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Tallis in Wonderland

Could The Universe Give A Toss?

Raymond Tallis thinks about probability and the frozen world of quantum mechanics.

The other week at the Hay Festival in Wales, I gave a talk ‘Has Physics Killed Philosophy?’, arguing that physicists need philosophers. Afterwards, I had a conversation with a remarkable man, Raja Panjwani, who, in addition to being trained in physics and philosophy, is an international chess champion. We got to talking about one of the most striking and disconcerting features of quantum physics: the replacement of causation by probability. At the sub-atomic level, the last vestige of ‘A causes B’ is replaced by patterns of events whose statistics can be predicted with stunning precision, although – outside of the ‘many worlds’ interpretation of quantum mechanics, in which everything happens in some world or other – no particular quantum event is obliged to occur. However, there is a constraint on the frequency of certain outcomes within a given range of values over large numbers of events, this frequency being what the most famous quantum equations predict. Raja, perhaps sensing that I was getting out of my depth, turned the conversation to the staple of probability theorists – the tossing of a coin – which subsequently provoked the thoughts that follow. The confusions, I am confident, are mine, not his.

Imposing Patterns on Events

When you toss a coin, there are two possible outcomes – heads (H) or tails (T). No outcome should influence its successor: there is no causal pressure exerted by Toss 1 on Toss 2, as there is, say, from the movement of the thumb to the movement of the coin, so the chances of H on a particular occasion are the same irrespective of whether its predecessor was H or T. Improbable sequences – such as 100 straight Hs – do not defy or even bend the laws of mechanics. But if the outcome of Toss 1 does not influence the outcome of Toss 2, such that there is no gathering causal pressure for a T to follow a long run of Hs, why don’t we easily accept that the series H, H, H… could be extended indefinitely? Why would an unbroken sequence of 100 Hs raise our suspicion of a bent or even two-headed coin?

Let us look a bit closer at the properties of a genuinely random sequence. As we extend the series of tosses, the number of possible patterns increases enormously, but the proportion of those that are significant runs of Hs or Ts are vanishingly small. There is a 1:4 chance of HH (the other possibilities being HT, TH, and TT), but 25 Hs in succession would be expected to occur by chance only once in 33,554,432 throws. The longer any run of Hs or Ts, the less frequently it will occur; so the most likely outcomes will be those in which runs of Hs or Ts are soon broken up. This is how we reconcile the 50/50 chance of getting H on a particular toss, irrespective of what has gone before, with the growing suspicion that appropriately greets a very long series of Hs and the mounting expectation of a T.

This is all basic stuff; but let us dig a little deeper. We’ll start by focussing on the expectation that has been the ruin of many a gambler. The key point relates to the history-so-far of Hs. It is this history that makes us feel that the coin sooner or later will feel obliged to come up T. We must not, however, see the history-so-far as a kind of pressure bringing about affirmative action for Ts, so that they match the number of Hs: a history of coin-tosses is not in itself an event, even less a cause. Random sequences do not have the kind of reality, even less the causal efficacy, that individual events have. A sequence, in short, is neither an event nor a cause that can influence what follows it. This may seem counter-intuitive, but it’s true, because 50/50 equipoise or symmetry is an intrinsic property of the (idealised) coin, and that’s not something affected by its history.

What makes a sequence seem like a cause is our subjective expectation, which turns a lengthening run of Hs into the idea of a kind of pressure to produce a T. Our expectation is, however, in no sense a force ‘out there’. Rather, as David Hume pointed out, our habits of expectation often translate ‘how things usually pan out’ into ‘how they are obliged to pan out’.

While it is clear that our subjective assessment of probability is not ‘out there’, we still retain the idea of there being objective probabilities ‘out there’ based on the expected relative frequencies of certain kinds of events or sequences of events. However, even probability understood in this way cannot entirely shake off their mental dependence. This is because a sequence of events is not ‘out there’.

Firstly, it is only by remembering past tosses, and gathering them up into a series, that we are able to place actual sequences into a fraction with a denominator corresponding to the sum total of possible sequences – a 1 in 33,554,432 chance of 25 tosses all turning up heads, for instance. (Moreover, collecting tosses for the sequence will require ring-fencing of the population we are drawing from: the series we have just started, or all the tosses in the history of the world, or something in between.) It is the gathering together of tosses that tells us that certain combinations ought to be common or rare, so that we should expect them to occur frequently or infrequently. But the present existence of no-longer-existent tosses is entirely mental. They are not even present by proxy as a cause of a present state of affairs, because, as we have said, Toss 1 does not have any influence on Toss 2.

