Stunned

I had a bit of a shock. I was chatting to a generally very sensible fellow, when he came out with the "atheism is a faith" bit, accompanied by a bit of mild invective.

Since he was normally given to being very sensible, I asked what made him think that. Turns that we had different definitions of atheism - he understood it to be an absolute, categorical denial of the existence of any gods; in that situation, his claim has at least some basis, and of course some dictionaries do support that definition.

We then had a brief but very reasonable discussion.

It turns out that, under the definition I gave (a-theism, absence of belief in gods), he is an atheist. Indeed, our positions are extremely close (we're both agnostic on knowledge of existence, but both lack belief).

I pointed to some online definitions of atheism to make it clear my definition was not esoteric.

I was really glad I didn't over-react to that old chestnut. He was being quite reasonable, within the scope of his definition.

Not everyone who brings up the "atheism is a faith" thing is a fundie.

A little tidbit I should have already known

I was reading online about a game I bought while in the US and which I have only just had time to take a peek at. Someone made a point about many parts of the game being based on the fact that a right triangle with sides of 7 and 4 units has a hypotenuse with length very close to 8.

Well, 7² + 4² = 8² + 1

so the hypotenuse is close to 8, as suggested: √(7² + 4²) ≅ 8.

In fact, I knew √65 to be very close to 8 ¹/₁₆

(if x is not too small, √(x² + 1) ≅ x + ¹/_2x).

(Note that (x + ¹/_2x)² = x² + 1 + ¹/_4x² , and if x >> 1, the final term is quite small )

So that's an error of around 1/128, or about 0.8%; pretty good, since the game aims for much less accuracy than that in general.

But then I thought about the fact that 16 in the denominator was a bit too small, and I wondered about how much. I realized straight away that it was in fact about a sixteenth too small. That is, it occurred to me that √65 is very close to 8 ¹/_{(16 + ¹/16}).

A little light went off in my head, so I hauled out my calculator.

Try this with me, if you have a calculator handy:
Take the square root of 65. (You should see 8.06225...)

Now subtract 8 (the bit we know).
Take the reciprocal (¹/x). You get 16 and a bit.
Subtract 16 and take the reciprocal. Looks like you get the same number back...

What is this number? A tiny bit of algebra shows it's 8 + √65.

So far, that may seem like a trivial curiosity. But this happens all over.
For example, you get the same thing with any positive integer, x;
√(x² + 1) + x is a number like that "16 and a bit", where
you can keep subtracting that integer part and taking the reciprocal.

That is, expressions like 8 ¹/_{(16 + ¹/(16+ ...))} come up lots of times (and recognizing that I'd hit one of these objects was what made the light go off).

Take √10 for example - it's 3 ¹/_{(6 + ¹/(6+...))}

And you don't just get it with roots of 1 more than a perfect square. As I said before, it happens all over.

We've hit continued fractions. They come up a fair bit in mathematics, and they appear in numerous places where rational approximation comes in - I remember playing with them when dealing with asymptotic approximations in statistics, for example. There's a much nicer notation (see the wikipedia article), so if you're playing with them you're not stuck with endless layers of fraction running down the page.

So, for example, the sequence 8, 8 ¹/₁₆, 8 ¹/_{(16 + ¹/16)}, ... 8 ¹/_{(16 + ¹/(16+ 1/16...))} would be rendered as:
8, [8; 16], [8; 16, 16], ... [8; 16, 16, ...]

Similarly, √10 is [3; 6, 6, 6, ...].

The well known continued fraction for √2 falls into this class: [1; 2, 2, 2...].

Compute a few terms in that sequence with me:
1, 1.5, 1.4, 1 ⁵/₁₂ = 1.416666... , ...

already we're quite close - and it continues to jump about either side of √2, getting closer and closer.

For larger numbers, the convergence is much faster. The general continued fraction for √(x² + 1) is [x; 2x, 2x, 2x, ...].

Try seeing if you can work out what is going on with square roots with different offsets from a perfect square.

So anyway not only is there a handy way of computing square roots that are close to perfect squares, there's a handy way to improve the calculation if it wasn't as accurate as you needed.

There are many beautiful things related to continued fractions. Take a look over at MathWorld if you've a mind for some boggling factoids.

What fun.

