Tuesday, June 01, 2010

Mathematical Mysteries

Someone emailed me asking if I could make sense of a couple of mathematically oriented matters.
The first was the strange pervasive quality of the “Golden Number”, as exemplified by this YouTube piece:
http://www.youtube.com/watch?v=PjrK96wasDk
The second was some of the amazing symmetries and regularities one can find in arithmetic operations. For example:
1 x 8 + 1 = 9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 9876543
12345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321
Here are my comments on the issues raised:

1. The Golden Number:
I pondered this question as far back as 1968 when my maths teacher gave me a colourful coffee table book on maths. It had a section on the Golden ratio and the amazing number of seemingly disconnected connections in which it arose. Why was this theme running through nature?
The Golden number arises where ever one finds the following quadratic equation:
x^2 - x -1 = 0 (read x^2 as “x squared”)
This equation returns a solution for x of value G, where G is the “golden number” = 1.618….etc. Thus any phenomenon where this equation is entailed will return a golden number. This immediately explains why the value of G appears in some contexts, Viz:
1. For example, a requirement of snail shells is that during growth of the shell the snail need not adapt to any fundamental changes in the distribution of the weight of the shell – hence, big snails look exactly like small snails; that is snail shells are “self similar”. This requirement, it can be shown, entails the above quadratic and therefore the value of G.
2. Another example is the proportions of A4 paper. These proportions are such as to fulfill the requirement that cutting a piece of paper in half produces two pieces with the same proportion as the original. Once again, given this requirement of proportionate similarity, the same quadratic arises, thus entailing G.
3. It is also clear why the Fibonacci series returns increasingly better approximations for G as the series progresses In this series we require n1+n2 = n3. That is, we generate the next number n3 in the series by adding the two previous numbers in the series. Now, we are interested in the ratio of n3 to n2; in other words the value n3/n2, because this value, as the series proceeds, tends toward G.
Now: n3/n2 = (n1 + n2) / n2, by the definition of the Fibonacci series.
But: (n1 + n2)/ n2 = 1 + n1/n2
Thus: n3/n2 = 1 + n1/n2
Or: n3/n2 = 1 + 1 / (n2/n1)
In this latter equation n3/n2 is a better approximation to G than n2/n1. This means that the left hand side of the last equation converges to G as the series advances. Therefore, for large numbers n3/n2 ~ n2/n1. Therefore we can write the last equation as:
G = 1 + 1/G
Which transforms to the quadratic we are looking for:
G^2 – G – 1 = 0.
The ubiquity of G, or rather the quadratic x^2 – x – 1 = 0, seems to be on par with other themes running through nature, like the Boltzmann distribution, the Gaussian bell curve, the power law, and even the value of pi (see my recent blog on the subject here http://quantumnonlinearity.blogspot.com/2010/05/back-of-envelop-mathematical-model.html). So if the ubiquity of G is mysterious then so are these other mathematical forms. We trace back the existence of these mathematical themes to their origins in very similar starting conditions, conditions which thus lead to isomorphic mathematical outcomes. But once we spot the isomorphic underlying conditions the existence of the theme seems less mysterious. For example, random walk underlies the bell curve; consequently where ever we find randomness we will not be surprised to find a Gaussian bell curve. Likewise, any phenomenon governed by the quadratic x^2 – x – 1 = 0 will give us a golden ratio. But having said that I have to admit that in the case of G, the underlying isomorphism’s are often not very clear and thus the presence of G can be rather mysterious; for example I don’t know why the genetic algorithm generates logarithmic spiral patterns in plant flowers – perhaps it something to do with growth self-similarity as in the snail shell. Perhaps a biologist could tell us.
Actually, one might claim that randomness (which is implicated in the Gaussian bell curve) is itself a very mysterious phenomenon: Koestler put it well in his book “The Roots of Coincidence”. After remarking on the remarkable fact that such diverse things ranging from nineteenth century German soldiers being lethally kicked by their horses to “Dog bites man” reports in New York, all obey the same statistical curves, Koestler goes on to say:
But does it (probability theory) really lead to an understanding? How do those German Army horses adjust the frequency of their lethal kicks to the requirements of the Poisson equation? How do the dogs in New York know that their daily ration of biting is exhausted? How does the roulette ball know that in the long run zero must come up once in thirty-seven times, if the casino is to be kept going? The soothing explanation that the countless minute influences on horses, dogs or roulette balls must in the long run "cancel out", is in fact begging the question. It cannot answer the hoary paradox resulting from the fact that the outcome of the croupier's throw is not causally related to the outcome of previous throws: that if red came up twenty-eight times in a row (which, I believe, is the longest series ever recorded), the chances of it coming up yet once more are still fifty-fifty.
It was this passage that help spur me into studying randomness so intensely in the 1970s.
In my opinion whichever way we look we find a profound mystery; namely, the profound mystery that contingent things – that is, things with seemingly no logical necessity or aseity – simply exist. The issue I have with the video you linked to is that in common with much transatlantic “Intelligent Design” thinking, only some phenomena are singled out and identified as betraying signs of design; the insinuation is that some things are somehow “natural” and therefore do not need Divine design, creation and sustenance. But to me it’s either all “natural” or its all “supernatural” – I’ll take the risk and plump for the latter as the sheer contingency of existence, no matter how mathematically patterned, will forever remain utterly logically unwarranted (what I refer to as the great “Logical Hiatus”), and thus in my view a revelation of God’s creative providence.
The problem we often have with mathematics is that sometimes it looks like a kind of magic: It reminds me of that “think of a number” game where somebody in due course appears to “conjure up” the number you first thought of – it may at first seem that some sort of underlying magic is at work but in fact it is the underlying logic unperceived by us whose consequences surprise us. Likewise, the ubiquity of G at first looks very mysterious, as if some divine magician has come along and tweaked the universe here and there thus betraying his presence by way of this magic number. It’s all very reminiscent of Arthur C Clarke’s “Tycho Magnetic Anomaly” in 2001 Space Odyssey, where a mysterious and yet clearly artificial object is found on the Moon; something very “unnatural” is discovered in a vista of “naturalness”. This 2001 scenario is basically the transatlantic paradigm of intelligence design; that is of a God who is manifest largely in various inexplicable logical discontinuities found in the cosmos. But in my opinion this is hardly the right model for God: The Judeo-Christian view of God is not one of an entity who is just responsible for the overtly anomalous: For as soon as we recognise the underlying and unifying mathematical logic at work in the cosmos we no longer see a patchwork creation punctuated by arbitrary “supernatural” anomalies. Instead these underlying logical themes open out into an even bigger mystery; the mystery of a much more totalizing kind of intelligence; an entity of an entirely different genus to the part-time intervening tinkerer who is on a par with alien beings.

