Who, exactly, intervenes?

Les Jenkins, over at Stupid Evil Bastard[1], comments about the absurdity, at so many levels, of being told to pray to God for help with the stock market. But there’s another level of absurdity that Les didn’t comment on. He quotes from this article about one Cindy Jacobs, on the Christian Bonehead Broadcasting Network program The 700 Club:

In January of this year, Cindy Jacobs was in a worship service when the Lord spoke to her, “Cindy, the strongman over America doesn’t live in Washington, DC — the strongman lives in New York City! Call My people to pray for the economy.”

This word so shook Cindy; she knew she had to call the people of God to converge on New York City the week of October 29 for an emergency prayer rally to cry out against economic collapse in the midst of shaking.

The Lord further said, “October 29 was Black Tuesday, the day the stock market crashed, and Satan wants to do it again.” We must be proactive in prayer. At the beginning of the year many intercessors began to hear from the Lord that without divine intervention, a major shaking was coming to Wall Street.

As Bill Cosby, as Noah, used to say: “Riiiight.”

Les concentrated, in his critique, on the hypocrisy: whether God cares about money, whether these people are being idolaters... but I’m looking at more fundamental problems with Ms Jacobs’s schizotypal fantasy. To wit:

• God wants this fixed. But the way God approaches fixing it is to have a chat with Ms Jacobs and tell her to hold a prayer session? Wait, who will they be praying to? God, we presume. Bit of a loop here, wot?
• Satan wants to undo the financial system — of course it’s Satan; we should have known. But God, who is all-powerful and can do anything, needs our help to thwart Bush’s Satan’s evil plan. Our help, lowly, powerless mortals that we be. Imagine.
• And what is it that they’ll be praying for? Divine intervention, Ms Jacobs says. Divine intervention by who? Well, God, of course, who else?

So, Ms Jacobs would have us believe that God came to her and said that he wanted to fix the financial mess that Satan is creating, but that the only way he can do that is if she gets a bunch of folk together to ask him to intervene and fix it. I guess there’s some rule that he’s not allowed to do it otherwise. They have to ask really hard, of course, and they have to give money to “the cause” [wink-wink].

Riiiight.

Of course, if there really were a God as Ms Jacobs and her lot describe, and he really did care about fixing things, well, he would just fix them, without all the ridiculous and self-serving mumbo-jumbo.

[It’s also very easy to come around after the problems have surfaced and tell us that God warned you about them months ago. They do that sort of thing all the time, and people continue to be sucked in by it.]
 

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[1] I know I’ve said this a coupla times before, but man, do I love that blog name!

Interior decoration

On Morning Edition the other day, NPR reported that as the housing market has softened, so has the market for new furniture:

If the tumultuous economy is this year’s top “kitchen-table” issue, that isn’t good news for people who sell kitchen tables, coffee tables and other furniture. The furniture industry is experiencing a tough year — September sales were down about 10 percent from a year ago. The mood was glum at the world’s largest furniture industry trade show in High Point, N.C.

Nothing surprising here, but one thing was said that I want to comment on. A minute or so from the end, they say that companies are pushing “accessories” to people who are shying away from buying new furniture:

NPR: Nan Feldman’s company, Badash Crystal, imports vases and other decorative accessories. She promotes them as a way to freshen a room if you can’t afford to refurnish it.

Feldman: This is a piece that I call my cheap and good. Even at $10... feel the weight.

NPR: This is a $10 crystal bowl.

Feldman: Right. A woman can put that bowl with bath soaps in her bedroom, so that it adds a little inspiration to your furniture or to your home decor.

A man, on the other hand, such as I... well, you’d better keep that bowl away from us or we’ll likely wash our feet in it, or use it to pee in when we’ve had too much beer. Maybe it’d be a good place for us to put our used socks.

To be truthful, I’d probably not put bath soap in it if I were using it to decorate the bedroom. Some dried flowers instead, perhaps, possibly mixed with some aromatics (cinnamon is always nice). Or in the dining room I might toss in an assortment of small, colourful citrus fruits — lemons, limes, and tangerines might look and smell nice.

How about layers of coloured sand? Depending upon the shape and depth of the bowl, that could be very interesting, and could be changed with the seasons.

Some men are interested in nice decor too. And some of us are even straight.

 

While we’re talking about sexism, let me mention a slogan I saw a lot of in rural Virginia last weekend: “I’m voting for the chick!” Great. How about that?: a slogan that in one breath both supports the Palin/McCain... uh, sorry... McCain/Palin ticket and belittles Ms Palin as “the chick.”

I do wonder what people are thinking, sometimes.

Food service for the masses

I’ve been meaning to write about this since the Dublin IETF meeting. At that meeting there was a great deal of side discussion about people’s food restrictions and the availability — or lack thereof — of meals that attendees with food restrictions could/would eat.

I have to give two pieces of background information before we go on:

• For the most part, the meeting was not providing food for the attendees.[1] The hotel stay included breakfast, and people generally had to get lunch and dinner on their own.
• The meeting was at a conference center well outside the city. Breakfast and lunch really were limited, because of time, to restaurants at the conference center. Shuttle buses were available for going into Dublin proper for dinner.

Most of the discussion started with complaints from various folks about the difficulty in finding suitable food, especially in comparison with other meeting locations we’ve used. Often, other participants — who did not share the subject restrictions — would basically say, “Don’t be silly,” and would point out all the options that were available.

Now, it’s my opinion that someone saying, “I had a very hard time finding food that I could [or was willing to] eat,” is right by definition, and telling him that he’s wrong is pointless (as opposed to suggesting restaurants he might not have tried, which is... pointful?). So that’s not what I’m going to discuss here.

What I do find interesting is consideration of the extent to which it’s the meeting’s responsibility to ensure that various restrictions and combinations thereof can be accommodated.

We can sort food restrictions into a few categories, here in no particular order:

• Medical
• Religious
• Moral/philosophical, but not “religious”
• Preference

Within those, there are restrictions that are more or less common. Vegetarian diets, for example, are common, while potato-free diets are uncommon (but not unknown: there are people who are allergic to plants in the nightshade family).

It seems clear that medical issues are paramount. The problem with them in this context is that there’s a wide variety — it’s likely that no single issue, beyond the generic “low fat” sort of thing — is prevalent enough that it can be separately considered.

I think we can also agree that “preference” comes last in priority. You don’t like carrots? Oh, well. Most preferences can be easily worked around or are subsumed by broader categories. For example, I don’t like steak or roast beef, but I can often avoid that by eating chicken or fish, or asking for a vegetarian meal.

As to the remaining two, people generally seem to place religious issues above other non-medical reasons for dietary restrictions. I’m not sure that’s fair or reasonable. Is it really more “legitimate” for someone to be vegetarian because he’s Hindu than because he’s morally opposed to killing animals, or because he simply thinks it’s healthier?

On the other hand, religious restrictions provide a convenient way to define a few common sets: vegetarian, kosher, and halal cover most of it. Of course, it shouldn’t be a surprise that kosher food, to pick one, isn’t readily available around the Citywest Conference Center. That said, people who are strict about keeping kosher generally bring food with them, and others may be willing to compromise, when they travel, on vegetarian options.

It seems that the primary difficulty was for people with medical restrictions to find food, and I find it interesting that that would be any harder outside of Dublin than anywhere else.[2] Perhaps it was that way because our meetings are generally in or near downtown areas of cities, and this was unusual in being in the suburbs. Maybe it would be just as difficult in the suburbs of Minneapolis, Dallas, Pittsburgh, or Chicago.