(Two posts in one day! OMFFSM)

Unconsciously annoying

Here's a peeve that I've been seeing all over the place the last couple of weeks:

"I'll leave that to your conscious"

. . . That's conscience, dammit.

You can't quite see them waving, yet.

Images of planets orbiting other stars.

58 down, two to go.

Stevens is behind Begich by 814 votes. With mostly Begich-heavy count left, Stevens is not going to pass him.

The only question remaining: whether Begich can (roughly) double that lead and avoid a petition for a recount (not that a recount would flip it).

A year (and a month)

With all the stuff that happened just after I got back from the US and then the hurried preparations for lecturing this subject I am still busy with, I entirely missed my blogging anniversary [for this blog, at least - I also have a long running, if recently neglected, personal blog that's been going for about 5 years].

Yep, I started this blog in October 2007, a year and a month ago.

Many thanks to my modest band of readers. Hi!

This is also my 150th post, so that's averaging about 3 posts a week.

Volume is down (and traffic with it) since with all the lack of time for much of anything but work lately (hmm... I think I have some kids around here somewhere), blogging is the thing that has had to drop off a bit.

The other thing is the fire mostly isn't there right now. There's plenty to get worked up about, but I just haven't have enough anger to go around the last few months, nor the time to deal with a more reasoned argument. I can't even keep up with science news (at last glance, my science news aggregator had about 800 unread articles).

[I have lots of ideas for things to write about, but by the time I find an hour to write a decent post, it has become out of date. I also have a number of topics that won't go out of date in a hurry, but they would take much longer to write.]

So volume is down and will probably stay that way for some weeks yet. But I am still here.

The joy and relief around here is palpable

I'm sitting in a large building at the moment, and echoing from different parts of the building I hear cheers, laughter, loud conversation as the news filters through the building.

I've never heard people so bubbly, excited, and at the same time, relieved at a US election. Obama's election victory seems to have energized almost everyone.

One colleague said to me "What I have now is hope".

Which about sums it up, I guess.

Christian charity rejects donation from evil roleplayers

Many roleplaying conventions hold charity auctions, which are often very strongly supported by the convention-goers.

Ogre Cave reports that Gen Con (a major roleplaying convention) raised over $17,000 at its annual charity auction, held in honour of (recently deceased) Gary Gygax.

Their chosen charity, Christian Children’s Fund (apparently a favorite of Gygax) learned that sales of Dungeons & Dragons materials were part of the auction and turned the money down.

Apparently saving children isn't really so important after all, if the money might have come from people who like to roll dice and make up stories.

Fisher House Foundation accepted the evil, tainted money, apparently without reservations. So far, the unstinting wrath of the almighty has failed to fall upon them, but obviously it's only a matter of time.

Southern Comfort?

I no more envy people that find faith comforting than I envy people that find a double shot of whiskey in their morning coffee comforting.


... I might find it understandable, but that's not at all the same thing.

Speed reading

Mid-last week I was asked if I could teach a segment of a subject.

I said "I'll need to check, but it will probably be okay. When does it start?"

"In a week."

Nice to have, you know, preparation time.

poetic PZ

"...your finest thoughts are made from well-ordered meat" - PZ Myers

What a fantastic way of putting it.


(Yeah, yeah, I know, you all read Pharyngula already. Tough.)

You can only rev your base up enough to vote once

What's the point in going to the extreme end of your base over and over, while losing the undecideds, independents and any crossover Dems?

I mean, really? If you rev up your base enough to vote, and you're doing public financing anyway, every time you go back to the well, you're losing votes. And its not just McCain that doesn't seem to understand the message.

The deluge of money to Tinklenberg in the MN house race, and the swing in the NC senate race from Dole to Hagan (Dole was ahead until she started attacking Hagan over actually meeting with atheists) seem to indicate that sufficiently extreme reactions will motivate people to support your opponent much more than it helps you.

So right wingnuts, here's the lowdown: once you convince your base to actually go out and vote for you, there's little point in going further - if you get them three times as worked up, they don't get to vote for you three times. They just get out the white hoods and the burning crosses. And then the decent, ordinary people, a whole heaping lot of them, suddenly start finding a few bucks for the other guy...

What's up with that? Has the right wing lost the ability to count to one?

