What does a blind mathematician do? (No, this is not a joke.)

T. V. Raman, a blind computer scientist who was trained as a mathematician, writes Thinking Of Mathematics —An Essay On Eyes-free Computing.

I think that even sighted mathematicians will get something from this, because the main issues for a visually impaired mathematician are that they cannot read or write in the usual way, and many of us do work in a situation where reading or writing is not available to us. Much of my best work gets done while walking to or from school, which is why I refuse to take SEPTA even though it would be faster. Plus, I get exercise that way. I've often taken to calling my own cell phone and dictating the solution to a problem into my voice mail. But this clearly isn't the same thing, because in the end I write things up in the traditional way.

Not surprisingly, Raman seems to find that the largest difficulties come in trying to communicate with other mathematicians, although this is becoming less of an issue as mathematics moves online, especially with the proliferation of TeX. (But this raises a question for me: often I write TeX that isn't strictly correct, but compiles anyway, and gives the right output on the page. How do systems like Raman's AS TE R (Audio System for TEchnical Readings, his Ph. D. thesis) handle this?

Paulos on "The Book"

Yesterday and today I read John Allen Paulos' Irreligion: A Mathematician Explains Why the Arguments for God Just Don't Add Up . A quite good book overall, although Paulos doesn't really cover any new ground here. (Paulos is a mathematician but the book is mostly devoid of mathematical content.) However, the book is a lot more lighthearted than some of the snarky anti-religion books that have been out there lately (say, Dawkins' The God Delusion, or Hitchens' God Is Not Great). Worth reading, although I'm not sure if it's worth paying $20 sure, since it's quite short.

But I just wanted to share the following:

Although an atheist, Erdos often referred to an imaginary book in which God has inscribed all the most beautiful mathematical proofs. Whenever he thought that a proof or argument led to a particularly exquisite epiphany, he'd say, "This one's from the book." (Alas, none of the arguments for the existence of God are even close to being in God's book.)

I suspect that some of the people who came up with such arguments thought they were proofs from the book, though. Indeed, are there any long arguments for the existence of God, or are all they all the sort of thing that can be written in half a page or so?

(And although Erdos' book is imaginary, Martin Aigner and Gunter Ziegler's Proofs from THE BOOK is real.)

Arthur Benjamin's mental arithmetic

Arthur Benjamin is a research mathematician (I've actually mentioned him before, although I didn't realize that until I looked at his web page and saw that the title of one his papers looked familiar...) and also a "mathemagician" -- he has a stage show in which he does mental calculations. See this 15-minute video of his show at ted.com.

He starts out by squaring some two-digit numbers... this didn't impress me much, because I could almost keep up with him. (And 37 squared is especially easy for me. One of my favorite coffeehouses in Cambridge was the 1369 Coffee House, and at some point I noticed that that was 37 squared. So I'll always remember that one.) Squaring two-digit numbers is just a feat of memory. Three- and four-digit numbers, though... that's a bit more impressive. And of course I'm harder to impress in this area than the average person.

One trick that might not be obvious how it works: he asks four people to each find a number 8649x (the 8649 was 93 squared, from the number-squaring part of the show) for some three-digit integer x, and give him six of the seven digits in any order; he says which digit is left out. How does this work? 8649 is divisible by 9. So the sum of the digits of 8649x must be divisible by 9. So, for example, say he gets handed 2, 2, 2, 7, 9, 3; these add up to 25, so the missing digit must be 2, to make 27? (How could he tell apart a missing zero and a missing nine? I suspect there's a workaround but I don't know what it is; the number 93 was given by someone in the audience, so I don't think it's just memory.)

He also asks people for the year, month, and day which they were born and gives the date; I found myself trying to play along but I can't do the Doomsday algorithm quite that fast... and I suspect he uses something similar. (I noticed that he asked three separate questions: first the year, then the month, then the day. This gives some extra time. I know this trick well; when a student asks a question I haven't previously thought about, I repeat it. I suspect I'm not the only one.)

The impressive part, for me, is not the ability to do mental arithmetic -- I suspect most mathematicians could, if they practiced -- but the ability to keep up an engaging stage show at the same time.

(The video is on ted.com, which shows talks from an annual conference entitled "Technology, Entertainment, and Design"; there look to be quite a few other interesting videos on there as well.

