Nov 24 1994, 7:04 pm
PI is a concept, by which we measure our brains.
This is another long post, so here's a quick summary of the contents to allow you to give up in disgust or continue reading regardless, as the case may be:
Please excuse any lapses of style or composition. Unless I post this as it came out, warts and all, I'll never post it.
First of all, let me apologise for being deliberately over-provocative in my challenge for proofs that pi would retain any significance in worlds governed by radically non-Euclidean geometries. It was my hope to provoke a slightly deeper degree of mentation and your responses have not disappointed me in this respect.
Reading through these responses, all of which re-iterate that "pi is pi", I was struck by the apparent lack of agreement as to what this actually meant or implied.
Needless to say, depending on how you restrict your context, all of you are right. What I have been trying to demonstrate in this thread is the possibility of adopting a wider context, in which all these partial truths cease to be absolute, without any loss to anybody. However, we'll get to that in a moment.
In the meantime, thanks to your efforts, I think I now can re-phrase my original objection to Matt W's objection to Sagan's plot in a way which makes no difference from my point of view, but is likely to be accepted to the assembled realist readership. As a recap for any new readers: the plot involved a discovery of a message from the creator of the universe buried deep in the expansion of the value of pi; to this Matt replied that no such messages were possible because "pi was pi". Ignoring the already agreed other reasons why the plot is nonsensical, this objection struck me as wrong.
Let's re-phrase the plot. A message from the creator of the universe is found in the expansion of the value of the most pervasive constant we know, which in our maths happens to be pi. Since at least some of you agree that it is possible (not necessary, but possible) that universes governed by other metrics (and/or any other deviant parameters) may have all-pervasive constant(s) other than 3.14159... Hence a creator, presumably having a choice of such parameters could choose a set which generates an all-pervasive constant with a particular feature of the creator's choice. From this point of view, Matt's objection that "pi is pi" misses the point. Whether or pi is synonymous with the value 3.14159... is neither here nor there. In the wider context, in which the creator had a choice of the all-pervasive value, it still might make sense to examine pi (leaving aside the statistical objections, of course). It doesn't even matter that I cannot prove that other universes may have other dominant constants, the very possibility is sufficient.
Once you put it like this, it becomes immediately obvious how Sagan hit upon the plot in the first place. Some years back, radio-astronomers decided to listen to potential messages from other civilisations on the wavelength of 21cm (or thereabouts) because it was believed the most pervasive radio frequency in the universe. (Not sure whether they still think it is – makes no difference.) I bet you that Sagan asked himself where one could look for a message from the creator and it occurred to him that it would be hard-wired into our world... e.g. a constant... and what is the most pervasive constant there is? Pi of course!
OK... I promise not to labour this any more. Before turning my attention to more interesting matters, allow me a historical digression.
Most of the "hard-core" realist argument so far hinged on the fact that pi is nowadays defined as the notorious Taylor series. This is of course an accident of fate. In the first place, pi could have been called twippple and Matt would be saying "twipple is twipple, period!". So far so trivial. However, all concepts have a history and pi was originally defined as the C/D ratio. By strict rules of taxonomy, that is where the label should remain attached and should it be shown that the ratio can have other values in other geometries, the Taylor series should be called something else (e.g. eupi, for Euclidean pi). You don't hold with taxonomists? Well fair enough. But suppose alternate geometries were thought of before the value of eupi got established by abstract means. It seems natural to me that the label "pi" would have shed any connections with a constant value and Matt W would have been arguing that "pi is pi – a ratio, not some silly constant". But that as may be... Mathematicians attach the label to 3.14159... and stuff the uneducated masses who still hold with the C/D fallacy: quality outweighs quantity! :-) Interestingly enough, the label attachment is now considered absolute and permanent, despite the fact that it has already been peeled off once (off C/D) and stuck on the Taylor series. Still, all of this is just an aside.
More seriously... I found it fascinating that some of you are unhappy about the fact that in my abs(x)+abs(y) metric, the concept of pi apparently breaks into two parts – the C/D ratio is now 4, but the value for which exp(-xi)=-1 is 3.14159... This unhappiness mostly takes the form of wishing to restrict pi to Euclidean geometry. But why? Let me give you my "vision thing".
Pi, as already noted, is for me a complex concept. I sort of "visualise it as a blob with a number of "arms" – one being the C/D ratio others relating to other definitions of the Euclidean pi, the blob itself being probably formed by the Taylor series. I don't expect any of you to disagree so far. Now, add a new complex dimension (well a whole set of dimensions) representing changes in the metric. The Euclidean geometry is now just a slice through a higher dimensioned space. Let's move this slice by varying metrics. As we have already established, once we reach abs(x)+abs(y) we shall see that the original connected region of pi has broken into two. But this is of course an illusion – it is much more appropriate to shift a mental gear and to say that the blob of pi bifurcates somewhere along the metrics dimensions. The apparently isolated regions are still connected to our original blob and still form a part of the same unified concept. Frankly I don't give a monkey whether you call this higher dimensional concept "pi" (as I am inclined to do for reasons explained in my historical digression), or "non-eupi" or "twipple" – the value of 3.14159... is simply one aspect of this overall complex concept, which is a sort of super-pi. It bothers me not at all that some branches of this higher-dimensioned blob cause the value of super-pi to vary and other don't. My pi is a concept – not a fixed value with a label irremovably attached to it.
