Not Infinity, just Astronomical Figures

By now, I presume that many will have heard the myth of how the man who invented the chessboard requested payment for it from the Emperor of India by doubling a grain of rice for each square. With 64 squares, 1 becomes 2, becomes 4, becomes 8 becomes 16 and rapidly doubling up 64 times becomes an enormous figure - 9.223372036854776e+18 - more rice than could be grown in the Empire over many centuries. Even so, this figure only doubling 64 times pales into insignificance by card shuffling numbers...

The UK TV programme, QI, recently delighted in showing us something that had 'never been done before' by informing us that, statistically, no two packs of cards have ever been shuffled into exactly the same running order twice. The reasoning behind this is that, with 52 cards in the pack, the possible combinations that govern this, amounts to the sum of 52x51x50x49x48x47x46x45x44x43x42 - well, you get the picture; which turns out to be a figure so huge *(8.065817517094388e+67) that it is very unlikely that shuffling a single pack constantly since the beginning of time would have started to produce repeats in the running order until now.

A comparable phenomenon exists with the possible combinations of so few musical notes which, together with tempo, time signature, volume, pitch, duration, harmony and timbre etc (despite many very similar 'four chord trick' songs), means that we have not quite run out of tunes yet. As Eric Morecambe once famously said with great comedic effect to Andre Previn, "I am playing all the right notes, but not necessarily in the right order."

Similarly, if a burgeoning list of new nouns were not being formed at an alarming rate these days, even with a static word set, it would still be possible to make completely new and never before heard compound/complex sentences using only the familiar words we all know - e.g.: The flamingo looked on whilst the inebriated mongoose led a flock of sheep into the temple at the foot of the billowing Mt. Vesuvius. We do not even need to make up a nonsense story; indeed, the vast majority of the sentences in this entire discourse will be previously unwritten - and certainly not together or in this order. This is why there are so many books for comparatively so few words.

The Egg and Sperm Race:

Many years ago, I learned that the chances of a woman becoming pregnant in any given cycle are surprisingly low. A healthy man's ejaculation contains more than 100 million sperm but less than 40% are considered suitably undamaged or with sufficient motility at the start of their journey and, of these, many millions will die, or get lost and swim off in the wrong direction before getting anywhere near the egg. After navigating their long and treacherous journey through the cervical mucus, the uterus and the fallopian tubes, around 50 sperm eventually get to where the egg should be waiting and is hopefully available during the very narrow conception window of a few days a month - and only one sperm makes it. It is indeed miraculous we ever conceive. However, this is just the start; cell division must take place and the embryo must attach itself to the uterine lining around six days after fertilisation, where then follows the risk of miscarriage at any time over the proceeding months.

Despite the dangers, pregnancy does occur every day and some of us end up being born; every living person being the result of a completely unbroken chain of births, requiring two parents, way back through their entire ancestry. Whilst someone was always going to be born when your parents got together at that exact moment, the chances of it being your sperm that met that egg - leading to 'you', are phenomenal.

Although the average age of conception is now much older and was previously much younger, if we take a nominal but consistent average of people being around 25 years old before they have children, going back just 20 generations (500 years) requires 1,048,576 people, 40 generations (1000 years) requires 1,099,511,627,776 people, but by 64 generations (1600 years) 1.844674407370955e+19 are needed, which is apparently a figure much greater than all the people ever to have lived on earth and suggests that a high degree of incest must be in operation down the ages. With this in mind, most of us are already related. With the largely static nature of early civilisations, it is actually estimated that 80% of all marriages in history have been between second cousins or closer, although this figure has reduced dramatically in more recent times with global mobility. Such practice was rife amongst royalty; Victoria-Albert and William-Mary being just two examples and indeed Charles Darwin and his wife Emma were first cousins. As children of first-cousin marriages have an increased risk of genetic disorder through lack of genetic variability, it is interesting that Darwin should choose to pursue such a topic. I wonder if humanity will see a rapid advance due to higher variability. But I digress.

A conundrum

Being born might be unlikely but if the genome is only made up of four letters, GATC, which has its twin helix structure upon which to arrange its complex order then, although it is an astronomical figure, which keeps turning out completely different people, there must be a finite number of possible different combinations.

This poses a fascinating question. Whereas the genetic information (data) is within each cell to be able to theoretically reproduce an infinite number of clones of a particular individual, the actual number of individuals available has to be finite. Although even monozygotic twins are not genetically identical due to mutations and errors on various strands of the genome, even if we were all alike, once you factor in the possible errors - even only one per person, this perpetuates the possible numbers of possible folk somewhat exponentially, but is it not still a finite number? Does this then mean that certain 'people' are recycled or that someone else may live with exactly the same genetic structure? I bet the reincarnation proponents will have a field day with that one. [This is my ignorance at work here; I am not sure of the accuracy/figures for above bit yet - or whether, once 'we're all coffee,' diversity would forever expand, not dilute gene pool]

The arguments for massive numbers and randomness remain the subjects that both sides of the argument put forward for 'proving' whether or not our origins are chance or design. So how do these vast numbers work out for us in our every day lives?