Is there a dog?: Maths

Supertasks and the impossibility of change

Mon, 20 Mar 2006 03:46:51 GMT

In the last post on infinity I looked briefly at the idea of the implication of a perfect being contained within the concept of infinite sets. This being has to be able to perform a supertask, or a task which involves infinitely many steps, within a finite period.

We saw in the old man what could possibly be that perfect being, but we could imagine a lot of different examples. Achilles in one of Zeno's paradoxes is another, in the fact that in order to move at all he has to traverse an infinite number of finite intervals in a finite period. For those unfamiliar with the paradox, suppose you are Achilles. Now being a runner you want to run a race, let's say a hundred metres. But in order to run a hundred metres, you must first run 50 metres. Which is all well and good, but in order to get from the halfway point to the end of the race, you must first run half the remaining distance, leaving another 25 metres to run. Zeno's conclusion was that we could carry on dividing this interval up for ever, leaving poor old Achilles stuck, like Tantalus, unable to reach the desired goal. As a corollary to this, he also concluded that all motion, and indeed all change, was impossible.

This is a paradox which most mathematicians think of as having been resolved by the invention of the differential calculus, but which still presents a dilemma. The fact that we can move and observe others moving is irrelevant to the gist of the paradox, which is that our model of the real number system is a poor fit with our understanding of the universe. In fact, theoretical physics postulates a basic unit of length (the so-called Planck length), and argues that our understanding of space and time collapses after attempting to divide space into smaller units than this. So Zeno's idea of infinite division will not work, after all.

Another supertask we may consider is that proposed by the mathematician David Hilbert, in his paradox of the grand hotel. Suppose you have a hotel with an infinite number of rooms. These rooms are all full - you're having a good tourist season. So then another tourist shows up, asking for a room. You are stuck, until you have a smart idea - ask all the guests from Room 1 onwards to move into the next room along, so the guest from Room 1 moves into Room 2, the guest in Room 2 moves into Room 3, and so on. This of course will leave Room 1 free for your tourist to move into, and because there is no end to the rooms in your hotel, every guest will have another room to move into. (Let's agree to ignore the fact that this will take eternity to organise, shall we? )

In fact, you could shift an infinite number of new guests into your hotel, in the following way: Shift the guest in Room n into Room 2n, so that the guest from Room 1 moves into Room 2, the guest in Room 2 moves into 4, and so on. Again, this will never fail to work, as you have no end to the number of rooms, and you will have freed up an infinite number of rooms for your new guests (they just move into the odd-numbered rooms).

Once again, ignoring the fact that this is a supertask, the nice thing about this example is that it shows up clearly how infinity disobeys our intuition. If there were only a finite number of rooms, by taking a guest out of any one of them, we have changed the number of rooms occupied. In this hotel, however, we haven't. An infinite set can lose any finite number of elements, and will remain infinite. In fact, it can lose an infinite number of elements and it may remain infinite - that is, the number of elements has not changed in the slightest.

There are really two notions at war here, and it remains to be seen which will come out on top. More to come...

The Ghost in the Machine

Thu, 16 Mar 2006 10:24:40 GMT

The set of numbers we use to count - the set {0,1,2,3,4,...} - is the most natural of the sets of numbers we use, and for this reason mathematicians refer to it as the set of "natural numbers". It is also, however, taken to be infinite, as my mum revealed to me as a youngster. This can cause some problems - well at least one, which I will explore a little here...

Firstly, an example I remember being used by a maths lecturer many years ago went something like this: Imagine you are walking in the woods and come across an old man - really, really old, with a huge grey beard, skin like parchment and deep sunken eyes. He is sitting on a rock, a look of intense concentration in his eyes and a manic air of purpose about him. As you approach, you realise he is counting, backwards, "...5, 4, 3, 2, 1, 0!", and with that he leaps off the rock, grabs you by the shoulders and shouts, "I did it! I did it! I counted backwards all the way from infinity!!" Now of course the curious mind, wondering about the veracity of the old man's incredible (if a little strange) claim, will immediately want to ask the question, "So...when did you start counting?"

