[FOM] The empirical foundations of deductive logic and ...
Richard Haney
rfhaney at yahoo.com
Tue Sep 20 18:05:17 EDT 2005
Quoting Vladimir Sazonov <V.Sazonov at csc.liv.ac.uk>:
> .... In particular, unrestricted application
> of modus ponens leads to a contradiction. (Intuitively, a
> "small error" is accumulating with multiple application
> of modus ponens and giving rise to a contradiction.
If I understand the example correctly, this is quite similar to the
example of "creeping error" introduced by the logic of
> Premise: A man with no hair at all is bald.
>
> Premise: If a man has only n hairs on his head is bald, then so is a
man
> with n+1 hairs.
>
> Conclusion: A man with 50,000 hairs on his head is bald.
as you quote from Charles Silver in your comments of
http://www.cs.nyu.edu/pipermail/fom/2005-May/008959.html
(Variations of the argument exist elsewhere as well.)
Similar "creeping error" examples can be found in Carl G. Hempel's
article "On the nature of mathematical truth", *Amer. Math. Monthly*,
Dec 1945, vol. 52, pp. 543-556, in his discussion of errors of identity
of colors vis-¨¤-vis the transitivity of identity, his discussion of 3
microbes + 2 microbes = 6 microbes, and his discussion of Boyles law
vis-¨¤-vis arithmetic.
Another area where "creeping error" is a factor is, say, in measuring
and adding up the angles of a triangle, for example. It seems there
can be no end to "creeping error" examples, whether the underlying
physical quantities are discrete or essentially continuous.
Examples I have used (elsewhere) similar to Hempel's example of
microbes are failures in the additivity (or even just repeatability of
counting) of collections of clouds, or of bodies of water, or of
rabbits, or of magnetic bits on the surface of a computer storage disk.
In physics "proper" those kinds of exceptions would be noted and
incorporated into the comprehensive theoretical accounting of empirical
matters. But in mathematics it seems the preference is to pretend that
those sorts of exceptions don't exits, or, at least, are outside of the
domain of mathematical subject matter. So mathematicians create a
special world of abstractions so that they (the mathematicians) don't
have to deal with "messy" exceptions.
Thus, it seems there may be a need for a better (logical?) theory of
applications of mathematics to the empirical world outside the
framework of formalistic mathematics.
(Incidentally, "physical mathematics" with stones may perhaps be
considered non-formalistic mathematics; I use the phrase "formalistic
mathematics" because I consider "intuitive" ideas and "content" in
mathematics to be at least as important as the formalism, at least for
the purposes of motivation and applications, even though the intuitive
ideas are not necessarily precise or formalizable.)
I should add that I would be especially interested in _systematic_,
_comprehensive_ scientific studies of my previously indicated empirical
questions concerning deductive logic and the axiomatic method.
An example of scientific studies I have in mind, broadly speaking, is,
for example, the kind of comprehensive survey articles dealing with
empirical verification of theories as found in the "Special Section:
Einstein's legacy" published in *Science*, 11 Feb. 2005, vol. 307, no.
5711, pp. 865-890.
Of course, individual, "isolated" examples can add up to eventually
make a comprehensive whole. However, the spirit of systematic,
comprehensive scientific study as found in a statistician's design of
comprehensive, efficient, minimally biased experiments is part of what
I have in mind.
It may be worth noting that physicist's readily consider experimental
physics to be at least as important as theoretical physics. But in
mathematics the empirical origin and empirical relevance of
mathematical ideas seems to have become an orphaned subject that
mathematicians often don't readily want to acknowledge or deal with.
In fact, Howard Eves book *Foundations and Fundamental Concepts of
Mathematics*, 3rd ed. (Dover, Mineola, NY, 1997; PWS-Kent Publ. Co.,
Boston, MA, 1990), essentially a historical account of the axiomatic
method, makes the point that ancient Greek mathematicians as well as
modern mathematicians generally have a disdain for empirical matters.
It seems as if such a strong disinclination for mathematical empiricism
exists, at least in modern times, out of "political" deference to
philosophical rationalists and mathematical Platonists and simply out
of a desire to continue the tradition and alleged "snobbishness" (my
word) of ancient Greeks in regard to empirical matters, which it seems
were regarded in ancient Greece as the subject matter for only slaves
to deal with.
Perhaps a better, perhaps formalizable, theory of empirical modeling is
needed. Such a theory might be analogous to logicians' model theories.
The theory would probably need to include a theory of experimental
errors. However, note that physicists are able to conduct experimental
physics without a formal modeling theory. And I am inclined to agree
with others that formality in mathematics can kill the intuitive
motivation. Note that many aspects of mathematics seem to have
standard empirical interpretations. To remove those intuitive,
implicit interpretations might also kill the intuitive motivation.
(However, I am inclined to think that making such implicit
interpretations explicit is likely to be quite helpful.) The important
things are accuracy, reliability, precision, and efficacy (i.e.,
usefulness). But formality sometimes seems necessary to remove
ambiguity.
An example of a mathematical concept that seems to have a standard
empirical interpretation (or a standard class of empirical
interpretations) is the general concept of conditional expectation
(based on the Radon-Nikodym theorem). While textbooks often do a good
job of clarifying this concept for probability spaces generated by
countable partitions, I have not yet found a textbook that shows that
the general concept does indeed capture the intuitive,
applications-oriented idea of conditional expectation. So as a result
it seems that an implicit assumption is made that it is OK to use the
general concept as being an accurate representation of the intuitive,
empirically-oriented idea of conditional expectation. At the very
least, it seems it would be a good idea to prove (if possible) that no
other generalization of the countable-partition version is possible.
And if it is not possible to prove that, the question arises: Why not
use such an alternative generalization instead as such an accurate
representation (if such an alternative can be exhibited)?
Richard Haney
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