PKT message, 'Treatise On Probability' Revisited

Yesterday, I had occasion to revisit Keynes' 'Treatise on Probability', in which he challenged the Frequentist View of Probability and which Bertrand Russell described late in life as "an extremely able" work on the subject matter. In short order, I formulated a simple thought experiment whereby elementary physics are shown to debunk the Coin-flipping version of the Frequentist View.

In view of the importance which Keynes attached to the subject matter, and considering the essential role which the Frequentist View of Probability plays in much of modern economic theorizing, I am sketching below a brief outline of my argument.

1. An Internet site offers the following definition of the 'Relative Frequency Interpretation of Proabability': "Recall that we defined the probability of event A as P(A) = # of ways A can occur/Total # of ways anything can occur. This is called the classical probability definition. Another way to interpret probability is as the long-run relative frequency (long-run fraction) of the event. That is, if I flip a fair coin hundreds and hundreds of times, the fraction of heads will be very close to 0.5. The more I repeat the experiment, the closer to 0.5 the relative frequency will be. This is the same result the classical definition gives us. The relative frequency interpretation of probability works especially well for repeatable events, e.g., flipping a coin, rolling dice, drawing cards etc."

2. "Why," I asked a supporter of the Frequentist View of Probability on an Internet Forum, "is the set of possible outcomes in the toss of a perfectly balanced coin specified as H + T, where H = Heads and T = Tails, rather than H + T + E, where E = Edge?"

3. He replied: "In abstract mathematics, the sample space is {H, T}, but in the real world it's {H, T, E}."

4. My response was as follows:

"Is that so?

a. Consider a perfectly balanced coin suspended in a contraption at the top of a perfect vacuum such that it is in perfect alignment with the vertical.

b. Now let the contraption's grip on the coin be relaxed so that the coin falls down.

c. Given the assumed vacuum and absent any other 'non-coin' related factors, why should the coin not land on its edge?"

5. Within a few minutes, a physics-literate third forumite provided the following input:

"If you assume that the coin "falls" then you must also assume that there is a gravitational force pulling it down. When the coin reaches the bottom or end of its fall the nearest point of the coin will stop first transmitting the energy of the coin into the "bottom" from the bottom of the coin to the top of the coin. I was trying to prove that the coin would fall over because the coin would try to compress from bottom up and since the coin can not compress or absorb all the energy released when it came to rest it would fall over, but I was wrong. It would remain on its edge. This could not be proven in a laboratory setting because you would need a perfectly uniform coin so that the energy released was perfectly released in a uniform way."

6. To which I replied:

"Thanks!

It may have taken you all of five minutes to reason your way through the physics of the thought experiment."

Which is readily shown to imply that the H/T Frequency Distribution is a function of the non-coin aspects of Coin-tossing.

7. As background, I had posted Harrod's summary of the key points made by Keynes in 'Treatise on Probability' (The Life Of John Maynard Keynes, Penguin Books, 1978, pp. 772-75). With reference thereto, another forumite asked: "What was Keynes's point?"

8. I referenced the above summary and added:

"Now, why did Keynes - and Whitehead, Ramsey, and Russell - either reject or question the logical adequacy of the Frequency Approach?

"The answer can be formulated in any number of ways - the key point at issue concerns the Humean view of the relationship between theory and practice which, in the case of economics, Keynes summarized as follows in 1922:

The Theory of Economics does not furnish a body of settled conclusions immediately applicable to a policy. It is a method rather than a doctrine, an apparatus of the mind, a technique of thinking, which helps its possessor to draw correct conclusions.

The distinction drawn here by Keynes between 'theory' and 'practice' mirrors that between pure and applied geometry - a distinction which has disappeared from much of modern physics and, apparently, has yet to appear on the NDS [Neo-Darwinian Synthesis] stage.

As for the Coin-tossing approach to the logical foundations of the Probability Calculus, it is nonsensical in the sense that an attempt to formulate the propositions of analytic geometry in terms of empirically observed lines, circles, and points is nonsensical."

The thrust of this argument is familiar to Gang8 and PKTers - until yesterday, I had not thought of relating it to Keynes' challenge to the Frequentist View of Probability.

Gunnar