[FOM] The empirical foundations of deductive logic and the axiomatic method (& applications to, e.g., ODEs & stochastic processes)

[Richard Haney]

I am generally rather amazed and somewhat puzzled by the degree to which mathematicians typically have confidence in (deductive) logic to obtain new mathematical "knowledge". This perplexing state of affairs seems especially poignant, for example, in the area of the mathematical theory of stochastic processes.

Apparently, ancient Greek mathematicians, and the world in general, gained confidence in deductive logic and the axiomatic method when it was discovered empirically that conclusions deduced from empirically true hypotheses (typically of Euclidean geometry) invariably turned out themselves to be empirically true within the accuracy of empirical methods and obvious interpretations available at that time. As a result, a great economy of effort in empirical testing of ideas was achieved. This "empirical validity" of deductive logic might be regarded as a law of nature much as Newton's law of gravity or any other empirical law of nature. (Philosophically, such "laws" might be regarded as complicated, subjective acts of "human-pattern-invention-and-matching" to nature and might not be essentially a part of "external" nature itself. This view gets into complicated questions as to "what is objective reality?", and is highly related to cultural psychology, cultural conditioning, and the
 psychological mechanisms of "perception", but such questions are a side issue here.) As with Newton's law of gravity, this empirical validity of deduction might be expected to fail as to accuracy and/or precision under certain extreme (or perhaps not so extreme) conditions.

I am wondering whether there has been any specific scientific or philosophical study, especially in modern times, of such empirical questions (as such) concerning deductive logic and the axiomatic method.

The axiom of choice seems justifiable as an addition to set-theoretic axioms if the resulting mathematics becomes more "manageable" and useful for purposes of empirical modeling and related analysis. Otherwise, it seems no more relevant scientifically (and epistemologically) than such questions as "How many angels can dance on the head of a pin?". In such a view, the addition of the axiom of choice might be regarded in the same "conceptual-manageability-and-usefulness" framework as the addition of negative numbers, irrational numbers, imaginary numbers, and so on, to the modern conceptual framework of mathematics. (Incidentally, I suspect that "conceptual-manageability-and-usefulness" is what most mathematicians have in mind by the use of the word "elegant".)

The axiom of choice is apparently useful for deducing the (formalistically nominal) "existence" of solutions of certain ordinary differential equations where more "constructive" methods do not seem to be available, but I am unsure whether such formalistically nominal existence makes the resulting mathematics more manageable or useful conceptually (or otherwise). However, analysis of solutions to ordinary differential equations has proven to be extremely useful to empirically-based science -- for example in the computation and analysis of planetary and satellite orbits and spacecraft trajectories.

But the mathematical theories of stochastic processes seem to be much more "far-fetched" in terms of empirically-based science than are the mathematical theories of ordinary differential equations. The uses of functional analysis, for example, seem to be extremely elaborate, devious and intricate in applications to the mathematical theories of stochastic processes. And empirical questions seem to be much more complicated due to the generally less obvious empirical testability of such theories in practice.

So I would also like to know to what extent theoreticians and practitioners have empirically verified the "conceptual-manageability-and-usefulness" of such mathematical theories of stochastic processes in actual applications. Specifically, what empirical basis is there for confidence in such highly intricate mathematical theories of stochastic processes in actual applications?

This second question as to applications to the mathematical theories of stochastic processes seems to be more immediately practical. But the first question as to studies in the empirical foundations of deductive logic and the axiomatic method seems to be much more far-reaching in scope.

Richard Haney

 


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