[...]
Del:
>
>It is a basic Newtonian principle (first law) that velocity of an object
>remains
>constant unless the object is subject to a force. Force is, of course,
>quantified as ma (that's the substance of the second law), acceleration
>being defined in terms of change in
>velocity. The principle in question thus reduces to: velocity of an
>object remains constant unless the object is subject to a change in
>velocity. Well, *that's* hardly news. And in practice, how is it
>determined that a force is being applied? A change in a vector. Uh oh.
>That all looks about as empty and tautological as the 'survival of the
>fittest' case. In fact, Holton (_Introduction to Concepts and Theories
>in Physical Science_, 2e, p118, 119) terms all this a "vicious circle".
>
>Does that count against Newtonian physics? No, says Brian - and I think
>he's exactly right on that. But Brian's rescue of Newtonian physics
>seems to rest on the claim that e.g., the first law is not a
>*definition* of F, but, rather, that F, m, and a can be independently
>determined, and when so determined, we simply discover that lo and
>behold, the independent value of F is always identical to the product of
>the independent values of m and a.
>
>Well, (i) I don't believe that Brian is right here, and (ii) I don't
>think that's how Newton got F=ma either. (Actually, "F = ma" never
>shows up in the _Principia_ - see Holton.) I think that we are dealing
>with a definition here - or at least an interdependent system of
>definitions. And if so, then if the 'tautology' charge (even were it
>correct) is damaging against Darwin, it would be equally damaging
>against Newton.
>
It is true that "F = ma" never appears in the Principia. One (trivial)
reason is that Newton did not write his definitions or laws in
concise mathematical form as we are used to today. But his word
statement of the second law is equivalent to "F = ma" for the
special case when the mass is constant. The general (original)
statement was in terms of the linear momentum G = mv and can
be expressed in equation form as
F = d(mv)/dt
for constant mass:
F = mdv/dt = ma
I've never actually read through the Principia. Glancing through
the first few pages just now I notice 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.
Let's consider an example. At this moment I'm sitting in my chair
and I swear that there is a (rather large) force being exerted by
this chair on my butt. I know the force is there, I can feel it.
But yet, as nearly as I can tell, my butt is not acceleratiing
in the direction of this force :). What can the concept of force
defined merely as mass times acceleration tell me about a force
that is clearly present on an object that is not accelerating?
Another example. I attach a spring to the wall and pull on it,
holding it a some constant displacement. Again there is clearly
a force being exerted on my hand even though nothing is
accelerating.
Before I go on let me remind everyone that I introduced this as
a *nontrivial* example, where nontrivial means of course that
not everyone is going to agree. Heck, I don't even agree with
myself all the time on this. What swayed me to my current view
is the brilliant discussion of exactly these points by Richard
Feynman in <The Feynman Lectures on Physics>, Volume I chapter 12:
"Characteristics of Force". Following is a lengthy quote
from the first section "What is a force?" of chapter 12:
===================================================
Let us ask, "What is the meaning of the physical laws of
Newton, which we write as _F = ma_? What is the meaning
of force, mass, and acceleration?" Well, we can intuitively
sense the meaning of mass, and we can _define_ acceleration
if we know the meaning of position and time. We shall not
discuss those meanings, but shall concentrate on the new
concept of _force_. The answer is equally simple: "If a
body is accelerating, then there is a force on it." 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.
Suppose we have a law that says that the conservation of
momentum is valid if the sum of all external forces is zero;
then the question arises, "What does it _mean_, that the
sum of all external forces is zero?" A pleasant way to define
that statement would be: "When the total momentum is constant,
then the sum of the external forces is zero." There must be
something wrong with that, because it is just not saying
anything new. 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. We could
also define force to mean that a moving object with no force
acting on it continues to move with constant velocity in
a straight line. If we then observe an object _not_ moving
in a straight line with a constant velocity, we might say
that there is a force on it. Now such things certainly cannot
be the content of physics, because they are definitions going
in a circle. The Newtonian statement above, however, seems
to be the most precise definition of force, and one that
appeals to the mathematician; nevertheless, it is completely
useless, because no prediction whatsoever can be made from
a definition. One might sit in an armchair all day long and
define words at will, but to find out what happens when two
balls push against each other, or when a weight is hung on
a spring, is another matter altogether, because the way
bodies _behave_ is something completely outside any choice
of definitions.
For example, if we were to choose to say that an object left
to itself keeps its position and does not move, then when
we see something drifting, we could say that must be due to
a "gorce"--a gorce is the rate of change of position. Now
we have a wonderful new law, everything stands still except
when a gorce is acting. You see, that would be analogous to
the above definition of force, and it would contain no
information. 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 were not
completely described by Newton or by anybody 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, i.e., if we study the
characteristics of force as a program of interest, then we
shall find that forces have some simplicity; the law is good
program for analyzing nature, it is a suggestion that the
forces will be simple.
Now the first example of such forces was the complete law
of gravitation, which was given by Newton, and in stating the
law he answered the question, "What is the force?" If there
were nothing but gravitation, then the combination of this
law and the force law (second law of motion) would be a
complete theory, but there is much more than gravitation,
and we want to use Newton's laws in many different situations.
Therefore in order to proceed we have to tell something about
the properties of force. -- Feynman
=========================================================
Well, I was tempted to type more because Feynman's discussion
is so good. He goes on to give some "properties of force" and
then discusses several different types of forces. While I'm
at it, I'll go ahead and give the last paragraph of the first
section:
===========================================================
In the same way, 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.
By establishing suitable postulates it is always possible to
make a system of mathematics, just as Euclid did, but we cannot
make a mathematics of the world, because sooner or later we
have to find out whether the axioms are valid for the objects
of nature. Thus we immediately get involved with these
complicated and "dirty" objects of nature, but with approximations
ever increasing in accuracy. -- Feynman
===========================================================
[...]
Del:
>
>Back to the main issue - natural selection and tautology. There may be
>some difficulty for Darwin somewhere in the region here, but if there
>is, the attempts to locate it in a tautology charge may not be quite on
>target.
>
agreed.
Brian Harper
Associate Professor
Applied Mechanics
The Ohio State University
Feynman's Second Law:
"everything stands still except when a gorce is acting"