House

I've been watching a lot of House lately. Here's how that goes:

Hey, look, we are introduced to someone who feels a little queasy. Maybe...oh, nope. Their friend/uncle/neighbourhood shopkeeper just keeled over and had a seizure instead. How clever.

[title sequence]

Let's guess at the disease, shall we? Could it be...vasculitis? No, no, it doesn't present like this. Well, then she must be on drugs! (Interrupt medical proceedings for search for drugs at patient's home.) Check it out! We found some drugs! Not the kind we thought she was taking, but still, drugs. (Confront patient.) Oh, she lied! She lied! She was taking these drugs we found all along! (Treat patient.) She got bet...oops. She crashed. Now what?

Let's check our standbys. Well, we ruled out vasculitis in the beginning, so...Lupus? No? What about paraneoplastic syndrome? Yay!

That sounds pretty bad, doesn't it? It's highly addictive, though. I understand why people are so into medical shows. I certainly feel like I'm learning an awful lot! Show me a sick person, and I'll diagnose that sucker! Also, I'll intubate 'em.

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)
  

7: Caro's Book of Poker Tells, Mike Caro

The title pretty much sums it up. Poker tells. This is still fall-out from Shirley and Tony's poker evening.

6: The Professor, the Banker, and the Suicide King: Inside the Richest Poker Game of All Time, Michael Craig

This was on super-extra-mega sale at Amazon. Now I know why.

The jacket informs me that Mr. Craig has written for such excellent publications as Cigar Aficionado. Jesus. Couldn't they have warned me? I guess they tried, but I didn't read the cover blurb. Had I been clever, and expanded the "See all Editorial Reviews" link on Amazon, I would have learned that Michael Craig is the author of The 50 Best (and Worst) Business Deals of All Time and The 5 Minute Investor. (Warning: cheap joke ahead.) My investing in this book was one of the worst business deals of all time.

The writing is men's magazine quality, with an annoying sensationalist style and limited vocabulary. The author repeats and repeats and repeats, states opinions as fact, and treats rather commonplace happenings as rare and exciting.

The material is interesting, but forty pages would have been enough. If you feel like you want a fluffy poker read and have read the good ones already, it's worth checking out from the library, but only after reading All In and Alvarez's The Biggest Game in Town.

5: The Athenian Oracle: A Selection (1892), John Underhill, Walter Besant

This is a fantastic selection of excerpts from the world's first newspaper (depending on your definition, etc.), the 17th century London-based Athenian Oracle run by John Dunton. The paper's most interesting feature was The Athenian Oracle, a Q&A column that was part Ann Landers, part Straight Dope. Questioners covered courtship, science, religion, folklore, law, and many other aspects of 17th century life in England and, occasionally, the colonies: there are two tobacco-related questions from Virginia.

Amusing bit from the introduction:

The truth is, Dunton was intensely interested in love-making and in politics--pursuits which take up considerably more time than the young man who wishes to succeed in the world can well afford to spare.

The chapter on popular seventeenth century science is particularly good, including an (accurate) explanation of rainbows and (a most inaccurate) one of "[why children are] oftener like the father than the mother". Bitchin' readin', y'all!

There's an appendix with ads from The Athenian Mercury. Cool, cool, cool!

Difficult Problems

It's fun to think about very difficult problems in mathematics. There is no danger of solving such a problem, so the stress that is associated with easy problems ("if I don't solve this as fast as the guy down the hall, I must be an idiot") is absent. Difficult problems also lead to a large amount of dry wit. This may be due to their difficulty, or due to the people that tend to think about them. I am not sure.

Today I happened upon a thread on open problems in mathematics that are easy to explain on the xkcd maths forum. I didn't feel like doing work, so that was an ace discovery. Two fun problems were mentioned: P?=NP and Ramsey numbers.

P?=NP

P?=NP asks whether a certain class of problems that are "very difficult" are, in reality, merely "rather difficult". Or, closer to the truth, whether problems for which a proposed solution can be checked "rather quickly" can also be solved "rather quickly". "Rather quickly" can still mean several billion times the lifetime of the universe, but it is well defined and quite a bit faster than "rather slowly".

A poster in that thread wrote this rather charming bit:

Re: Open Problems That Are Easy to Explain
Post by EstLladon on Tue Apr 01, 2008 9:15 am UTC

It is slightly off-topic and probably not useful, but lately I like to think about P=NP problem like this: suppose there is an art critic, who can by looking at a picture immediately say whether it is good or bad. Can he draw a good picture? Sure he can - he can just draw a picture, if he does not like it he throws it away. Doing that a lot of times he will at some point draw a good picture (this is mathematical world, so there is only a finite number of possible pictures of given size). But this takes a lot of time. Can he do it faster? (like drawing a small but good piece, and then trying to draw another one trying not to ruin it :) )

That was cute enough to remind me of a paper I read recently. In 2002, William Gasarch decided it would be a good idea to take a poll among mathematicians and computer scientists to check the prevailing opinion: does P=NP or does it not? Can the art critic do it faster, or not? The author's introduction is a kind of "eh, this problem is too hard to actually think about right now":

The P=?NP problem has been open since the early 1970’s. Many people-hours have been spent thinking about it and many (perhaps irrelevant) things are known. Are we making progress? This is hard to say for sure. When will it be solved? Also hard to say. Lacking the mathematical tools to answer these questions rigorously we turn to rather non-rigorous means—polling. Almost all respondents are people whose opinions can be taken seriously.

It contains some wonderful bits like this one, courtesy of Ron Fagin (IBM Almaden Research Center):

I’ve proven (at least twice) that NP does not equal co-NP (and hence P does not equal NP). I’ve also proven (also at least twice) that NP equals co-NP. My most recent proof that NP does not equal co-NP occurred about a week ago as I write this, and the proof survived for about half an hour (not quite long enough for me to run it by someone else). My longest-surviving proof that NP does not equal co-NP (about 5 years ago) survived for about 3 days and fooled some very smart people into believing it.

and my favourite, from Stuart Kurtz (U of Chicago):

Knowing Ketan Mulmuley, I live in fear that the solution will be via algebraic geometry, and it will come soon enough that I’ll be expected to understand it. An alternative nightmare is that the undergraduate who solves it will publish his solution in French.

Ramsey Numbers

Mathworld explains Ramsey numbers better than I can, and instead of pilfering their article, I'll just direct you there.

Now that you know what they are (or not, but don't worry, all you need to know is that R(x,y) is some integer that is infernally more difficult to find as x and y grow even a tiny bit), you can appreciate this Paul Erdos quotation, courtesy of that same xkcd thread & Wikipedia:

Imagine an alien force, vastly more powerful than us landing on Earth and demanding the value of R(5, 5) or they will destroy our planet. In that case, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they asked for R(6, 6), we should attempt to destroy the aliens.

So that's the kind of stuff I read. Anyway: I'm three books behind or so. One of these days I'll post about them.