Product rules for more complicated products

Michael Hardy wrote a paper entitled Combinatorics of Partial Derivatives. He gives Leibniz' rule for the differentiation of a product in the rather unfamiliar form

which I am calling unfamiliar not because it's a particularly deep result, but because it's just not something one usually writes out. It could be proven by induction from the more "standard" product rule (uv)' = u'v + uv'.

But why don't we teach this to freshmen? Sure, the notation might be a bit of a barrier; I get the sense that a lot of my students learn the Σ notation for the first time when we teach them about infinite sequences and series, at the end of the second-semester calculus course; of course they learn about derivatives, including multiple derivatives, sometime in the first semester. (If it is true that they are seeing the Σ notation for the first time then, it doesn't seem quite fair, because then we're asking them to assimilate this weird notation and some real understanding of the concept of infinity at the same time. Indeed, at Penn we often tell them not to worry so much about the notation.) But ignoring the notational difficulties, fix a value of k -- say, 4 -- and get

so basically we notice two things: there are four primes in each term, and the coefficients are the binomial coefficients, which are familiar to most students.

One doesn't take the fourth derivative of a product that often; but even knowing that might be preferable to

Also, one can expand a rule like this to products of more than two terms; we have

Again, this doesn't come up that often, and I don't want to try to write it for derivatives of products of an arbitrary number of factors. Still, the idea is fairly natural but how many freshmen would even know that

?

I really don't know the answer to this -- but products of three factors are not incredibly rare, and the rule here is quite simple -- just take the derivative of each factor in turn, and sum up the results. There's even a nice "physical" interpretation of it -- how much does the volume of a three-dimensional box change as we change its various dimensions?

The coefficients seem kind of arbitrary, though; the point of Hardy's paper is that if things get recast in terms of partial derivatives they go away, both here and in Faa di Bruno's formula for the derivative of a composition of functions. One way to think of this is to imagine that, in the product rule, we have a "prime 1", a "prime 2", and so on up to a "prime k" if we're taking kth derivatives; we attach these primes to the variables in all possible ways, sum up the results, and then "collapse" them by reading all the primes to just mean ordinary differentiation.

Reference
Hardy, Michael, "Combinatorics of Partial Derivatives", Electronic
Journal of Combinatorics, 13 (2006), #R1.
http://www.combinatorics.org/Volume_13/PDF/v13i1r1.pdf

End quintile disparity!

A fictional "presidential candidate" on The Simpsons said something about how the top fifth of Americans consume sixty percent of the resources and the bottom two fifths consume only one eighth of the resources, leading to his slogan "end quintile disparity".

You just don't see that many statistics jokes.

In which I do not endorse a presidential candidate

So I usually don't talk about politics here. And for the moment, this blog will refrain from endorsing a presidential candidate. This is mainly because I haven't thought too hard about the presidential elections, because the Pennsylvania primary isn't until April and the nominations will probably be decided by then; the election that really matters for me is the general election, and I don't want to get too attached to a particular candidate right now since they may not be in the general election. (Here's an interesting interview with William Poundstone on different methods of voting, via Slashdot.)

But it occurs to me that one thing that we should be against in a presidential candidate is pigheadedness of the sort that George W. "Stay The Course" Bush has shown. New information becomes available, and this is something that any presidential candidate -- well, really any president -- should take into account. They should not become wedded to their positions if new information becomes available. (By the way, I'm not saying that I want a president who decides what to do based on opinion polling that tells them whether they'll be able to keep their jobs. I want someone with principles -- but these principles should include a willingness to change their mind.) I don't want to go so far as to say "path-dependence is the scourge of history", but I'll say it inside quotes.

The fissured ceramics come out...

Mark C. Chu-Carroll dissects a psuedomathematical proof of God's existence. Here come the fissured ceramics...

One of Chu-Carroll's big points here is that mathematical notation should be used as a tool for communication, not just a way to make the author look smarter.

Scott Aaronson had a recent post Ten Signs a Claimed Mathematical Breakthrough is Wrong which is also of interest. His first criterion for identifying a flawed "breakthrough" -- "The authors don’t use TeX." -- seems in itself quite strong. Of course, this is not because TeX forces the author to be a good mathematician (wouldn't it be nice if such a program existed?). Rather, the use of TeX is something of a shibboleth; people who haven't bothered to learn TeX are, at the present time, probably outside the mainstream of the mathematical community. I'm not saying that this automatically means the author is wrong -- there's no reason why advances can't come out of left field -- but that's a strike against such an author that they have to overcome. Using nonstandard nomenclature or notation -- which Aaronson doesn't mention -- is in the same general area.