Reduction and Elimination in Philosophy and the Sciences

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Relating Theories. Models and Structural Properties in Intertheoretic Reduction — Rafaela Hillerbrand
choosing a certain temperature scale. Then the formal
analogy between the thermodynamic ideal gas law – a
combination of Boyle's (Mariotte's) law and Charle's (Gay-
Lussac's) law – and the statistical law relating pressure,
volume, and mean kinetic energy allows correlating
thermodynamic with statistical pressure as well as
thermodynamic temperature with the mean kinetic energy
and the number of degrees of freedom of the individual
gas particles. Only by settling for a concrete temperature
scale, are we able to identify Rydberg constant and thus
Boltzmann constant; only by choosing a concrete
realization of statistical mechanics were we able to relate
(statistical) pressure to volume and mean kinetic energy.
Note that the need to invoke the latter model was also
noted by S.W. Yi (2003).
The (not purely) formal analogy between the model
equations allows identification of the corresponding
quantities, yielding the well-known bridge concept that
relates thermodynamic temperature to the mean kinetic
energy of microscopic particles per degree of freedom.
Note that we follow here a distinction introduced by L.
Sklar (1993): Bridge rules may merely correlate the
involved quantities, or they can identify a concept in T with
a corresponding one in t. Following Yi (2003), any
identification of terms between various theories is to be
rejected as a metaphysically heavily loaded concept with
many nontrivial and far from obvious assumptions on how
theoretical terms can make sense outside the theory they
are embedded in. Nonetheless, Sklar is right when he
points out that without further specification the term
correlation is so vague that it begs the question as to how
the terms are actually related. The preceding analysis
showed a way out of this dilemma: It is not terms within
theories that are mapped in one way or another, but in the
narrow setting of concrete models, various descriptive
terms can be identified on (not merely) formal analogies
without the metaphysical ballast bothering us if the
identification were on the level of the involved theories.
3. Reducing structural properties
Even assuming that the commonly suggested correspondence
rules successfully reduce models of t to models of
T, this is not yet the end of the story of reduction to be told
here. Not only the observational vocabulary stated in theoretical
terms like temperature or pressure that take on
specific numerical values needs to be correlated; also
(parts of) the structural properties of T have to be mapped
to those of t.
Structural properties refer to those properties of
theories that do not turn on arbitrary choice of units, like
the choice of a certain temperature scale, but concern
intrinsic features of a system. Consider the thermodynamic
concept of quasistatic changes, meaning that the system
goes through a sequence of states that are infinitesimally
close to (thermodynamic) equilibrium. It is not
straightforward what the equivalent of a quasistatic
transformation in statistical mechanics might be (cp. Frigg
2008). However, implicit correspondence rules map this
structural property of thermodynamics to that of statistical
mechanics. The claim for quasistatic transformation within
the thermodynamic framework translates to the
requirement that on the microscopic level the relaxation
time tpa of the particles is much smaller than the typical
time scale tav at which changes occur in the coarsegrained,
averaged quantities. Hence the microscopic
condition corresponding to the thermodynamic
requirement of quasistatic changes is: tpa >> tav, implying
tpa/tav → 0.
Following R. Batterman (2002), this limiting
procedure is a special kind of explanation common within
physics, a so-called asymptotic explanation. As C. Pincock
(2007) noted these belong to the broader class of abstract
explanation appealing primarily to the ``formal relational
features of a physical system'' and thus account to what I
have referred to as structural properties.
It is indispensable that reduction accounts also for
(some of) the abstract explanations of the different
theories involved. One obvious objection against this claim
contends that the content of a theory is identified with its
empirical content, embedded in the observables that take
on numerical values. However even then some of the
structural properties need to be bridged. Any model or
theory makes predictions only within some range of
applicability. Beyond theory-external conditions, there are
always specifications of the range of applicability internal
to the theory or the model under consideration. In
specifying this range of applicability, we fall back on the
structural properties of the theory. Thermodynamics, for
example, makes predictions about the state of a system if
the undergone changes are quasistatic. A successful
reduction requires that at least those structural properties
of the reduced theory required to specify the range of
applicability are connected to the reducing theory, and vice
versa.
Concluding this section, it is worth noting that there
is a genuine difference in how the connection of the
descriptive vocabulary and the structural properties of two
theories describing the same physical system are treated
within the sciences. While for the former explicit
correspondence or bridge rules are stated – as for
example in relating the microscopic and the
thermodynamic temperature discussed above – the
relation between the formal relational features of two
mathematical descriptions is mostly given implicitly and
often remains among the `tacit knowledge' of the
scientists, shared by the practice of doing a specific
scientific research.
4. Conclusion
This paper argued that even when well established theories
exist, the reduction might not be an intertheoretic one.
Rather a concrete model of a theory T is correlated with a
model of theory t in a way that qualifies as reductionist.
Our discussion of the alleged reduction of thermodynamic
to statistical mechanics thus explicitly showed how some
of Kitcher's classical criticism of Nagelian reduction within
biology translate into the more formal sciences. With a
view to the debate within philosophy of physics, the central
role of models as regards intertheoretic reduction can be
taken as a hint to not only ``take thermodynamics less
seriously'' (Callender 2001), but also take statistical mechanics
as a theory less serious.
Turning to more general debates within philosophy
of science, both points made in this paper – the central
role of models in the process of reduction and the
necessity to also connect (some of the) structural
properties of T and t – reveal a more complex picture of
scientific progress as commonly recognized within
philosophy of science. Although the raised points do not
refute the hopes and fears advanced in the microscopic,
reducing theories like molecular biology or
nanotechnology, they do raise serious doubts as regards
the common view that `microscopic' theories are generally
more embracing.