The Space of Possibilities

May 3, 2012

The mathematics of inferential statistics is based on the logic of random sampling: the inferences we make in inferential statistics work on the assumption that the data we are inferring from is randomly sampled from the population we are inferring to – that every member of the population has an equal chance of ending up in our dataset. Obviously this usually isn’t the case; but that’s the assumption, and the further our actual sampling practice deviates from that ideal situation, the less likely our inferences are to have any validity.

In much inferential statistics, the population we are sampling from is an actual population of cases, which could in principle be observed directly if we only had the money, time, staff, access, etc. etc. Here the ideal situation is to create a sampling frame that lists all the cases in the population, randomly select a subset of cases from the sampling frame, and then collect data from those cases we’ve selected. In practice, of course, most data collection doesn’t work this way – instead researchers pick a convenience sample of some kind (sometimes lazily, sometimes unavoidably) and then try to make the argument that this sampling method is unlikely to be strongly biased in any relevant way.

Sometimes, however, the population from which we draw our sample is not an actual population of cases that happen for contingent practical reasons to be beyond the reach of observation. Sometimes the population from which we draw our sample is a purely theoretical entity – a population of possible circumstances, from which actuality has drawn, or realised, one specific instance. Thus our actual historical present is a ‘sample’ from a ‘population’ of possible realities, and the generalisations we aim to make from our sample is a generalisation to the space of possibilities, rather than simply to some aspect of crass and meagre fact.

When we make claims that are predictive of future events, not merely of future observations of present events, we are, tacitly or overtly, engaged in this endeavour. To predict the future is to select one possible reality out of a space of possibilities, and to attribute a likelihood to this prediction is to engage in the statistical practice of assigning probability figures to a range of estimates of underlying population parameters – or, equivalently, to give probability figures to a range of estimates of future sample statistics ‘drawn from’ that underlying population. I may try to articulate this point with more precision in a future post – I’d like to spend more time on Bayesian vs. frequentist approaches to probability. And there is, of course, a ‘metaphysical’ question as to whether such a ‘population’ ‘really exists’, or whether the ‘samples’ themselves are the only reality, and the ‘population’ a speculative theoretical entity derived from our experience of those samples. Functionally, however, these stances are identical: and by my pragmatist lights, to note such functional equivalence is to collapse the two possibilities together for most theoretical purposes.

When we speak of universal natural laws, then, we are stating that a given fact – the law in question – will be true in the entire range of possible worlds that might, in the future, be actualised in reality. (Whether this ‘possibility’ should be understood in ontological or epistemological terms is beside the point). For some, it is the role of science to make such predictions: on this erroneous stance, science attempts to identify universal features of reality, and any uncertainty that accrues to scientific results is the uncertainty of epistemological weakness, rather than ontological variation. Here, for example, is a video of Richard Feynman making fun of social science for its inability to formulate universal laws of history:

To take this attitude is to misunderstand the nature not just of social science, but of science in general. Science is not characterised by a quest for certainty or for permanence, but is rather characterised by an ongoing collective process of hypothesis formation and assessment, based on specific collectively accepted evidentiary standards. The conclusions of science cannot be certain, because they must always be vulnerable to refutation in the light of empirical evidence and the application of community norms of argument. Similarly, the phenomena examined by science need not be necessary, or even ongoing. A scientific endeavour can be entirely descriptive, of the most local and variable phenomena imaginable, so long as the process of description is subject to the appropriate communal evidentiary norms. It can, similarly, be explanatory without being predictive, for we can analyse the causes of the phenomena we observe without being able reliably to predict those causes’ future impacts and interactions. The set of phenomena regarding which long-term or even short-term reliably predictive hypotheses can be formed is smaller than the set of phenomena that can be studied empirically using the relevant community norms of hypothesis formation and assessment.

The social sciences often approach this limit case of the purely descriptive. Social reality is enormously variegated – and often there is little in the way of testable general claims that can be taken from a study of any given social phenomenon. But prediction is nevertheless sometimes the goal of social science. When the social sciences aim to study social phenomena, the ‘laws’ they aspire to uncover are always local and limited in scope – and when we form a hypothesis, this hypothesis applies within a certain local limit and no further. Where to draw the line – where to locate this limit – is a qualitative question that the community of social scientists must always bear in mind, but the existence of this limit in no way renders the endeavour ‘unscientific’.

When we make a social-scientific prediction, then, we are making a claim about what future reality will drawn from the space of possibility. We do not know the scope of this space – nor do we have any reason to regard the principle of selection as random or unbiased – indeed, we have strong reasons to believe the contrary. Further, the nature of social reality is such that we can and do aspire to intervene in this selection – to attempt to influence what possibilities are realised. As social scientists we sometimes aim to predict what outcomes will be drawn from this space of possibilities – and such a prediction can only be made within the framework of a broader, historically informed judgement of the narrower space, within the space of possibilities, that we aspire to model.

But we should also be aware of other, unrealised but potentially realisable social possibilities, beyond the set of possibilities we are modelling at any given moment. Part of the function of the scrupulous social scientist is to describe this space of possibilities itself – to describe not just regularities, but also the possible variety from within which those local regularities are drawn. We cannot know the limits to the space of possibilities – no sampling frame of possible societies exists. But we can explore what the ‘samples’ themselves – existing and historical societies and behaviours – tell us about the scope of that hypothetical space.

This latter task is where social science intersects with political practice. The understanding of the likely behaviour of social reality is important for political practice – but so too is a sense of the larger space of possibilities from which our own past and present societies have been drawn, and from which alternative futures could be drawn, or made, if we only had the political ability to do so.

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