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structures. That is, the functionals are mappings from functions to mathematical
structures, rather than mappings from physical entities to those mathematical
structures. The point is that certain functionals are physical magnitudes. The
final assumption is (4) that to each scientific law not only is there some set
of possibilities ! associated with that law, but there is also a special kind of
property, £¦ (which depends on a functional magnitude ¦), which holds or
fails to hold of the possibilities in!. We shall refer to the property £¦ as a
functional property. So here is the idea of how it is that laws exclude or narrow
down possibilities:
(LRP) (1) For each law L there is some property £¦ expressed with the aid
of a functional ¦ which holds or fails to hold for the members of a
certain set of possibilities, and (2) L implies that some possibilities
fail to have that property.
According to (2), if L is a scientific law, then for some possibility ± of a set
of possibilities !, L Ò! ¬£¦(±).5 It may also happen that some law excludes
all the possibilities that are associated with it, but that is not generally so
for all laws. Furthermore, a law L may actually guarantee that some of the
possibilities have the functional property £¦, that is, for some possibility ²,
L Ò! £¦(²).
On our proposal, it is not the possibilities themselves that are implied
(or not) by the law L; it is the statements £¦(±) (that the possibility ± has the
functional property £¦) that are implied (or not) by L.
There is little space to show in any detail how typical laws reduce possibili-
ties. Any law which can be expressed as a special case of Hamilton s Principle
of Least Action easily falls under the concept expressed by (LRP). The reason
is that such applications proceed by specifying a particular Lagrangean (T U)
for some physical system, where T is the kinetic energy and U the potential
energy of that system. A set of curves f, g, & is specified between two points in
a state space. Think of these curves as the possibilities, and there is a functional
called the action for the particular Lagrangean L, which is given by
t1
¨ L
[f]= +" (q, dq / dt,t)dt
t0
(where the integral is taken along the curve f). With the help of the functional
¨, we can define a functional property £¨ of the possibilities (curves f, g, & )
as follows:
Laws, explanations and the reduction of possibilities 179
£¨ holds of the curve f if and only if f is a curve for which ¨, the action, is
an extremal  that is, ¨(f) is either a maximum or a minimum.
In all these  Hamiltonian cases, possibilities are ruled out or excluded if
and only if it is implied that they fail to satisfy the functional property £¨.
That is, they fail to be extremal paths.
Here is a simple example of this kind of case. For a free particle moving
in Euclidean three space, the potential, U, is 0, so that the Lagrangean for
the free particle is L = T = mr2/2. If one uses generalized coordinates,
L = m/2(q12 + q22 + q32). The Lagrange Euler equations then yield that the
generalized momentum p = "L/"qi is constant [dp/dt = d("L/"qi) = 0], since
the Lagrangean is not a function of the generalized coordinates qi. This simple
result can be expressed by saying that straight lines are the extremals of the
action of free particles.6 That is, the law of inertia in this formulation rules
out any possible path that is not a straight line.
A law need not be formulated using Hamilton s Principle of Least Action in
order to show how it excludes possibilities. Here is a sketch of how a traditional
presentation of Newtonian Gravitation Theory can also do the job.7 Assume
among other things that the force acting upon a body is a central force with
the potential V = -±/r with ± positive, so that the force is attractive, central
and proportional to the inverse square of the distance. Using polar coordinates,
the equation of the orbit is given by
r(¸) = »(1 + µ)/[1 + µcos (¸  ¸0)]
where the constant » is defined to be æøLæø2/[m±(1 + µ)], where L is the total
angular momentum of the planetary body, and ± in this, the gravitational
case, is Gm1m2. This equation is the focal equation of a conic section, with
eccentricity µ. (Recall that any conic section can be described with the aid of
a line, the directrix, a fixed point F (a focus) and the ratio of the distance r
between the body and the focus F, to the distance between F and the directrix.)
The following table is a standard result for the types of conic that satisfy the
equation of the orbit:
µ = 0 » > 0 circle
µ = 1 » > 0 parabola
µ > 1 » > 0 or »
Consequently, assuming that the gravitational force is attractive, central
and inverse square, it will follow that there are four possibilities left open. The
idea conveyed by (LRP), that scientific laws exclude certain possibilities, is
180 Arnold Koslow
illustrated in the gravitational case this way: the possibilities or curves for
planetary orbits consist at the very least of differentiable trajectories, and the
Newtonian Law of Gravitation narrows down those possibilities to just the
conics. The reduction or narrowing down can be cast in the same terms that
were used for all the Hamilton Least Action cases, only in this case a simpler
functional property is available: let r(¸) represent the path of a body in polar
coordinates, and the functional property £¦( ) holds of any curve r(¸) if and
only if there are some », µ, ¸ and ¸0,such that r(¸) = »(1 + µ)/[1 + µcos (¸  ¸0)].
Clearly, the requirement that the gravitational force is central, attractive and
inverse square will rule out all non-conic curves X(¸), because these require-
ments on the force function imply that ¬£¦(X(¸)).
The gravitational case shows in a clear way that, although laws reduce the
possibilities, they need not reduce them to all but one. In the gravitational
case, several possibilities (the various conics) may be left, and in the case
of some laws, the so-called impossibility laws, all of the possibilities may be
ruled out. An explanation of the elliptical orbit of Mars narrows down the pos-
sible orbits from the conics to the ellipse. This further reduction beyond that
furnished by the law of gravitation is obtained through further information
in the explanation  say the values of » and µ, as in our discussion, or in the
specification of definite values of other parameters that would pick out the
ellipses from the other types of conics.8
6 Explanations and the reduction of possibilities
Do explanations rule out possibilities? From the facticity condition on explana-
tion,9 it follows that those explanations that are either explanations of laws or
involve laws as part of the explanation of something else will certainly exclude
some possibilities. The reason is that by facticity, in either of the two cases, [ Pobierz całość w formacie PDF ]