8: Unknown Quantity: A Real and Imaginary History of Algebra, John Derbyshire

Robert lent Prime Obsession to me a year or two ago, and now this.

Derbyshire writes excellently. Apparently he's a bit of a conservative crackpot (and my republican friend told me this, so it's gotta be pretty bad), but he sure can write. ("Yeah, she's ugly, but she sure can cook.")

So. Random things I learned:


Cardano was pretty much an interesting bad-ass. We all know he solved the cubic, blablablah, but did we all know he wrote Consolation, a book of advice for the sorrowing? And that Hamlet's soliloquy reminds one (if one has happened to have read a little-known sixteenth century work on sorrow) of some remarks on sleep in that book? No, we didn't.

Cardanas: that's Dutch for the universal joint axle that transfers power from the engine to the drive axle in a car. It's named after Cardano. We didn't know that either. Now we do, and we're better humans for it.

Finally, Cardano was an early acceptor (well, kinda) of complex numbers. This quotation from his Ars magna is lovely:

Putting aside the mental tortures involved, multiply 5+√-15 by 5-√-15, making 25-(-15), which [latter] is +15. Hence this product is 40. [...] This is truly sophisticated.

You've got to be impressed with that last sentence. (Also with my ability to display square roots in HTML.) Up there with Snoop Dogg's response to the suggestion that rappers' derogatory references to women are similar to Don Imus' calling women's basketball players "nappy-headed hos":

It's a completely different scenario, [rappers] are not talking about no collegiate basketball girls who have made it to the next level in education and sports. We're talking about hos that's in the 'hood that ain't doing shit, that's trying to get a nigga for his money. These are two separate things.

Rob, who pointed the above out to me had this to say: "My only reaction: Who thought, hey, this is a bit confusing, but Snoop Dogg can probably clarify matters?" (Are you reading this, Rob?) Anyway—that last sentence is gold again.

Enough about Cardano and Snoop Dogg. Unknown Quantity contains many, many pages that mention neither. What do these pages mention? Some of this:

In a section dealing with mathematics in the Han dynasty in China, this marvelous bit of unexplained fact:

A calendar was duly produced, based on the usual 19-year cycle.

There is a usual 19-year cycle? It's clearly not usual enough for me to be familiar with.

In a bit about Sylow's three theorems and p-subgroups:

And it is infallibly the case, at any point in time, that somewhere in the world is a university math department with a rock band calling themselves "Sylow and his p-subgroup."

On the difference between the Berlin (rigorous and pure) and Göttingen (imaginative and geometrical) schools of thought in the second half of the nineteenth century:

Weierstrass and Riemann exemplify the two styles. Weierstrass, of the Berlin school, could not blow his nose without offering a meticulous eight-page proof of the event's necessity. Riemann, on the other hand, threw out astonishing visions of functions roaming wildly over the complex plane, of curved spaces, and of self-intersecting surfaces, pausing occasionally to drop in a hurried proof where protocol demanded it.

After asking a rhetorical question of the "who knew..." form:

Well, I knew, having plotted that curve with pencil, graph paper, and slide rule during my youthful obsession with Cundy and Rollett's Mathematical Models, which gives full coverage to plane curves as well as three-dimensional figures.

The reader who at this point might be beginning to suspect that the author's adolescence was a social failure would not be very seriously mistaken. In partial defense of my younger self, though, I should like to say that the now-lost practices of careful numerical calculation and graphical plotting offer—offered—peculiar and intense satisfactions.

Books added to my reading list due to mentions and references:

• Oystein Ore, Cardano, the Gambling Scholar
  
• Dionys Burger, Sphereland
  
• Ian Stewart, Flatterland
  
• A. K. Dewdney, The Planiverse
  
• Augustus De Morgan, A Budget of Paradoxes
  
• Allyn Jackson, As If Summoned from the Void: The Life of Alexandre Grothendiecke in Notices of the AMS, 2004
  
• Eugene Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Sciences (essay)