How Big Is Infinity? Dec 1995

Anyone who has studied mathematics, physics or engineering to any degree knows that the world as we comprehend it is analog in nature. That is to say that the things we see around us vary smoothly and continuously in value, and can be represented mathematically by suitable well behaved functions. This, of course, is as opposed to the digital nature of computer software and data, which vary only in discrete amounts. Indeed, with the increasing use of digital sound, graphics and video, our world of ideas and communication is rapidly becoming all digital, as opposed to analog.

But wait, is the world really analog? Can it really be sub divided ad infinitum? Is the nature of space time really smooth and continuous in the same way as an abstract mathematical function?

The very nature of quantum theory suggests that this is not so. The uncertainty principle leads us to say that below the 'Planck Limit' of (as I recall) 10 ^ -33 m, nothing sensible can be said about measures of distance - in a similar way there is a lower limit to intervals of time. This implies that perhaps the world is really digital after all, and that 'regular' mathematics is only an approximation to the real world.

Of course there are branches of (discrete) mathematics that deal with digital constructs rather than continuous ones, but in general these are regarded as a sideline to the real thing. There is however, (at least) one exception to this rule which I came across in a public lecture at the Newton Maths Institute last year; that is Regge Calculus.

I hasten to add that I am no expert in this field, but roughly speaking Regge Calculus attempts to recast general relativity in a discrete form. It does this by constructing space time out of small chunks which piece together following some sort of reasonable rules about what happens at the edges. It struck me as bearing more than a passing resemblance to a technique known as Finite Element Analysis, which is used in my old domain (CAD) to analyse physical structures by breaking them down into small piece-wise linear elements. The lecturer (Ruth Williams from the dept. of Applied Maths and Theoretical Physics at Cambridge) agreed that this was an apt simile. I spoke to her briefly about the light that this might shine on the analog/digital nature of things, and she told me that there was a physicist in the US who had been making considerable play on this topic (T.D.Lee, at Columbia). Up to this point, I had regarded any ramblings my mind had taken along the 'digital reality' road as being amusing but harmless; now I decided to at least take the idea a bit more seriously.

 Perhaps I should be slightly more expansive than my usual cryptic approach, and mention at this point that there have been other attempts to model reality on a 'digital' basis - I seem to recall Feynman and Wheeler playing around with ideas around the world as a 'big computer', and others (possibly Wolfram?) have suggested that the world is a 'software process' running on an invisible underlying digital system.

But enough of that, I don't want to get bogged down in detail, tis the big ideas that interest me <g>.

At this point I want to review why we have mathematics at all (other than to provide employment for mathematicians, that is). Maths is not an end in its own right, although it often seems that way. Maths is a way of formalising our understanding of the world in terms of physical theories. Of course the dividing line between maths and physics has all but vanished in these days of string theories and knot theories. In fact, I have it on good authority that much of the most interesting maths is now being driven by the young theoretical physicists, who are used to advancing at a faster pace than the old maths plodders (please, no insult intended). The point I am making is not new, and has been chewed over many times by philosophers of science and maths historians (e.g. Morris Kline: Mathematics - the loss of certainty). There has always been a danger that pure maths can diverge from reality sufficiently far that it becomes a sterile exercise, though judging how far you can drift is the mark of the good mathematician.

It was thinking along these lines that led the 'Intuitionists' to mistrust much of the body of pure mathematics sufficiently to try a break away movement. Truth to tell, they overdid things somewhat, and most modern maths survives happily without them. However, it is sometimes instructive to question our 'self evident' truths along the lines followed by the intuitionists. In particular, one thing that bothered them was (as I recall) certain types of infinite sets. They raised a reasonable objection that something was not well founded if there could be no defined method of constructing it. This follows the philosophical argument that an object is defined as much by its history as by its present state - an argument that has influenced thinking in differing disciplines, including computer science.

Now, to return to the digital nature of things. If indeed, reality cannot be sub divided indefinitely, what are we make of infinitesimals? Now I enjoy a good epsilon as much as anyone (this is intended for the maths readers among you), but how far should we trust them? Of course, I hardly intend to try and bring down the whole structure of Calculus, it works, its fine, no problem. If someone says that something is Riemann integrable, then I know exactly what that means because there is a well defined theory of limits that back it up. But just because that is so, does this mean that the limits 'need' to be strictly infinitesimal - whatever that means. Normally, infinities both big and small mean roughly - 'whatever number you can think of, I can think of a bigger/smaller one'. This is evidently true, but that does not necessarily mean that the nature of reality follows along indefinitely. If something is correct to 1 part in 10^30 then its going to be very hard to spot the difference.

