Reduction and Elimination in Philosophy and the Sciences

Reduction and Reductionism in Physics
Rico Gutschmidt, Bonn, Germany
The good old standard definition of reduction is penned by
(Nagel 1961) and demands for a theory to be reduced to
another that the laws of the first one can be logically
deduced by the laws of the latter with the help of bridge
laws connecting the different languages of the theories. A
theory reduced in this way should then be in principle
superfluous - if all their laws are, given the bridge laws,
logically contained in the reducing theory, it is in a strict
sense not required anymore in our physical description of
the world.
But things are not that easy. As (Feyerabend 1962)
has shown, this concept is somewhat naïve and there are
no interesting examples of reduction in the Nagelian
sense. Feyerabend’s point is based mainly on two
objections. First, the links between physical theories are
mostly an approximate derivation of laws rather than their
deduction - and there is a great and in the debate of
reduction largely overlooked difference between derivation
and deduction. And second, the conception of the bridge
laws is rather vague: Feyerabend argues that the terms of
different theories satisfy not only no identity relation, which
could be expressed in bridge laws, but are actually
incommensurable and not comparable whatsoever.
Let us take a closer look on these two assertions.
First, there is the mathematical problem of approximate
derivation: Within physics it seems to be a well established
practise to derive laws “only” approximately. But what does
this mean in the context of intertheoretic relations? To take
an example from the context of gravitation, according to
Newton’s law of gravitation Galileo’s law of falling bodies is
strictly speaking false: The acceleration increases instead
of being constant. Hence these theories contradict each
other, and therefore a deduction is simply impossible and
any “derivation” of Galileo’s law from Newton’s law of
gravitation must thus contain some contra-to-fact
assumptions. Such assumptions can in this case be and
are widely in physical derivations hidden in limiting
processes where some parameter, which is not zero or
infinite within the law to be derived, is taken to be zero or
infinite. In our case the distance to earth of the falling body
compared to the earth’s radius is taken to be zero - but the
law derived under this assumption is strictly speaking only
valid for bodies laying down on the earth’s surface, while
Galileo’s law is about falling bodies. Thus the common
mode of speaking that this derivation delivers approximate
validity only for small distances covers the fact that we
haven’t deduced Galileo’s law but rather established a
comparison between the two theories under certain
circumstances: This is all we can say about “approximate
derivations” here and similarly elsewhere.
Nevertheless, Galileo’s law is superfluous, not
because of being deduced, but rather because Newtonian
physics can also describe falling bodies, in a similar
manner as Galileo’s law as shown in the comparative
limiting process. But if “reducing” theories are more
complex it is far from certain that they are able to
reproduce any statement of a theory to be reduced.
Comparisons between theories in the sense of
approximate derivation seem to be just comparisons of
mathematical structure and not of concrete explanations of
phenomena - and the possibility to compare mathematical
structure does not include that the “reducing” theory is able
128
at all to deal with the phenomena explained by the theory
to be reduced as it is the case in our simple example. If we
are able to deduce the laws of a theory we are
automatically able to explain their phenomena but we can’t
expect to be able to do that by virtue of comparisons
between mathematical structures as we will see in the
closing part of this presentation, which discusses the case
of general relativity in this respect.
This point doesn’t catch one’s eye if one considers
simple cases like that of Galileo’s law and has therefore
widely been overlooked within the debate of reduction. But
intertheoretic relations in physics actually are in many
cases nothing but comparisons between mathematical
structures: A look at the details shows, that e.g. in the case
of the Newtonian theory of gravitation and general relativity
the intertheoretic relation is much more complicated than a
limiting process and far from being well established. A
mathematical relation in a precise manner between these
theories is given e.g. in (Scheibe 1999) in terms of a
topological comparison between sets of models of these
theories formulated axiomatically (cf. p. 59-108 for the
case of general relativity). And while there are only single
cases in which explanations of concrete phenomena can
be compared (e.g. the planet’s orbits, cf. p. 89-101),
Scheibe’s “reduction” of the whole Newtonian theory of
gravitation is not completely worked out and can either
way be no deduction but nothing but a very subtle
comparison between mathematical structure.
The second of Feyerabend’s objections concerns
language and the incommensurability of the vocabulary of
different theories. In our context, the equation of motion
within general relativity is the geodesic equation for neutral
test particles whereas Newton’s law of gravitation
describes a force between two masses. We thus are
concerned with two entirely different concepts and the
identification of the Newtonian gravitational potential with
Christoffel Symbols, which can be found in physics
textbooks (cf. e.g. (Misner et al. 1973), chapter 12), is a
“component manipulation” (l.c., p. 290) rather than a basis
for a deduction. Even more concrete, in the example of the
planet’s orbits their description within the Schwarzschild
solution deals with test particles without influence to the
overall curvature and thus without gravitational masses
whereas their Newtonian description is based on forces
between just these masses. Therefore, these concepts
cannot be related by any simple identification and we have
to concede that we cannot establish reductions via logical
deduction with the help of bridge laws.
But nevertheless theories need not to be
incommensurable – we can of course compare the
concepts of different theories. But this is in general a
difficult and not straight forward comparison process far
from being able to establish bridge laws suiting for a
logical deduction. We can relate the terms of two theories
with the help of special case studies and prove that e.g.
the Newtonian potential is somehow related to the
Christoffel symbols, but such case studies are no selfevident
processes and again lead to a comparing relation
rather than a deduction, and such a comparison on its own
doesn’t make any theory superfluous.