Is it reasonable for those making the meeting arrangements to be expected to check out the availability of food that accommodates some list of restrictions, and only to select venues that pass the test? Who manages the list; who decides what goes on it? We can come up with a starter: vegetarian, kosher/halal, gluten-free, dairy-free, low-fat. Do we include vegan? Nightshade-free? Who decides?

Or is the responsibility purely on the participants to do their homework and make sure that they have what they need to eat?
 

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[1] Yes, I know that the meeting isn’t feeding anyone; if you were going to cavil at that, look up synecdoche. For extra credit, compare and contrast it with metonymy. That should keep you off the street for a little while.

[2] In general, it’s not always as straightforward as it might seem. Nearly anyone can make do for a meal, or even a day, but a stressful week full of meetings means that one needs proper nutrition, not just a bowl of lettuce. Also, some restrictions require avoiding things that aren’t obvious: a gluten-free diet, for example, means more than just skipping bread and wheat-based pasta; many things can contain traces of gluten proteins that are sufficient to cause problems. Sauces may be thickened with wheat; soy sauce usually contains wheat; white vinegar may be distilled from wheat, and is a common ingredient in foods. Other troublesome ingredients include modified food starch, maltodextrin, and other things with “malt” in the name. It can be a struggle.

The Real Virginia

One week to go....

Verrrrry interesting...

I’d like to finish up the paradox series with a bit of mathematical humour, using proof by contradiction to show that all natural numbers are interesting.

Let U be the set of all uninteresting natural numbers:

U := { n ∈ N | n is not interesting }

Because the set of natural numbers is well ordered, any non-empty subset of the natural numbers must have a smallest element. But the “smallest uninteresting natural number” is certainly interesting! So there can be no smallest element of U, and, thus, U must be the empty set:

U = {}

∴   ∀ n ∈ N, n is interesting

A bit more mathematical humour to go along with that:

There are 3 kinds of people in the world: those who can count, and those who can’t.

There are 10 kinds of people in the world: those who understand the binary system, and those who don’t.

Aleph-null bottles of beer on the wall,
Aleph-null bottles of beer,
Take one down, pass it around,
Aleph-null bottles of beer on the wall...

And you thought mathematicians couldn’t be funny. Well!

Carnivals!

What would you do if you tried out for a TV game show and they turned you down? Well, of course, there’s only one thing to do: strap something to yourself, walk into a law office, and tell people it’s a bomb:

Ten floors of a building in San Francisco’s Financial District were evacuated Monday afternoon after a man caused a bomb scare by walking into a law firm with a suspicious device strapped to him, police said.

San Francisco police spokesman Sgt. Neville Gittens said a man in his 30s walked into a law office on the 30th floor of a building at 275 Battery St. around 12:40 p.m.

Witnesses said the man said he was angry because he had not been selected to appear on the TV game show, “The Price is Right.”

Pointers to this fortnight’s blog carnivals:

• Skeptics’ Circle #98.
• Carnival of the Liberals #76.
• Carnival of Feminists #67.
• Carnival of Education #193.
• Carnival of Education #194.
• Carnival of Mathematics #42, including “Modus ponens, modus tollens, and proof by contradiction”.
• Carnival of the Godless #103.

New blog template

I’ve been using a heavily modified Blogger template, so when Blogger came out with their newer “layout” system (basically, new templates with pluggable “widget” modules, allowing you to do a good deal of modification without actually having to change your template) I stayed with my template instead of migrating.

I have a (hidden) test blog, which I use when I have a post with a complicated layout and I want to play with it before I publish it. The composer’s preview function is too rudimentary; I want to see what it looks like with my actual blog template.

Last weekend, I spent some time — quite a bit of time, actually — making the test blog look like the real blog, but using the layout features. I could do a lot of it with the widgets, but I did have to modify the template, still. Anyway, I got it working, and have now migrated the real blog to that setup also. Even with the work done on the test one, it took quite a bit more work to get everything right.

But right I think it be, now. The new layout should look very, very close to the old template version. There are a few differences that I couldn’t reasonably change, but that don’t matter (such as the placement of the labels at the bottom of the posts), a few differences that I like (it no longer says “Name:” and “Location:” in the header), and some new features in the sidebar — the category (label) archive list, and the “recent comments” section.

Please let me know if something doesn’t look right or work right. And tell me if you have comments or suggestions about the layout. Just leave a (moderated) comment on this or any post, and if it’s about the layout I’ll probably not publish it, but I will appreciate the feedback.

Three “guest bloggers”

You know, sometimes it just way easier to let other people do the blogging, especially when they say things so devastatingly well. And so, this Friday I’ll send you to three treatises that are just like that: terrific material well written, giving me more time to be lazy get some work done.

Item one is from a bloke who calls himself Honeybakedham. He writes in response to the McCain/Palin criticism of anyone opposing them as not being “real Americans”. Honeybakedham and others like him, he says, are certainly real Americans:

I believe that any two people, being of different sexes or the same, who are in love and desire to be married, have a fundamental right to be so. And I am an American.

I am opposed to the war in Iraq. And I am an American.

I believe in protecting a woman’s right to reproductive freedom. And I am an American.

[...]

I am a “REAL” American.

And I believe that any aspiring candidate to any high office, regardless of ideology, does not deserve that office if they cannot recognize that we are all Americans, regardless of our own political views or philosophies.

Item two is a personal plea from a friend of mine, Thom Watson, who asks for help from everyone, in and out of California, in defeating Proposition 8 and allowing him and his partner of five years to retain the option to marry:

There is an unfair ballot proposition in California that, if passed, will take away my fundamental rights. This is really important to me. Will you help me defeat Proposition 8?

Jeff and I have been together for five years. We love and support each other in the same way as families all over the country; we share the same joys and the same sorrows, we have the same dreams and the same fears. We intend to spend our lives together, and we hope to be married next year. The California Supreme Court ruled earlier this year that it is unconstitutional in California to deny us the right to marry, just as it was the first court to rule, in 1948, that laws prohibiting interracial marriages also were unconstitutional. It is the constitutional duty of the court, in fact, to safeguard the rights of minorities, and that is what the California Supreme Court did.

California’s Proposition 8, however, now would take away our constitutional right to marry. It would take this right away only for same-sex couples and it would write discrimination directly into the state constitution.

Item three is another post against Proposition 8, this one from a straight woman in California who goes by the moniker Green Yogurt. She doesn’t understand why there’s a problem here; she “doesn’t get it”:

There are a slew of commercials about this. Here’s the gist of them: little pigtailed girl runs up to her mom (who naturally, is in the kitchen), and says, “Hey, guess what I learned in school today! A king can marry a king and when I grow up, I can marry a princess!” Cut to the mom looking horrified.

Probably no surprise to anyone who knows me, that “I don’t get it.” These commercials are aiming to say two things:

1. Children shouldn’t know about gay people.
2. Gay people shouldn’t be allowed to marry.

I understand neither of these. Should children not know about handicapped people? Should white children not know about black children? Should children not know anyone who is in any way different from them? Different religion? Correct me if I’m wrong here, but I thought there was something about stimulating children’s brains by exposing them to different things.

Especially if you live in California, please help my friend Thom, help my other friends out there, help actor George Takei, help all Californians — and, ultimately, help the rest of real America — ensure that any two people who are in love and want to get married... can.

Spammers in court again

More catching up; today, week-old news that a U.S. district court has frozen the assets of a group of spammers and ordered them to stop their operations:

The Federal Trade Commission won a preliminary legal victory against what it called one of the largest spam gangs on the Internet, persuading a federal court in Chicago on Tuesday to freeze the group’s assets and order the spam network to shut down.

