PROBABILITY theory is a quaint little piece of mathematics. It is about sets of non-negative numbers that are attached to actual and possible physical events, that sum to 1 and that obey certain rules. It has numerous practical applications.
So does the flat-Earth theory: for instance, it’s an excellent approximation when laying out your garden.
Science abandoned the misconception that Earth extends over an infinite plane, or has edges, millennia ago. Probability insinuated itself into physics relatively recently, yet the idea that the world actually follows probabilistic rules is even more misleading than saying Earth is flat. Terms such as “likely”, “probable”, “typical” and “random”, and statements assigning probabilities to physical events are incapable of saying anything about what actually will happen.
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We are so familiar with probability statements that we rarely wonder what “x has a probability of ½” actually asserts about the world. Most physicists think that it means something like: “If the experiment is repeated infinitely often, half of the time the outcome will be x.” Yet no one repeats an experiment infinitely often. And from that statement about an infinite number of outcomes, nothing follows about any finite number of outcomes. You cannot even define probability statements as being about what will happen in the long run. They only say what will probably happen in the long run.
“Probability and randomness are large sledgehammers to crack some small eggs”
The awful secret at the heart of probability theory is that physical events either happen or they don’t: there’s no such thing in nature as probably happening. Probability statements aren’t factual assertions at all.
The theory of probability as a whole is irretrievably “normative”: it says what ought to happen in certain circumstances and then presents us with a set of instructions. It is normative because it commands that very high probabilities, such as “the probability of x is near 1″, should be treated almost as if they were “x will happen”. But such a normative rule has no place in a scientific theory, especially not in physics. “There was a 99 per cent chance of sunny weather yesterday” does not mean “It was sunny”.
It all began quite innocently. Probability and associated ideas such as randomness didn’t originally have any deep scientific purpose. They were invented in the 16th and 17th centuries by people who wanted to win money at games of chance.
Gaming the system
To discover the best strategies for playing such games, they modelled them mathematically. True games of chance are driven by chancy physical processes such as throwing dice or shuffling cards. These have to be unpredictable (having no known pattern) yet equitable (not favouring any player over another). The three-card trick, for example, does not qualify: the conjurer deals the cards unpredictably (to the onlooker) but not equitably. A roulette wheel that indicates each of its numbers in turn, meanwhile, behaves equitably but predictably, so equally cannot be used to play a real game of roulette.
Earth was known to be spherical long before physics could explain how that was possible. Similarly, before game theory, mathematics could not yet accommodate an unpredictable, equitable sequence of numbers, so game theorists had to invent mathematical randomness and probability. They analysed games as if the chancy elements were generated by “randomisers”: abstract devices generating random sequences, with uniform probability. Such sequences are indeed unpredictable and equitable – but also have other, quite counter-intuitive properties.
For a start, no finite sequence can be truly random. To expect fairly tossed dice to be less likely to come up with a double after a long sequence of doubles is a falsehood known as the gambler’s fallacy. But if you know that a finite sequence is equitable – it has an equal number of 1s and 0s, say – then towards the end, knowing what came before does make it easier to predict what must come next.
A second objection is that because classical physics is deterministic, no classical mechanism can generate a truly random sequence. So why did game theory work? Why was it able to distinguish useful maxims, such as “never draw to an inside straight” in poker, from dangerous ones such as the gambler’s fallacy? And why, later, did it enable true predictions in countless applications, such as Brownian motion, statistical mechanics and evolutionary theory? We would be surprised if the four of spades appeared in the laws of physics. Yet probability, which has the same provenance as the four of spades but is nonsensical physically, seems to have done just that.
The key is that in all of these applications, randomness is a very large sledgehammer used to crack the egg of modelling fair dice, or Brownian jiggling with no particular pattern, or mutations with no intentional design. The conditions that are required to model these situations are awkward to express mathematically, whereas the condition of randomness is easy, given probability theory. It is unphysical and far too strong, but no matter. One can argue that replacing the dice with a mathematical randomiser would not change the strategy of an ideally rational dice player – but only if the player assumes that pesky normative rule that a very high probability of something happening should be treated as a statement that it will happen.
So the early game theorists never did quite succeed at finding ways of winning at games of chance: they only found ways of probably winning. They connected those with reality by supposing the normative rule that “very probably winning” almost equates to “winning”. But every gambler knows that probably winning alone will not pay the rent. Physically, it can be very unlike actually winning. We must therefore ask what it is about the physical world that nevertheless makes obeying that normative rule rational.
You may have wondered when I mentioned the determinism of classical physics whether quantum theory solves the problem. It does, but not in the way one might expect. Because quantum physics is deterministic too. Indeterminism – what Einstein called “God playing dice” – is an absurdity introduced to deny the implication that quantum theory describes many parallel universes. But it turns out that under deterministic, multi-universe quantum theory, the normative rule follows from ordinary, non-probabilistic normative assumptions such as “if x is preferable to y, and y to z, then x is preferable to z“.
You could conceive of Earth as being literally flat, as people once did, and that falsehood might never adversely affect you. But it would also be quite capable of destroying our entire species, because it is incompatible with developing technology to avert, say, asteroid strikes. Similarly, conceiving of the world as being literally probabilistic may not prevent you from developing quantum technology. But because the world isn’t probabilistic, it could well prevent you from developing a successor to quantum theory. In particular, constructor theory – the framework that I have advocated for fundamental physics, within which I expect successors to quantum theory to be developed – is deeply incompatible with physical randomness.
It is easy to accept that probability is part of the world, just as it’s easy to imagine Earth as flat when in your garden. But this is no guide to what the world is really like, and what the laws of nature actually are.
This article appeared in print under the headline “Definitely not maybe”
