Introduction
The famous mathematician Henri Poincare once wrote of the probability calculus: “if this calculus be condemned, then the whole of the sciences must also be condemned” (Poincare 1902, 186).
Indeed, every branch of science makes extensive use of probability in some form or other. Quantum mechanics is well known for making heavy use of probability. The second law of thermodynamics is a statistical law and, formulated one way, states that the entropy of a closed system is most likely to increase. In statistical mechanics, a probability distribution known as the micro-canonical distribution is used to make predictions concerning the macro-properties of gases. In evolutionary theory, the concept of fitness is often defined in terms of a probability function (one such definition says that fitness is expected number of offspring). Probability also plays central roles in natural selection, drift, and macro-evolutionary models. The theory of three-dimensional random walks plays an important role in the biomechanical theory of a diverse range of rubbery materials: from the resilin in tendons that help flap insect wings, to arteries near the heart that regulate blood flow. In ecology and conservation biology, we see concepts like the expected time to extinction of a population. In economics, stochastic differential equations are used extensively to model all sorts of quantities: from inflation to investment flows, interest rates, and unemployment figures. And all of the sciences make extensive use of probability in the form of statistical inference: from hypothesis testing, to model selection, to parameter estimation, to confirmation, to confidence intervals. In science, probability is truly everywhere.
But the sciences do not have exclusive rights to probability theory. Probability also plays an important role in our everyday reasoning. It figures prominently in our formal theories of decision making and game playing.
In fact, probability is so pervasive in our lives that we may even be tempted to say that “the most important questions of life, are indeed for the most part only problems of probability,” as Pierre-Simon Laplace once did (Laplace 1814, 1).In philosophy of probability, there are two main questions that we are concerned with. The first question is: what is the correct mathematical theory of probability? Orthodoxy has it that this question was laid to rest by Andrei Kolmogorov in 1933. But as we shall see in §2, this is far from true; there are many competing formal theories of probability, and it is not clear that we can single one of these out as the correct formal theory of probability.
These formal theories of probability tell us how probabilities behave, how to calculate probabilities from other probabilities, but they do not tell us what probabilities are. This leads us to the second central question in philosophy of probability: just what are probabilities? Or put another way: what do probability statements mean. Do probability claims merely reflect facts about human ignorance? Or do they represent facts about the world? If so, which facts? In §3, we will see some of the various ways in which philosophers have tried to answer this question. Such answers are typically called interpretations of probability, or philosophical theories of probability.
These two central questions are by no means independent of each other. What the correct formal theory of probability is clearly constrains the space of philosophical interpretations. But it is not a one-way street. As we shall see, the philosophical theory of probability has a significant impact on the formal theory of probability too.
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