Probability: A Philosophical Perspective, Part II
In my last essay on probability I highlighted two ways probability can be interpreted. First it can be a measure of uncertainty in a deterministic world. Second it can be viewed as an accurate model of a genuinely ‘non-deterministic world’. This essay will critique both of them, and their implications for God, Scepticism, Science and more.
Um, remind me what this is all about again?
Here’s a brief recap.
A deterministic model might be a system governed by classical mechanics, perhaps a dice roll, but we model it with a non-deterministic model of the world (probability) where the probabilities measure our degree of uncertainty. The non-deterministic model is where there is genuine randomness at play, i.e. multiple observed realities are actually possible. The non deterministic model is essentially the rationale behind the current mathematical foundations of probability.
The Deterministic Model
The deterministic model has conceptual flaws. Identifying uncertainty with a number is not a well-defined concept. The process by which we create such as model is to observe many outcomes and hypothesise a distribution. As mentioned last time, in a deterministic world this model creates astonishingly poor predictions for each individual outcomes; it merely preserves some useful properties for aggregate results. For instance, in the ‘deterministic’ world the dice was always going to roll a 6, but the model only gives it a weighting of 1/6. This is a measure of our uncertainty about the underlying mechanisms: i.e. when we roll a dice, our best estimate given our lack of knowledge is a weighting of 1/6th/
(It is unsurprising that the predictions are poor given the nature of the model and that we assume the world is deterministic!)
Let us review the process by which we created the model. First, we observe. Second, we suppose some ‘likely’ distribution. Third, we check to see if the model seems consistent with the data. This is a process rather similar to Scientific investigations, where we create a mathematical framework which we keep because it is useful (in a very human way!) but where the concepts involved are ill-defined and philosophically un-rigorous. As a fun example of this, I asked my friend (who was in the international physics Olympiad and knows his physics!) definitions of physical concepts until we came round in a full circle to show the ideas were circular. With some introspection, the understanding we think we have in science is merely of the sort of learning that X goes to Y goes to Z, and then being validated (/reassured) of our reasoning by comparing it to accepted results and consistent with past observations (which relied on the same process).
Less controversially, an atheist will feel confused at a theist understanding God, the scientist may get confused as the Platonist ‘understanding’ this realm of forms etc. I think these all point to the fact that when humans understand a model, we can think we are understanding it even when the model (may be)/is meaningless. In the above examples they can’t both be right!?]
Okay, the uncertainty model isn’t satisfactory. Uncertainty is more an emotion than a number; besides what does it mean to say that you have ‘0.6’ knowledge of a system? At best, you want a replacement measure of uncertainty, where supposing you knew the rules you could create a measure of similarity between the rules and the model. Unfortunately, if you knew the model, the exercise is pointless, but if you don’t the concept remains ill-defined.
This isn’t to deny the use of probability! I remain thoroughly sceptical of Science without denying its use — although the concept of use relies on quite a few assumptions itself!
The Non-Deterministic Model
What about a non-deterministic system? Unfortunately, this too is poorly defined! We define it mathematically easily enough, but the concepts of numbers weighted in a sample space is not a good philosophical understanding of what it is. We invoke vague notions of ‘possibilities’, yet our idea of possibilities comes out of what our brain considers feasible combinations of things. Presumably that is because our brain has a model of the world which some things are incompatible with, and the remaining are not incompatible, but this is to dodge the question. Worryingly, our understanding of a sample space with a measure relies on potentially deterministic ideas for our understanding. The intuitive understanding normally invokes the law of large numbers, and that with a large enough sample size the probabilities and the relative frequencies are the same (as sample -> infinity). It is immediately apparent that such as system might be deterministic, and claims that it are not then have to define this idea of non-determinism again.
Of course, here I am putting much of the burden of proof on non-determinists which may be unfair. Human conception of causality makes deterministic systems a natural extension, but our understanding of causality may not be justified. Where would that leave us??
