FOM: More on Arist. and Problem of Unity

[Robert Tragesser]

Abstract: Explaining how the breakdown into
paradoxes of a fundamental Aristotelian logic
led to Galileo's applications of mathematics
as a (sort of) logic.

        Some further problems about the unity
of the sciences,  as well as the unity of particular
sciences, and the Aristotelian frame(s) have occurred
to me these perhaps more relevant to the phom
fom connection. (I'm still not clear about when to
write f.o.m.)  Because it concerns the sense mathe-
matics had for Galileo when he (contrary to one
thread of the Aristotelian spirit) applied it to
"nature" not as a kind of accidental and useful
application (which sense of application the
Aristotelian could permit),  but as rooted in
the essence or arche of things. 
        Aristotle introduced what Randall (I think)
called a distributive conception of being, which
I take to mean that arche,  "first principles" are
distributed among beings in the primary sense,
individual substances.  Aristotle comes to this by
attacking the "cosmologists" who sought to
institute arche for the whole of being.
        The cosmologists typically exploited
what were taken to be real _contrarieties_ as
the driving forces of being in the cosmos. (Please
excuse short-cuts and unqualified assertions. . .
for I want to point to the essential things.)
        Aristotle used the "reason of contrarieties",
but not as principles in things;  rather as logic.
        So the contrariety (actual-potential) was
considered logical rather than metaphysical.  In
particular,  we reason about change (including
locomotion) in terms of contrariety,  INSTEAD OF
GEOMETRIC ARITHMETIC.  This wasn't very good for
understanding either violent motions or combinations
of motions.  But the logic of contrariety fell apart
entirely when it was applied to circular motion.
        This break-down emerges dramatically in the
opening pages of (pseudo-)Aristotle's _Mechanics_.
        There the problem is to understand seemingly
miraculous effects of levers and pullies (the very light
moving the very heavy).  These effects are reduced to
circular motions (motions of wheels).  The problem
is to understand the motion of wheels in terms of
the logic of contrariety.  Then the motions of wheels
appear paradoxical,  e.g.,  move forward by moving 
backward. (There are some moments when it seems that
this paradoxical character is levered in as an 
explanation of the miracle of levers!)
        (By the way,  there is the implication
here that the understanding of the circular
motions of the heavens is in deep trouble -- 
incomprehensible rather than wonderfully intelligible!)
        In TWO NEW SCIENCES and TWO WORLD SYSTEMS,
Galileo begins with attacks on Aristotle's logic
of contrariety with a view to installing geometric
arithmetic as the proper logic of motion.  
        Hence the idea of a mathematics as a logic,
a tool for soundly reasoning about things in order
to arrive at understanding.  (One finds something of
this emphasis in Weyl's 1940 lecture to the Philosophical
Association.)
        The book of nature [=nature in its logical aspect]
is written in the language of geometric-arithmetic.
        compare with
        The book of nature [nature in its logical aspect]
is written in the language of contrariety.
        By introducing the descriptively far more fluid
logic of geometric-arithmetic,  Galileo was able to unify
the sublunary and superlunary regions.
        Here the elaboration is perhaps more interesting than
the point of Aristotle having a real hard time with unity.
        it is worth pointing out that Galileo (and we) have
 just a hard a time as Aristotle had in understanding how a
logic could powerfully apply to effects without their being
really in the effects correlates of the terms of the logic.
(For Aristotle -- on one good reading -- contrariety is
in the logic,  not in things. . .)
        robert tragesser