Reduction and Elimination in Philosophy and the Sciences

All in all, we have seen that reduction via deduction
has no interesting examples because intertheoretic
relations typically are no deductions but comparisons
between both mathematical structure and terms differing
completely in its usage. So, if we want to define a relation
of reduction we cannot rely on deduction.
Having in mind that in our investigation we are
looking for a concept of reduction that is able to support
claims of reductionism, one of the most interesting
answers to that problem is that of (Schaffner 1967). For
him a theory is not reduced to another if their laws are
logically deduced, but if it is possible to deduce a new
corrected theory from the reducing one which is formulated
in the latter’s vocabulary and strongly analogous to the
original one to be reduced. In a similar manner, reduction
is defined in (Hooker 1981), and recently in (Bickle 1998)
as the so called new wave reduction, which is a result of
merging Schaffner’s and Hooker’s concept of analogy with
the structuralistic approach to physical theories, as e.g.
(Endicott 2001) points out.
Let us have a closer look at these concepts. At first,
Schaffner’s definition rests on a very vague and not clearly
defined “strong analogue”-relation between the original
theory to be reduced and a “corrected theory” deduced
from the reducing one. This relation can surely be made
more precise within the structuralistic approach: As stated
in (Moulines 1984), a reduction in structuralistic terms
yields a “mathematical relationship between two sets of
structures” within a “scheme of reduction” which “does not
require semantic predicate-by-predicate connections nor
deducibility of statements” (l. c., p. 54-55). So this
approach delivers indeed a very sophisticated concept of
comparing theories that avoids the difficulties of a concept
based on deduction, but can on the other hand be nothing
but a comparing relation between both concepts and laws.
Moreover, this account yields a comparison between two
independent theories in such a way, that “we could have a
reductive relationship between two theories that are
completely alien to each other” (ibid.).
Such a comparing relation now substantiates only
reductionistic claims, if it is a comparison between
concrete explanations as in our simple case above, but
that seems not to be the case if the theories involved are
more complex as we will see below considering general
relativity. A topological comparison in the sense of
(Scheibe 1999), which is possible also between theories
“that are completely alien to each other”, doesn’t make any
theory superfluous and hence cannot on its own support
reductionistic claims. It is much easier to establish a
mathematical-conceptual comparing relation between two
theories than to show that the one can explain the
phenomena which are typically explained by the other.
Now, if it is possible to deduce a “corrected theory” from a
reducing one, this would show that the latter is able to
cope with the phenomena described by the theory to be
reduced. However, comparing relations can be established
between theories without deducing a corrected theory or
explaining phenomena, and there are indeed theories (as
general relativity) not permitting such a deduction or
explanation despite being comparable to another one (for
instance to Newton’s theory of gravitation) – hence the
Schaffner-Hooker-Bickle account of reduction seems not to
be adequate.
Reduction and Reductionism in Physics — Rico Gutschmidt
Guided from these observations, I’d like to propose
the following two definitions. First it seems to be
appropriate to call most of the intertheoretic relations in
physics a relation of compatibility: One actually can
compare two independent theories with each other and
mostly such comparisons show that the involved theories
are via approximate derivation and related concepts
compatible to each other. This term doesn’t evoke any
reductionistic claim and isn’t meant to do so. If we want to
find, secondly, a definition of a relation of reduction in such
a way that a reduced theory is in principle superfluous, it
seems that we have to refresh an idea of (Kemeny and
Oppenheim 1956). Their definition of reduction is based on
the explanation of phenomena (“observable data” in their
terms, cf. p.13): Any phenomenon explainable by means
of the theory to be reduced must be explainable by the
reducing theory. If, furthermore, the explanations of the
reducing theory are in a sense better and if the theories
involved are compatible and therefore in a way related to
each other, it seems legitimate to say that the one is
reduced to the other. A theory reduced in this sense is
indeed superfluous: “Their” phenomena are explained
better by another theory, to which it is compatible (and
there is no need to go a long way round via “corrected
theories”). This is now surely the case e.g. for Galileo’s law
of falling bodies or Kepler’s laws of planetary motion but
not for Newton’s theory of gravitation: We will now see that
we have “only” compatibility here.
The reason is that in spite of having comparable
laws as shown e.g. in (Misner et al. 1973) or (Scheibe
1999), there are many phenomena explained by
Newtonian physics but not by general relativity, because
no one solved the field equations for them. While the twobody
problem is directly solved by Newton’s law, it has
(and as a matter of fact can have) only numerical solutions
within general relativity. And while the orbits of the planets
can be described as geodesics within the Schwarzschild
solution, their interaction as described by Newtonian
physics is not yet explained by general relativity for there
are no solutions of the field equations for moving
gravitational sources. Similarly, there are no general
relativistic explanations for complex formations as star
clusters or spiral galaxies either, while they can too be
handled with Newtonian physics. It is surely possible to
claim that one will find relativistic explanations of these
phenomena one day, but because of the difficult and
abstract character of the field equations in contrast to the
high applicability of Newton’s theory we can also put the
possibility of such explanations in question. And indeed, as
a matter of fact numerical simulations of such phenomena
on the basis of the field equations depend due to their
complex structure on the heuristic help of Newton’s law of
gravitation (within the so-called post-Newtonian
approximation). Therefore, Newton’s law of gravitation can
be improved by general relativity, but is not superfluous – it
is in our terms not reduced to general relativity despite
compatibility.
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