Dorky Poll: Dubious Proofs

Over at Good Math, Bad Math, Mark explains “Proof by Contradiction,” a common mathematical technique that doesn’t translate all that well to politics. Whenever proof techniques come up, I always think about one of the very few things I remember from my graduate class on Math Methods.

We were talking about some sort of complex analysis technique– I don’t remember what it was– and the professor was drawing diagrams and doing contour integrals on the board to demonstrate whatever it was that he was talking about, and at one point he drifted into Proof by Invocation:

“So we integrate along this curve, and we see that we just integrated through the singularity. But we know that everybody– including Feynman!– has done this, so we continue on…”

I guess it all worked out in the end, but that was a singularly unconvincing step in the lecture. (I wrote the comment in the margin of my notebook, which is why I remember it to this day. And no, Jonathan Vos Post was not guest-lecturing that day.)

So here’s the dorky question:

What’s your favorite dubious proof technique?

What justifications have you seen people use, either in class or in research papers, that just don’t make any sense if you think about them?

Comments

  1. #1 Not Walter Rudin
    November 15, 2007

    “Trivial.”

  2. #2 Simple Country Physicist
    November 15, 2007

    I always called that proof by association. Which is why I don’t do proofs in physics lectures; I do mathematical demonstrations.

    On a more serious note, I have found that most physicists have problems with contour choices in demonstrations. Students seem to have this problem frequently when the contour is different in lecture from text, and many lecturers have trouble with following the text so they either use prestidigitation or they figure out a different contour (the honest ones?)

    Back on a lighter note, Feynman was a bit notorious about being cavalier in expounding why he used a particular choice of contour so those who ape his choices are doubly in the dark.

  3. #3 Stephen
    November 15, 2007

    “…but this is just another infinity, so let’s throw it away.”

  4. #4 Eric Lund
    November 15, 2007

    A dubious technique that is used far too often in plasma physics courses is a posteriori justification of assumptions. Most commonly, the lecturer assumes that the wave amplitude is small, grinds through the calculation, and finds–voila!–that the amplitude of the predicted wave is indeed small.

    Proof by assertion would be a close second. I have seen far too many manuscripts, and referee reports, in which the author states a position (in the case of referee reports, that something the authors have said is false) without showing or citing evidence in favor of said position.

  5. #5 Peter Woit
    November 15, 2007

    Proof by circular cross-reference:

    In the proof of a theorem, make a crucial step in the proof a lemma, which you then announce you will be proving later on in the course. Later on, prove the lemma by using the original theorem.

    I deny having ever (intentionally…) used this method in my own teaching.

  6. #6 such.ire
    November 15, 2007

    I like to call the “it’s trivial” proof the “proof by intimidation.”

  7. #7 Jonathan Vos Post
    November 15, 2007

    Invalid statistical arguments.

    When I was an undergraduate at Caltech, the war in Vietnam was raging. To flunk out meant getting drafted and shipped to the jungle. Stephen King wrote exquistely about this in “Hearts in Atlantis.”

    The Draft featured a pseudorandom assignment of birthdays to numbers from 1 to 366. They’d first draft the 18-year-olds whose birthday was #1, then those for whom it was #2, and so forth.

    So if your number was low, you were more likely to get drafted. If your number was high, they might not get to you.

    The night before the numbers were assigned, we stayed up late arguing about this. Someone demanded that I predict what number my birthday would be assigned.

    I said : “The mean is roughly 366/2 = 183. So I will probably be #183.”

    The next day the numbers were published. I was #183. “Didn’t look like a valid proof to me last night,” one fellow student said. “But somehow, in the light of day, it seems to have worked well enough for all practical purposes…”

  8. #8 John Novak
    November 15, 2007

    Engineering: “…By inspection.”

    Computer Science” “Assume without loss of generality….”

    (Sometimes there is in fact no loss of generality, but you do actually need to set that up properly.)

  9. #9 KeithB
    November 15, 2007

    “The rest of the proof is so simple, we can just call in the nighttime janitor to finish it for us, so we won’t bother.”

  10. #10 onymous
    November 15, 2007

    Your post suggests another dorky poll: what’s your favorite weird/funny/nonsensical/clever comment from a professor that you liked so much you wrote it down in your notes?

  11. #11 onymous
    November 15, 2007

    And now I remember that one of my favorite comments written in a notebook is also a dubious proof, of the “it’s trivial” variety. On how to treat overlapping divergences in quantum field theory:

    “I’m talking on a Scientific American level about some integrals here… but it’s clearly going to work.”

