Reduction and Elimination in Philosophy and the Sciences

Relating Theories. Models and Structural Properties in
Intertheoretic Reduction
Rafaela Hillerbrand, Oxford, England, UK
1. Introduction
The Russian doll model of scientific progress is very appealing:
When a new and more profound theory is able to
reproduce and refine the results of one or several wellestablished
theories or even exceeds the scope of the old
theories, this is seen as a clear instance of scientific progress.
The older theories tii , i=1,2, …, nest in the new and
i
more profound theory T just like a Russian doll nests inside
the next bigger one; the old theory or theories are
said to be reduced to T. For simplicity, this paper considers
the case n=1; tii =t. Not only within the philosophical litera-
i
ture, but also among many scientists and non-scientists
alike such a reduction from t to T is perceived as a central
part of progress in science.
In many instances, the reducing theory T is a more
fine-grained, `microscopic' description of the system under
consideration: For instance, in the wake of Lucas critique
(Lucas 1976), microeconomics aims at founding large
parts of macroeconomics; molecular biology strives to
explain classical genetics; … While the `microscopic'
theories are seen as fundamental, the coarse-grained
ones – macroeconomics just as classical genetics – are
often disdainfully referred to as `mere phenomenological'.
The alleged success of fine-grained theories in reduction
explains at least partly the great hopes and fears
associated with advances on micro-sciences like molecular
biology or nanoscience (cp. Schmidt 2004).
However, reducing one theory to another is not a
piece of cake and closer inspection reveals a plethora of
unsettled questions. Likewise, all examples mentioned
above have been subjected to heavy doubts as to whether
they indeed fulfill the criteria of reduction. These criteria
are commonly equated with the ones given by E. Nagel
(1974). I follow this notion and identify reduction roughly
with Nagelian reduction.
Despite various criticisms, the paradigm of
successful reduction of an alleged phenomenological to a
microscopic theory remains the merging of
thermodynamics in statistical mechanics. My arguments
will be developed along these two theories. By choosing a
highly mathematized science like physics, I hope to
provide arguments that can be carried over to other, less
formal sciences in a straightforward way. In particular, I
want to point to two omissions of the classical account on
intertheoretic reduction: Firstly, it is often not theories that
are reduced; rather, models deriving from adequate
theories are related in a way that may be called
`reductionist'. Secondly, the common view on reduction
focuses on different descriptive entities appearing in the
mathematical formulation of the theories t and T. These
entities – the theories' furniture of the world – are
correlated via so-called correspondence (or bridge)
principles. The structural properties of the theories are
commonly overlooked, whereas I will contend that a
successful reduction must at least correlate some of the
structural properties of the theories t and T.
2. A Tale of Two Models: Models as
Mediators in Intertheoretic Reduction
The core idea of Nagel-type reductions is that some theory
T reduces another t only if the laws of t are (logically) derivable
from those of T. In the case of thermodynamics and
statistical mechanics just like in many other instances of
intertheoretic reduction, the descriptive vocabulary of T
and t differ. Terms like entropy or temperature, for example,
are defined in very dissimilar ways in both theories –
one speaks of heterogeneous reduction.
For heterogeneous reductions, the requirement of
connectability involves the provision of correspondence (or
bridge) rules connecting the vocabulary of T to the one of
t. Within the philosophy of physics, the debate on nature
and status of the bridge rules results in a heated debate on
what it actually means to reduce thermodynamics to
statistical mechanics. The original approach of Nagel and
others has been dismissed as too simplistic and Nagel's
requirements for a successful reduction turned out too
stringent a criterion. The only aspiration we can reasonably
hope for is that statistical mechanics gives us an
approximation of the laws of thermodynamics (e.g.
Callender 2001, Frigg 2008, cp. Schaffner 1976): T does
not actually reduce t, but reduces a modified version t'. For
instance, from a statistical theory no strict universal laws
given by thermodynamics can be deduced. Consider a
system characterized by intensive state variables.
Statistical physics tells us that the corresponding extensive
variables can be only specified as mean values. No matter
how sharp this mean value is for a macroscopic system, in
the statistical approach the extensive variable never
becomes a state variable as this is a non-stochastic
variable.
How exactly the approximated theory t' that
connects to T via correspondence laws actually relates to
the original theory t, raises serious questions. In this paper,
I want to contend that it is not t that is reduced to T: not
theories reduce or become reduced – rather a concrete
model of T can be related to a model of t in such a way
that the connection between these models qualifies as a
reduction. Only for concrete models does the notion of
reduction make sense.
Take as an example the bridging of the concepts of
temperature in statistical mechanics and in
thermodynamics. To determine the correspondence
principles, concrete models of the considered physical
system are set up – a model deriving from statistical
mechanics, another from thermodynamics. Let us begin
with the former and focus on an ideal gas. The model
considers gas particles confined to a container. This allows
deriving an explicit formula for the pressure of the gas via
the force the particles exert on the idealized and
rectangular walls of the container. By averaging, we obtain
a formula relating the (microscopic) pressure of the gas to
the volume of the container and the mean kinetic energy of
the gas particles.
Conversely from thermodynamics, deriving a
concrete model that allows to specify the temperature of a
concrete system amounts, amongst other things, to
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