The G*d*mn Particle

Another fabulous anecdote from the RJLipton Blog, regarding the Higgs Boson (what's popularly called "The God Particle"):

Higgs himself believes neither the particle nor the mechanism should carry his sole name, and was happy that he, Brout, Englert, and the three authors of another 1964 paper (Gerald Guranik, Carl Hagen, and Tom Kibble) were all awarded the 2010 J.J. Sakurai Prize for this work. He may have gotten his wish, as the popular name “The God Particle” has stuck to the boson. This is the title of a 1993 book by Nobel prize-winning physicist Leon Lederman and science writer Dick Teresi.

According to Higgs, Lederman had wanted to title the book The G*d*mm Particle to emphasize how elusive the boson was. His publisher declined to have a swear word in the title, but thought it fine to use just “God.” However, they could have settled on the Orthodox Jewish practice of writing “G-d” to avoid situations where the fully-written name might be erased or discarded. The title The G-d Particle could then be read with Lederman’s original meaning or not. Higgs is said to join many scientists regretting the “God Particle” name, more from concern over hype than irreverence.

I love this story so much. First, it finally makes sense of that stupid name in a way that eluded me until now. Secondly, it again shows that the "real" existential experience of scientific problem-solving is more generally one of a desperate, teeth-grinding, curse-filled battle (and not so much a dainty and refined observation of museum-like beauty).

Divine grace is a marketing pitch you use to sell something to the public. It's not something seen in the real world, or actual live math work, very much.

On that note, happy holidays from MadMath! :-)

Dyson Quote

An excellent MadMath-approved quote from Freeman Dyson:

If science ceases to be a rebellion against authority, then it does not deserve the talents of our brightest children.

The Peanut Butter Protocol

Here's something that pops up in math/computer science that I honestly just HATE so much (I got sufficiently riled up while describing it to my girlfriend tonight that I thought it would make a perfect blog post). On the question of "How do you introduce programming concepts to students for the very first time (possibly children)?", a very common answer is "Ask them to give the steps for making a peanut-butter and jelly sandwich!" (or something very similar). For example, whenever this comes up on Slashdot, the responses are predominantly along the lines of "love this... lovely... hilarious" (link). But I'm completely contrarian about it.

Of course, the point is basically a "gotcha" exercise: the students say "scoop out the peanut butter" and you go "what!? look, now I'm batting the jar-top with my hand, because you didn't tell me to pick up the knife, and you didn't tell me to screw off the jar-top," etc., etc. etc. If I was a student, and this my first encounter with computer programming, then it would instantaneously sour me on the whole subject, maybe permanently: the task is inherently ambiguous, impossible, unfair, and a trick to apparently set up the respondents for ridicule and embarrassment.

The primary problem (in my opinion) is that's very much not how mathematics or computer programming work. What we must do in practice is to start with an agreed-upon set of atomic operations, which we may call "definitions" or "axioms" or a "function library", depending on the context. Of course, the power of your elementary pieces is variable, depending on the abstraction level at which you're operating. But the real work of programming or proof-building is in how we connect these well-known (and well-defined) basic building blocks in a way that constructs something new, useful, and interesting.

So the "peanut butter sandwich" task is thoroughly and painfully unfair without presenting the allowed operations up front: Am I supposed to say "pick up the knife" or "wrap your fingers around the knife, apply opposing force with thumb, lift forearm" or "bend index finger 5 degrees, now 10 degrees, now 15 degrees..." (it's sort of irrelevant, because without well-defined operations, the presenter can always pick some lower-granularity abstraction and create a "gotcha!" moment). The demonstration does manage to get across the idea that you will be "working with small operations", and also that "unexpected bugs will happen" -- but in my mind, neither of those are essential or even very important. The essence of any creative work is in taking well-known basic tools and building something greater from them than previously existed, and that's something that I think almost anyone can understand and justifiably take satisfaction from.

(P.S. a counter-offer: Rudimentary programming like LOGO. Write on the board 3 allowed operations: (1) turn left, (2) turn right, and (3) step forward. Now direct me how to get from one corner of the room, around some desks, and out the door -- possibly listing the whole instruction set in advance of testing it. Something like that.)

Arguing Infinite Decimals

Recently the RJLipton blog had two interesting and contentious posts about people who dispute Cantor's diagonal argument (that real numbers have different cardinality than natural numbers), which I'm pretty sure generated more comments than anything else to date on the blog. Apparently this is one of the more popular topics for math-cranks to extensively argue that they've proven the other way -- read for yourself here and here.

I wish that I had the opportunity to address issues like this in the classes I teach, but unfortunately at the moment I don't have any such opportunity. It would be nice to have a venue to refine the argument with a fresh audience every so often, and to work to ferret out the criticisms that arise. If we do so, with a disputatious subject like this (namely: the first few times a student deals with infinite sets and their counterintuitive by-products), then I think it's extra-important that we carefully lay out initial definitions at the start, break down the argument into very atomic numbered steps (so that we can refine discussion and disputes as they come up later), and also give explicit justifications for each step.