The group, which used several names but was known among spam-fighting organizations as HerbalKing, sent billions of unsolicited messages to Internet users over the last 20 months, promoting replica watches and a variety of pharmaceuticals, including weight-loss drugs and herbal pills that supposedly enhanced the male anatomy, according to the commission.

“This is pretty major. At one point these guys delivered up to one-third of all spam,” said Richard Cox, chief information officer at SpamHaus, a nonprofit antispam research group.

(Here’s the FTC’s press release on the case.)

This gang is at the forefront of spam technology, with a large zombie network, or “botnet”, and worldwide operations — the investigators cite the group’s connections to Australia, New Zealand, India, China, Russian, and Canada, in addition to the United States. The FTC worked with the FBI and with their counterparts in Australia and New Zealand (one of the principals, Lance Atkinson, is from there).

The group sells “medicine”, both real and fake, including “male enhancement” pills containing sildenafil (Viagra), hoodia remedies, and prescription drugs shipped from India. According to the Times, the FTC says that the spam operation “cleared $400,000 in Visa charges in one month alone.” Think about that: how many people out there are responding to this spam by actually buying the products? Do you wonder why there’s so much spam? Do you shake your head and say that no one pays attention to this stuff? Think again.

If things work the way the FTC would like, we’ll be seeing less spam about these things now. That seems unlikely, though, except for the briefest transition period as others take over. I agree with Trend Micro’s Paul Ferguson, quoted in USA Today: “Someone else will fill the void. While it’s great they caught these guys, the last time a major spam king was busted, the spam increased.” I don’t know that I specifically expect it to increase because of this, but there’s just too much money in spamming for one prosecution and one injunction to stop much.

Also, this isn’t a conviction, but only a temporary injunction — the order is for them to stop their operations while the court case proceeds. There’s no guarantee that it’ll end in a conviction.

Nevertheless, I will stress that it’s great they caught these guys, and it’s another case that shows that the CAN-SPAM Act of 2003 is effective, even with its flaws.

Diversity in leadership

I’m catching up on some New York Times articles that’ve been in my browser tabs for too long. One that’s struck me is titled “Quiet Political Shifts as More Blacks Are Elected”. It talks about something that’s really quite intuitive: as we get more African-American leaders in mainstream politics, winning elections as mayors and legislators, it becomes less remarkable to have a black candidate — even in a mostly white community.

Blacks account for less than 1 percent of the population in this small suburban district near the Massachusetts border. But none of that seemed to matter to the people here at an annual fall festival this month.

[...]

And next month, Mrs. Levesque is expected to win re-election to her seat in the New Hampshire House of Representatives, where she represents one of the whitest districts in one of the whitest states in the nation. She is part of a new generation of black elected officials who are wooing white voters and winning local elections in predominantly white districts across the country.

Political analysts say such electoral gains are quietly changing the political landscape, increasing the number of black lawmakers adept at crossing color lines as well as the ranks of white voters who are familiar, and increasingly comfortable, with black political leadership.

Of course, that stands to reason: the more we see things, the more “normal” they become. And that’s the focus we have to take as we work against racism and other prejudice: get people used to having “them” around — whether “they” be black, Latino, gay, female, or whatever — so that it becomes a matter of course that they show up in all parts of our communities. A monochromatic community is a closed one; diversity begets openness and acceptance.

And, lest we think that it’s just in ultra-liberal northeastern states, or crazy places such as California, it’s happening all over:

But over the last 10 years, about 200 black politicians have won positions once held by whites in legislatures and city halls in states like New Hampshire, Iowa, Kentucky, Minnesota, Missouri, North Carolina and Tennessee.

In 2007, about 30 percent of the nation’s 622 black state legislators represented predominantly white districts, up from about 16 percent in 2001, according to data collected by the Joint Center for Political and Economic Studies, a research group based in Washington that has kept statistics on black elected officials for nearly 40 years.

Political scientists and local officials also point to an increase in the number of black mayors who represent predominantly white cities in places like Asheville, N.C., population 74,000, and Columbus, Ohio, population 748,000. According to a study conducted by Zoltan L. Hajnal, a political scientist at the University of California, San Diego, about 40 percent of Americans have lived in or near cities that have elected black mayors or in states with black governors.

Of course, we’re not there yet, as we can see from the various sexist, racist, and xenophobic things we’ve heard, and we continue to hear, in this year’s political campaigns. The Times article points one thing out, despite the above situations in New Hampshire and elsewhere:

Most black elected officials, however, still represent predominantly black communities. And Dr. Hajnal and other analysts say racial animosity toward black candidates still exists and may affect the results of local and national elections, including the race for president. But he said such feelings were declining.

It’s the decline that makes me feel good about where things are going. As we stand less than two weeks away from the likely election of a black president, and even as I know there’ll be many people in this country who will — who do — hate him just for his race, even as I see lies spread that he’s an Arab, a Muslim, a terrorist (as though those all go hand in hand), I see how far we’ve come even in my 51 years. In the time since Rosa Parks, since Medgar Evers, since Martin Luther King, Jr, we’ve come a long way toward Dr King’s dream.

And even as a white man, I feel very good about that.

Autumn, the wind blows colder than in summer

It seems that some time last week, someone said “Fall!”, and the trees around the Croton reservoir complied.

Teach your children well

I had a conversation yesterday with a Libertarian on the subject of home schooling. The gist of the Libertarian’s argument was that parents have the right to teach their children (or have them taught; the point wasn’t just about home schooling, but also about the option of choosing schools and having a selection to choose from) according to their beliefs.

It strikes me that there are really two aspects to this:

1. Teaching belief-based things that are contrary to accepted science or history or whatever. Examples of this would be teaching Biblical creation instead of evolution, teaching that the Holocaust never actually happened, teaching astrology as a serious subject, or teaching that the ratio of the circumference of a circle to its diameter is exactly 3.
2. Considering your children to be your children, possessions to which you have “rights”, as opposed to considering them to be part of society as a whole and the future leaders of that society.

The first has been argued by so many by now that I have little to add. It’s the second part that’s particularly connected to the Libertarian philosophy, the general concept that individual rights are paramount.

The trouble is that when we’re talking about one’s children, we’re not talking about individual rights any more. Children can’t choose for themselves what rights they want to exercise or how they want to be reared — or taught. Should it be exclusively up to the parents to decide every aspect of the preparation of a child for its future role in society? Or does society have some say in that too? And if society does have a say, should demanding a “standard” educational curriculum be a reasonable thing?

My parents were not qualified to teach me, themselves, the things I learnt in school. That’s not to say anything against them; quite the opposite: their goal was to have my brothers and me go farther than they did, to learn more, go on to college (the first generation of my family to do so), and get “professional” jobs, where we used that knowledge. They encouraged us to “do well in school,” and never would have imagined second-guessing what we were taught, nor trying to do it themselves.

It worked as they’d intended. And, perhaps more importantly, my brothers and I are very glad they did things that way. I’m thrilled to have had a good education and to have learned the best that we know in mathematics and science and technology, as well as in the other subjects. I’m pleased with the opportunities it gave me. I like the life it’s enabled for me, and I thank my parents for making that their most important goal.

Well, one might say, indeed, but that was my parents’ choice, and it should be their choice to make.

But what about me?

Had my parents made a different choice when I was too young to have a say in it, my life would be very different. Of course, that’s the case with many things, and I’m not suggesting that the state should micromanage all aspects of child-rearing. What I am suggesting is that the state has a reasonable case for setting standards that have to be met, and that the rights of society to expect parents to prepare their children for adulthood and of the children to be properly prepared sometimes override the parents’ desires for control.

