[FOM] Is there a compendium of examples of the "power" of deductive logic?
Vladimir Sazonov
V.Sazonov at csc.liv.ac.uk
Tue Dec 13 13:24:01 EST 2005
Quoting Richard Haney <rfhaney at yahoo.com>:
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.
A rather simple consideration:
As to the concept of "mass", I also had the impression that there is a
logical gap in its "definition" as it is usually done in physics. For
measuring length, time, speed, velocity and acceleration we have some
basic physical procedures and mathematical definitions. The mass, and
force concepts seemingly require some additional insight of rather
unclear logical character, and here is a gap.
But I remember lectures of Yuri Kulakov in Novosibirsk University
on "Foundations of Physics" (197?-198?) - an interesting, really
foundational approach - where he introduced the second low of
Newton (f=ma) without explicit using the concept of mass or force
as the following experimentally checked statement about accelerations
a(ij), where i ranges over accelerators (springs, gravity or
electromagnetic fields, etc.) and j ranges over bodies being
accelerated:
determinant of any 2x2 matrix
|a(i j) a(i j')|
|a(i'j) a(i'j')|
is always = 0.
Then, in the infinite matrix {a(i,j)} of all possible accelerations
we evidently have CONSTANT coefficients of proportionality between
lines (accelerators) and between rows (bodies). By choosing
some standard, "unit" accelerator i_0 and body j_0, this
will assign a force f_i and mass m_j values (coefficients) to each
accelerator i and body j. Then the above second low of mechanics can
be rewritten in the usual form as f=ma, or f_i = m_j * a(i,j).
Thus, the concepts of mass and force arise quite understandably and
AFTER this low is stated and DUE TO this low, NOT BEFORE it, as it
is usually presented. This way of presentation seems to me quite
"logical".
In the next step we can experimentally check the low of
mass additivity: m_{j+k}=m_j + m_k where j+k is the compound
body, and the same for the force additivity. These additivity
lows do not follow logically from (and do not precede to)
the above Newton's low. So, we can imagine a world where the mass
and force are not additive... Just to fantasize...
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".
I agree, but for physicists the formal deduction is not of so high
value as for mathematicians for whom the mathematical rigour (+ intuition
and imagination, of course), but not truth! is the heart of mathematics.
If anybody will tell us that P=NP is "true" (what does it ever mean?
or, let, consistent with PA), will it change anything in our mathematical
behaviour? Moreover, mathematicians will always doubt in any such (declared
by an Oracle) "truth" T still trying to formally prove either T or \not T
or to prove a consistency result relative to some respectful (not true!) formal
theories like ZFC.
Vladimir Sazonov
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