[FOM] The empirical foundations of deductive logic and the axiomatic method (& applications to, e.g., ODEs & stochastic processes) [repost, with revision and expansion, due to technical problems with original posting]

[Richard Haney]

[Because of technical problems with the original posting I am reposting
here and also taking the opportunity to revise and expand.]

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.  A great
economy of effort was thus made possible in that the need for extensive
empirical testing of every new deductive conclusion was greatly
reduced.  This was a tremendous and very impressive accomplishment, and
as a result, deductive logic and the axiomatic method has become
indelibly embedded in our culture ever since.  In fact, deductive logic
as a methodology seems historically to have been viewed sometimes as
superior to empirical determination for developing new knowledge, even
to the exclusion of empirical evidence (or even of empirical
relevance).

While the nature of mathematics in ancient times was closely connected
with intuitive, empirical verification of the facts discussed,
mathematics in modern times is often extremely abstract and seemingly
far removed from empirical verification and intuitiveness.  This fact
seems to make the philosophical issues of mathematics much more
complex.

The validity of deductive logic (as a methodology for deriving
empirically testable facts) seems to be an empirically-based general
fact similar to the laws of conservation of mass-energy and of linear
and of angular momentum.  We seem not to know of any exceptions to
these "laws"; however, because they are empirically-based, there
remains the possibility -- true to scientific experience generally
regarding supposed "laws" of nature -- that these laws could be invalid
or false in some extreme (or perhaps not so extreme) circumstances not
yet specifically explicated and tested.  Newton's law of gravity is
such an example in the history of science.  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.


Does anyone know of any specific scientific study, especially in modern
times, of such empirical questions concerning deductive logic and the
axiomatic method?  Any insightful comments on this issue would be
especially of interest.


In a more specific area of application, the axiom of choice (and its
conditionally equivalent theorems) 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?".

The renown, highly regarded mathematicians, Dellacherie and Meyer, in
their book *Probabilities and potential*, state unabashedly, to my
astonishment, that they regard the axiom of choice as "intuitive".  I
am not sure exactly what they mean by that statement, but it seems as
if they are perhaps declaring some sort of metaphysical act of faith in
the axiom of choice by their statement.  There is a question as to
whether the axiom of choice can be regarded as some sort of universal,
objective "truth" (in some sense) that most people can readily agree
with.  As for myself, I certainly do not regard the axiom of choice as
intuitive in the sense that I commonly understand "intuitive".  It
seems to me that the only scientifically acceptable way to regard the
axiom of choice is in the formalistic context of whether the notion is
useful in a scientific sense of helping to unify (i.e., helping to
manage) mathematics for the purposes of empirical applications outside
the framework of formalistic, abstract mathematics.  This is the same
sort of scientific, "conceptual-manageability-and-usefulness" criterion
as for justifying the introduction 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.  However, 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.
 Human lives and the success of various space flights and explorations
depend on the theoretical soundness and empirical reliability of
analysis and various techniques concerning ordinary differential
equations.

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.  Even the various techniques for proving the
mathematical existence of Weiner measure (or of Brownian motion) seem
extremely elaborate, devious and intricate.  

(These books may be of particular interest here:

Robert B. Ash & Melvin F. Gardner: *Topics in Stochastic Processes*
(Academic Press, SF, 1975);

Robert B. Ash: *Real Analysis and probability* (Academic Press, SF,
1972);

Frank B. Knight: *Essentials of Brownian Motion and Diffusion*
(American Mathematical Society, Providence, RI, 1981; series:
Mathematical surveys, no. 18);

David Williams: *Diffusions, Markov Processes, and Martingales, vol. 1:
Foundations* (Wiley, NY, 1979);

and the online book *Stochastic Integration and Stochastic Differential
Equations* [at http://www.ma.utexas.edu/users/kbi/SDE/C_1.html , or
more specifically, in pdf form at
http://www.ma.utexas.edu/users/kbi/SDE/C_00.pdf ] by Klaus Bichteler.)

In particular, although apparently the easiest to follow as to logical
structure, Ash's proof of the existence of Weiner measure is rather
long and essentially spread out through several books.  Certainly, the
kind of proof we see here seems extremely far removed from the sort of
intuitive, easy, confidence-building proofs involving empirically
related concepts that ancient Greek mathematicians developed for
Euclidean geometry.  And Bichteler's online book is especially useful
for appreciating how elaborate, intricate, and "devious" functional
analytic considerations can become in developing mathematical theory in
this matter.  In fact, in spite of an excellent, intuitive discussion
and motivation as to applications and basic theoretical considerations,
Bichteler's book seems to show in candid detail just how so far removed
the logic of such mathematical theory seems to be from the empirical
world outside the framework of formalistic, abstract mathematics.

