Probability for a Deterministic World

We need to reconceive probability for deterministic systems. In doing so we run up against the limits of introspection

Watch out below!

Probabilities are most suited for non deterministic systems. We all know how probabilities work for systems with genuine randomness – although things are not as clear cut as they seem, see this article – but it is not at all clear what they mean for deterministic systems.

Why do we need probability in deterministic systems?

Consider a game of monopoly where the dice is rolled by a computer. You may think this is random but it’s not. An algorithmically generated code, which is deterministic, generates the dice roll.

Even a dice arguably is a deterministic system, we just lack the knowledge of the bumpiness of the ground or the air currents. Yet the system seems to be well modelled by probabilistic models regardless of whether it is deterministic.

Probability as Uncertainty

In fact the probabilistic models of deterministic systems do rather terribly. For instance, you weight the probability of a three showing up as a one-sixth, yet in hindsight it was a two.

When I refer to ‘strategy’ or ‘coping’ below, that is because probabilistic models are thinking tools designed to make sense of and help us live in the world. Hence as a model, even if wrong, it can be a successful ‘strategy’.

In these cases, probabilities capture the essence of uncertainty. The distributions generated by genuinely random distributions can be thought of as optimal coping strategies under uncertainty. In the absence of knowing the causal mechanisms at play, probabilities approximate uncertainty and suggest ‘strategies’ which are consistent with that uncertainty. Yes, the dice was predetermined to end up a two, but in the absence of that knowledge assuming you’ll get a two is inconsistent with the uncertainty present.

Uncertainty and probability are indistinguishable

Say I observe a dice being rolled. From my perspective, I cannot know whether I observed a truly deterministic system where I would always see 2,4,5,3,3…. or whether I saw a probabilistic system which just happened to provide those values.

Furthermore, we posit the probability density function (e.g probability one sixth for each number on a dice roll) based on empirical data. Thus we see at root we have constructed a probability distribution merely to be consistent in some sense with a large amount of empirical data. We might consider every single dice roll in history as one continuous experiment 1,4,3,4,6,2,1,1,… in the which our assigning of probabilities such that the outcome is consistent with that probability distribution is clearly observably identical the string being deterministic

Quantum physics

A big thing in Quantum Physics is the proofs that it has inherently random parts. Yet when probability distributions are observably identical as determinism plus uncertainty, then we can no longer claim this. In other words, uncertainty and genuine randomness with probabilities have identical observable effects, and so the scientific method cannot distinguish between them.

The limits of our mind

Our conceptions of probability and of determinism are not well suited to deep theorising. Our understanding of probability is based in having a model of the world in which multiple outcomes are considered possible; for instance I can think of different consequences of skipping lectures and visualise and imagine them. My understanding of determinism is when my model conceives of a necessary causal chain (for instance, my conception of gravity). Yet by now it should be clear either can lead to the other. I can have a fundamentally probabilistic view in which my deterministic view of gravity is merely a result forming a probability distribution by observing the tendencies of things to go the ground. As this happens more our certainty grows. Or we might say we have certain deterministic principles but we have uncertainty. For instance, when I am young I can conceive of travelling to other galaxies or going back in time as possible, whereas now I am older I might say these violate several deterministic principles (say, the speed of light)

It then becomes unclear what we even mean by these concepts! I struggle to define determinism or probabilistic, and challenge you to either without being circular. I also challenge you to not now feel a deep sense of confusion as to what these concepts mean.

Have we created concepts to fudge our way in the world, and are now running up against the limits of our mind in examining them?

The author for this article was Ethan Horsfall. He studies economics at Cambridge University and has a passion for Philosophy and Mathematics

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