Today I will talk about probability, because most philosophers interpret “probability” in logical rather than mathematical terms. When most people think “probable,” they have such a narrow definition in mind that they are foreclosed from comprehending the deeper implications of some philosophical theories. I hope that readers might forgive me for the presumptuous title of the post, but this has been a communication-stopper for discussions of process, emergence, time, continuity, and teleology for far too long.
For instance, a friend of mind, when hearing that I planned to write this, quipped that A.N. Whitehead has frequently been misunderstood--and I would add C.S. Peirce as well--because he was writing as a mathematician, an algebraist especially, and thus readers not trained in mathematics may misunderstand the subtleties of concepts of a mathematical origin. For example, "continuity" is a mathematical concept that most misinterpret as meaning "discrete continuum." Process philosophy cannot be understood unless one has the mathematical concept of continuity, else they do not fully grasp what it means for a process or time to be continuous. Moreover, in my own work, John Dewey describes an event in terms of frequency structures in Experience and Nature, and I wonder if one reason that reviewers seem puzzled over my articulations of Dewey is perhaps because they do not realize the pertinence of probability structures. Last, but not least, those interested in emergence would have a deeper insight into "emergent teleology," which is a form of probable causation, is they realized that a telos may be described through a probability function. In my exchanges with Levi Bryant, he appeared to miss this point entirely, as he accused myself and others of Platonism.
In philosophy, there are three common definitions of probability: logical, physical, and technological or practical. A proposition is logically possible if it does not violate the laws of logic, which conventionally include the law of non-contradiction, law of excluded middle, and law of identity. Different logical systems may include more laws or modify standard interpretations of the three basic laws. A proposition is physically possible if it does not violate the laws of physics, which are not logically necessary, but are often understood to pertain necessarily in the actual world. A proposition is technologically or practically possible if it is physically possible, and its realization is merely a matter of knowledge, technique, materials, etc.
None of these three common philosophical definitions of possibility pertain to mathematical possibility in anything other than a trivial and uninformative way. Mathematical possibility is about the structure of possibility regardless of whether it pertains in any actual world, and is a species of logical possibility. Mathematical possibility, or more commonly denoted “probability” is about the frequency of an event given some axiomatic assumptions that define a “space” or what philosophers might call a collection of “possible worlds.”
Assuming that we are talking about a continuity of events (a “continuous distribution” in math-speak), we can visual a “structure” of possibility by graphing it. The most well known such graph is the “bell curve,” which might also be called a Gaussian or normal distribution. What the graph tries to communicate is that that most events occur and display whatever traits are being measured around some normal range indicated by the peak of the graph. The further something is from normal, whatever that is, the far less likely it is to occur, which is included by the rapidly declining slope or “tails” of the graph.
However, this is not the only kind of probability distribution. Consider a poisson distribution. The graph is skewed. In a poisson distribution, it is not necessary that the frequency of events be symmetrically “balanced” around some point or common occurence.
My point is not to teach a lesson in probability. My point is to indicate that most people unfamiliar with mathematics think “uniform” or “normal” probability distributions when asked to think about probability. They think in terms of coin-tosses or of the one-time odds of their favorite football time winning a match. But those are singularly calculated events. If we were to model the likelihood that a team wins any given match against various opponents, the graph is not likely to look uniform. Probability has a structure. In mathematics, one way to capture and visualize that structure is through graphing probability distributions as shown. Moreover, the more conditions that are added, especially erratic conditions, the more gnarly a probability distribution may become—very far from intuitive. Many commonly recurring probability distributions have names, as do the examples I give, wherein to name a probability structure is to indicate what the structure is, e.g., the relative frequency of events.
How does this compare to logical possibility? Logical possibility says nothing about the structure of probability, e.g., the frequency of an event occuring given certain conditions. Only the mathematical concept of possibility indicates anything about structure, e.g., a function represented as a probability distribution.
When I write about “probability structures,” peer interlocutors, reviewers of my articles, etc. rarely have any idea what I mean by it. It took me quite awhile to realize that knowledge of probability and statistics, and thus working knowledge of probability structures, is uncommon among philosophers. We should change that and require some discussion in any advanced logic class, I propose, else it is likely that only philosophers in certain subfields will have very valuable concepts at their command.
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