2. Mathematics as a Human a Construction?
I’m not so sure that it is right to think about mathematics as a purely human invention. I’m slanting toward the view that “invention” is kind of “discovery” and vice versa. In platonic space every structure/configuration exists in the sense of being a possibility awaiting realisation – it is then down to us to drag that structure out of potentiality and to reify it in material form as either symbols on paper, a computer program, or as concepts in our head.
In fact consider the case of the story writer. As I have said before an average size book has about 30 to the power of a million possible arrangements of text characters. Think of the creative act of story invention as one of dragging out into the real world a single combination of text from those many possible combinations! Achieving such a feat presupposes that the combinatorial hardware exists to take up one of those many possible platonic states. Thus, the “invention”, “discovery” or “reification” of one of those combinations is conditional upon a pre-existing substrate and is thus very much bound up with the “hardware” providence has supplied in the first place in order that these platonic entities may be reified.

3. Regularity and Symmetry in Mathematical Patterns.
Here we have some mathematical operations that have produced patterns of symmetry and regularity – in other words patterns of “high order”. Now let’s think about the opposite; namely mathematical operations that generate disorder. To this end take a number like 2455, square it and then take out the middle digit. Take the first four digits of the square and square that. Once again select the middle digit. Repeat the process many times. One finds that the sequence of selected digits looks very much like a random sequence of numbers (Probably only approximately so) with no discernable pattern. Hence, mathematical operations can generate both order and disorder.
I think of mathematical operations as physical systems, systems that manipulate tokens; basically they are algorithms that can be programmed on a computer. I’m not sure whether or not this algorithmic view of maths is valid for some of the more abstract meta-mathematical thoughts (Like Godel’s theorem), but it is clear that a large class of mathematics comes under the heading of algorithmics and certainly the examples you give are algorithmic in as much as they could be generated on a computer with the right programming.
Anyway, in the context of algorithmic mathematics the ordered patterns you observe make sense to me. One way of making sense of these patterns is to start with the observed patterns themselves rather than the process that generates them. These patterns are highly organized. Highly organised patterns are readily compressible because they don’t display the variety and complexity of disordered patterns and thus they don’t contain a large amount of data. Hence they can be “compressed” by expressing them as simple rules of generation. Expressing such a pattern as the product of some kind of rule based mathematical procedure is a way of compressing that pattern. It is therefore not surprising that elementary arithmetic procedures generate highly organized patterns because those procedures are in effect ways of compressing the ordered patterns they generate. Thus, the symmetry and order you observe is really not surprising. What I have more difficulty with is the fact that simple arithmetic procedures, as we have seen, can also “compress” some disordered sequences in sense that they can generate disordered patterns in a relatively short time. This actually created a paradox for me and I proposed a solution in one of my blog entries – see this link: http://quantumnonlinearity.blogspot.com/2009/10/proposed-problem-solution.html

1 comment:

Vinnes said...

Very nice post Tim. Thank you.