang Pauli that a mediocre speaker's or author's work was "not even wrong" seems to be mostly an urban legend. I could not find a clear path to a specific incident using Google, Google Print, or Amazon books. What I did find was a passage in a letter from Pauli to Einstein in which he obliquely raises the general concept of not even being wrong. In this letter he doesn't use it as a sarcastic criticism of a specific person. The anecdote may have been inspired by this letter, but if so, Pauli may have been a more polite person than the anecdote implies.

  21. With respect to Greg Kuperberg's interesting post on what famous scientists are supposed to have said, versus what they actually said, I can supply the following example.

    Back in the mid-1970s, UofC physicist Valentine Telegdi was fond of quoting the following maxim to his graduate students, which he attributed to Dirac: "Golden eras occur when ordinary people can make extraordinary contributions."

    Our UW QSE Group like this Dirac quote so much that we featured it prominently in yesterday's QSE Journal entry.

    But the question is, did Dirac ever say it, or say anything resembling it? As best we can determine, what Dirac actually said is summarized in the appended BiBTeX reference.

    The point being, that what Dirac said was very interesting, but not quotable. It seems that many (most?) stories and sayings undergo a "polishing" process that improves their aesthetic merit, at the cost of historical accuracy.

    Me, I enjoy both versions: raw and polished.


    editor = {H. Hora and J. R. Shepanski},
    booktitle = {Directions in Physics},
    author = {P. A. M. Dirac},
    title = {The Development of Quantum Mechanics},
    chapter = 1,
    publisher = {Wiley-Interscience, New York},
    year = 1978,
    pages = {6},
    mynote = {Lectures delivered during a 1975 visit to Australia and New Zealand. "[In the eary days of quantum mechanics\ldots ] It was a good description to say that it was a game, a very interesting game one could play. Whenever one solved one of the little problems, one could write a paper about it. It was very easy in those days for any second-rate physicist to do first-rate work. There has not been such a glorious time since. It is very difficult now for a first-rate physicist to do second-rate work."},}

  22. Apologies for the broken link ... hopefully this Daily QSE Journal link will work (it's mainly for engineers and history-of-science fans).

  23. Regtarding John sidles' comment of "polishing" quotes: If you look at Rene Descartes' "meditations", you'll notice he never explicitly writes "I think therefore I am". It is implied from his writings, of course, but it was polished too...


  24. Indeed you will not find that phrase in the meditations, but rather in the fourth part of his Discourse on Method. And while it has been translated from the french "je pense, donc je suis" it didn't receive -- nor need -- much polish in the process.