  12. #12 Jonathan
    November 15, 2007

    The Smart Man Proof: Euler (or Gauss, Reimann etc.) did X and he was far smarter than anyone in this room, therefore we conclude X.
    Jokingly (?) used by a number theory professor of mine.

  13. #13 Brad Holden
    November 15, 2007

    There is always “proof by successive publication”.

  14. #14 Natalie
    November 15, 2007

    While I don’t have one, my husband (a statistician) was complaining about one the other day after attending a math conference. It summarizes something along the lines of “Zero is neither positive or negative, unless you’re really smart and you need it to be one of those to fit your pattern.”

  15. #15 Chris Koeberle
    November 15, 2007

    My favorite classroom demonstration proof element is the proof by absolute value: “Oh, I must have made a sign error back there somewhere, so we’ll just negate this.”

  16. #16 Colin McFaul
    November 15, 2007

    onymous @11:

    “Some of you probably learned this in elementary school.”

    Referring to Gauss’s Law, in an upper-level undergrad E&M course at the University of Chicago. There were a few professors there that were particularly talented at producing such quotes.

    Earlier this week at a seminar, in response to a question, the speaker started stammering something about a German guy in Dresden. The questioner interrupted, “Well, there’s lots of German guys in Dresden.”

  17. #17 Jonathan Vos Post
    November 15, 2007

    Baby-boomer Cosmology proof by the “‘Scuse me while I kiss the Sky” Lemma.

    “A singular Conundrum: How Odd is Our Universe”, Adrian Cho, Science, Vol.317, 28 Sep 2007, 1848-1850.

    “… Fanfare for the common universe

    “Like a Jimi Hendrix power chord, the CMB reverberates through time. The harmonies in the electromagnetic echo reveeal the state of the universe when tyhe chord was struck immediately after the big bang…. WMAP researchers broke it down much as a musical chord can be broken into musical pitches…. Researchers measured the strengths of hundreds of harmonics and plotted them… Amd the cacophony, however, scientists detected some distinctive harmonies…”

    See also the Moody Blues methodology:

    “And you can fly
    High as a kite if you want to
    Faster than light if you want to
    Speeding through the universe
    Thinking is the best way to travel”
    [Mike Pinder, The Moody Blues]

    The latter has the spectral corollary:

    The Word (Graeme Edge)

    This garden universe vibrates complete.
    Some we get a sound so sweet.
    Vibrations reach on up to become light,
    And then through gamma, out of sight.
    Between the eyes and ears there lie,
    The sounds of colour and the light of a sigh.
    And to hear the sun, what a thing to believe.
    But it’s all around if we could but perceive.
    To know ultra-violet, infra-red and X-rays,
    Beauty to find in so many ways.
    Two notes of the chord, that’s our fluoroscope.
    But to reach the chord is our life’s hope.
    And to name the chord is important to some.
    So they give a word, and the word is OM.

  18. #18 Mark P
    November 15, 2007

    No example, but it reminds me of a cartoon I saw long ago: professor in classroom at board says, “It’s obvious,” then retires to his office to work on it for a long time. Returns to class and says, “As I thought, it is obvious!” Anyone know which cartoonist did that one?

  19. #19 onymous
    November 15, 2007

    #18: I don’t know the cartoon, but I have heard this anecdote told about a famous mathematician (which one seems to depend on who tells it). He is giving a seminar when someone interrupts and asks a question. He replies “oh, that is obvious!” and continues with the talk. A few minutes later he stops, confused, and thinks in silence for a long time. Then he says “yes, it is obvious” and again proceeds with the talk. A few minutes later he stops once again, thinks for a still longer period of time, and then says “it is wrong.”

  20. #20 CCPhysicist
    November 15, 2007

    Proof by Emphatic Assertion (a combination of #4b and #6).

    Most commonly heard at national meetings.

  21. #21 Harry Abernathy
    November 15, 2007

    Going back to my high school days, I remember a classmate in geometry who would always prove theorems by using the theorem itself in the proof.

    As for good professor comments: my physical chemistry professor was the type who spent the entire class doing derivations on the board, with his back to the class and mumbling. One day, a student asked a question about the derivation he was doing. The professor stopped, turned around, and said, “You’re kidding, right?” He then turned back around and kept writing.

  22. #22 Alejandro
    November 15, 2007

    Not having any direct contributions to the list (because of course all proofs by me and the people I associate with are 100% sound ;) ), I will contribute the best way to describe a proof that commits any of the listed sins:

    “That is a proof of genus one. It has a hole in it.”

    The generalization to proofs with n holes is left as an exercise for the reader.

