A more intellectual offering this week. It was non-fiction night at my writing group again. The more long-standing of my readers may remember my piece on the history of zero last year. This time round, I thought I might take a look at infinity …
To Infinity And Beyond
The story of infinity has, as you might expect, a middle but no recognisable beginning or end. In its various guises, it weaves its way through time and space, interacting with human history whenever the fancy takes it, and going off on its own adventures for the rest of – and presumably, the bulk of – its existence.
What I have written here is not an attempt at the history of infinity, nor have I striven to paint a complete picture of the complex, yet simple, idea. Rather this is a snapshot of my reflections about, and responses to, the intriguing concept. A piece about infinity.
I begin this way because I wish to point out that it cannot, of course, be a piece of infinity, because that would entail knowing how big infinity is. The construction of a fraction of infinity – indeed, of anything – can only be made by measuring the whole, and if one knew precisely how large infinity is, then it would, by definition, no longer be infinity. Infinity is infinity because it has no limitations or boundaries.
I offer this perplexing paragraph as an introduction because it sums up precisely the conundrums (or should that be ‘conundra’?), fears and ‘incomprehensibles’ that surround the concept of infinity, making it a thoroughly intriguing and challenging idea.
The first recorded writing about infinity comes from an ancient Greek philosopher, Anaximander, but the earliest use of it as a mathematical concept is attributable to Zeno of Elea, who you may remember had a lot to say about zero and the nature of tortoises. In fact, many of his ponderings seem to have included both zero and infinity as essential ingredients in understanding – or misunderstanding – the universe. For even in pre-Socratic circles, there were already a number of different kinds of infinity being defined.
The trouble with infinity is that it defies definition as much as it does measurement, making it a dangerous concept to play with. It is not a straight-forward mathematical notion, veering at will into the worlds of philosophy, metaphysics and spirituality. It just will not behave.
Why, then, bother to include it in such a precise discipline, one may ask. Surely, we’d be better off restricting the numerical world to numbers that actually have tangible meaning? Unfortunately, the blessed construct keeps cropping up of its own accord. For example, one cannot ask perfectly innocent questions such as what is the largest number?, or what do you get if you divide any number by zero?, without automatically arriving at the necessity to answer ‘Infinity’, even though the solution will appear to be imprecise.
Dealing with infinity is rather like trying to get a cat into a bag. Unlike Schrodinger’s cat who was apparently well-trained enough to sit inside a box (albeit managing to be both alive and dead at the same time), more normal cats have an in-built aversion to being contained, and will inevitably either employ shape-shifting qualities to leak through even the smallest opening, or use their excellent teleporting skills to transport themselves to the other side of a closed door, thus rendering it impossible to trap them where they do not wish to be. Infinity behaves in a similar fashion.
Doing mathematics without infinity is impossible, but doing them with it, is almost as difficult. Mathematicians, therefore, cheat. They make useable definitions of infinity, they create categories of infinity and they lie about what it really is.
For example, in the Jain mathematical text, written approximately in the fourth century BCE, all numbers are divided into three categories: enumerable, innumerable and infinite. These categories are further subdivided into three. Enumerable includes lowest, intermediate and highest; innumerable is made up of nearly innumerable, truly innumerable and innumerably innumerable; whilst infinite numbers can be nearly infinite, truly infinite and infinitely infinite.
Such wonderful nonsense persists today, as academicians everywhere attempt to make the unassailable, manageable, in order to practise their craft. The nomenclature of ordinal and cardinal infinites in set theory, the use of countably infinite integers as opposed to the infinite set of uncountable real numbers, and the subtle invention of hyperreal numbers which include infinite numbers of different sizes – are all magnificent illusions which enable mathematicians to do what they do.
Mostly, they seem happy with their tricks, but scepticism about this approach has brought about an extreme form of mathematical philosophy called finitism, with all the turf-wars that entails.
But if you think mathematicians have a problem, spare a thought for the physicists. After all, they have defined their branch of academia as the science of measurement, so they can get really ‘antsy’ when infinity shows up. Some take the simplistic view that if you can’t measure something, it can’t exist, however theoretically valid it might be. Mostly, they propose that using infinite series and the like is tolerable if the end result is physically meaningful.
These are the physicists who aren’t that keen on quantum theory. As you might imagine, infinity – in all its glorious manifestations – inhabits this world in abundance. When it appears as the inevitable consequence of a calculation, it is quickly hijacked and made into something more acceptable. A process known as ‘Normalisation’!
However, cosmologists inevitably come to the rescue, spilling the indefinable all over their infinite universe, proclaiming the impossibility of normalising, for example, a black hole and asking difficult questions such as how many stars are there? and how big is the universe?
Because here it becomes nonsensical to insist on limitations. Wherever one puts the end of the universe, there must be something beyond it. Enter the magical world of Topology – that magnificent branch of mathematics that deals with consistent properties of certain objects that remain after transformation, such as connectedness and continuity. We’ve probably all encountered the Möbius strip, that tantalising bit of paper with its single surface and one edge.
These adventurous ideas have allowed cosmologists to point out that the two-dimensional surface of the earth is finite yet it has no edge. Perhaps, they suggest, the universe has a similar topology.
I pondered this one on the park walk with Rosie, my collie, this morning. She very sensibly retorted that if the universe were ball-shaped, there would have to be a dog lurking somewhere nearby, with jaws big enough to clamp its teeth around said object – and where would your finite universe be then?