Computational Complexity and other fun stuff in math and computer science from Lance Fortnow and Bill Gasarch
Sunday, March 23, 2014
The answer is either 0,1, or on the board
I have heard (and later told people) that the in a math course if you don't know the answer you should guess either 0 or 1 or something on the board. This works quite often.
I have heard that in a course on history of theater you should guess either
the theater burned down or prostitution. For example, the first musical
was The Black Crook and it happened because of a fire (see the pointer).
In upper level cell biology the guess is If only we could solve the membrane problem.
In a math talk you can always ask is the converse true? or Didn't Gauss prove that?
In computer science when someone asks me about a problem I say Its probably NP-complete.
In Christian Bible Study a good answer is either Salvation or Jesus. These are referred to as Sunday school answers.
If you know what the usual things to say in other fields is, please comment.
A variant on this in mathematics is that all you need to know to be a mathematician is how to add 0, multiply by 1, and (only if you're an applied mathematician) integrate by parts. It's shocking how often these are the things that you do...ReplyDelete
How about: "I don't know, and I'm not going to waste your time BSing."ReplyDelete
How do I solve problem XY?
-> Use a library function.
What's wrong with code XY?
-> Buffer overflow.
Does the NSA do XY?
Carl Jacobi Man muss immer umkehren (one must always invert)ReplyDelete
For a discrete math class I would add 'choose at random' to your list :-)ReplyDelete
In applied econometrics the guess is "if only we could find a good instrument(al variable)"ReplyDelete
In Physics, "Yes, but only in a vacuum."ReplyDelete
In Stephen Potter’s spoof self-help book One-upmanship (at least, I think that’s the one), you are advised that you can authoritatively contradict any assertion with the claim “Yes, but not in the south”ReplyDelete
In further regard to Carl Jacobi's Man muss immer umkehren, fans of complexity theory, quantum computing, and mathematical history will alike find plenty to enjoy in Edward Van Vleck's AMS Presidential Address "Current tendencies of mathematical research" (1915), which was reprinted in a Bulletin of the AMS theme issue of the same title (vol 37(1), 2000).ReplyDelete
At the risk of appreciating Lance's question too seriously, Van Vleck extends Jacobi's to question to (in effect) Man muss immer umkehren, naturalisieren, verallgemeinern ("One must always invert, naturalize, generalize"). And there are plenty of examples of this in Van Vleck's own lecture. E.g. Van Vleck asserts
"Extension to a countably infinite number of variables commonly affords opportunity for the finest sagacity and insight. An excellent program for work could be found in extension of almost any finite theory. It is your own Professor Moore whom I have heard glowingly preach that to every finite theory there must correspond, under proper limitations, a general transcendental theory with an infinite number of variables."
By inverting, naturalizing, and generalizing Van Vleck's recommendation, what do we obtain but Cartan, Eilenberg, Mac Lane, and Grothendieck's great program of 20th century algebraic geometry? In which Van Vleck's advice is inverted to extend the continuous to the discrete, and naturalized per the category-theory formalizm of Eilenberg and Mac Lane, and generalized per Zariski's topology and Grothendieck's sheaves and motives?
Needless to say, there remain plenty of opportunities to apply Van Vleck's rule in the 21st century. E.g., when we invert, naturalize, generalize the principles of (for example) the great article "Non-abelian anyons and topological quantum computation" (by Nayak, Simon, Stern, Freedman, and Das Sarma, 2008, arXiv:0707.1889v2), then we obtain a program that looks a lot like the scientifically successful "Quantum Hall effect as an electrical resistance standard" (Jeckelmann and Jeanneret, 2003) as instantiated in the mises en pratique of the technologically transformational "new SI Program" of the Bureau International des Poids et Mesures.
Van Vleck's invert, naturalize, and generalize principle substantially informs both our engineering appreciation of Grothendieck's work and the rising mathematical tide of our quantum system engineering Green Sheets.
These questions are sufficiently fundamental that it would be regrettable if everyone felt alike in regard to them. And so the above historical and technological perspective is offered in the same subversive spirit as Donald Saari's introduction to the Bulletin of the AMS/Van Vleck theme issue:
"It is hoped that these articles and reviews will encourage readers to enter the dusty, far reaches of a mathematics library to thumb through old Bulletin issues. I promise you a delightful afternoon."
Similarly, we can reasonably foresee that young researchers who creatively embrace Van Vleck's advice to "invert, naturalize, and generalize" will enjoy delightful 21st century careers.
I have heard a story told (probably apocryphal) of a theoretical physicist who always went to experimental talks, sat in the front row, and fell asleep at the beginning of the talk. At the end, he would wake up, and always ask the same question: "What would it be like if you ran you experiment at low temperature?" To which the speaker almost invariably replied, "Ah! That is a very interesting question..." and would expound upon the implications. Of course, one time the response was, "But the whole point of my talk was that we ran the experiment at low temperature." So our theorist said, "Oh, but what I meant was, what if you ran it at really really low temperature?" To which the speaker replied, "Ah! That is a very interesting question..."ReplyDelete
I once heard the story about a mathematician's wife who always felt excluded at dinner party's. Her husband gave her secret signs so she could say eitherReplyDelete
1) But does this also work in the infinite case?
2) Didn't Gauss do this? (indeed!)
3) Do you really need a Banach space for that?
My favorite trick when I *have* to ask something at a talk I didn't understand is to take any assumption and ask what would happen if the assumption wasn't valid.
Hartmanis' Alternative How does [open CT postulate] alter when oracles are [introduced/excluded]?ReplyDelete