We have to be careful how far we take that, to be sure. But your children are not possessions over which you have property rights. Children have their own rights that sometimes need to be protected in spite of their parents.

Sunday morning

How many?

It’s that time again: time for my local public radio station to have its “membership drive”, when they interrupt their programs constantly with interminable harangues about why you should give them money. The thing is, I do give them money, but that doesn’t stop me from having to lose Morning Edition and All Things Considered three times a year to this membership drive. They call it that because it drives one away from the radio.

No worries, though. I use these times to catch up on “podcasts”, which appears to be what we’ve taken to calling “mp3 files” when they’re talk instead of music. I have quite a stash of them just waiting for my commutes during membership drives, or time spent in airports and airplanes. Mostly, they’re episodes of This American Life — I have a lot of them queued up — along with certain other selections that I thought would be interesting, bits from Fresh Air, Bob Edwards Weekend, and WNYC talk shows hosted by Brian Lehrer and Leonard Lopate.

Yesterday, I listened to some Bob Edwards Weekend pieces, talks with Jackson Browne, Julian Barnes, and Spike Lee. And, wouldn’t you know it, it’s turned into another language gripe.

Not because of anything Mr Browne, Mr Barnes, or Mr Lee said, mind. It was triggered by this announcement during a pause in the program:

Bob Edwards Weekend is supported in part by the Bill and Melinda Gates Foundation: dedicated to the idea that all people deserve the chance to live a healthy, productive life.

I found myself mentally paraphrasing Nathan Hale: “I regret," I thought, “that I have but one life to share with all my countrymen.”

Now, I have to say that I am one of those who stickles for not using “they” as a singular pronoun. I suppose I’m willing to accept it when it’s the only reasonable way to say something without being awkward or stilted. Most of the time, though, what one wants to say can be done properly, matched in number and still gender-neutral,[1] with a bit of re-wording. Usually that re-wording involves putting the whole sentence in plural.

And that’d be one correct solution here, even though there is not a “they” to be seen: “dedicated to the idea that all people deserve the chance to live healthy, productive lives.” The thing is that it’s such an obvious way to write it... this one isn’t hard at all; there’s not even a danger of having gender sneak into that sentence.

The difficulty is usually with a sentence like, “Everyone must show his or her driver’s license before entering the secure area.” The lazy way to avoid the tedious “his or her”[2] is to turn it into “their”, even though one needs a singular pronoun here. Substituting “one’s” is even worse than using “their”: it’s correct in number, but so horridly stilted and un-idiomatic. But the right fix is simple, if one isn’t so worried about minimizing changes, and is willing to do slightly more re-writing: “All travellers [or all persons, or whatever] must show their driver’s licenses before entering the secure area.”

For the foundation’s slogan, by the way, there’s also a second approach. We can replace “all people” with “each person”, or, more simply, “everyone”: “dedicated to the idea that everyone deserves the chance to live a healthy, productive life.” Now the singular works fine.

As with most of these complaints, it’s not just that I’m picking at nits. There’s no problem understanding that slogan as written, even with the inconsistency in number. But there are times when being sloppy with it creates ambiguity or otherwise confuses the reader. We owe it to our readers to allow them to save their effort for understanding clever turns of phrase and interesting nuances. We shouldn’t make them work so hard for nothing, to fill the gap created by our ineptitude.

If we get used to doing it wrong, we’ll soon be doing it wrong without ever knowing it.
 

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[1] And we have a far easier time of making things gender-neutral than do people who speak other languages. In English, we don’t have gender for inanimate objects (two genders in Spanish and French, three in German), and we don’t have to match the adjectives or the pronouns to the gender of the nouns. Actually, I don’t worry so much about making my language gender-neutral. Most things just work out that way without a lot of effort, and at other times I just like to mix it around, sometimes making something feminine that would have traditionally been said with “he”.

[2] I don’t mind an occasional “he or she” or “his or her”; it’s when we get into endless strings of them — a trap that’s easy to be caught in — that I want to jump off a cliff rather than read further.

Modus ponens, modus tollens, and proof by contradiction

In a recent post, I gave a (non-rigorous, sketchy) argument that shows there can’t be a one-to-one correspondence between the natural numbers and the real numbers. There’s a cliché that one “can’t prove a negative,” or, more specifically, that it’s not possible to prove that something doesn’t exist. But it is. We do it by assuming that what we want to prove is false — assume the positive to prove the negative — and showing that the assumption results in a contradiction.

Let’s look at how a conditional statement works, logically. Suppose I tell you that the following statement is true:

If it’s raining, then I’m at home.

If you live a few blocks away, when tomorrow comes you can look outside and see whether it’s raining. If it is, then you can conclude that I’ll be at home. But what if it’s clear and sunny? Does that mean I’ll be out?

It does not. My statement said nothing about what happens when it doesn’t rain. In plain English, of course, there’s other context, and semantic expectations about what we say. But considering it as a purely logical construct, you know nothing about my whereabouts if it’s sunny.

Now, what if you live across the country? You can’t look outside, but you can phone me — let’s assume that I’ll always answer the phone if I’m home. If you call and I’m not there, what can you conclude? And what if you call and I am there?

Well, there are four possibilities:

1. I’m home, and it’s sunny.
2. I’m home, and it’s raining.
3. I’m out, and it’s sunny.
4. I’m out, and it’s raining.

It should be clear that the only one that contradicts my statement is 4. As I said earlier, I’ve said nothing at all about what happens when it’s sunny, so 1 and 3 are consistent. And 2 is exactly what I described in my statement.

So, if you wanted to “prove” that it’s not raining where I live, you could see if I’m at home, or out. If I’m out, you have your proof that it’s not raining, since possibility 4 has been eliminated as being inconsistent with what we know to be true.

To put this into symbolic logic, let’s call P the statement that “it’s raining”, and let Q be the statement “I’m at home.”

If P then Q.

...or, in symbolic logic notation:

P → Q

Looking at the examples above, including the cross-country example, we can see these logical equivalencies:

P → Q

~Q → ~P   (the contrapositive)

~P ∨ Q

All three statements have the same truth values, and are logically equivalent. And to add a bit of cool terminology to this, we have two methods of proof to name, which take advantage of the above equivalences:

• Modus Ponens:

  given P → Q
  given P
  therefore Q

• Modus Tollens:

  given P → Q
  given ~Q
  therefore ~P

Proof by contradiction is, then, an application of modus tollens: if we show that some proposition P implies an impossible conclusion Q, then because we know ~Q, modus tollens tells us that ~P must be true (that is, our assumption was wrong, and P must be false).

These types of proofs are widely used in mathematics, and I’ll give a nice example here,[1] of a proof that there is no largest prime number.[2]

Definition: A prime number is a natural number p such that the only factors of p are 1 and p.

Proposition: There exists a “largest” prime number. Formally, there exists a prime number pmax, such that for all prime numbers pi, pi ≤ pmax.

If such a number exists, then, of course, the set of prime numbers, P is finite. We can therefore compute the number np, which is the product of all prime numbers.

np  :=

∏ p∈P

p

Consider np+1. Clearly, np+1 > pmax. Also, when we divide np+1 by any prime number other than 1, we get a remainder of 1, meaning that np+1 has no prime factors. Since we can reduce any factor down to prime factors, this means that np+1 has no factors other than 1 and np+1, and it is, therefore, prime. So:

np+1 is a prime number and np+1 > pmax.

