Thursday, June 16, 2011

A Fermi Challenge

Here’s an example I love, and which illustrates my problem with sceptics who are too keen on ‘argument from authority’. There's more about it here. The great physicist Enrico Fermi was fond of setting problems that seem difficult to solve, but which can be approached by a mixture of logic and common sense (and particularly without any appeal to ‘scientific’ knowledge or complex mathematics). One such problem was that of estimating the number of piano tuners in Chicago given only the population of the city. This seemed impossible to his students, but is fairly straightforward to resolve as follows.
  • From the almanac (the only authority required), Chicago has a population of about 3 million people.
  • Assume that an average family contains four members (Fermi was teaching in the 1930s when families were that large).
  • Therefore the number of families in Chicago is about 750,000.
  • If one in five families owns a piano, there will be 150,000 pianos in Chicago.
  • If the average piano tuner serviced four pianos every day of the week for five days rested on weekends, and had a two week holiday during the summer, then in one year (52 weeks) he would service 1,000 pianos. 150,000/(4 x 5 x 50) = 150, so that there must be about 150 piano tuners in Chicago.
Why do I like it? It does not rest on any appeal to authority, and depends on assumptions that are either common knowledge, or located in easily verifiable sources. Thus it is essentially difficult to refute. ‘Scientific’ claims, by contrast, rest on complex arguments based on previous scientific results, published in difficult-to-access sources, supported by a community who may, for all we know, be conspiring against the truth. Thus all ‘scientific’ claims, like appeals to authority in general, are easy to refute.

As an exercise, what is the simplest argument against the Young Earth Creationism - the theory that God created the world about 5,700 years ago? Most people now accept that God created the world (together with time and space, and the number system) more than 14 billion years ago. But why do they accept that? What is the simplest argument for the generally accepted age of the universe that you could write down on a credit card, and which you could explain to a 9 year old? You will find that this article, which is completely incomprehensible, is absolutely no use. I offer a prize of a £5 Waitrose voucher to the first decent reply to my challenge.


David Brightly said...

I suppose the simplest argument for the age of the universe would go like this.

1. We would start with the observation that distant galaxies are receding from us at velocities (v) that are proportional to their distance (d) from us. So v=Hd and H (Hubble's constant) is measured as 70 km/sec per megaparsec. A parsec is just an astronomical unit of distance equal to 3.1x10**13 km.

2. Assuming that recessional velocities are constant, a galaxy receding from us at speed v and now at distance d from us would at time t=d/v in the past have been coincident with us. But d/v is just 1/H. The reciprocal of H turns out to be 14x10**9 years. Hence all the matter in the universe was in the same place 14 billion years ago and that's when the universe supposedly began in the 'Big Bang'.

The second part of the argument is quite elementary. The first is highly 'scientific'---it depends on lots of theory about firstly, the Doppler shift in wavelength of light emitted by a moving source, not in itself too hard to follow, and secondly about how to measure the distance of remote galaxies, which is a horrendously complicated and theory-laden business. So I think your £5 will be safe.

But why would anyone expect a simple explanation for the age of the universe along the lines of the Fermi puzzle? It's not as if it's something any of us have much experience of or intuitions about.

If the Fermi argument is difficult to refute and the age of the universe argument is easy, then I'm not sure I get your notion of 'ease of refutation'. I'd say the former is easy---you have assumed each piano gets tuned once per year---and the latter hard, precisely because it hinges on tricky issues in astronomical distance measurement. Of course, the assumption of constant recessional speed is readily rejected.

The Fermi puzzle reminds me of the old Cambridge entrance papers in physics. A couple of problems I've always remembered are Estimate the power of a windmill, and Estimate the change in the length of the day if all the world's shipping sailed round the equator from west to east. They were fun.

Edward Ockham said...

Turning your argument on its head, so that we start with the hypothesis. The hypothesis is that if there was a huge explosion in the beginning, the objects that were hurled out the fastest would have travelled the farthest. Assuming their speed was constant, the speed would be proportional to the current distance of the object.

E.g. if one rock is hurled at a constant 20mph, another at 10mph, you would expect, after one hour, that the first rock was 20 miles away and the first only 10 miles. And that (according to your 1) is exactly what we do observe.

