Thursday, August 09, 2007

Dangerous Knowledge

The Sunday Times loved it, making it the Pick of the Week (BBC4, Wednesday, 10.05). “There is no space here to describe exactly what was so tricky about the work of Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing but this fascinating documentary describes how trying to solve the biggest problems in maths or physics drove the four men to their deaths.” Right.

Ninety minutes to cover transfinite set theory, entropy, the incompleteness theorems and non-computability is a tall order. The programme settled for not really explaining the great men's work in any detail, focusing instead on the thesis that it was the radical and unsettling nature of their theoretical contributions which drove them to insanity and/or death.

There is evidently a problem with this thesis. At any one time thousands of students are studying all of these concepts, in mathematical logic or physics courses. There is no evidence whatsoever of a corresponding widespread inclination to mental illness or suicide.

A more likely, although less telegenic theory is that each of these individuals was vulnerable/mentally-unstable, and each was subject to unbearable social pressure from unsupportive colleagues (and in the case of Turing, state repression). Unsurprising they buckled, as many people do.

Cantor was said to have been driven to schizophrenia by his failure to prove or disprove the continuum hypothesis. The programme failed to explain to its audience what this hypothesis was and I was irritated. This is really pretty simple stuff. After the programme, and over her objections, I explained it to Clare.

“The problem here is to work out the size of various sets of things. For example, the size of the set {apple, orange, pear} is three, because we can create a rule which assigns 1 to apple, 2 to orange, 3 to pear. Then we run out of new things to count so the size of the set is 3.

“If we think of the set 1, 2, 3, 4, and so on, this is called the natural numbers and we can call its size, for the time being, 'infinity'. OK so far?” (Yes, it’s still working).

“Counting infinite sets is more difficult, but more interesting than with finite sets. Take the set of even numbers {2, 4, 6, 8, 10, ...}. We can create a correspondence like this:

1 <-> 2
2 <-> 4
3 <-> 6
n <-> 2n

Every even number is somewhere on the list, and this shows that the size of the natural numbers and the size of the set of even numbers is the same. It’s the same infinity, right?”

Shuffling in the chair and a nod to continue.

“Now we come to the fractions. At first sight it seems obvious there must be many more fractions than natural numbers. After all, there is an infinity of fractions between 1 and 2. But appearances can be deceptive.

“Imagine a big grid or spreadsheet with the numbers 1, 2, 3, ... along the top and also down the side. In each cell, write down the fraction with the fraction-top being the top column number and the fraction-bottom the side row number. So, if you go three along and four down, in that cell you write 3/4. Do you agree that every fraction can be found in this infinite spreadsheet somewhere?”

She thinks about it. “Yes.”

“Right, imagine an ant starting in the top corner 1/1. The ant walks diagonally backwards and forwards, visiting each cell and unspooling a tape measure with squares marked 1, 2, 3, 4, .... As the ant visits each cell, she writes down the fraction in that cell. After a little while, the start of the tape measure looks like this.

index <-> fraction as written by the ant
1 <-> 1/1
2 <-> 2/1
3 <-> 1/2
4 <-> 1/3
5 <-> 2/2
6 <-> 3/1
7 <-> 2/3

“Do you agree that the ant will eventually meet every fraction (some multiple times, as for example 1/1 = 2/2 and so on, although we could delete copies if it mattered)?”

“Yes, that’s quite clever.”

The ant will take an infinite amount of time to complete her tour of the spreadsheet, but when she’s finished, we pick up the tape measure and let it hang. There’s the list of all the fractions, each lined up with a number from the list of natural numbers. So despite the infinity of fractions between any two natural numbers, and indeed, between any two fractions, the total number of fractions is just the same as the total number of natural numbers. It’s the same infinity.”

Here it all starts to unravel.

“No, that can’t be right. You’ve got a fixed gap between any two natural numbers, like 2 or 3, but you can always fit in more fractions.”

“Yes, that’s true, but the way we set the ant to work, that doesn’t matter. We always get to any fraction eventually, and then it’s position is fixed on the list. I agree, the list is out of order, but that’s inevitable.” (Given that the fractions are dense - but I didn’t say that).

“No, I don’t agree with that way of counting.”

I pause for breathe and continue.

“OK. Think of the set of natural numbers, {1, 2, 3, 4, ...}. Now consider a new set - call it the ‘Clare Set’. This is the natural numbers with just the number 10 deleted. It’s {1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, ...}.

"Now is the ‘Clare Set’ bigger than, the same size as, or smaller than the natural numbers?”

“It’s smaller. Obviously. It hasn’t got 10 in it.”

“But both sets are infinite. So, is your ‘Clare Set’ infinity a bit smaller?”

“That’s why no-one wants to be a mathematician. You’re always hair splitting and inventing stupid rules. I wouldn’t do it that way.”

“But that’s the way that it works - you have to make some assumptions to analyse infinity at all, and this is the most natural assumption - it’s a simple extension from the finite case.”

“You’re so conventional. Try thinking for yourself, not just parroting what you read in books.”

So there you are: game over.

I never got around to the famous diagonal argument, to show that the Reals are uncountable and therefore a ‘larger’ infinity. I wasn’t able to explain the power set construction, whereby from any set (finite or infinite) you can always create the power set of strictly greater size. So I couldn’t finally get around to the continuum hypothesis - the suggestion that the size of the Reals (which is equal to the size of the power set of the natural numbers) was indeed the ‘next infinity up’ from the countable infinity of the natural numbers, rather than there being any further 'interpolating' infinities between them.

And the great mathematical drama whereby it was shown that the continuum hypothesis is undecidable within standard set theory also remained unexplained. This is the reason Cantor couldn't prove or disprove it.

I concluded that it’s just naive to think this stuff can be packaged for even an educated non-mathematical audience ‘out there’.

And to think that Clare used to come round to my flat before we were married for maths lessons. How I have failed her!