Isotropy
Tuesday, January 03, 2006
In Chapter 2 of "Model Theory", Wilfred Hodges lists three ways that mathematicians use formulas:
"First, a mathematician writes the equation 'y=4x*x'. By writing this equation one names a set of points in the plane.... As a model theorist would put it, the equation defines a 2-ary relation on the reals....
Or second, a mathematician writes down the laws
- ∀ x, y, z, x ≤ y and y ≤ z imply x ≤ z;
- ∀ x,y exactly one of x ≤ y, y ≤ x, x=y holds.
Third, a mathematician defines a homomorphism from a group G to a group H to be a map from G to H such that x=y*z implies f(x) = f(y)*f(z). Here the equation x=y*z defines a class of maps."
From this starting point, how should a model theorist regard the skein relations of knot theory?
These specify a small area of a knot where differences are allowed - outside the displayed area, the three knots must be identical.
Thursday, September 15, 2005
Why Religious Freedom?
Michael Newdow is back in court, still trying to strip "under God" from the Pledge of Allegiance. Whatever - but this issue got me wondering again, why is religious freedom important? It IS important, but WHY is it important? It tends to get squashed into the same elevator as "free speech", and people ride that all the way to "freedom of conscience" as the truly important freedom.
But that's bunk: we use our conscience much more often, and with less cause or effect, than we speak uncomfortable truths, and most of us more often than we practice our religion. And the government impinges on the conscience of pacifists, death penalty opponents, and anti-abortion crusaders every day. You do, in fact, have to pay your taxes even if your conscience says "No" to war, or if the police you pay for refuse to arrest abortionists.
Religious freedom is important for the simple reason that most people believe in and many people pledge allegiance to a Higher Power than the state. The government can react only a few ways to this fact: ignore it (what Newdow would have us do), embrace it (and become some kind of theocracy, and beat down the people on the wrong side), or acknowledge it as a potential conflict of interest and yield when necessary. Religious freedom is important because some troublemakers are so infused with God that they will defy Caesar openly if pushed too far. There are not enough re-education camps in all the fantasies of Stalin to house all of us who believe our actions have cosmic moral import.
Either people have the right to pledge their highest allegiance to a Creator, or they don't. If the Creator exists, clearly we have the right to believe we owe everything to Him. And if the Creator exists, the possibility of an *enforceable* obligation to Him exists. Since the question of Existence is still open, the government can't stop people from trying to ally themselves with a Higher Power.
From this perspective, what exactly IS the religious freedom of an atheist? Explicit rejection of the idea of any enforceable higher obligation does not allow an atheist to choose his own conscience as the Higher Power. The religious person can find them self saying to the government, "If you make me do this, you are forcing me to choose a finite prison or an eternal punishment." The atheist cannot find himself trapped in this dilemma. So why should his claim to religious freedom be as strong as any other? After all, logically it should mean less to him....
Tuesday, March 29, 2005
Primogeniture is Rational, Too!
Since my last post, six months ago, was on a stupidly clever way of picking a ruler, here's one with a stronger intellectual pedigree: Primogeniture. Why? It implements a depth-first search of the ancestry tree. Computer science meets Political Science, and we don't even bring voting into it. Now that's a smart system!
Tuesday, November 02, 2004
A Pox on IRV
Instant Runoff Voting (the soothing name for "Ranked Choice Voting") is not a big issue anywhere, but it's fun to write about on Election Day. Basically, everybody ranks the candidates in order of preference. The first-place votes are counted. If nobody has a majority, the lowest candidate is dropped, and his second-place votes are allocated to the remaining candidates....
"Hang on!"
What?
"You mean some people get to vote twice?"
Well...yeah.
"But only if they're stubborn or dumb enough to pick a candidate that comes in last?"
Um...yeah.
"Why doesn't everybody else get to vote twice?"
Um....it's not twice - it's Instant!
"Yeah, but some people get counted more often than other people? What's that about? Is that constitutional? What happened to one person, one vote?"
Well, see, the people who powerfully, deeply care about the minor-party candidates and fringe issues should also get to vote for a major-party candidate, because...um...because....
"Because they say so?"
...because otherwise, they are disenfranchised by their candidate's poor showing! See?
"That sounds pretty weak."
That's because you don't underst....
"Hey, here's an idea! Why don't we just count everybody's second vote? Isn't that more fair? And isn't it just about guarenteed to give us a majority by round 2?"
Guarenteed? How's that?
"Well, if you've got three candidates, you'll have a majority by round 2. No way to avoid it - do the math. If you've got four candidates, you'll at worst have one candidate with exactly half the votes. For N candidates, it takes N/2 rounds to reach 50% - that's faster than IRV, which looks like it can take N-1 rounds in the worst case. And it's easier to implement - you just count two marks on every ballot, rather than checking which ones to count the second, and which to count the first."
I guess so.
"So, seriously, isn't that more fair?"
Uh...maybe. But I don't understand IRV proponent's concept of "fairness" anyway, so don't ask me....