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? 
Again, any insightful comments on this issue would be especially of
interest.

Richard Haney

P.S.:  In regard to physicist's concepts regarding mathematical "laws"
of nature, it seems as if many physicists are attempting to obtain some
sort of unified, absolutely accurate mathematical description of nature
(e.g., in the vein of grand unified theories).  In fact, it often seems
as if such physicists have an abiding faith that such an anticipated
absolutely accurate, perhaps "complete", mathematical description of
nature is some sort of ultimate, metaphysically objective, realizable
entity.  I submit that all "laws" of nature are essentially
complicated, subjective acts of "human-pattern-invention-and-matching"
to nature and that, if they are in some absolute sense essentially a
part of "external" nature itself, we will never know.  Such a
metaphysical tendency by physicists seems to be in the same vein as the
physicists' earlier, now-discarded idea of "ether".  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".  In other words, all such
issues are highly relative to our human perspective.  But of course, if
such an abiding faith helps motivate the search for better descriptions
of nature, what's the harm in it as long as we recognize that all human
theorizing consists essentially of complicated, subjective acts of
"human-pattern-invention-and-matching" of one sort or another, and is
thus subject to human foibles, even many we may not have thought of?

P.P.S.:  I am also interested in exploring the possibilities for
gaining new understanding of the particular histories of fundamental
concepts (e.g., deductive logic) that arose in prehistoric times, much
in the same way that cosmologists are able to "tease" likely facts from
the vast milieu of data regarding the early history of the universe or
that geneticists are able to "tease" likely histories of species from
the vast milieu of genetic and other data presently available.

P.P.P.S.:  It seems to me that the success of popular forms of set
theory rests primarily on the probably very, very ancient and
well-tested linguistic invention and idea of sets or collections of
objects, at least as that ancient idea is applied to finite
collections, however large.  It seems that the systematic exploration
and development of primitive, elementary, empirical laws regarding
collections of objects could provide better insight for unifying very
abstract mathematics with the empirical world outside the framework of
formalistic, abstract mathematics.  

I submit that, even if such elementary facts may seem so elementary
that no-one seems to want to discuss them from a scientific
perspective, such elementary facts can give better insight into, for
example, why many people may want to adopt the axiom of choice.

Here is one such amazing, but very elementary fact:  every time you
take five stones and physically add three more stones to the
collection, and then count the resulting collection, you will always
count eight stones.  That's a fact that is completely "prior" to any
notion of arithmetic systems; it doesn't even depend on a number
system; you could use a string of beads or a separate collection of
stones for counting in a one-to-one fashion.  Another fact is that, if
you start with three stones and then add five stones, you again get
eight stones; thus you get a kind of empirical, physical commutativity.
 An even more basic fact in this area is that many objects of our
experience have some kind of permanence; in particular, if you count a
collection of rocks and get five, you will usually get the same number
when you do the same "experiment" again with that collection.  Another
fact is that, as suggested above, the counting results do not depend on
what reference collection (e.g., beads) you use for counting purposes
as long as you have enough of them for the experiment; there is a kind
of equivalence of reference collections.

And so by using similarly empirical explorations as to the
psychological and cultural origins of the axiom of choice, it seems we
could probably get better insight into what role (if any) the axiom of
choice should play in mathematics.

P.P.P.P.S:  Incidentally, when I speak of "empirical", "empirically",
"empirically testable", "empirically relevant", etc., I usually mean
empirical, empirically, empirically testable, empirically relevant,
etc., outside the framework of formalistic mathematics.  (It can become
rather cumbersome and monotonous to keep adding "outside the framework
of formalistic mathematics".)

Of course, there is also another sense in which deduction is empirical:
checking a proof is also empirical, but usually one thorough check is a
sufficient empirical test (provided there are no procedural errors in
the process); any repetition of the test, like a rerun of a computer
program using the same data, will only yield the same result.  This
kind of empirical test is similar to counting stones; one such test or
"experiment" is usually sufficient.  So it seems that this kind of
empirical test is quite different from the usual empirical testing in
the empirically-based sciences, where the relevant data and the
circumstances vary from one test (or observation) to another.

And so it seems that mathematics, at least mathematics in a formalistic
sense, might be defined as the science of deterministically repeatable
processes.  (Even probability theory is itself deterministic in this
sense; the theorems themselves don't change randomly from day to day,
although of course revisions of theory can be different.)

Of course, there can also be alternative deductive proofs, but, still,
one thorough check of such a proof would ordinarily be sufficient.


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