Abstraction and improbability

I’ve been dipping into George Berkeley’s philosophy recently, mainly because his mind-only view of reality resonates with some other thinkers whose ideas on the matter of matter have impressed me over the years, such as Arthur Eddington, Werner Heisenberg and William Dembski.

One of the first insights I’ve gained from him is his aversion to the whole practice of abstraction, which was becoming a feature of science then, and was set to do so even more in our time. That is, “abstraction” as opposed to “generalization,” the latter being the concentration on some aspect or trait of reality to discover principles of nature, but the former being the false conclusion that such generalisations have an independent existence of their own.

One simple example of this is current at Peaceful Science, on a thread entited “The Inevitability of Improbability,” in which the aim is to deliver ignorant creationists of their misapprehensions about probability theory.The ultimate aim, hinted from time to time, is to show that to regard the existence of life as we know it as in any way surprising is ignorant.

“Chance favours even the unlikely” is the message driven home by a succession of scientific types there, including even a couple of Christian ones. Though it is not insignificant that most of them are not believers, and the non-necessity of God to creation appears to be a barely concealed subtext. To a man, though, they seem to fall into the trap of which Berkeley warns, that is of abtracting and then reifying chance.

The thread revolves around the old chestnut of a “fair” lottery with a million tickets, limited to one per purchaser. The chance of any individual winning is very improbable – one in a million – but somebody has to win. Ergo, unlikely things happen all the time.

But they don’t, actually: as I pointed out in a comment, the Planet Venus has never, and will never, win that lottery. That would be improbable. And the reason is that the planet Venus, like each person going down to their local shop and buying a lottery ticket, is a genuine entity within a complete, complex, reality which does not include planets participating in lotteries, whereas the 1 million [lottery ticket buyers] is a mathematical abstraction which represents always and only our ignorance of the reality behind it.

I’ll return to how that matters after a diversion into another example. But I’ll set you thinking by saying that, statistically, in the circumstances of the example given, each [lottery ticket buyer] has a 1 in a million probability of winning. But in the real world, the people who have purchased tickets most definitely do not stand an equal chance of winning.

I was the first member of my family ever to go to university. When one of the teachers announced the formation of a preparation group for those applying to Oxbridge, I was tempted, on the basis that if I didn’t go for it, and ended up somewhere crummy, I might regret it for life. However, though I was bright, I wasn’t that bright, when I considered how many thousands of university applicants there were compared to the small number of places at Cambridge.

In those days we had to complete a form with six choices of university course, so to blow one of those on something with near-impossible odds seemed foolish. But then someone told me that, of those who actually went as far as applying to Cambridge, one in four got in. That risk seemed, and was, a lot more manageable, and I put my name forward.

What was happening here? My initial ignorance about whether it was worth applying was at first turned to pessimism by a crude calculation of how many people were applying to all universities, compared to the number who would get to Cambridge. My knowledge was increased in a specific way by learning the 1:4 statistic, which altered my predictions. But in reality, as opposed to in statistical theory, neither of those numbers was a predictor of whether I had a “chance” of getting into Cambridge.

And the reason was that, in the real world, students were selected by human beings on various choice criteria, and not by chance. Now, I had to rely on my crude statistics because I didn’t know, and still don’t know, exactly what the selection criteria were. Part of it, of course, consisted of exam results, which in my particular case I already had before applying (not that common in the English system). But there was no particular minimum target as there is now.

Another criterion was, perhaps, a report from my school. A mixed bag there, potentially – the careers officer, who had once taken me briefly for chemistry, my weakest subject, for some reason took a shine to me and recommended me to apply to his old college. The headmaster, on the other hand, didn’t like me.

So there was an interview up at Pembroke College, and who knows what the Director of Studies’ criteria were? Maybe he judged my academic potential, or thought I had a pretty face, or liked the fact that I professed an interest in ornithology (far beyond what I actually had).

But the fact is that, if I had not fitted his particular selection criteria, I could have applied year after year for ever and never got in, “25% probability” notwithstanding. That is, unless the selection criteria changed to give me a place just to shut me up. I had exactly the same chance of admission before I thought of the “impossible odds,” and before I judged that they were actually manageable. Given the criteria and my match to them (presumably), my admission was inevitable.

If you like, as an abstract [Cambridge applicant] I had a 25% probability of admission. As real kid, competing against three other real kids, we did not in reality have an equal chance. I happened to fit the criteria, and was bound to be chosen; my nameless rivals didn’t, and were doomed to failure as soon as they filled in the application form, if not before.

Now, what about that simple lottery question? A “fair” lottery is set up to keep its causal chains invisible both to the punters and the organizers, though without doubt there are causal chains within it.. The one in a million probability is based purely on the assumption of our state of ignorance. In order to do the maths, one million individual people have been abstracted into a uniform category called [lottery ticket holder].

But to talk, then, about each possible outcome being as unlikely as the others is to claim that our ignorance is actually a reflection of the real state of the world. In other words, it is a claim that there is a real force called chance, which is indiscriminate, and which through the mechanisms of the lottery, is as likely to select Jill Smith or Peter Piper or anyone else as the winner.

But is that the kind of universe we actually live in? And it is, in fact, the nature of the universe itself that is the crucial factor, as George Berkeley well realised, and not the mathematics of statistics. To ignore that is to beg the question. The “inevitability of improbability” is simply not true in a universe that does not contain a mysterious abstract force called chance, which acts statistically on equally abstract entities like [lottery ticket holders] or [university applicants], but instead is governed by a God who sustains all things in their being, who governs specific events providentially by his divine love and justice, and in whom both Jill Smith and Peter Piper live, and move, and have their being – and are accountable to him.

In that universe, although God’s thinking might be as opaque to us as Dr Pratt of Pembroke College’s was to the teenage me, there is only one possible winner of this lottery. 999,999 other [lottery ticket purchasers] were never going to win, and even in advance of the draw, Jill Smith’s success (say) was not an improbable event, but an inevitable one because it was caused by providence, not by chance. Those two are alternative views of reality, rival worldviews. But they both encompass the same probability theory, because is a theory of human ignorance, not of the indeterminacy of reality.

From another angle, you could repeat the lottery twenty million times, and there would be no likelihood that everyone would win the roughly same number of times. Except there’s a caveat on that – if you did repeat it that often, in reality as opposed to on paper or in a computer simulation, you would no longer be dealing with abstractions, but with an altered real situation, in which God’s providential activity was equally governing the twenty million outcomes for his own reasons.

For this reason, the idea that if time goes on long enough, any outcome is statistically possible is an argument based on abstractions and infinities that do not, actually, exist. It is also reasoning from one of two alternative worldviews, not from probability theory, properly understood. The statistics may have practical applications in compensating for our lack of detailed knowledge, but the real world, under providence, is simply acting on different principles – principles in which there are plenty of surprising realities, but no abstractions.

Now, follow the conclusion of this. In practice, we have varying degrees of ignorance about future real events, and in particular we have no insights into the choices God makes in his providential government. And because of that ignorance, probability theory is not only useful, but correct. It is correct, that is, provided we see it as a measure of human ignorance rather of the causal powers of chance.