Here quantum physics leaps into the fray with the Copenhagen interpretation of quantum physics. I am not a physicist so don’t understand enough to feel confident in this area (i.e. take the following with a pinch of salt). The Copenhagen interpretation (along with a supposed proof that quantum mechanics is non deterministic which is very subtle and I don’t understand it!) says that quantum objects don’t have definite physical properties before being measured (hello Wikipedia), and have to be modelled with probabilistic models.
There is quite a subtle point here. The properties are not defined when the object until measurement. Does that mean that there were a host of alternative possibilities or simply that it is impossible to model it as a deterministic system. Given that we have two bits of data, that certain properties are undefined before observation, and that groups of these objects show a probability distribution-like pattern when observed is a truism — we fit PDFs to fit data which shows regular similarities like this. Does that mean any individual particle could have had a different property? It may not be any specific property before observation, but does that really imply that it could have had a range of properties after observation? I don’t think the implication is there; rather we know that the property is undetermined until event A, after which a property is well defined. From that it is impossible to know whether the object could have had a multiplicity of outcomes after the observations or not. We only have one observation of sets of events in this universe, so can hardly test this!
Probability is meaningless?
We are led inescapably that probability theory is meaningless as it isn’t properly defined, and the vague intuition for its basis does not stand up to scrutiny.
A closer investigation of science and the process of understanding sheds some light on why this might be the case and why probability theory seems to be similar to science (in its conceptual understanding, not in its mathematical basis) in that it is useful despite the concepts being poorly defined. At best, we observe outcomes, interpolate a distribution which is helpful, and then consider it some vague notion of uncertainty or non-determinism.
What are the implications of this?
A large part of philosophy and everyday life involves the counterfactual, which is basically a conceivable outcome. I.e. we don’t reject it as contradictory. (What is a conceivable outcomes changes, for instance someone who has never understood/studied politics and economics is more likely to view a ‘successful implementation’ of socialism as a conceivable outcome; those who have only read Hayek will find it harder to conceive of a successful state intervention).
With probability being ill-defined, formal arguments in most areas of thought are left in quicksand, as these rely on some idea of a conceivable outcomes defined through probability models.
The flipside is that our woolly treatment of counterfactuals and what can be considered a counterfactual (or have a probability assigned to it) has led to absurd arguments. For instance, those who think we could actually prove one way or the other that the universe is well designed for life/prove that it could happen by chance by applying probabilistic weightings for things like creation of universes and universal physical constants. Obviously this is absurd, as we have no idea if there could be a range of physical constants or different universes, or what different forms of ‘consciousness’ could emerge, so can hardly apply probabilistic weightings. Thus, many arguments both for and against God are ill-founded.
Probability and scepticism
More important philosophically is application to scepticism. Ever since Descartes’ ‘evil demon’ (there have actually been many earlier recorded instances of extreme scepticism) philosophers have pained themselves over whether the world exists. Many innovative approaches in the 20th century looked at the meaning of words and whether the sceptical questions were well defined. I would add that the very conception of counterfactual outside a very narrow range is ill defined. At best, humans understand counterfactuals within the social environments and physical landscapes we live in; applications beyond these typically involve appeal to these for intuition but then remove the idea of counterfactual well removed from the original source of the idea. Hence, they give us the feeling of understanding from the appeal to the intuition, a notion of how different types of things interact under this framework, and then we feel when we apply this framework in a situation far removed from the original that we understand an argument which is actually meaningless.
Out of our depths?
More broadly, in science and probability and elsewhere, humans are ill-suited to the tasks we set ourselves. We manipulate symbols and are satisfied when we have the feeling of understanding. I don’t doubt that we are capable of manipulating ideas, but also that when we derive our feeling of understanding from intuition in one scenario, we might mistakenly apply it in another situation. Even writing this I appeal to physical notions for the placement of ideas and the interactions of those ideas.
The mind is a very strange thing and I doubt I will every understand how ideas and thoughts work. The scientific approach has heinous assumptions, but does ‘work’ (with a woolly conception of ‘work’) and might be no more than this. If this is true then reading too much into science is a waste of time. Perhaps this is true for probability also.