To what degree is statistics -- the mathematics of probability -- relevant for the humane sciences, such as economics?
J.T. Fraser observes that
"Probablistic laws that govern the behavior of aggregates of objects assumed to be indistinquishable occupy an intermediate position between the absolute unpredictability of chaos . . . and the predictability and retrodictability of deterministic processes" (Time, Conflict, and Human Values, 60).
Chaos is the state of the pre-quantum universe, probability is the state of the quantum universe, and determinism is the state of the macrophysical universe, with material goal-orientedness the state of life, and material and symbolic goal-orientedness the state of humans. So probability is most appropriate for the quantum physical level of reality, and to that level of complexity (each level is exponentially more complex than the one preceeding it in Fraser's formulation). Yet, in addition to quantum physical particle-waves, and because quantum physical particle-waves do,
"macroscopic physical systems from pebbles to galaxies also follow probablistic laws, as do biological species and as does human conduct both individually and collectively" (60).
The bottom line here is that probability occurs between chaos and determinism, and is thus most relevant to systems three levels of complexity below the human -- and for levels of complexity below an economy. This means that statistics is relevant for understanding humans and the economy, but it is less relevant the further you get away from that level of complexity. Thus it is actually less relevant to use statistics at the level of economics than it is for humans. This may seem odd, since it would seem that humans in aggregate are behaving in more statistical fashion. There is a probability that people will do X in situation Y. But wait, I actually just describe humans in a statistical fashion, not the economy. An economy cannot be understood by aggregating humans, as an economy in fact emerges from the repeated interactions of a wide variety of human beings acting on their values of the moment. That cannot be aggregated. Thus, statistics becomes at best difficult. At the least, it suggests that we need a different kind of statistical approach than we have perhaps used in the past. At worst, it suggests that we may never have the math to explain economic processes, since the economy is more complex than the humans in it -- and we certainly do not have a mathematics of human behavior!
So where does that leave us? There are no doubt models that are appropriate to understanding the economy -- but I suspect they are only just now coming about, with complex adaptive systems work, agent based modeling, etc. More, I suspect that they are only ever going to be so abstract that they are useless for anything other than creating general understanding of certain kinds of patterns. The economy is less predictable than are individual human actions. That does not mean we cannot understand the economy, or other social systems, only that we cannot ever predict outcomes. Unless actions are taken to simplify the economy, of course, in which case the system does become more predictable -- but also less stable as a result. Humans can be made more predictable by destroying our psyches, making us more animal-like, and living things can be made more predictable by killing them and turning them into chemicals. But it should be clear that reducing complexity should not be our goal.
Math can thus only ever explain the economy in the sense that it can highlight general principles and show certain patterns. It is even able to help us understand subprocesses within an economy. But for the truest, fullest understanding, we have to have a level of description that approaches the economy in complexity. Math is good at describing the lowest levels of complexity, but not the highest. It cannot even describe living things, let alone processes one or two levels of complexity above life's complexity.
In the end, the true economic scientist will recognize that a full understanding of the economy is futile. Language, which is at the human level of complexity, by definition is then not complex enough to explain the economy. It is unpredictable, but not impossible to interact in. Its chaos is not that of the lowest level (though it contains it), but the apparent chaos of that which is deeply incomprehensible to beings less complex than it.