Reduction and Elimination in Philosophy and the Sciences

Limiting Frequencies in Scientific Reductions
Wolfgang Pietsch, Munich, Germany
1. Introduction
Limiting frequencies have been largely discredited in the
philosophical discussions on the interpretation of probability.
A crucial reason is that frequency interpretations are
usually classified as objective accounts of probability which
stands in obvious contrast to the non-empirical notion of
infinite frequencies. In this essay I will argue that notwithstanding
these conceptual difficulties, limiting frequencies
are an indispensable tool in theory reductions.
Section two will be concerned with a classification of
reduction according to its methodological role: This role
can be ontological, i.e. concerned with unifying
phenomena that were originally thought to be of different
nature. Or reduction can have an epistemological function
in making a theory simpler and thereby better applicable –
e.g. when the general theory of relativity is simplified to
Newton's theory of gravitation, which suffices to treat most
phenomena concerned with the motion of planets.
In Section three an important case of such an
epistemological reduction is examined, which will be
termed statistical reduction.. In statistical reductions the
concepts of the higher-level theory necessarily involve a
large amount of entities of the lower-level theory.
Therefore the probability calculus is essential for this type
of reduction. Furthermore, it turns out that such reductions
usually require the limit of an infinite number of lower-level
entities, i.e. limiting or infinite frequencies.
Finally, Section four will discuss how probability in
statistical reductions should be interpreted. The described
use of limiting frequencies generally calls for further
research into the conceptual difficulties of this notion: Can
the limit be empirically justified by means of certain
averaging procedures, e.g. time or ensemble averages?
What is the relation between such different types of
averages? Are there subjective and epistemic elements
involved and if so, which role do they play? Due to their
indispensable role in theory reduction, infinite frequencies
cannot simply be dismissed, as often happens in
philosophical discussions on probability.
2. Reduction as an Epistemological
Enterprise
There has long been the sense that reduction comes in
two different kinds, although the dividing line has been
drawn in several quite different ways. Ernest Nagel (1974)
distinguished homogeneous from inhomogeneous reductions:
In the former, all concepts of the reduced theory are
already contained in the reducing theory. In the later, the
reduced theory employs additional concepts beyond those
of the reducing theory, which leads to the need to establish
bridge laws between these new concepts and the reducing
theory. Another way, in which a distinction has been made,
is between interlevel and intralevel reductions. Interlevel
reductions concern theories on different levels of fundamentality
– for example chemistry and physics – while
intralevel reductions refer to the same level, for example
the relation between Newton's theory of gravity and the
general theory of relativity. When the reduced theory is not
eliminated in favor of the reducing theory – which is mostly
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the case in interlevel reductions – one speaks of synchronic
reductions. When the reduced theory is given up
after a successful reduction, this is called a diachronic
reduction.
In what follows we will adopt a classification that
somewhat differs from these suggestions. We will
distinguish two types according to the function of the
reduction: (i) reduction as an ontological enterprise and (ii)
reduction as an epistemological enterprise 1 . As with the
other distinctions this is not meant to suggest, that one can
always unambiguously classify a reduction as one of these
cases. Rather, we always deal with a mixture, where each
of the types is more or less clearly present.
(i) In ontological reductions, phenomena that were originally
thought to be quite different in nature are traced back
to the same mechanism. Examples for this type are the
derivation of optics from electrodynamics or the unification
of Galileo's law of falling bodies and Kepler's laws for the
planetary motion within the general framework of Newton's
theory of gravity. Usually, these are intralevel and diachronic
reductions, where the original theories do not survive.
(ii) Reduction as an epistemological enterprise deals with
those cases where considerable simplification is required
in order to apply theories in a specific context. Examples
for this type are the reduction of thermodynamics to manyparticle
mechanics or the reduction of macro- to microeconomics.
These epistemological reductions need not always
be fully worked out. In many cases only certain elements
of the higher-level theory can actually be reduced to concepts
of the lower-level theory, i.e. there are emergent
aspects in the higher-level theory – at least for the time
being. Most epistemological reductions are 'imperfect' in
this way: the reduction of psychology to neuroscience, of
biology to chemistry, or of chemistry to physics. Usually
these reductions are interlevel reductions, but not always:
a counterexample is the reduction of geometrical optics to
wave optics. Since the different theories in an epistemological
reduction each retain their significance in particular
contexts, mostly both theories are kept, i.e. the reduction is
synchronic.
For the rest of the paper, the focus will be on
reduction as an epistemological enterprise. Whenever a
theory is simplified in order to make it better applicable,
this can in principle be interpreted as a reduction of a
simpler framework to a more complicated and general one.
Two important types can be further distinguished, that I will
call (a) parametric and (b) statistical reduction. This
classification is not necessarily exhaustive, one can well
imagine other types of simplification.
(a) In parametric reduction the limit of a parameter of the
reducing theory is taken in order to make the equations
mathematically better manageable. This type is often said
to be typical for reduction in physics (Nickles 1973, Batterman
2002) and indeed many physical reductions seem
1 The distinction between the ontological and epistemological role of reduction
is certainly not novel (compare e.g. Hoyningen-Huene 2007, 181-183). However,
there are subtle differences between our treatment and that in the literature.