At the other end of the scale, there is much interest in whether the universe will continue to expand 'forever', or collapse, or just sit on the sidelines, with unfortunately most of the evidence on the boring side (which is why cosmologists keep looking for more dark matter). Mind you, these are tricky concepts. It has been pointed out (Penrose or Barrow & Tipler, can't remember which) that even if the universe were to collapse, then relativistic time dilation arguments can be used to say that the inhabitants would never 'see' the end, so for them it would continue 'forever' anyway. As I say, tricky. So what if the universe is finite, that does not set an upper bound on the set of cardinal numbers. And anyway, there may be more universes outside our own, even an infinite number (sigh). All this I cannot deny, but I am not trying to destroy infinity, merely to say that it should be treated with caution, and to look at what it is and what it is not.

For those of you without a mathematical background, I now have some bad news. There is not just one 'infinity', but lots of them. This has been known for a long time, since back in the days of Cantor, who showed that the set of 'real' numbers was 'bigger' than the set of integers. The 'ordinary' infinity (1,2,3....) is known as Aleph-null, and the next as Aleph-one, which is also the 'size' of the number of mathematical points on a line. But hold, there is also Aleph-two, Aleph-three,... yes you guessed it, there is an infinite number of Alephs (I think). But wait, is that an Aleph-one infinite number of Alephs, or... is there any end to this process at all? I don't know the answer to that, my knowledge of infinities stops somewhere around here, but I strongly suspect that the answer is no. In that case, we cannot even actually define how many infinities there are with a closed statement. This is all very reminiscent of 'Godel Escher and Bach', there is something Godelian about infinities which I mistrust.

To revert to the intuitionist ideas, perhaps we should be happy to accept infinities if we can define a method for constructing them. But then, what does that mean? If I write a computer program (or an algorithm, to avoid syntactic arguments) which says

loop while true: X=X+1, then that 'constructs' an infinite number, but I don't know of any computer that will do it for me. In fact, I would probably need a computer the size of the universe, and even then...

I know that I am perhaps being simplistic in this approach, but I just want to throw a few stones at infinities and see how they bounce off. It seems to me, after all this cogitating, that 'infinity' is not a 'thing' or an 'object' or a 'place', it is instead a 'process'. Perhaps this betrays my computer background, where there has been much theory and thought on the data/program, attribute/method duality in software, and structure/timing in hardware, the last of which can be illustrated using Petri nets, I seem to recall. Of course, a process can be coded (in computer terms), and then can be regarded as data, which just goes to show how tricky this stuff is.

Now the question that started me thinking on these lines - do we really need an infinite number of parallel universes if we adopt the many universe (or Multiverse in current parlance) interpretation of quantum theory? In my piece on 'Why God Invented QM', I pointed out that the Multiverse is a very elegant way of providing parallel versions of universes that would mostly be very boring, but occasionally highly interesting. I also explained that this method bears quite a strong resemblance to a technique known as parallel fault simulation. In both cases, the number of variants can reduce as well as grow when events take place, and according to David Deutsch, this means that the Multiverse, though needing infinite variants, stays stable in 'size'.

 Now I want to argue with these infinite variants. Of course, quantum theory is built on infinitely valued functions, so this naturally leads to an infinite number in the Multiverse. But hold on, do we really need them? If the Multiverse approach tells us that whenever there is an event in a universe, it splits into different variants, and if there is an underlying discrete nature implied by the Planck limit, then should not the number of variants be finite? - large, huge even, but finite. Just because we have some mathematics that works very well should not lead us to automatically assume that it is correctly based. All maths is an approximation. Continuous mathematics is very elegant and very successful, but it just could be a very good approximation to 'reality'.

Now I should add a large number of health warnings. I have been using arguments based on quantum theory to criticise some quantum theories - am I justified?

Then again, it is clear that current quantum theories are not the final omega point of science. It is well known that relativity and quantum theory are incompatible, at least in their current formulations. This is why lots of people are hunting the magic TOE (Theory of Everything), and why there has been much interest in String Theories, which could possibly point in the right direction. But then, this argument could be used to support what I have been drivelling on about if some discrete formulation should prove to be the way forward. I am myself far from clear what the Multiverse interpretation tells us - it is tempting to think that it provides a 'classical' way out of the quantum wood by using variants rather than mystical clouds of superposed states. Unfortunately, there is lots of evidence for the mystical side, not the least of which being superconductivity - which displays room sized quantum phenomenon. And the mathematics remains much the same whichever the interpretation.

So, having displayed much ignorance, and having sown plenty of doubt on my own arguments, I would just like to leave you with a couple of questions on your mind - Should we should trust these darn infinities that we casually employ quite as much as we do? And is the real world really continuous or discrete?