Thinking about science upon entering the field. Part II: What are 'probabilities'?

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Probabilities are everywhere. They come up in our conversations when we talk about making plans. They are there in our games as “chances”, “odds”, and “risks”. We use them informally when we make decisions about our health and well–being. And, in a more formal sense, we use them in science when we make inferences about data. Indeed, probabilities are so common that they can at times seem almost familiar.

But just what exactly are we talking about when we talk about “probabilities”? When I say, for instance, that the probability that a tossed coin will land heads up is 50%, am I saying something about that coin’s disposition to produce a certain outcome, or am I only expressing the degree to which I believe that that outcome might occur? Do probabilities exist out there in the real world as things that we can measure, or are they just in our minds as opinions and beliefs?

The short answer seems to be, yes, probabilities are both. They have an objective and a subjective element to them. This duality has apparently been there from the start, at a time when formal probability concepts were first developed in the seventeenth century. According to the philosopher Ian Hacking, during the Renaissance, the term “probable” was taken to mean “approved by some authority” rather than by evidence. It was not until the Enlightenment, when early Empiricists first began looking to Nature for “signs” to support casual associations, that “probable” came to mean “having the power to elicit a change”. Hence, “approval by testimony” from people and institutions was superseded by evidential observations. Transforming signs into evidence helped to advance what we might call frequentists–based induction, which was formalized as a mathematical concept in the Port Royal Logic in 1662.

Of course subjective probabilities have hardly disappeared, and in fact, it may be argued that we have seen their resurgence in the popularity of Bayesian– or conditional–based statistical inquiry. That being said, however, I am not sure that understanding how the term “probability” developed gets us much closer to understanding what probabilities really are.

It seems that in order to make progress here, we must talk about cause and effect. Namely, we need to discuss whether probabilities are like physical laws that define an event, or whether they are contrivances that we use to describe things after the fact. If they are descriptions based on the past, then what rational do we have for extending our inferences into the future? Is there any legitimate guarantee that future events will proceed at the same frequency as their predecessors? And even if they do, then for how long?

On the other hand, we might ask, if probabilities are only descriptive then what makes them so regular? Why does a tossed coin land heads up one-half of the time, almost as though it had some property that we might call its “probability”. Moreover, how are probabilities such as this determined? Could it be that we really are living in a clock–work universe, and that even our uncertainty is defined by deterministic processes? These questions are perhaps beyond what sciences and mathematics is able to answer. But maybe that is okay. This seems to be fertile ground for philosophical inquiry, which might provide insights where they are needed most.