  23. #23 Chad Orzel
    November 15, 2007

    I didn’t think of this initially, but another old favorite is Proof by Inaccessible Citation. You just say “As shown in Ref. 17,…” where Ref. 17 directs the interested reader to a set of stone tablets atop a peak in the Himalayas, guarded by a mythical beast with the head of a lion, the wings of an eagle, and the body of a turtle, who will ask you five questions…

    Or pretty much any journal without a web interface for their back issues, these days.

  24. #24 capella
    November 15, 2007

    My PhD advisor was fond of saying “As you learned in Freshman Physics…”. He seems to think these courses are several orders of magnitude more rigorous than they actually are.

  25. #25 Harry Abernathy
    November 15, 2007

    One of the funniest discussions of this sort of stuff is the satirical “The uses of fallacy” by Paul Dunmore. It was originally published in the New Zealand Mathematics Magazine (7:15, 1970), but you can find it online in many places, including

    http://www.komplexify.com/math/humor_pure/UsesOfFallacy.html

    It describes many of the techniques already posted here.

  26. #26 Bob Oldendorf
    November 15, 2007

    The SJ Harris cartoon, of the scientists writing out the proof on the blackboard, with the critical step:

    “Then a miracle occurs”

  27. #27 Coin
    November 16, 2007

    I think my favorite dubious proof method has to be “the conjecture is fairly old and has received a lot of attention, and has been neither proved or disproved, therefore clearly it must be [ true | not true ] because if it were possible to [ disprove | prove ] it then surely someone would have done so by now”.

    The trick to this one is that this is, while not proof, still often a pretty strong argument. But it’s still not valid, because using it depends on you actually being able to tell whether proving or disproving that particular conjecture would be “harder” (in other words, whether protracted inability to prove implies truth or falsehood). And of course you don’t know which one is harder unless you’ve actually proved or disproved it.

    Of course, knowing this is not valid somehow does not seem to stop me from using this argument from time to time. Incidentally, string theory doesn’t work and P != NP…

    A related trick is (there’s probably an actual term for this on a list of logical fallacies somewhere) what I might call the “proof by necessity”, where some conjecture is eventually assumed to be true because an entire science somewhere has been built up around assuming it to be true, using it to prove things, etc. The Riemann Hypothesis / the Malcadena conjecture / etc has to be true, because if it isn’t, then a lot of people are screwed. By the way, did I mention that P != NP?

  28. #28 Coin
    November 16, 2007

    what’s your favorite weird/funny/nonsensical/clever comment from a professor that you liked so much you wrote it down in your notes?

    Hm, well, neither of these are funny, but one way or another the two sentences that still stand out in my mind above all the others from college were (from two different classes taken the same semester, incidentally):

    “A proof is a social process.” (Randomly announced by the professor as part of a prelude to a proof he was about to provide, in which for some reason I don’t clearly remember he felt compelled to emphasize that in providing the proof, his job wasn’t so much to establish the absolute truth of the theorem as it was just to one way or another convince the people in the room that it’s true. Kind of relevant to this thread, actually…)

    and

    “A type is a set.” (Referring here to types in programming languages… though this one may not seem like any particularly deep insight out of the context in which it came up, actually.)

  29. #29
    November 16, 2007

    I seem to remember reading a story about Feynman – perhaps in Surely You’re Joking – which stated he never learned contour integrals as he was always able to do whatever integrals he needed by differentiating under the integral sign, until one day his friend gave him an integral which had to be evaluated by contour integration and Feynman couldn’t do it. So, if this story happens to be true, then Feynman didn’t contour integrate through a singularity, and so the above Proof by Invocation is invalid!

    As for my favourite professorial comment, in my Modern Physics class the professor was leading into Schroedinger’s wave equation by talking about the usual equation for waves on strings and contrasting the two, and as he was talking about them, he paused for a moment, and said, “I was about to say, ‘But an electron is not a string’, but there are people who might disagree with that.”

  30. #30 HennepinCountyLawyer
    November 17, 2007

    #19

    I heard that one from my college roommate, who heard it from his father, a math professor.

  31. #31 Josh
    November 19, 2007

    I had a DEQ professor who would always use his magical future sight abilities to decide “Okay, we’re going to call this big mess of things the ‘initial velocity’… you’ll see why soon!” and then after some arduous derivation, we get an ugly equation, and, by golly, there’s our mass of things that we’re arbitrarily decided to call initial velocity earlier! It was astounding.

    Luckily, he was a nice guy and would do that on our homework, too… “When you get the result, note that is actually equal to the initial velocity!”

    Right now I have a math prof who is actually pretty good about things. She’ll actually refuse to speculate on things that she doesn’t know! It’s astounding.