God Plays Dice

In proving the fundamental theorem of arithmetic to my students, I was establishing the fact that any number has a factorization into primes. The proof goes as follows:

By way of contradiction, say there are positive integers without prime factorizations. Then there is a smallest such integer; call it N. N is not prime, because then it would have a prime factorization. So N has some divisor a such that 1 < a < N, and we can write N = ab for some integers a, b greater than 1. By assumption, N was the smallest integer without a prime factorization, so a and b have prime factorizations and we can concatenate these to get a prime factorization of N.

I want to bring your attention to the bolded "we can write", which I definitely said while presenting the proof. (The other language might not be exactly what I used.)

Sure, we can write that. But we can also write "2 + 2 = 5". Or we could have just written "N = ab" at the beginning When a mathematician says "we can write X" for some statement X, they mean something like "X is true, for suitable values of some variables which might be contained in X that we haven't mentioned yet, and which we'll talk about now."

In short, mathematicians are only capable of writing true things, or so we'd want people to think from our writing. If only it were so easy!

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?

I just came across this article: Florian Cajori, History of symbols for n-factorial. Isis, Vol. 3, No. 3 (Summer, 1921), pp. 414-418. Available from JSTOR, if you have access. Cajori was the author of A History of Mathematical Notations, which is the canonical source on the subject of the history of mathematical notations; I will confess I have never seen a copy of his book.

I didn't realize how many historical notations there have been for the factorial. n! is of course the most common one these days. Γ(n+1) is seen sometimes, although I personally find it a bit perverse to use this notation if you know that n is a positive integer.

Supposedly Gauss used Π(n). Someone named Henry Warburton used 1^n|1, a special case of
a^n|1 = a(a+1)...(a+(n-1)). (This is a variant of the Pochhammer symbol. It's not clear to me what the 1 in the superscript means.) Other notations include a bar over the number and writing the number inside a half-box (with lines on the left and below). Augustus de Morgan is mildly famous for not using a symbol, and once said: "Among the worst of barabarisms is that of introducing symbols which are quite new in mathematical, but perfectly understood in common, language. Writers have borrowed from the Germans the abbreviation n! to signify 1.2.3.(n - 1).n, which gives their pages the appearance of expressing surprise and admiration that 2, 3, 4, &c. should be found in mathematical results." (I'm copying this from Earliest Uses of Symbols in Mathematics, although I learned it from somebody's office door. I found the Cajori article while looking for this quote just now.)

Apparently adopting any sort of symbol was resisted by some people, at least in their more elementary writings (textbooks for undergraduates and the like), because they didn't want to overload their students with symbols. I'm not sure if I agree with this for the factorial. A little thought experiment, though -- why don't we have a symbol for 1 + 2 + ... + n? (Although a bit of reflection convinces me that the reason is because we have an explicit formula for this, namely n(n + 1)/2.) But n! probably arises more often.

While I'm on the subject, you should read Knuth's Two notes on notation (Amer. Math. Monthly 99 (1992), no. 5, 403--422; arXiv:math/9205211v1), which suggests the notation [P] for "1 if P is true, 0 if P is false"; this turns out to be a quite useful generalization of the Kronecker delta. It also suggests notation for the Stirling cycle and subset numbers (those are, um, the Stirling numbers of the first and second kinds, respectively? or the second and first kinds? See, those names are better.)

Happy New Year!

Karl Fogel attempts to pay a bill for 0ドル, because of course you have to pay bills for zero, because otherwise the companies that issue them keep sending them.

In this case, the bill was for the purchase of a book from the Mathematical Association of America, so he wrote a check for e^iπ+1 dollars. They didn't deposit it, because "check needs to be wrote out in U. S. dollars", as they put it.

I can see a more legitimate reason for rejecting the check. Usually, when writing a check, one puts, say, "3.14" in the little box on the right, and "Three and 14/100" on the line. The reason for writing out the value of the check in both figured and words is for redundancy. (Although then why don't we write "three dollars and fourteen cents" on the line? I suppose redundancy doesn't matter quite as much when we're talking about sub-dollar amounts.)

So you might say he should have written "e to the i π plus one dollars" on the line. But even that seems a bit suspect, because e, i, and π are themselves bits of mathematical notation. It seems that he really should have written something like

"The base of the exponential function, raised to the product of the imaginary unit and the ratio of a circle's circumference to its diameter, plus one"

for the number of dollars he wanted. Of course, this is the sort of thing that makes it obvious why having a compact mathematical notation is a good idea. I am not enough of a mathematical historian to have looked at the way things used to be written, but from what I understand this is the sort of thing they would have written five centuries ago, and I can't imagine working like that.

Unfortunately, the fact that we have such a good mathematical notation creates another problem -- people think that they can just put a bunch of symbols on a page and not explain what they mean by them, and that's "mathematics". Terry Tao, at his blog, has lots of writing advice; of particular interest in this discussion is his advice to take advantage of the English language. Here he gives a couple dozen ways to say that two statements are true, which are logically equivalent but have a wide variety of connotations. To take two examples of his examples at random, "P(x) is true. Unfortunately, Q(y) is also true." and "P is satisfied by x. Similarly, Q is satisfied by y." might be logically equivalent but are philosophically (psychologically, emotionally, morally -- what's the right word here?) quite distinct. I think that mathematicians as a whole are not sensitive enough to the connotations of their words; this is useful when doing formal mathematics but not so useful when trying to express the results of it. Perhaps we kneel too much at the altar of Bourbaki.