The Axiomatic Formulation of Probability
When I first heard of the axiomatic formulation of probability, it was presented to me as "resolving" the dispute between the frequentist interpretation of probability and the subjective interpretation. (Interestingly, John Maynard Keynes was one of the leading proponents of the subjective interpretation, while Richard von Mises was a prominent champion of the frequentist interpretation. So Keynes had running disputes with both of the von Mises brothers.) And the Wikipedia page just linked to describes the axiomatic formulation of probability as a "rival" to the frequentist interpretation.
But to see the axiomatic formulation as rival to those other theories is a serious mistake (as Kolmogorov himself seemed to recognize -- see below). The frequentist and subjective theories of probability are concerned with the relation of probabilistic statements to the real world. They are essentially asking, "If we say, for instance, that the odds of rain today are 30%, what exactly do we mean, and on what basis do we mean it?"
What the axiomatic formulation does is simply set such philosophical questions aside, and formulate a probability theory as a purely abstract mathematical system arising from certain axioms, with no concern at all about how this system might be connected to the real world. And I think that this was the right direction for mathematics to take: mathematicians could focus on their strength -- developing a consistent formal system -- and leave the question of how it applies to reality to others.
As mentioned earlier, Kolmogorov himself recognized this, saying, "The basis for the applicability of the results of the mathematical theory of probability to real 'random phenomena' must depend on some form of the frequency concept of probability, the unavoidable nature of which has been established by von Mises in a spirited manner."
The type of mistake made in thinking that the axiomatic formulation of probability could resolve the dispute between the existing interpretations of how probability applies to the real world is a common one. It consists in mistaking the development of some formalism for as a solution to a vexing philosophical issue. People who think that the mathematical theory of limits solved Zeno's paradoxes are making this mistake. Similarly someone who claims that the development of different formalisms called "logics" empirically demonstrates that logic itself, philosophically speaking, is not unitary, is also making the same mistake.
But to see the axiomatic formulation as rival to those other theories is a serious mistake (as Kolmogorov himself seemed to recognize -- see below). The frequentist and subjective theories of probability are concerned with the relation of probabilistic statements to the real world. They are essentially asking, "If we say, for instance, that the odds of rain today are 30%, what exactly do we mean, and on what basis do we mean it?"
What the axiomatic formulation does is simply set such philosophical questions aside, and formulate a probability theory as a purely abstract mathematical system arising from certain axioms, with no concern at all about how this system might be connected to the real world. And I think that this was the right direction for mathematics to take: mathematicians could focus on their strength -- developing a consistent formal system -- and leave the question of how it applies to reality to others.
As mentioned earlier, Kolmogorov himself recognized this, saying, "The basis for the applicability of the results of the mathematical theory of probability to real 'random phenomena' must depend on some form of the frequency concept of probability, the unavoidable nature of which has been established by von Mises in a spirited manner."
The type of mistake made in thinking that the axiomatic formulation of probability could resolve the dispute between the existing interpretations of how probability applies to the real world is a common one. It consists in mistaking the development of some formalism for as a solution to a vexing philosophical issue. People who think that the mathematical theory of limits solved Zeno's paradoxes are making this mistake. Similarly someone who claims that the development of different formalisms called "logics" empirically demonstrates that logic itself, philosophically speaking, is not unitary, is also making the same mistake.
Gene, how would you resolve Zeno's paradoxes, then, assuming you think a successful resolution has been found?
ReplyDeleteIf this topic interests you, see Atomism and Its Critics.
DeleteThe book seems to be out-of-print, and it's really expensive from used book sellers. Any alternate recommendations?
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