This contradicts our proposition, proving that our proposition was false. Therefore:

There does not exist a “largest” prime number. Formally, there does not exist a prime number pmax, such that for all prime numbers pi, pi ≤ pmax.

 

---

[1] The example assumes, as usual, the axiom of choice. Quite a critical bit, that. We’re safe, there, because the customary set theory these days is Zermelo-Fraenkel with the axiom of choice.

[2] Note that, while no largest prime number exists, there is a largest number that we know to be prime. In fact, since I first wrote this up, a new largest known prime has been discovered: 243,112,609-1, a phenomenally large number with around 13 million digits. Large though that be, there are nevertheless many more that are larger still — a countably infinite number of them, in fact.

There’s a small error, by the way, in the last paragraph of the Science News article (at least at the time of this writing); did you catch it?:

Current cryptographic systems rely on the challenge of factoring large primes. This task is distinct from verifying primeness, but the root difficulty is the same: limited computing power. Through this prize, “we maintain a pulse on what people might be able to do in breaking cryptosystems,” Noll says.

Factoring prime numbers is, of course, trivial, by definition. What they mean to say is that cryptographic systems rely on the challenge of factoring products of large primes. If p and q are large prime numbers, it’s very difficult to factor (p * q) into p and q.

On campaign finance

A friend of mine recently asked a friend of his, someone who he describes as “involved in politics,” for advice about how to decide on political donations. A key part of his question is whether it’s useful to donate to a campaign that’s already “rolling in money,” and whether donating “this late in the campaign” matters.

The response suggested considering things in three steps. First, and most obviously, decide whom you want to support from the point of view of ideology and policy. Second, of those who made the ideological cut, consider who has a reasonable chance to win — why spend money on someone who’s polling in single-digit percentages?

And third, consider what they’ll do with the money:

[...] it’s astounding how much money candidates can burn through extremely quickly — particularly for presidential candidates and senate candidates running in expensive media markets like NY, Calif. Donating to most candidates now would help, most likely. But, I would look at any candidate’s latest financial report first. Go to fec.gov [the Federal Election Commission web site] to see what their cash on hand is. If they are in debt, you can help. If they have a six- or seven-figure balance, they may well end up in a surplus. If your money goes to their surplus, they might use it as seed money for the next race. Or they might give some to other candidates whom you might or might not support. As to DNC and party committees in general, you just need to have faith they will use it in a way you like as you really don’t have control over how it will be used.

It’s certainly not surprising that party committees can use donation money for whatever they want, and for that reason I’d stick with donating to specific candidates. What bothers me here is that the candidates may turn over their surplus to someone else. I suppose it would be unwieldy to try to give a large number of small contributions back to the donors. On the other hand, donors to the campaigns of Mike Huckabee and Hillary Clinton would not necessarily be pleased to have their money handed to John McCain and Barack Obama, respectively.

But perhaps more to the point, the system lends itself to abuse by quiet collusion. Perhaps a close race is expected between Republican Ferdinand Wingnut and Democrat Taxon Spennd. Mr Spennd finds three other Democrats to run against him in the primary, people with varying positions that are more centrist and more fiscally conservative than his. Through the primary season, the four raise money and support, amassing a larger total than Mr Spennd could do on his own, and pulling some Republican support over to the more conservative of the four Democrats.

But just before the primary, the two least popular of the four withdraw, giving their money and support to Taxon Spennd. He easily defeats his remaining “opponent,” who then also throws his resources to Mr Spennd, giving the candidate a significant edge over Mr Wingnut, both in money and in support, compared with what he could have gotten on his own.

It’s undoubtedly illegal, sure. But if it’s done with a quiet agreement among the four, who would know?

And then there’s this:

One last thought — If you give over $200, your name address, occupation, and employer are collected and will find [their] way onto FEC reports. That’s the law. Your friends and neighbors and business associates can look you up by name and see who you gave to.

I found that interesting: I hadn’t realized that the threshold for public disclosure was only $200; I thought it was $2000. There’s an interesting conflict, here, between “free speech” (using the courts’ consistent decisions that donating money is a form of political speech, a characterization I disagree with) and the right to anonymity, or at least to privacy.

I see, for example, that one Ned Leiba (no relation of which I’m aware), a CPA in Stockton, CA, gave a total of $1300 to Ron Paul’s campaign. That Mr Leiba appears to be a partner in his own accounting firm... but what if he worked for someone else? What if his bosses had been Giuliani supporters and vehemently disliked Ron Paul? What if he feared for his job if they learned that he donated to Mr Paul. These small contributions are well below the levels where we really need to worry about special interests... mightn’t the publication of their details create a “chilling effect”?

I’d rather see campaign spending (and timeframes) limited to the point where donations don’t really matter. Barack Obama has raised more than $460 million so far in his presidential campaign, according to the FEC. That’s just an insane amount of money to spend (and it’s not even considering the paid leave of absence he’s taken from his job — remember, he’s a Senator, but he hasn’t actually been doing that job for about half the time he’s had it — in order to run for president). We should not be allowing that much money and time to be spent on political campaigns.

It’s strict spending limits, more than donation limits, that can quell these excesses. And the spending limits have to apply to all political advertising, whether it directly supports particular candidates or not (ads saying “McCain is Wrong” and “America Out of Iraq”, as well as “Vote Obama!”).

Unfortunately, we can’t have any real reform in our campaign spending as long as the courts consider money to be speech.

A different approach to the bailout arrives

As I’ve said before, I’m not an economist. So when, during the bailout hearings and after, I thought Henry Paulson, Ben Bernanke, and gang were going about things in an odd way, well, I figured that they know better than I. And so I didn’t say anything here. But I did talk about my concerns with friends. Here’s an email message I sent on the afternoon of September 25th:

Date: Thu, 25 Sep 2008 15:39:38 -0400
From: Barry Leiba
To: [redacted]
Subject: Re: $700 billion?

What I don’t get is why they want to do it this way, by buying up dubious assets of failed companies. Here: when you invest, is that what you do? Or do you invest in companies that are successful, or that you think soon will be? I’m not an economist, and maybe there’s a really glaring reason not to handle it like this, but I wonder why, instead of allocating $700B as they’re proposing, they don’t allocate, say, half that, and spend it on investments in the companies that’ve been doing things right, encouraging others to invest in them and encouraging them to further prosper and to fill the gap left by the dead ones.

Why not?

Barry

Of course, that was just a silly idea.

But, wait. Um... what are they doing now?:

Two weeks after persuading Congress to let it spend $700 billion to buy distressed securities tied to mortgages, the Bush administration has put that idea aside in favor of a new approach that would have the government inject capital directly into the nation’s banks — in effect, partially nationalizing the industry.

As recently as Sept. 23, senior officials had publicly derided proposals by Democrats to have the government take ownership stakes in banks.

The Treasury Department’s surprising turnaround on the issue of buying stock in banks, which has now become its primary focus, has raised questions about whether the administration squandered valuable time in trying to sell Congress on a plan that officials had failed to think through in advance.

It has also raised questions about whether the administration’s deep philosophical aversion to government ownership in private companies hindered its ability to look at all options for stabilizing the markets.

OK, so... if I thought of that two weeks ago, surely someone in the upper-echelon financial circles, someone looking into how to fix this mess, someone advising the administration and the legislators also thought of it.

Why did it take another two weeks for it to surface?

Legislators with spine

Not everyone will agree in general with the title characterization, but here:

LONDON — After nearly 12 hours of debate over two days, the House of Lords overwhelmingly rejected a Labor Government proposal on Monday evening that would have allowed the authorities to hold terrorism suspects for 42 days without being charged.