Immaterial Logic

What’s more, coin tosses have had to be shorn of their material features and classified simply as H or T in order to be gathered up into a sequence that feeds the calculations of probability of what we think is going to happen in future. Importantly, those future possibilities have to be defined as the branches of a fork, as the mere instantiation of the logical alternatives ‘H’ or ‘T’. This crash-dieting of a physical event to one of its characteristics also warrants further examination.

Any actual coin-toss must have numerous features additional to, and irrelevant to, the dichotomy H or T: when the coin lands head up it does so via a unique trajectory, and is propelled by a unique force to a particular height, to land on the ground, all courtesy of a particular individual whom I will refrain from calling a ‘tosser’. None of these additional elements are criteria for H or T. But in order for there to be an H, a toss has to occur, and in order to occur, it has to be more than H. Without these additional features, the coin couldn’t land either H or T, in fact. Furthermore, for the coin to fall either H or T, something has to be bent in order to break the 50/50 equipoise or symmetry between H and T: not necessarily a bent coin, but necessarily a bent event. (Since nobody knows how to bend the event, no caller has an advantage, so the ethics are not bent: contingent influences are inescapable, but that’s OK if they’re hidden and cannot be manipulated.) But this only highlights the fact that describing any particular toss as H or T is to strip it of numerous features necessary for the full-blown event to happen – to be an H or a T. More broadly, material events in a material world cannot be reduced to forking branches of possible outcomes; just as a victory for a football team like Arsenal is not just a featureless ‘V(ictory)’ as opposed to a featureless ‘D(efeat)’.

Any specific toss that instantiates ‘H’ or ‘T’ will have a vanishingly small probability of occurring as that specific event. Actual events, specified precisely in advance, are highly improbable. The circumstances that produce a real event, even a little one like a coin falling H, are in fact unique, because each event has unique characteristics. The more fine-grained the description of an event, the more the improbability of that event increases. If events had a million either/or features, and each of the features had a 50/50 chance of happening, then each event’s individual probability would be 1 in 21,000,000, the denominator being a number greater than that of the number of atoms in the universe. And there is in principle no limit to the grain of the description, as no description captures an actual event completely. However, this uniqueness and improbability applies equally to all Hs and all Ts – any actual H is as unlikely as any actual T – which is why H and T are equally likely to occur. In short, probabilities apply not to specific actual events but to types of events – reduced, in the cases of tosses, to the dichotomous possibilities H or T. The mathematics of ‘Either H or T’ applies only to a future reduced to branching logical possibilities: a material future reduced to a logical one.

While the probability of 50/50 Hs and Ts is built into the job description of coin-tossing, its realisation – and the apparent pressure for it to be realised – is in events that are in possible futures reduced to either H or T. In addition, the past also has to be present in the gathering up of these strictly stand-alone events into a (retrospective) series pointing to this prospective future. The mobilisation of all three tenses of time – which do not have a foothold on the material world – itself betrays that with probabilities we are a long way from the material world. Material events are what they are, and not what they were or will be.

Improbable Realities

In short, the mathematical logic of probability deals with events slimmed down to forks of logical branches, and draws on retrospective and prospective views that have no place in the material world. Could this be a source of some of the problems quantum mechanics has with time and change? When causation is replaced entirely by probability, defined logically or mathematically, there is no obligation for anything in particular to happen, because actual events – macroscopic, real events like real coin-tosses – are beyond the reach of probability. A 50/50 probability of an event such as H is not a 50/50 probability of any real, messy, fat event. Real events are necessarily more than logically defined possibilities (though they can be logically reduced to them), and they do not exist in sequences that encompass past and present.

Many physicists trying to unite probabilistic quantum mechanics with general relativity lose time and change altogether, instead envisaging a frozen 4D universe in which nothing happens. Physicist Carlo Rovelli has even welcomed the possibility that quantum mechanics will become “a theory of the relations between variables, rather than the theory of the evolution of variables in time” (‘Forget Time’, FQXi Essay, 24th August 2008): in short, a theory of the eternal relations between kinds of possibilities rather than between actual events in time. If this were true, we would be justified in concluding not only that the universe couldn’t give a toss about us, but that it couldn’t give an actual toss. The lack of contamination by actual events is the necessary condition of the purity of a mathematical vision of the world based upon probabilities. As mentioned, a minority of physicists invoke a ‘many worlds’ version of quantum mechanics, in which every fork of possibility is taken. This seems a very expensive way of melting a universe frozen as a consequence of replacing causation with probability. This may be why a few physicists now think physics need philosophy; although many more would add “like a hole in the head.”

© Prof. Raymond Tallis 2013

Raymond Tallis’s new books are Reflections of a Metaphysical Flaneur (Acumen), and (edited with Jacky Davis), NHS SOS: How the NHS Was Betrayed and How We Can Save It (One World).