There are only 12 posts on Jim Propp’s blog Mathematical Enchantments so far, and they are all superb. Propp is a professor at the University of Massachusetts Lowell, and his blog is different from a lot of blogs I read. He only posts once a month, on the 17th, and his posts are quite long and in depth. Some math bloggers who take the long post approach can veer towards academic-ese, sometimes ending up with posts that read a bit like journal articles. (There’s nothing wrong with that, and those posts can be very valuable to researchers interested in those specific topics.) Propp’s posts don’t feel long, they feel leisurely. He takes a topic and slowly unfolds it, trying to make it accessible to people regardless of their mathematical background.
I decided to write about Mathematical Enchantments now because of his most recent post, The Curious Incident of the Boasting Frenchman. I try to hide it, but I have a strong curmudgeonly streak deep down inside, and one of my inner curmudgeon’s pet peeves is when people uncritically report that Fermat had a proof of Fermat’s Last Theorem that would not fit in the margin of his copy of Diophantus’ Arithmetica. In this regard, Propp seems to be a kindred spirit, though he is not very curmudgeonly about it.
Fermat’s Last Theorem is the statement that there are no positive integers x, y, and z so that xn+yn=zn for n an integer greater than 2. Fermat did indeed claim that he had a proof which could not fit in the margin, but the proof we know was only completed in the 1990s by English mathematician Andrew Wiles using techniques far removed from any Fermat would have known. Was Fermat bluffing, was he mistaken, or did he indeed have a proof that we have just failed to recreate for more than 300 years? Propp explores those options and includes information about the mathematical culture Fermat was a part of at the time. Along the way he walks us through the idea of infinite descent, one of my favorite proof techniques.
(Serendipitously, Pat Ballew just published a post about “almost integers,” numbers that are surprisingly close to being integers, like eπ√163, which is close enough to 262,537,412,640,768,744 to fool a lot of calculators. Ballew doesn’t mention them in his post, but some of my favorite almost integers are ones that come from Fermat “near misses,” numbers thatare close to satisfying Fermat’s equation for some n other than 1 or 2. Harvard mathematician Noam Elkies has a page of Fermat near misses for your perusal.)
One of the things I like about Mathematical Enchantments is that Propp often writes about subjects I’ve seen a hundred times, but I still learn from his posts. He often brings more depth or historical context to the discussion than the topics usually get. For example, too many proofs that 0.999…=1 rely on intimidation. Instead, Propp digs into how the expression 0.999… can mean anything at all, going back to Archimedes and forward to the hyperreals for different interpretations of infinitesimals. The math he covers is always interesting but never feels like a party trick, even though one of his posts is literally about a party trick—the trick of harnessing projective geometry and symmetry to rotate children through paintball teams optimally at a birthday party.
Each post on Mathematical Enchantments takes some time to read and digest, but they are worth the effort. Besides your own edification, they are great examples of mathematical exposition done both accessibly and deeply, and I know I am always looking for inspiration in that direction.