Thursday, September 16, 2004
A Special Day
Today I turn 33. As they say in Mexico, "la edad de Cristo".
Today is also Rosh Hashanah, starting the year 5765.
The High Holy Days are a time for re-evaluating your life. I'm not Jewish, but this is a pensive-making coincidence if there ever was one.
May the Lord mark you down for a good year....
Friday, September 03, 2004
It's for Science...Really
I love sites like What's Better? and Hot or Not?. Recently I found Pick the Hottie, a perfect combination of the two: the front page presents two pictures, and asks you to click the more-attractive person. Each "hottie" has his/her own page, showing their win-loss record and photos of recent competitors ("see who I've lost to"). Here's an example. The nice thing about this is you two different pieces of information: the lifetime win-ratio, and a sample of recent wins and losses. That means you can find plenty of plainer Janes that at least one visitor liked better than sexy Susan.
Okay, so the science bit: find the hottest Hottie (without cheating by looking at the Top Ten list). For a lot of search problems, we can only do head-to-head comparisons of candidates. We can't figure out the best solution from scratch, and we can't try every possible candidate (so many candidates...so little time....) We can only search through examples that we actually have, and hope we find the best one. This is called "heuristic search". One technique in heuristic search is "hill climbing" - you look at head-to-head comparisons between some candidates and your current "best girl". If any are better, pick the best new candidate. Repeat. The nice thing about this technique is that you're situation can only improve or stay the same - you'll never make a choice that leaves you worse off.
The problem with hill climbing is beautifully illustrated by "Pick the Hottie" - you want to find the best-looking person on the site? You can look at the Top 10 list, of course, but try this: start with anybody on the site, and hill climb - look at who they lost to recently, then pick the hottest hottie that beat them. Look at who they lost to, and pick the hottest...etc. You'll reach a "local" maximum - a hottie that hasn't recently lost to anybody with a higher score. You will very, very rarely reach the very top, though. There aren't enough direct recent comparisons between the top scorers to produce an increasing-only path from, say, the 50th-hottest picture to the 1st.
Saturday, August 28, 2004
The Parable of the Astrodome
Let me offer another way of thinking about the religion question: gather a thousand of your closest friends. Crowd them together in the middle of a giant featureless round room. Give everybody a beeper and announce "when the beeper goes off, line up to face the same direction as your neighbor!" One of two things will happen:
A: If there really is nothing to orient by....
When the beepers go off, people will clump together facing the same direction, but there will be breaks - little errors in lining up will accumulate until some folks have to choose between facing one direction and facing another. The crowd will develop regions (called domains) where everybody is lined up, with sharp boundaries between the domains where people aren't able to locally reconcile their decisions.
B: If there is an external focus....
If there are field lines, or home plate, or a scoreboard still sitting on the wall of the stadium, some folks will get the idea to line up facing the external marker. This will dramatically cut down on the number of domains and make lining up easier.
What's my point? If there's an outside authority that helps each individual make a decision, the entire crowd can coordinate itself better. This is not random smacking on my part - it's exactly how iron atoms line themselves up in the presence or absence of a magnetic field. Nothing that I've said here is anything other than mathematics and physics of crowd behavior, but you can make the analogy to the value of religion easily enough.
Notice the question of the validity of the authority is left unanswered - its actually very easy to have this kind of discussion without insulting people. I can't imagine how Dawkins thinks it's fitter to constantly berate and aggravate religious folks. Such extravagant, baroque wastefulness on his part....
"There's More Light Over Here"
Richard Dawkins would be more interesting if he would reach a different conclusion once in a while. He starts with a good question:Can natural selection explain the pervasiveness of religion? But he also starts with this statement:
As a Darwinian, the aspect of religion that catches my attention is its profligate wastefulness, its extravagant display of baroque uselessness.He reminds me of the man who looks for his keys under the streetlamp, rather than over where he lost them, on the grounds that he can see better in the light.
Dawkins mentions a few intriguing solutions, only to cheat them of proper consideration. Could reassurance against death actually extend life?
"This could be true or it could be false....but I shall not pursue the matter."Could religion be a case where group-selection works, in contradiction of Dawkins' noted hostility toward it?
"This is an interesting line of theory to pursue, but I shall not do so here."He cheats this way because he has limited space and needs to get to the acceptable answer: religion is a pointless sham caused by a malfunctioning aspect of selected behavior - in this case, the "fact" that children "[o]bey without question" what adults tell them, and pass it to their own children. He has a nice explanation of why moths fly into candles on the way to explaining that religious people are also headed for the flames.
Granted, he's writing for Free Inquiry, so he knows his audience wants to hear that religious people are maladapted for long-term survival. It's always a pleasure to read that right thinking folks like yourself are the Chosen People, even if it's only natural selection that's doing the choosing. And it's so frequently the same sermon, no matter who the preacher is - they are wrong, evil, wasteful, and stupid, and we are destined to come out on top in the long run....
Do You Take Sugar?
One lump or two?
If you went to high school in the 80's, this horror show will make you feel like retirement is right around the corner....