On one level, the old man is simply deluded, naturally: one cannot count the entire set of natural numbers in a finite time period. But the problem goes deeper than that. With any finite number, we could imagine a situation like this, finding an old man counting backwards from this number to zero, with the simple proviso that if the numbers got really big, he would have to have counted really fast to fit them in to one lifetime. So if numbers reflect some sort of objective reality, perhaps this is one way we can verify that they do in fact match up: if it is conceivable that there is some person, or being, or machine, that given enough time could perform the task of counting from a number to zero. Infinity, when matched against this criterion, is not a number at all: it is a concept which has no possible reflection in our experience of the universe. All we have done is imagined it: if you like, we have reasoned it into existence. But mathematics accepts this existence without question. Proofs in almost every area of mathematics rely on the process of 'induction', whereby we essentially agree to enumerate a set of things all the way up to infinity.

So there's one problem, in a nutshell. Incidentally, one of the implications of accepting this notion of infinity could be that we also have to accept the existence of a timeless being inhabiting the ideal (platonic) realm, who would be able to count forever. Interestingly, some writers have explicitly identified this being as God, and many mathematicians throughout history have seen themselves as being on a mission God has given them. One writer who has advocated removing this being from maths altogether is Brian Rotman, in the book Ad Infinitum: The Ghost in Turing's Machine.

However, in my next thought on this topic, I'll let this pass and allow this existence - as they say, for the sake of argument. Until then, happy imagining...

Eternity is a long time...

Mon, 13 Mar 2006 09:06:41 GMT

Well, as promised I will say something about infinity now. It struck me after reading some comments and looking at the arguments I referred to earlier that many people have just one picture of what infinity is. And fair enough - most people who are not mathematically inclined will not ever need any more than that. In fact the argument could be made that noone really needs any idea of infinity at all, and again fair enough. But since this is my space and I get to say what I like I can write this sort of entry and you either read it or not, as you like. Permit me to try to inspire you...

The earliest time I can recall thinking about the infinite was when I was about 6 or 7 or so and I was just starting to get interested in numbers. My mum naturally enough wanted to encourage this sort of thing and so she got me to try to think of bigger and bigger numbers. I got to a million, then a billion, then a trillion and she was able to explain what these were to my satisfaction. But what came after that? I wanted to know. Well, then there is a thousand trillion, then a million trillion, then a billion trillion, then a trillion trillion, she said. And what's after that? asked I, thinking that perhaps I was beginning to push the boundaries of my mum's presumed boundless knowledge. Well, she said after a pause, then there is ... a trillion trillion plus one.

Oh I said, now for some reason a little crestfallen. Was that it? You just add one? Well, yes, mum said, but you can keep on adding one again and again for ever. You could keep on doing that for a trillion trillion years and never run out of numbers. That's what we call infinity.

I thought, that's pretty cool. Numbers just go on, for ever and ever. That was the first time I came also to contemplate the idea of eternity. See, as my family are Catholic we'd had plenty of talk about eternity before without me really being awake to the idea. A long time to a kid at that age is about 20 minutes, or maybe less (like the time it took to sit through a sermon in church). Until that point, eternity was just a longer time than that. Like maybe a whole hour, or even two. Now I was beginning to wake up to the notion in this quote from Hendrik Willem van Loon:

High in the North in a land called Svithjod there is a mountain.
It is a hundred miles long and a hundred miles high
and once every thousand years a little bird comes to this mountain to sharpen its beak.
When the mountain has thus been worn away a single day of eternity will have passed.

Similarly, I began to think about the far reaches of space, stretching out my imagination forever among the stars into a universe that never ends. This is about when I began to take an interest in science fiction, which remains to this day (the interesting sci-fi, not space opera or Star Wars).

All of the above is what got me interested in infinity. Now, years later it finally occurs to me that this was perhaps in part what accounts for my interest in maths. Perhaps I wished to explore the idea, to see how we could justify having such an outrageous thought. And indeed it does underly all mathematical thinking, to the point where some philosophers of mathematics have stated the need to rework mathematics from the ground up, to eliminate all reference to the idea of infinity. The reason? I'll get into that later.

For now, let's be content to think of this as a pretty cool idea. Next post, I'll have a look at why infinity can actually create some serious problems, if you really think about it...