The vote was 309 to 118, with many Labor members joining Conservatives, Liberal Democrats and independents in opposition.

The bill, pushed hard by PM Gordon Brown, only barely passed the House of Commons a few months ago (by a nine-vote margin). In a country that possibly has the most surveillance cameras per square mile of any on Earth, legislators have drawn a line at adding another two weeks to the four weeks that those accused of terrorism may already be locked up without charges.

Let’s note a few things about the current 28-day period:

1. It’s already the longest in the “free world.” In contrast, even our vile “USA PATRIOT Act” only permits 7 days.
2. Note that this is talking about people accused of terrorism. They haven’t yet actually been determined to be terrorists, and people are falsely accused all the time.
3. Note that this is talking about people who have not been charged with any crime. In other words, the police haven’t enough evidence — they may have nothing at all except a suspicion, perhaps something from a third-hand report or an anonymous tip. File charges against the accused, if you want to hold them.

Three cheers for the British legislators for standing up against their leader for what’s right and decent. Three cheers to them for not being the sheep that our Congress have so often been, agreeing, far, far too quickly to clamp down on civil liberties with the USA PATRIOT Act; agreeing to cede their power to declare war; allowing themselves to be bullied by the Bush administration into so many bad policies.

Lord West, the Home Office minister, isn’t pleased with the decision. Addressing the comparison of the already-long 28 day detention period with what’s allowed elsewhere:

“I don’t really care what any other country does,” Lord West, the Home Office minister, said during a powerful speech in support of the bill in the House of Lords on Monday evening. “You’re bloody lucky to live in this country,” he added.

Luckier after this decision than had it gone otherwise, I’d say.

A SIDS study

The idea of babies dying in their sleep, unexpectedly and without explanation, frightens and horrifies us. Of course, the idea of adults dying that way does too, or should... but when it’s babies, few-month-old bundles of cuteness and love, we find it especially jarring. So we give it a name — SIDS, Sudden Infant Death Syndrome — and we grope for explanations, trying to find out what the problem is and to stop it.

One problem with doing that is, of course, that we’ve just given a name to something vague. We may talk of a SIDS “diagnosis”, but it’s not so in the same sense as “liver cancer”. A diagnosis of SIDS means, “Your baby died, and we don’t know why,” a non-diagnosis, really. We assume that there are enough cases that are from the same cause that we can isolate at least one cause of at least some cases.

But it makes studying it difficult. And, of course, making it more difficult are other factors, such as the relative infrequency of SIDS and the fact that there are serious ethical issues preventing many of the ways medical issues are typically studied.

The result is that the studies we have are anecdotal, and we’re using a trial-and-error approach to it. Or better to call it a trial-and-progress approach, because progress has been made: there was a 56% reduction in the incidence of SIDS between 1992 and 2003.

There’s a recent study, published in the Archives of Pediatrics & Adolescent Medicine that adds itself to the mix. Unfortunately, one needs a subscription to read it, so I have to go by the news articles, which are notoriously spotty on reports of scientific studies. The New York Times says this, for example:

Since 1992 the rate of SIDS deaths has dropped by more than half, to about one death per 2,000 live births from 2.4 per 1,000. The decline is linked to a national “Back to Sleep” campaign that promotes putting babies on their back instead of their stomach, which has been shown to lower the risk of sudden death.

That’s a reduction from .24% to .05%. In what sense is that merely “more than half”? I suspected that it’s an error, and that at least one of the numbers is wrong.

Checking other reports turns up that the Times article tried to normalize the numbers and made an error in the process: it should have said 2.4 per 2000, not 1000. So the correct numbers are .53 SIDS deaths per 1000 live births in 2003, and 1.2 per 1000 in 1992, .053% vs .12% eleven years earlier.

Putting that in other terms, it means that in 1992 the rate was one SIDS death in about 833 live births, and in 2003 it was one in about 1887. We’ll come back to this after looking at what WebMD says about the study:

One theory is that SIDS is caused by the buildup of carbon dioxide when infants with inadequate sleep arousal responses re-breathe exhaled air trapped by bedding or proximity to other sleeping family members.

With this idea in mind, Li and colleagues from the research division of the California-based managed health care group Kaiser Permanente hypothesized that increasing airflow near a sleeping baby would help protect against sudden infant death syndrome.

To test the theory, they interviewed the mothers of 185 babies who died of SIDS in Northern California and Los Angeles County from 1997 to 2000. The average length of time between the SIDS death and the interview was 3.8 months.

The mothers of 312 children matched for age, area of residence, and socioeconomic and ethnic background to the SIDS victims were also interviewed.

Sleeping in a room with an open window was found to reduce the risk of SIDS by 36%, while sleeping with a fan in the room was associated with a 72% reduction in risk.

And the NY Times adds that the benefits of a fan along with the previously identified “safe sleeping environment” are “not statistically significant”, though there’s a 16% reduction in risk. How can that be?

It’s that statement that really points out what the problems with this sort of study are, and where its limitations lie. Let’s go back to the numbers I computed earlier: in 2003 you’d have to look at 1887 live births to statistically expect one SIDS death.

The researchers interviewed parents of 185 SIDS victims in the late 1990s. Even if we assume that the rate of death was 1 in 1000 then (somewhere between the 1992 and 2003 rates), that means we’d need 185,000 live births to expect that many SIDS deaths. If we looked at 18,500 live births, we’d expect to find only 18 deaths — still enough to do statistics on this. With 1,850, two deaths would be against the odds.

But that’s still six times the number of survivors they actually looked at.

If you talk with only 312 parents of living children, even if they had ignored all the “safe sleeping environment” recommendations, even if that group had the 1992 death rate applied to it, you’d only expect a 37.5% probability of a single SIDS death.

That means that things have to be pretty glaring before they have any statistical significance with a sample that small, and even then I wouldn’t call it a “reduction in risk”. At best, we can say that there’s a correlation — but we have to be careful about confusing correlation with cause.

So, should we be running fans in babies’ rooms when they sleep? Sure, it seems harmless and there’s some small indication that it might help. Mostly, it needs more study, but the only way to study it is to recommend it, have parents do it widely for a few years, and then see what happens to the SIDS rate by, say, 2011.

We’ll get there... but until we can actually confirm causes of this, it’s going to remain difficult to study and hit-and-miss to prevent.

Carnivals!

In the early days of these pages, I opined that my ideal job position would be a U.S. Supreme Court justice. But that’s for adults, of course. I think the best job for a kid might just be goddess. No quotes here, because it’s an AP article, but the gist is that a some priests in Nepal have chosen Matani Shakya, age 3, from among other girls to be the new Kumari. The article doesn’t say what her duties will be, but she gets to live in the temple until puberty, at which point she’s not a goddess any more. But she’ll always have it to tell her grandchildren.

Pointers to this fortnight’s blog carnivals:

• Skeptics’ Circle #97.
• Carnival of the Liberals #75.
• Carnival of Feminists #66.
• Carnival of Education #191.
• Carnival of Education #192.
• Carnival of Mathematics #41, including “Countable and uncountable sets”, parts 1 and 2.
• Carnival of the Godless #102.

New York gets Randi

Last night, the New York City Skeptics hosted a guest lecture by famous skeptic James Randi, and it was a very entertaining evening. Mr Randi talked about faith healing, homeopathy, and assorted other related things. He showed clips of some of his old appearances on the Johnny Carson show, including one in which he exposed a “faith healer”, Peter Popoff, who got tips from his wife through a wireless earpiece — they discovered the transmission frequency and listened in, and overlaid the audio onto the video of the faith healer in action.

