Sunday, November 27, 2011


One could see this as a demotion of this blog, and one would not be completely wrong. I am doing this, in part, to send Google a message: censorship is not acceptable. Engage in it, even temporarily, and what you'll get in return is a permanent loss. This was going to be where I wrote about mathematical counterexamples. What it will be, instead, is a support blog for that side blog - a collection of footnotes to the footnotes. Not an exalted position in the world of my scribblings, but a useful one, nevertheless.

What this will be, instead, is a guide to the reading to be done by those reading those side blogs, both in the library and on the internet - or, if you're unusually rich for a present day nerd, in the comfort of your study, as you look over the books you just purchased. You'd better be wealthy, though, because $50 is cheap for a Math book, these days. Oh - and no, I did not just insult my readership. Nerd is a label to be worn with pride, and I've been proud to do so, myself, as long as I can remember. What a strange, upside down society we live in, when intelligence, imagination and responsibility are seen as things to be ashamed of, while somebody who pours so much ethanol down his throat at the end of the week that he ends up acting like one of the apes at the zoo is seen as a hero, somebody to be emulated.

I'll be giving ISBN and library catalog numbers (both Dewey and Library of Congress) for the books, and links to the sites, which will open in new windows, as links to bookmarked sites should. I don't know that I can provide links to places where people can purchase these books. I'm sure that I can find them, and I wouldn't object to the notion of doing so, but if I did, I'd be worried that somebody at Google might decide that this was a spam blog. Even if the company, itself, doesn't feel that way, what if I run into some admin who is a little quick on the trigger, and unwilling to listen to reason? Ever try getting in touch with Google? I don't mean to be unsympathetic, especially if you have exams coming up, need clarification on a point, and the books I'm mentioning will be out for the next few weeks, but at the moment, this is probably the best I can do. If that changes, I'll tell you, but you shouldn't expect that I will.

The word "bookmarking" conjures up images of, and dry lists of links, pages that give the reader nothing more than an opportunity to escape them - like little tables of contents for the Internet. The mention of book reviews, maybe, leaves one picturing some minimum effort affair in which somebody says something like "good read. liked his proofs and the way he used illustrations. four stars all the way, get a copy now and buy one for your friends, they'll thank you later" - something like a newspaper movie review, condensed further. Yes, a lot of sites work that way, which is insane, because the space we are given is free, and we can't possibly write enough in a hundred year's time to fill it all. I don't know about you, but if I was still up and running around after spending a hundred years writing the same blog, I'd be in a mood to mothball the thing and go on to do something else. So, why impose space restrictions on oneself more stringent than those imposed on writers in traditional print periodicals, who work in media in which space really does cost money? I don't do that, and feel that those who do haven't really tried to understand the new medium in which they are working, or the new era, one in which some providers are now offering unlimited bandwidth and space, with mild limitations on the filetypes allowed - no movies, but all of the text and photography one wishes.

I don't do that - give you the Reader's Digest version. I couldn't - the resulting density of links would probably get this automatically flagged as a splog if I did - and I wouldn't, because such a blog would be no more fun to write than it would be to read. What I'll do, instead, is do what book reviewers did in the days before anybody coined that foolish acronym TL;DR and talk about what I'd have you read, at length. Criticism, in a positive sense of the word.

"Just sites and books about counterexamples?" No, because that would be a very short reading list for a continuing blog. One will need a fair amount of mathematical maturity to even understand these counterexamples, so I'll also be discussing the books that help you gain that maturity. For the ten steps this blog has taken downward, then, it has taken one upward. A demotion overall, but not a complete demotion - fitting, I think, as a response to actions that while definitely wrong on balance, weren't wrong in their totality. The punishment fits the crime.

Monday, April 19, 2010


A counterexample, in Mathematics, is a mathematical object that, merely by existing, shows that a proposed theorem can not be valid, the failure of validity often being quite strikingly counterintuitive. The construction of these objects serves an intellectual purpose, in that they set limits on that which is provable. They serve an aesthetic purpose, in that they can be strikingly beautiful in their own way. They also serve an emotional purpose, in that they can be wonderfully effective in quieting whiny students, when the students confront the grader over the points they lost in making an invalid assumptions in some of the proofs they submitted, and say "but thaaaaat's obviouuuuuuus!", reducing the probability of the hapless grader developing an ulcer by the end of the week. Of course, it's a dead on certainty that he'll have one by the end of the year, but by then he should be able to go home and have his relatives nurse him back to health, just in time for another year of pulse pounding action in the back of the library, red pen in hand and papers stacked high. Yes, live the dream. Oh, dear G-d.

This is a companion to "Doing Your Homework", a blog on which I'll be working a variety of math problems.