This continues from the previous post, Risk Versus Uncertainty .
In our daily lives, we routinely look past surface appearances to arrive at deeper truths hidden beneath the surface. Science involves looking at falling apples and deducing the existence of gravity. However, empiricism prohibits us from going beneath the surface, and restricts us to the world of appearances. This handicaps our thoughts and theories, and is especially damaging in terms of understanding probability.
Section 3: The Blinders of Empiricism (part 1)
The Arabs have many words for sand, and the Eskimos have many words for snow. According to the Sapir-Whorf hypothesis, lack of vocabulary can prevent us from seeing the differences in sand and snow which can be seen by those who have the words to express the difference. There is no doubt that linguistic confusions form an important reason for a century of persistence of an erroneous argument which informs us that we know – despite our feelings to the contrary – the probability of every uncertain event. The impoverished vocabulary which prevents us from seeing the flaws in the argument is due to logical positivism, which denies meaning to several concepts essential to understanding probability. We have now developed the terminology required to explain the nature of the subtle and complex mistakes which have been made in the Ramsey – De Finetti – Savage argument which establishes the existence of subjective probabilities. We use this terminology to provide the philosophical foundations which underlie their argument. In the next section, we replicate their argument in a simplified model for subjective probability and explain why it does not prove what it claims to prove.
Proposition 1: Empiricism prohibits us from talking about single case ontic probability.
Discussion: Intuitively, even those who deny its existence, understand the meaning of the concept of ontic probability for a coin flip. However, for a single coin flip, there are no observations which can provide any empirical evidence in favor of, or against, the modal idea that even though the outcome of “tails” occurred, the other outcome of “heads” was possible. A strong empiricist tradition, starting from Hume, prohibits us from talking about such things. The statement that the probability of heads is 50% as the outcome of a single coin flip is not a “fact” about the real world which can be observed. Hume would have us burn this statement. Kant would call probability a “thing-in-itself” and tell us that it is futile to think about it. Wittgenstein admonishes us to not speak about the non-factual: “Whereof one cannot speak, thereof one must be silent.” It is significant that one of the founders of subjective probability, Frank Ramsey, was a close friend of Wittgenstein, and re-expressed this aphorism as: “What we can’t say we can’t say, and we can’t whistle it either.” About the other principal architect of subjective probability, Bruno de Finetti, Dawid (2004) writes that “de Finetti’s philosophical approach might be described as ‘Machian’ or ‘extreme positivist’. It ascribes meaning to only those events and quantities which are observable.”
Wrong ideas of dead philosophers which prohibited talk about the obvious intuitive meaning of probability have resulted in a century of confusion about the meaning of probability. The most important of these confusions is the dominant conception of probability as a limiting frequency
Proposition 2: The Ontic probability of an event p(G) can only be “observed” in external reality as the limiting frequency of an infinite sequence of repetitions.
Explanation: Many kinds of theoretical calculations can lead to computations of precise probabilities for events. Roulette, Dice, and quantum events have probabilities which can be computed via a theoretical model which generates the probabilities of all possible outcomes. All such calculations must necessarily be based on a theoretical model we have for different possible outcomes. The “observation” in Proposition 2 does not refer to a theoretical, model-based calculation, but rather some aspect of external reality which can be measured to equal the theoretical probability p. However, in external reality only one of many possible outcomes can be observed. For example, if G is the event of rainfall, some weather forecasting models lead to computed values of p(G), but in external reality we can only observe 1 or 0 – whether the rainfall occurs or not. The probability number itself cannot be observed in any single trial. We might try to extend to a finite sequence of repeated trials. However, probability theory itself informs us that on any finite sequence of Bernoulli trials, the probability of an exact match between the observed frequency of occurrence of G and theoretical probability p(G) is low, and converges to zero as the number of trials increases to infinity. The number p only reveals itself in external reality in the limiting frequency, as the number of trials goes to infinity.
It is worth emphasizing this result: The only possible way to measure the ontic probability p(G) in external reality is as a limiting frequency of an infinite sequence of trials. Lord Kelvin expressed dominant views about science and measurement in his dictum that: “when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind”. The limiting frequency is the only possible objective and scientific definition of probability, according to these positivistic conceptions of science. This accounts for the dominance and popularity of the limiting frequency definition, despite many difficulties associated with this definition. For a discussion of these difficulties, see Section 3.4 of Hajek (2012).
Proposition 3: Single Case Ontic Probability is not observable and is not measurable.
Discussion: By unobservable, I mean that a single case probability asserts the possibility of occurrence of multiple events, but we can actually observe only one of them. It is important to note that this may be, conceptually, a limitation of our human sensory capabilities. The TV Series Doctor Who depicts an ancient extra-terrestrial race of Time Lords, who can sense forks in the timelines running to the future. For example, the timelines may branch into a world in which heavy rainfall occurs, preventing or frustrating an invading force, and another in which heavy winds diverted the rainfall leading to a successful attack. Human beings can observe only one of the two possible outcomes, rendering probability unobservable to us. Lack of measurability of ontic probability refers to the impossibility of carrying out infinite trials, which is the only possible way of measuring this probability.
Next Post: Fallacies of Frequentism and Subjectivism (5/10) which is part 2 of Section 3 of my paper of Subjective Probability Does Not Exist.
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