What he said about that video that I thought was saddest — or maybe scariest — though not really surprising was that after the show was broadcast, hundreds of people called and wrote, wanting to know how to contact Popoff. Even after seeing the sham exposed, what people remembered was not the fakery but the “healing”. And, of course, fraud Popoff is still in business and making more money than ever, some 20 years later (I won’t link his web site here, but you can find it in the Wikipedia entry).

The audio of the talk should be available soon on the NYC Skeptics’ web site. I’ll try to remember to update this with the link when it’s there.

An over-simple look at leverage and risk

The financial mess is, as they say, complicated. With all the interrelationships among the institutions and the instruments and the investments, and with the different market mechanisms and other such things involved, it’s hard to understand it all. But there are two aspects that I do understand, and that, put together, explain a great deal of how it happened.

Excessive “leverage”

When you have to move a large stone that you don’t have the strength to budge by hand, you can use a lever, one of our basic tools, to make it possible. A pry bar gives you an advantage that your strength alone doesn’t.

In finance, leverage has a similar purpose: it allows you to buy something you don’t have the assets to buy.

Notice I said “assets”, not “money”. Most of us buy things on simple credit all the time. When we buy houses, we usually get mortgages, secured by a combination of the house itself and our regular income. The bank is reasonably confident that our incomes are adequate to let us make the payments, and if we fail, they can take the house and get most of their money back (possibly all, possibly more).

But suppose I should want to buy a $300,000 worth of stock in company X, but I only have $100,000 to spend. And suppose I have all this other business going on, my reputation is good, and I can convince you to sell me the stock for the money I don’t have. Or I can buy a $5 billion company when I only have $1 billion to put into it. That’s an oversimplification of leverage.

It’s the way the financial industry works. Very often, money is just moving around on paper, and isn’t really being transferred as money. And it works in part because of the other aspect:

Assumption of independence of risk

A company that sells insurance on houses all over the country can weather a storm in Florida, because even though it might have a lot of claims from there during hurricane season, it makes enough money from the rest of the country — and during years without bad storms — to cover it. The chances that there’d be a large number of claims from Florida, New York, Kansas, and California all at the same time are extremely small. But a company that only insured houses in Florida would have a much worse situation.

In the financial industry there are certainly independent risks as well. But with so many companies investing in the same things — and in each other’s success — the assumption of independence goes out the window. If company B puts money into “safe” instruments that depend on company A’s success, company C does the same with company B, and company D does it with company C... if company A falls, the others will go down in turn.

There’s also a related assumption at play:

Assumption of constant risk over time

Obviously, the longer a company insures your house, the more likely it is that you’ll make a claim over the life of the policy. But the chances of a claim from you in any given year is about the same from year to year — or, at least, any increase in risk that owes to the aging of the structure is known.

What happened here, though, was that as companies used their leverage to pile on more and more financial obligations that they couldn’t meet, the risk of failure went up dramatically. In addition, they piled on investments that were riskier to start with, having the dual effect of increasing the risk of the overall portfolio directly, and of increasing the risk of the other investments indirectly.

So: Everyone owes everyone else far too much, which means that their financial successes are too intertwined. They underestimate the risk involved because of invalid assumptions about the independence of the risks from each other and over time.

Can $700 billion of government money thrown into the system fix that? And prevent it from happening again in a few years?

Keep bailing; get more buckets

Well, so, the bailout financial recovery package was passed and signed into law, with much foofaraw, and it’s really done its job. I feel so silly for having been skeptical, seeing how the markets turned right around and have been going up and up since then.

[What? I... Oh, I see.] Um, excuse me; I’ve been told that I had the chart upside down. My mistake.

It seems that we had six straight days of continued crashing markets since Congress and the president signed $700 billion over to Henry Paulson, and the only “up” day was yesterday, caused by a worldwide interest-rate cut.

Yes, with things going from bad to worse after our legislators did their work, European countries started getting together to crank out their own bail-out, which included some issues among The Netherlands, Belgium, and Luxembourg, actions by Germany and Ireland to guarantee personal savings, and worries in Iceland of isolation and financial collapse (they’re not in the EU).

Follow that with a bailout plan in the UK yesterday that’s comparable to ours (PM Gordon Brown calls it even more extensive, saying, “We had to do more than just buy up assets.”), and then a coordinated cut in benchmark interest rates that finally had some effect:

The Federal Reserve, the European Central Bank, the Bank of England and the central banks of Canada and Sweden all reduced primary lending rates by a half percentage point. Switzerland also cut its benchmark rate, while the Bank of Japan endorsed the moves without changing its rates.

In another monetary first, the Chinese central bank joined the effort — without explicitly saying it was doing so — by reducing its key interest rate and lowering bank reserve requirements to free up cash for lending.

But it’s still not clear that all this will actually fix things.

And Secretary Paulson, now armed with a $700,000,000,000 bank account (doesn’t it look impressive when one actually writes all the zeroes?), tells us that we have to be patient, that this will all take time, that none of these actions will turn things around overnight:

“The turmoil will not end quickly and significant challenges remain ahead,” Paulson said as he outlined what the Treasury has done to start implementing the $700 billion financial rescue package Congress passed and the president signed last week. “Neither passage of this new law nor the implementation of these initiatives will bring an immediate end to current difficulties.”

That’s as may be, but didn’t Mr Paulson tell us, when asked why he needed the whole 7×1011 dollars at once, that it was a question of investor confidence, and the full amount was necessary to ensure confidence?

Confidence, it seems, is more related to cuts in interest rates than to hundreds of billions of dollars at the disposal of bureaucrats. Or maybe not to that either: the response to the rate cut might just be brief. In any case, no one appears to be very confident in Henry Paulson.

Least of all, me.

The last word goes to Emilio Botín, the chairman of Spain’s Banco Santander (pronounce it sahn-tahn-der), which is faring quite well through this whole mess. Señor Botín has three points of banking advice that should be obvious, but that seem to have escaped many in recent times (listen at 1:28 into the audio):

If you don’t fully understand an instrument, don’t buy it. If you will not buy for yourself a specific product, don’t try to sell it. If you don’t know very well your customers, don’t lend them any money.

Now, excuse me, while I put the numbers from my latest financial statement into my spreadsheet. The value of my investments went down 10% just in September (18% so far this year), but I remain... patient.

Countable and uncountable sets, part 2

When we last looked at countable and uncountable sets, we defined countable sets as those for which we can make a one-to-one correspondence between the set and the natural numbers, and that, therefore, have the same cardinal number as the natural numbers: . And we found that the even numbers, the whole numbers, and the integers are all countable.

Let’s look at one more, the set of rational numbers.

The rational numbers are the set of all fractions... that is, the set of all numbers that can be written as a/b, where a is an integer and b is a natural number. We seem to be going up by an order of magnitude here; surely there’s no one-to-one mapping from the natural numbers to the rationals, is there?

There is. To make it easier, we’ll observe that the existence of a one-to-one correspondence is transitive, so we can use the integers instead of the natural numbers (since we already have a one-to-one correspondence there). We’ll again map 0 to 0, and we’ll map each negative integer to a negative rational in the same manner as with the positive numbers. How we’ll map the positive numbers, then, needs some explanation.

