First, you said that
"...Newton introduced a number of things by way of definitions. For
example, mass, momentum and acceleration are all defined quantities.
The second law was stated differently, as a law or axiom rather than as
a definition of force. For the moment anyway, I have to believe that
Newton had a reason for doing this and that he did not consider his
second law as merely a definition of force."
We might need Ted Davis (is he still on here?) or some other historian
to set us straight on what Newton believed that he was actually doing.
But Newton did introduce his concept of force via definition:
"Definition IV. An impressed force is an action exerted upon a body in
order to change its state, either of rest or of uniform motion in a
right line." (Definitions and Scholium, _Principia_, Book I)
The second law, it seems to me, is a stipulation of how he wants to
quantify/measure force - the definition of a metric. In your original
message (the one I initially responded to) you said that "For Newton's
second law to have any significance it is necessary for all three
quantities (force, mass and acceleration) to be defined independently of
the law itself." Did you mean to claim that the concepts had to be
defined independently (even if the metric is stipulative) or did you
mean that all three had to be independently measurable, the equation
then being simply an empirical generalization? In any case, do you take
the definition of force above to be in terms independent of
acceleration?
Second, I'm not sure why you take the Feynman stuff to support your
general contention.
a. Feynman says (in your quote):
"That is what Newton's laws say, so
the most precise and beautiful definition of force imaginable might
simply be to say that force is the mass of an object times the
acceleration." I take it that you would claim that he is not using
'definition' in any precise sense here?
b. Feynman says (in your quote):
"If we have discovered a fundamental law, which
asserts that the force is equal to the mass times the acceleration, and
then _define_ the force to be the mass times the acceleration, we have
found out nothing."
I'm not sure that that makes good sense. If we
*discovered* that F = ma, we would (presumably on your view) have to
already have some independent concept and measure of F. What would it
mean to then subsequently *define* F as ma? Wouldn't we already
*have* a definition?
c. Feynman says (in your quote):
"The real content of Newton's laws is this: that the
force is supposed to have some _independent properties_, in addition to
the law _F = ma_; but the _specific independent properties that the
force has wer not completely described by Newton or by anyvbody else,
and therefore the physical law _F = ma_ is an incomplete law. It
implies that if we study the mass times the acceleration and call the
product the force ... then we shall find that forces have some
simplicity;... it is a suggestion that the forces will be simple."
One way of reading that is that "F=ma" is a gross definition, but that
the intricacies of its specific applications and extensions will have to
be worked out case by case (empirically). If that is what Feynman
intends, that would not necessarily change the definitional character of
the equation - only its detailed completeness. And saying that "if we
study the mass times the acceleration and *call* the product the force
..." (my emphasis) certainly sounds very like defining force in the
relevant terms.
The thrust of the Feynman passage, it seems to me, is that Newton begins
with some definitions, but that since confining ourselves to mere
definitions will never generate any actual *science*, we have to find a
content for those definitions. So, says Feynman, "the real
[*scientific*] content of Newton's laws" involves pursuing the F=ma
framework/structure into nature empirically, and in adopting that
framework/structure as foundamental guiding principle in doing one's
science, one is thereby commited to a picture of nature as ultimately
reducing to a simple underlying pattern/motif (specifically, that the
world is F=ma-ish). That sounds to me very like the Rothman approach I
mentioned last time, and very different from what I understood you to be
saying.
That impression is strengthened by the Feynman addendum you included. n
Feynman says:
"...we cannot just call F=ma a definition, deduce everything purely
mathematically, and make mechanics a mathematical theory, when mechanics
is a description of nature. [snip] [S]ooner or later we have to find out
whether the axioms are valid for the objects of nature. Thus we
immediately get involved with these compolicated and 'dirty' objects of
nature..."
That does not imply that F=ma is not definitional - only that we cannot
*stop* with a definition and its purely abstract derivations if we are
going to do real science. In particular, we have to see *if the system
applies* - i.e., as Feynman says, "whether the axioms are valid for the
objects of nature." But that is just the Rothman scheme.
So again, I'm not sure that Feynman really helps you here, unless I'm
misreading either you or him. Out of courtesy to me, I will ignore the
possible charge that I am misreading me.
Del
Del Ratzsch voice: (616) 957-6415
Philosophy Department (616) 451-4301 (home)
Calvin College e-mail: dratzsch@legacy.calvin.edu
Grand Rapids, MI 49546 fax: (616) 957-8551