[FOM] The empirical foundations of deductive logic and ... (& applications to stochastic processes)

[Richard Haney]

I am somewhat concerned about assuming the axiom of choice in its full
generality since the intuitive idea seems likely to be founded
(subliminally?) only on human experience with finite sets (however
indefinite or large) outside the framework of formalistic mathematics.

However, my main concern with the axiom of choice is not so much
whether the axiom is "true" or not (say, in some sense of a priori
"knowledge"), but rather that inclusion of the axiom of choice may be
simply littering the theoretical landscape of mathematics with theorems
that are irrelevant to the empirical world (i.e., applications) outside
the framework of formalistic mathematics.  So my original concerns in
this regard are these:  Does anyone know of any application of the
axiom of choice that is useful in, for example, the physical sciences
(say, in developing models and in analysis of such models)?  Does the
axiom of choice make applied mathematics easier to "manage¡±?  Does it
simplify any conceptual framework in applied mathematics?

My interest in Dellacherie and Meyer's book *Probabilities and
potential* is in using it for applications to stochastic processes in
the empirical world.  So my original concern was that adding the axiom
of choice may simply complicate the already complicated subject
unnecessarily.

But actually, the axiom of choice does have relevance to applicable
mathematics, at least in a proof-theoretic sense.  Vitali's theorem,
that there are non-measurable subsets of the real numbers, is a
consequence of the axiom of choice.  (It is a "non-constructive"
result.)  So if the axioms of set theory are consistent, then it would
be futile to try to prove that all subsets of the real numbers are
measurable, whether one adopts the axiom of choice or not.  (Of course,
if one's objective is to try to prove that the axioms of set theory are
inconsistent, one still might want to try to prove that all subsets of
the reals are measurable.)  So in this sense the axiom of choice is
very relevant to applicable mathematics (that is, mathematics
applicable to the empirical world outside the framework of formalistic,
abstract mathematics).  It provides some idea of where one should or
should not focus attention for the purposes of developing theory for
applications.

There still remains the question, however, whether the axiom of choice
plays a useful role in the applicable mathematical theory itself, say,
for the purposes of developing models of empirical phenomena and of
analyzing those models in regard to empirically meaningful phenomena.

My chief reason for pointing out the empirical basis for deductive
logic was to point out that, as far as relevance to applications to
empirical phenomena is concerned, there is reason to doubt the complete
reliability of deductive logic.  So when deductive proof become very
long, as in the mathematical theory of stochastic processes, there is
even more concern that the conclusions may be fallacious in actual
empirical applications, simply because, in such a long proof, there is
more opportunity for fallacies to creep into the theory, over and above
any procedural errors in applying the formal rules of inference.

However, the comparison of modern mathematics with Euclidean geometry
does also present the enigma that modern mathematics tends to be far
less intuitive and far more removed from empirical relevance than
Euclidean geometry, which seems to be largely the origin for confidence
in deductive logic.  So the question remains whether modern logic and
resulting mathematics are still entirely reliable for the purposes of
applications to empirical phenomena.  If the purpose of applied
mathematics is to suggest the existence of phenomena that might be
explored empirically, that is one use.  But if human lives rely on
deductive conclusions as applied to empirical phenomena, that is quite
another and requires a higher level of confidence in the empirical
validity of deductive conclusions.

Considering that the axiom of choice is generally used to prove
Vitali's theorem, I wonder if Vitali's theorem is undecidable without
the axiom of choice; in other words, could one simply assume (as an
axiom) that all subsets of any set are measurable?  (This, of course,
would require excluding the axiom of choice from the axioms, unless one
is willing to have an inconsistent theory, which, in order to be
useful, would probably require fiddling with and changing the rules of
inference.)

Richard Haney

P.S.: Good references for the axiom of choice and Vitali's theorem seem
to be
http://en.wikipedia.org/wiki/Axiom_of_choice
and
http://en.wikipedia.org/wiki/Vitali_theorem


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