Before you start fainting in coils (which I hope won't be a general reaction), let me point out that there is a very obvious precedent in the history of mathematics.
Once upon a time there was a geometry resting on postulates, of which one stated that if one had a plane, a straight line in the plane and a point in that plane which did not lie on the line, than only one straight line existed which passed through that point and did not intersect the original line. This was, by definition, a line parallel to the original one. This concept of a parallel line had an obvious consequence – the shortest distance between the two lines measured at any point of either of them was always the same: an inherently equivalent definition, which was in practice much more useful than the non-intersection one. Realists were happy with this postulate, because it was simply a matter of definition, but some other mathematicians kept having an unreasonable urge to derive this postulate from the rest of the postulates, thereby relegating it to the status of a theorem.
Then along came a pair of hooligans by names of Rieman and Lobachevski and upset the applecart by defining straight lines in curved spaces which left realists shrieking with rage. The problem, you see, was that the concept of parallel lines got bifurcated in geometries invented by these two undesirables. I can hear the equivalent of Matt W, patiently ( :-)) explaining to hapless Lobachevski that "parallel lines were parallel lines and just because gospodin Lobachevski could construct on a curved surface something that might be meaningfully called straight lines, but broke the basic concept of parallel lines" – well, it meant nothing at all, of course. He was right too. Take a sphere. As we all know, on a sphere, straight lines are defined as great circles and there are no parallel lines. Yet one can construct connected sets of points equidistant from a given straight line – and this set, while a line would not be a straight line. I.e. parallelism got divorced from its fundamental property of equidistance. In curved spaces the two aspect of the original concept break apart. Did this cause mathematicians to restrict the notion of a straight line to flat spaces? No. Did it stop us talking about "parallel lines" in curved spaces? No. I hope I don't need to labour the analogy. (I'll be told it's all wrong anyway, I am sure! :-))
Where do we go from here? Well, clearly enough there is a potential "fracture line" between the geometric definition of pi and the Taylor series one and others, which manifests itself once one starts shifting along the metrics axis. Perhaps that is the source of the uncertainty in the "pi is pi" camp as to whether all definitions of pi are strictly identical and hence interpretable in terms of each other, or whether the abstract ones have "nothing to do with geometry".
Could there be other fracture lines which would manifest themselves as bifurcations along some other complex axis? My intuition is that there are. Occurrences of pi in statistics may be one splittable area. Of course, I am well aware that at least some such occurrences could be due to the formulae indirectly referring to stochastic experiments in which geometric objects get tossed onto multidimensional spheres. Question, what happens in my Game of Life universe? Well, probably nothing much, but it would be nice to be sure. Maybe some other change in basic assumptions would achieve a bifurcation. E.g. break the assumption of absolute independence of "atomic" stochastic events (perhaps a la Sheldrake?). Would that do the trick? I don't know... But having read Lem, I can imagine somebody simulating artificial universes of arbitrary properties and observing AIs in these universes inventing maths suited for their worlds. What wouldn't I give to see the results!
Let's look at it from yet another angle... I asked several time whether it was going to be necessary to explain the infinitely unlikely coincidence of the Euclidean C/D having the same value as other derivations of the pi blob – a possibility one has to consider if it is true that the abstract ones have nothing to do with geometry. From my higher-dimensional point of view it is clear that the pi blob may split, but there is still no coincidence to explain.
In fact, I am inclined to think that the Euclidean metric is more fundamental than my abs(x)+abs(y) one – because what is two split regions in the latter, merges in the former. I am surprised that nobody offered this in response to my challenges the special status of our geometry. This leads me on to my final metamathical speculation...
Does Euclidean geometry achieve the maximal "size" of its pi blob, or could one construct a geometry (or maths) in which yet more regions, disjoint in our thinking, would be amalgamated into one whole. (Shades of the physicists' quest for ever grander unifications?) If such a theory could be constructed, would it be then somehow yet more fundamental than what we have at the moment? I don't know. But I find these questions damn interesting. And if you have been wondering why am I discussing all of this in r.a.s.w instead of in sci.math, the reason should by now be quite obvious.
Maths is indeed a scaffold we use to approximate our reality, but I doubt whether it is the only one possible such scaffold. Lem's "Odysseus of Ithaca" (in _Imaginary Magnitude_) is very persuasive in explaining that we may be constantly lopping off branches of development which cease to be available to us as we continue building our conceptual frameworks. I suspect that in reality there is a dark sea of maths, and we are only building a network of bridges over this sea, often creating our own landmarks. Hence it occurs to me to wonder, to what extend pi is so all-pervasive simply because we are conditioned to look out for it. For all we know Feigenbaum's constant is even more pervasive, but how many of us would recognise it popping up as a limit behaviour of a given function at a particular point?
Anyway, I've rambled on long enough. I thank those of you who are still reading this for your attention. I doubt that I persuaded any realists to lift up their eyes into the hills, but that is hardly surprising – the realist/nominalist/conceptualist tussle is hundreds of years old and better brains than mine have despaired at settling it once for all.
At the very least, I hope to have given one or two readers a feel for the advantages to be derived from taking a larger view than dictated by any particular, narrow definitions – of anything!