But there is the doubt about your 1. Also, doesn't this assume we are at the place where the explosion occurred? And what about the constant speed of recession? Wouldn't gravity tend to slow the speed down over time?

David Brightly said...

>> Also, doesn't this assume we are at the place where the explosion occurred? And what about the constant speed of recession? Wouldn't gravity tend to slow the speed down over time?

Certainly. But if we take the recessional evidence for granted, just as we take the population of Chicago for granted, what is it about the assumptions I am making, compared to the assumptions in the Fermi puzzle, that make the former argument easy to refute and the latter hard? I would say that they are on a par.

Edward Ockham said...

>>But if we take the recessional evidence for granted, just as we take the population of Chicago for granted

We had to look up the population of Chicago, I agree. But we felt no need to challenge that, and it is easily verifiable. Indeed, I have been a few times to Chicago and rode the elevated train through the suburbs, and can confirm the population is of that magnitude (give or take a few million).

There is no reasonable basis for challenging the population figure. But the recessional evidence could reasonably be doubted, because it rests on totally unverifiable assumptions (AFAIK).

David Brightly said...

OK, you are saying the man on the Clapham omnibus is not in a position to verify the recession evidence. He has to accept the opinion of astronomers. But if he wants to reject the conclusions of the experts he needs to explain why he thinks they are collectively mistaken or are conspiring against the public. This is not so easy, unless perhaps our man advocates the strong programme in the sociology of science.

Edward Ockham said...

>>OK, you are saying the man on the Clapham omnibus is not in a position to verify the recession evidence. He has to accept the opinion of astronomers.

I wasn't quite saying that. The problem is to explain the recession claim in a manner similar to the piano tuner example.

David Brightly said...

In what way are the arguments dissimilar? Both rely on theoretical knowledge of some sort---in one case the average behaviour of mid-20th century Chicago piano owners and tuners, and in the other, a large helping of astrophysics. Apart from sociological aspects---accessibility to the man in the street, say---where would you say the difference in 'manner' lies?

Edward Ockham said...

>>where would you say the difference in 'manner' lies

As I was saying, the difference is in the recession assumption. Unless I have misunderstood, this is an assumption that seems to depend on a significant chain of reasoning that hasn't been explained. Ordinary logic suggests that if something explodes, and if gravity exists, then they will be pulled back so that the parts stop expanding, then fall together to the original place (for reasons similar to the fact that a cricket ball hurled into the air falls back to earth).

Why doesn't this happen? Can it be explained in a way that the woman on the Tube can understand? If not, it's open to the fringe proponent to claim that it's a cover up, or to propose some strange alternative theory.

My only knowledge of this is from watching a Horizon program. There was some theory that proposed that some basic laws of physics changed over time, which presumes a more basic law explaining this change. It was also implied in the program that this explanation was special pleading. (I amy be wrong).

Edward Ockham said...

That said, Cantor's Theorem is frequently a target for the fringe, yet it is one of the simplest proofs in mathematics.

David Brightly said...

>> Why doesn't this happen?

Well, until quite recently, this was thought to be a possibility, though there is now evidence that the expansion is accelerating. Putting this to one side, we could explain to the woman on the tube that the outward motion is still ongoing, despite the huge length of time involved. The cricket ball is still on its way up, as it were. And there's another possibility: not everything that goes up need come down. Ignoring air resistance, a cricket ball thrown up with sufficient speed will escape the earth's gravity altogether and travel indefinitely far away. See here. Since our experience of gravity is limited to conditions near the Earth's surface the woman on the tube might be forgiven for not knowing this. But she might wonder how a photo like this is possible.

Edward Ockham said...

>>there is now evidence that the expansion is accelerating

This is exactly what I thought I had read. It was something on the lines of the laws of physics actually changing through time. But if they change through time, how can they be laws?

There is of course a 'wild card' or 'Hume card' that the fringe proponent can play, namely that our belief in physical laws is based on constant experience, but there is no logical reason why that experience should continue to hold. Do you remember Nelson Goodman and the 'grue' paradox.

In reply, I suppose the Ockhamist can appeal to the law of simplest explanation. (Except the law is not Ockham's, though he appeals to it, but rather Aristotelian and Scotistic).