[FOM] Is there a compendium of examples of the "power" of deductive logic?

Richard Haney rfhaney at yahoo.com
Mon Dec 12 20:52:05 EST 2005

In response to my question, which states in part:

>> So I am wondering about the possibility of stunning
>> examples that can serve as paradigms for the power
>> of logic in empirical applications.  In other words
>> I am looking for empirical evidence that demonstrates
>> unequivocally that deductive logic is indeed worthy
>> of study.

Neil Tennant wrote:

> How about Newton's derivation of Kepler's laws of
> planetary motion, using as premises the law of
> gravitation, the second law of motion, and whatever
> mathematical axioms are needed for the calculus
> involved?
> Another example, drawing on less technically demanding
> mathematics (if any at all) would be the deduction
> underlying the central Darwinian insight.  This is the
> insight that whenever the three conditions
>	Variability
>	Heritability
>	Differential reproduction
> hold ofr a population of self-reproducing entities,
> there will be
>	Adaptive evolution.
> (Note that this is the case even if the individuals
> never die.)

These examples indeed have the general flavor of what I am looking for.
 And it seems there is great potential for working these examples into
a very exemplary status.  However, it seems it may take quite a bit of
work to do so!

Your first example seems indeed to be a stunning example of a mixture
of deductive inference with empirical research.  It seems I probably
did study this example a few years ago in studying books on celestial
mechanics.  And I seem to recall that the deductive analysis was very
nice -- it was easy to fill in the details.  I should probably study
that example again. 

The example does seem likely to have the complication I alluded to
earlier, namely, because of the inexactness of measurements of
essentially continuous parameters, there will be a certain "fuzziness"
in the deduction; that is, the deduction probably is not a strictly
precise one.  But this drawback seems likely to be not much different
from the problem of empirically verifying that the interior angles of a
triangle sum to 180 degrees.  So as to "purity" -- i.e., freedom from
"fuzziness" -- your example might be on par with theorems of elementary
geometry.  The reason I also suggested "discrete and finite" empirical
phenomena is that such phenomena seem to offer the hope of avoiding
such fuzziness in the deduction; there is more hope that statements
about such phenomena are empirically either unequivocally true or
unequivocally false.

Your example is also closely related to the general mathematical theory
of "analytic mechanics".  I have perused a number of books on this
subject in the hopes of getting some sort of clear idea of a deductive
structure.  I found the supposed deductive structure to be very
puzzling at times.  It seems the theory gets stuck very early in the
development in considering the idea of "mass" and what role it should
play in what would be some sort of axiomatic framework for the theory. 
Of the books I examined, Corben & Stehle's *Classical mechanics* seemed
to have the best treatment of the analytical inferences, but even this
best of books (at that time) seemed to have deductive gaps in its
proofs where one is compelled to merely take certain statements on
faith and gloss over the gaps to proceed further.  Sometimes the gaps
are simply omissions of difficult details.  But sometimes the authors'
intent also seemed unclear.  However, there is much about theoretical
mechanics that does seem wonderful, if perhaps a bit still puzzling,
from a deductive standpoint.  Physicists do seem to have a different
concept from mathematicians as to what is an adequate deductive proof,
and my frustration with that reminds me of the popular song about a man
who wishes his wife could "think more like a man".

Raymond Wilder, in his book *Introduction to the Foundations of
Mathematics* (Second Edition at pp. 4-5) states that "[t]he treatment
of analytic mechanics published by Lagrange in 1788 has been considered
a masterpiece of logical perfection, moving from explicitly stated
primary propositions to the other propositions of the system."  I
haven't seen Lagrange's treatment, but if modern textbooks I have seen
on analytic mechanics are, from the standpoint of deductive exposition,
in any way considered to be as good as, or an improvement on,
Lagrange's treatment, it certainly seems there is more work that is
needed to put the theory into an exemplary status that is readily
appreciated by a wider audience.

As for your second proposal, it sounds tantalizing, but I would like to
see it framed in more convincing terms.  It seems like it may possibly
take some hard work or at least some fortuitous insights (or both) to
work out the details.

Richard Haney

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