Statistics as philosophy

This post is related to a point I often try to make (not that I am completely of a mind with the author, but much of what he's saying I identify with).

Fundamentally, statistics is different from mathematics, though it uses the tools of mathematics. Mathematics helps with the "what" (such as "given I want to measure this, what do I do?", but the "why" (such as in the sense of "why work that out, rather than something else") is somewhere other than mathematics.

This point is often lost on otherwise highly competent people. I have seen many good mathematicians come a cropper on it. Some of the worst explanations of statistics I have ever seen come not from people who have trouble with the mathematics in it, but from people who have no trouble with the mathematics at all. (I could name names, but I am feeling generous today.)

And it's often like that with students - I see it a lot. As a student, I had a similar experience to the poster I linked to - I was - manipulation-wise - reasonably competent at statistics. I could do the calculations, if it was reasonably clear what calculations were required. But I did two years of it and still didn't comprehend it. I didn't even comprehend that there was something to comprehend - after all, I could pass the subjects okay, so I must have 'got it' okay, even though it seemed sort of wishy-washy to me. But actually I didn't get it at all. It wasn't until I was some doing third-year subjects that it eventually clicked. I suddenly understood what all the previous subjects had been about. I got it. The material I had learned wasn't a bunch of different stuff all lumped together that was done the way it was purely by convention (though there are no shortage of conventions) - there was, in fact, a coherence to it all. It was all of a thing, it fitted together; the stuff I'd learned was the result of a limited collection of concepts applied to different problems. I could actually begin applying my understanding outside my direct learning, to problems I'd not seen before. I had a framework within which each new piece of knowledge fitted in with everything else.

I'm not certain how to even convey this understanding, though I try. Students recognise that I'm passionate, at least (or so they tell me), though I'm not sure that the "why" always comes across to more than a very few of them.

They wasted no time with That One

http://www.thatone08.com/

http://shop.cafepress.com/that-one

Happy Happy Joy Joy

I am one happy blogger. I have completely sorted out a little mathematical problem that's been plaguing me for ages. It's one where I already knew the result, but all the proofs I could construct were either too embarrassingly clunky to use (I mean, really, really awful), or elegant but handwavy in one place.

I realized last week we really needed this result for something I'm working on with my research student. I sat and thought about it for a while today and finally noticed that the one remaining bit of argument we needed to make was obvious if you just recast the whole problem as a count from a thinned Poisson process. The crazy thing was my old handwavy argument was in effect already doing that, I just had failed to recognize it for what it was. Now that it's been set up in the right way, all the handwavy aspects drop away, and a nice clean half-page argument based on already-known results establishes the result we need.

This is one of those moments when after the fact everything is so obvious that I feel inadequate for not having seen it much earlier, but for the moment the joy is undiminished, because what this small step gets us to is something dramatic (assuming showing a bunch of well-known-in-their-application-area people that what they've been saying and doing is completely wrong is dramatic).

Quote

"My basic objection to religion is not that it isn't true; I like plenty of things that aren't true. It's that religion grants its adherents malign, intoxicating and morally corrosive sensations. Destroying intellectual freedom is always evil, but only religion makes doing evil feel quite so good." - Philip Pullman

I haven't given up blogging - I'm just keeping very busy indeed. My mother even emailed me wondering what was wrong.

Drawing of the space habitat

It appears that a number of people are misunderstanding my description of the space habitat (on the order of the "invasion fleet being swallowed by a small dog" scale).

So here's a picture, approximately to scale:

(Edited to correct drawing the moon orbit too small.)

The large circle is the earth's orbit. The circle in the middle is the Sun. The little circle on the left, that's the Moon's orbit around Earth. (Earth itself is a teensy dot in the middle of that circle, about as big as the line marking the orbit of the Earth around the Sun. I drew the Earth in, but you probably can't see it in the small image). The relatively large circle on the right, that's the space habitat sitting at a Lagrange point (or rather, sitting with a Lagrange point at its centre). It is so phenomenally big it might not actually be stable here (I have not done the calculations to check).

(By comparison, the original Ringworld would be represented by the really big circle. Much larger.)