Write the positive integers, the tops of the fractions, as the headings of the columns of an infinite table. Write the natural numbers, the bottoms of the fractions, as the rows of the table. In each cell of the table, write the resulting fraction. We get this:

	1	2	3	4	5	6	...
1	1/1	2/1	3/1	4/1	5/1	6/1
2	1/2	2/2	3/2	4/2	5/2	6/2
3	1/3	2/3	3/3	4/3	5/3	6/3
4	1/4	2/4	3/4	4/4	5/4	6/4
...

If we traverse this table by upper-right to lower-left diagonals, skipping the fractions that are not reduced (and are, therefore, duplicates — 2/2 and 3/3 are the same as 1/1; 2/4 is the same as 1/2), we get this mapping:

1 maps to 1/1, 2 maps to 2/1, 3 maps to 1/2, 4 maps to 3/1,
5 maps to 1/3, 6 maps to 4/1, 7 maps to 3/2, 8 maps to 2/3,
9 maps to 1/4, 10 maps to 5/1, 11 maps to 1/5, ...

That mapping is easily proved to fit the bill: the rational numbers are, indeed, countable — their cardinal number is also .

Now, that one is really counter-intuitive. The rational numbers are densely ordered: between any two rational numbers, there’s another rational number. In fact, cascading that aspect infinitely, it means that between any two rational numbers, there’s a countably infinite number of rationals. The natural numbers, on the other hand, are not dense — there’s no natural number between 1 and 2, for instance. And, yet, they have the same cardinality

Think about what that means. The following sets all have the same cardinal number, — they are all, in a sense, the “same size”:

• The set of natural numbers.
• The set of integers.
• The set of rational numbers.
• The set of rational numbers between 1 and 2.
• The set of rational numbers between ¼ and ½
• For any two rational numbers, a < b, however small the difference between them, the set of rational numbers between a and b.

One more thing, then, which we’ll state without proving (it should be intuitively clear): any subset of a countable set is countable.

Let’s move on to the real numbers: the union of the rational numbers — those that can be represented as fractions of integers — and the irrational numbers, numbers such as √2, √3, and π, which can not be represented as fractions. Non-rigorously, it’s the set of all numbers that can be represented with (possibly infinite) decimal representations. 3 (or 3.0, or ...00003.0000..., a natural number). 3.1 (31/10, a rational number). 3.142857[142857]... (22/7, a rational number). 3.14159265... (π, an irrational number). And so on.

A rigorous definition of the set of real numbers involves definitions of properties and operations, and concepts of group theory that are beyond our scope here, so we’ll have to stay with the intuitive description.

What’s the cardinality of the set of real numbers? Is its cardinal number also ? Is the set of real numbers countable?

As it turns out, it’s not. The mathematician Georg Cantor is credited with proving that in 1874, but his more famous demonstration of it, the Cantor diagonal, was published more than 15 years later.

Here’s an understandable (I hope!) explanation of Cantor’s argument, using the “infinite decimal representation” intuition: Suppose we do have a one-to-one correspondence between the natural numbers and the real numbers. That means that for any natural number n, there is a corresponding real number r. Let’s call that mapping f, so f(n) = r. Since r can be written in a decimal representation as an infinite sequence of digits, we can look at the nth digit. So let’s construct a real number q this way: make the first digit different from the first digit in f(1); make the second digit different from the second digit in f(2); ... in general, make the nth digit different from the nth digit in f(n).

It should be clear that q must be different from any of the numbers that appear in {f(n), for all natural numbers n}, because it always differs from the nth number in the nth decimal place. It’s therefore not possible for any chosen f(n) to be a one-to-one mapping, because we can always find such an exception.

Another fact about cardinality is that for any set S, its power set (the set of its subsets), P(S), has a cardinal number that’s strictly greater than the cardinal number of S. This is intuitively true for finite sets, of course (try it, and remember that the empty set is a subset of any set), but Cantor’s diagonal argument proved it for all sets. It turns out, then, that the power set of the natural numbers has the same cardinality as does the set of real numbers. (It also means that there are at least a countably infinite number of cardinal numbers above , because we can keep taking the power set of the power set of the power set... to get larger and larger alephs.

What, then, is the cardinal number of the real numbers? Is there a concept of a “next” aleph, or is the set of alephs dense? If there is a next aleph, is that the cardinal number of the reals?

There is, in fact, a next aleph, and we have , , and so on, so that there’s an for any whole number n. Even beyond that, there are more, but we’re getting into other areas now, and it’ll get hopelessly confusing.

So, back to the other question at hand: is the cardinal number of the set of real numbers equal to ? Well, the fact is that we don’t know — we can’t prove it either way. We can say that the cardinality of the reals is 2 to the power of , but we can’t say what that is, in our sequence of alephs. We can, in fact, consistently assume the cardinality of the real numbers to be any , for any natural number n. I find that to be very cool.

Now, since we can assume any, if we choose to assume that the cardinality of the set of real numbers is we get Cantor’s continuum hypothesis (and, again, remembering that in all of this we’re also assuming the axiom of choice).

I think that’s all I want to say about countability and cardinality. If you think, as I do, that this is fascinating, read more about Georg Cantor and his work, and the work that came after.

“My friends...”

Aiiiiiiiiiiiiiiiiiieeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee!

Ethnic atmosphere

I wondered about this recently, and Zandria, at Keep Up With Me, happened to just mention it in a blog post. She’s talking about why she particularly likes a bar in her neighbourhood, a place called O’Connell’s, and she says this:

The bartenders. It’s an Irish bar, so all of the bartenders are Irish (I guess it must be a pre-requisite to get hired there). They’re nice, and they have cool accents. It’s nice to go to a place often enough [...]

The context in which I was thinking about it was for sushi bars, where people expect to see Japanese chefs and Japanese waitresses. And that’s what one sees; I don’t think I’ve ever had an experience otherwise.

Now, surely it would be illegal for a business to hire only from one ethnic group. I suppose lots of Americans could pass for Irish, if they had to, with a bit of dialect coaching. It might not be a valid condition of employment to be Irish, but being able to sound Irish certainly could be. Think of it as an acting job.

But could it be a valid condition of employment to look Japanese? Would customers expecting a Japanese man behind the sushi bar or at the hibachi table be put off by, say, a tall blonde woman who looked more Swedish than Asian? Would a Chinese man be close enough? And could the restaurant be sued for turning an applicant down for this reason?

There certainly are exceptions for this sort of thing in acting jobs. If a Broadway role requires a slim, young, light-skinned man, the producers certainly are allowed to reject someone too fat, too old, too dark, or too female. A hibachi chef can arguably be called an actor. A sushi chef or a waitress or a bartender, not so much.

Why, then, do I never see variations? Is it that non-Japanese don’t apply for the jobs? Is it that Japanese restaurants (or Irish bars, or whatever) are unlawfully discriminating, and getting away with it? Or am I wrong: is it really completely legal to filter these sorts of job applicants this way?

The Chinese taints have gone too far

We’ve seen unsafe products galore, from China, and tainted foods, and, most recently, tainted milk. It’s just gone over the edge, though. They’ve despoilt one of the staples of the modern diet, food without which one might as well pack it in.

There’s now tainted chocolate.

BEIJING (AFP) — British sweet maker Cadbury said Monday it ordered a recall of China-made chocolates over safety fears in the latest fallout from the ever-widening scandal over tainted Chinese dairy products.

The company issued the recalls in Hong Kong, Taiwan and Australia after internal tests "cast doubt" on the safety of chocolates made in the company’s Beijing plant, it said in a statement.

If chocolate isn’t safe, what will become of the world?

[No, really: this stuff is no laughing matter. China needs to be totally cut off in exporting to the rest of the world until it can fix its problems in ways that can be demonstrated and monitored.]