Here's another issue which I feel has the same flavor to it: the fact that 0.999... = 1 (or more generally, that any terminating decimal has two different, equivalent representations: the normal one, and a second one that ends with an endless sequence of "9"'s). Here's a suggestion on the careful way that I'd want to do it (again -- not having had this battle-plan encounter the enemy yet):

Definition of 0.999...
(a) The number has infinitely repeating digits.
(b) After every "9" digit, there is another "9".
(c) There is no end to the "9"'s.

Proof that 0.999... = 1 (by algebra)
(1) Let x = 0.999...
(2) Then 10x = 9.999... (multiply each side by 10)
(3) So 9x = 9 (subtract step 1 from step 2; note decimals cancel)
(4) Which means x = 1 (divide each side by 9)
(5) Therefore 0.999... = 1 (substitute from step 1)

And then when the arguments arise you can at least ask your interlocutor to focus on one single step or definition in which they think there's a logical gap.

Statistics vs. the Lottery

So apparently there's an article in Harper's (can't see the original; link below is commentary at Forbes) on the following subject -- Joan R. Ginther has been "outed" as a statistics professor with a PhD from Stanford, who possibly deduced the winning-ticket lottery distribution schedule in Texas, and has hit multi-million dollar jackpots 4 times in the last decade. Notes:

(1) While I don't see any assertion of any way in which this would be illegal, the overall tone is clearly one of how-dare-she-think-she-can-get-away-with-this. “When something this unlikely happens in a casino, you arrest ‘em first and ask questions later,” says a professor at the Institute for the Study of Gambling & Commercial Gaming at the University of Nevada, Reno.

(2) "The residents of Bishop, Texas seem to believe God was behind it all."

http://blogs.forbes.com/kiriblakeley/2011/07/21/meet-the-luckiest-woman-in-the-world/

Less Time to Learn

Hypothesis: The less time students have to learn, the higher their testing scores are.

This has been a suspicion of mine for a while now. For example, I find that my accelerated summer/winter modules (6-week courses) generally outperform my normal fall/spring modules (12-week courses) in the subject material, testing procedures, etc. I'm guessing that the major factors involved are (a) a greater focus and more connections with the given subject material, (b) fewer competing courses being taken at the same time, vying for mental attention, and (c) simply less time and opportunity to forget stuff from class to class, which I feel is a real issue for many of my students. (Countering factor might be: Maybe more dedicated students register for summer/winter courses?)

So this summer I had an excellent accidental experiment in this regard. I'm teaching two statistics classes in parallel on Mon/Wed and Tue/Thu nights. There was a weird burp in the schedule (specifically, the Mon Jul-4 holiday) that caused one class to be ahead of the other by one evening's lecture. So heading into the last test (partly on hypothesis tests and P-values), the Mon/Wed class was first introduced to the subject just 2 weekdays (48 hours) in advance of the test, whereas the Tue/Thu class had a whole week (7 days) to see P-values and study for the test (including, obviously, a whole weekend).

So I was rather concerned that the Mon/Wed class was being unfairly put upon, what with such a short window in which to study, and on Wednesday they did seem to struggle. But then to my surprise it turned out that the Tue/Thu class found what was basically the same test even more challenging, and got a significantly lower average score on the same assessment.

On Tau

So recently there were some popular news articles with titles like, "Mathematicians Want to Say Goodbye to Pi" -- first I've heard of it, and of course initially it sounded ridiculous (I guess that's the point of news-article title-writing, eh?) The gist of it is that in theory, when dealing with circles, it would easier to exchange the value pi = circumference/diameter for tau = circumference/radius, i.e., tau = 2*pi.

And actually, that very quickly hit me as something that would be very nice to have. It would make a lot of trigonometry and calculus easier. The number of radians in a circle would simply be tau (instead of 2*pi). Perhaps most important for me, circles are inherently defined by their radius (all points a given distance from the center), not by their diameter.

Now my first attempt at an objection was the formula for a circle's area, which would get ever-so slightly more complicated, switching from A = pi*r^2 to A = tau*r^2/2. But that's a small thing, and in fact it reminds you of the fundamental integral(r)=r^2/2 which is used to derive it in calculus (instead of a disappearing denominator trick, canceled by the constant 2*pi).

The other thing that just occurred to me -- and motivated this post -- is what it does to Euler's identity, e^(i*pi) = -1 (or however you want to move the terms around). Now, I may be an angry crank, but if I think deeply about this celebrated identity (it was voted "most beautiful formula" in the Mathematical Intelligencer, 1990; a post which I have taped on the wall over my computer), it's not terribly interesting; granted that the imaginary part of the exponential function is a rotation in the complex plane, and coincidentally pi happens to be half a circle, i.e., landing on the point (-1, 0). If we used tau more commonly, then the triviality would be more apparent: e^(i*tau) = 0, and no one would get as worked up about it anymore. Or maybe people would think it's even more "beautiful" then, hell, I don't know. :-)

Am I going to try to switch the thousands-year legacy of using pi to tau? Not me, man, I've got enough to do without quixotic crusades. But yeah, if I could pick different historical legacies the options for (1) switch pi to tau, and (2) switch electrical current signs (link), would be near the top of the list.

What do you think?

Edit: Of course, e^(i*tau) = 1 (not 0). [Knocks self on head.] Maybe that actually is more beautiful.