Now the habitat goes around the Sun in the same orbit as the Earth, offset by 60 degrees. It rotates about its own central axis once every 24 hours, giving it a day-night cycle. The ring would be tilted a little, so it doesn't shade itself most of the time - in fact the ring will precess as it goes. The artificial gravity induced by the rotation would be about 1g (it would, I think, vary somewhat between night and day, because at midnight you're orbiting a fair bit nearer to the Sun than you "should" for the center of mass of the habitat, and midday you're a bit further from the Sun).

Large space habitats

I was thinking about space habitats. It was probably triggered indirectly by the fact that I'm reading* a SF story at present (The Algebraist, Iain M Banks), though there's no direct connection to my resulting train of thought.

*when I'm travelling, I read. There's lots of waiting about and sitting in planes and stuff (13-14 hours across the Pacific, for starters) and I rarely feel well enough in flight to concentrate on actual work. Even if I only spend a fraction of that time reading, I will get through several books on a trip.

It occurred to me that there will be a particular radius of "ring"-type space station where one rotation in 24 hours would produce artificial gravity of about 1g (this will obviously be large, even without doing calculations).

If you put such a habitat at an L4- or L5- point on the earth-sun system, you'll get earth-level radiation from the sun (without the nice van Allen belts to protect you, unfortunately), with earth "days" and earth "gravity".

So how big is it? Well, I did the calculations, and I get a radius of 1.85 million km. This thing is huge - the earth-moon system would fit comfortably inside it.

It's not Ringworld-huge, by a long, long shot - on that scale, it's miniscule. But still, very very large indeed. If it were a thin ring about 100 km wide (it's about 11.5 million km around, so 100 km wide is indeed "thin"), it would have a "land" area roughly eight times that of earth (assuming people live only on the inner surface of the ring). For the present I'm assuming you'd have a series of interconnected domes or similar on the inner surface, which allows you to bring air, water and other resources in stages.

Assuming, that is, that I have done all the calculations correctly.

Many of the problems associated with a ringworld habitat go away - you don't need to worry about the orbit being unstable for example, though I guess if the mass gets large enough there may be some issues with the stability of the Lagrange points. The amount of material required is far, far smaller than for a ringworld - and the relatively more obtainable amounts of material mean much less hyper-engineering is required.

I haven't done any engineering calculations to work out stresses and such, so I don't know whether you could build a lot of the base structure from simple rock, or if you really need to go to strong metals or even unobtainium.

A nice little thought experiment, anyway. I don't recall seeing anything like this in a story. I'm not sure if that's because I don't read around enough or just that nobody has used an idea like this - but I cannot have been the first person to work this out, so I am curious to know if it has been used in a story somewhere.

[Edit: there's some nifty ringworld artwork to be found.]

Living in interesting times

Regular readers will have noticed a dramatic reduction in my posting frequency over the last month. This has been largely due to an upcoming trip to the USA and some other things keeping me busy; I will be in the USA next week.

I will be busy for a time after I return as well, so the sporadic posting will continue for at least a little while.

TV Science Fiction on humanism and nonbelief

Characters who lack belief are common in SF, yet - unusually for television - they are usually presented in a very positive light. Let's consider a few popular shows.

Star Trek
The characters in the various series are almost universally humanist, and the Federation is almost relentlessly humanist. Star Trek is famous for the sense of hope it conveys about the future, and I think that's largely connected to the humanist sentiment that runs through the entire franchise. In the original series the only main character with any apparent religious sensibilities is McCoy, who does refer to God, and does seem to have a religious background and appears to maintain some level of faith (though he does not appear to be observant of religion). The only main character with any real involvement in even quasi-religious ceremony is Spock, but that ceremony is disconnected from supernatural beliefs. In TNG, the main characters are if anything, more humanist. Worf, while raised by humans, appears to have had enough steeping in Klingon culture to have some degree of acceptance of Klingon religion (he does make reference to Sto'Vo'Kor, for example) and practices some Klingon rituals. None of the humans is particularly religious. Deep Space Nine presents a strongly faith-based culture (the Bajorans), but it is made clear in the show that the beings they base their religion on are not supernatural, simply very powerful aliens.

Firefly
The captain of the Firefly, Mal Reynolds, repeatedly discusses his lack of belief, and is consequently presented as an atheist. One of the other main characters, Book, is a "Shepherd", a kind of priest, but we almost immediately discover very strong indications that he is much more than a simple priest. In one 'episode' (in both senses) River fixes Book's bible, by removing or changing all the parts that make no sense - the book ends up in tatters. The main "spiritual" character is Inara, her religious sensibilities are more Eastern, and several times she is seen to minister to Book; to my recollection, one episode ends with what can only be described as a kind of benediction - Book kneels before her in misery while she places her hand on his head in a metaphorical blessing. Religion may be somewhat important in the wider culture (it features in several episodes, but is as frequently a source of hatred and manipulation as much as comfort), but it is not important in the lives of most of the most of the crew. A sense that people will be good or evil with or without religion clearly comes through. Firefly is somewhat more independent and libertarian in sentiment than the other shows, but many of its characters have a strong humanist bent.

Battlestar Galactica
This show is unusual in that it's an overtly religious society, though many characters are not particularly religious (and a few are openly doubtful). Doubters are not regarded as "evil". Religious and nonreligious characters generally seem broadly accepting of each other. The main religion of the humans is polytheistic, that of the Cylons is monotheistic. Several humans become much more religious over time, but one is of dubious sanity (starting out sane and atheist and becoming apparently insane and somehwat religious); in each case the increase in belief is associated with apparent evidence consistent with that belief (even though some of the resulting beliefs are contradictory).

Stargate
Stargate is another overtly "humanist" program, but it is much more explicit in its treatment of religion. There are three main "enemies" in the series - the Goa'uld, the Replicators and the Ori. The first and last are gods to their followers - false gods, but gods with great powers nonetheless, so it is little surprise that they have great followings. The heroes aim to convince their followers that those they worship are not gods, and ultimately to defeat the false gods. None of the main characters are religious (though Te'alc is initially a believer, he throws off his religion). The Ori in particular, mirror the worst aspects of fundamentalist, dominionist religiosity.

Dr Who
The Doctor himself is an avowed lover of humanity, and the show is unremittingly humanist. Religious themes do come into the show sometimes, but the explanation is generally more on the natural side than the supernatural.

All of these programs deal in some way with "constructed families"; Firefly is probably the most explicit of these, but in each of them, "family" is something you make, not somethng you are. 'Traditional' families are not a major aspect of any of the shows (on DS9, Chief O'Brien has a 'traditional' family ... and marriage problems); one parent families are common. Yet love and loyalty to friends and colleagues run very strongly in all of the programs. All four show quite clearly that morality and religious belief are largely orthogonal. Stargate is perhaps the strongest in its anti-faith message (it takes no clear position that all religions are false - but all the religions that have a substantive place on the show clearly are false, dangerous and evil). People who lack overt belief are common, and almost all of them are moral, heroic, loyal, loving... and most of all, human.

Science Fiction, and in particular, TV science fiction - because of it reaching a large and regular audience, has a small but significant influence on our society. Because it is forward looking, the influence is generally strongly progressive, and in the way it presents its major characters, generally presents atheism in an extremely positive way, far out of keeping with the common depiction of atheists in other programmes (who are often presented as cold, abrasive, misanthropic or amoral, when they're presented at all).

--

Book: What are we up to, sweetheart?
River: Fixing your Bible.
Book: I, um... What?
[River is working on a mangled bible. Passages have been crossed out or corrected. Loose pages lie about.]
River: Bible's broken. Contradictions, false logistics... doesn't make sense.
Book: No, no. You - you can't...
River: So we'll integrate non-progressional evolution theory with God's creation of Eden. Eleven inherent metaphoric parallels already there. Eleven. Important number. Prime number. One goes into the house of eleven eleven times, but always comes out one. [She looks him in the eye.] Noah's ark is a problem.
Book: Really?
River: We'll have to call it "early quantum state phenomenon". Only way to fit [laughing quietly] 5,000 species of mammals on the same boat... [She rips more pages out.]
Book: River, you don't... fix the Bible.
River: [Speaking gently.] It's broken. It doesn't make sense.
Book: It's not about... making sense. It's about believing in something. And letting that belief be real enough to change your life. It's about faith. You don't fix faith, River. It fixes you.
(Book tries to pull some of the ripped out pages from River's hand, but they tear.)
Book: You hang on to those then.

Firefly: Jaynestown