Reduction and Elimination in Philosophy and the Sciences
Reduction and Elimination in Philosophy and the Sciences
Reduction and Elimination in Philosophy and the Sciences
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
142<br />
Relat<strong>in</strong>g Theories. Models <strong>and</strong> Structural Properties <strong>in</strong> Inter<strong>the</strong>oretic <strong>Reduction</strong> — Rafaela Hillerbr<strong>and</strong><br />
choos<strong>in</strong>g a certa<strong>in</strong> temperature scale. Then <strong>the</strong> formal<br />
analogy between <strong>the</strong> <strong>the</strong>rmodynamic ideal gas law – a<br />
comb<strong>in</strong>ation of Boyle's (Mariotte's) law <strong>and</strong> Charle's (Gay-<br />
Lussac's) law – <strong>and</strong> <strong>the</strong> statistical law relat<strong>in</strong>g pressure,<br />
volume, <strong>and</strong> mean k<strong>in</strong>etic energy allows correlat<strong>in</strong>g<br />
<strong>the</strong>rmodynamic with statistical pressure as well as<br />
<strong>the</strong>rmodynamic temperature with <strong>the</strong> mean k<strong>in</strong>etic energy<br />
<strong>and</strong> <strong>the</strong> number of degrees of freedom of <strong>the</strong> <strong>in</strong>dividual<br />
gas particles. Only by settl<strong>in</strong>g for a concrete temperature<br />
scale, are we able to identify Rydberg constant <strong>and</strong> thus<br />
Boltzmann constant; only by choos<strong>in</strong>g a concrete<br />
realization of statistical mechanics were we able to relate<br />
(statistical) pressure to volume <strong>and</strong> mean k<strong>in</strong>etic energy.<br />
Note that <strong>the</strong> need to <strong>in</strong>voke <strong>the</strong> latter model was also<br />
noted by S.W. Yi (2003).<br />
The (not purely) formal analogy between <strong>the</strong> model<br />
equations allows identification of <strong>the</strong> correspond<strong>in</strong>g<br />
quantities, yield<strong>in</strong>g <strong>the</strong> well-known bridge concept that<br />
relates <strong>the</strong>rmodynamic temperature to <strong>the</strong> mean k<strong>in</strong>etic<br />
energy of microscopic particles per degree of freedom.<br />
Note that we follow here a dist<strong>in</strong>ction <strong>in</strong>troduced by L.<br />
Sklar (1993): Bridge rules may merely correlate <strong>the</strong><br />
<strong>in</strong>volved quantities, or <strong>the</strong>y can identify a concept <strong>in</strong> T with<br />
a correspond<strong>in</strong>g one <strong>in</strong> t. Follow<strong>in</strong>g Yi (2003), any<br />
identification of terms between various <strong>the</strong>ories is to be<br />
rejected as a metaphysically heavily loaded concept with<br />
many nontrivial <strong>and</strong> far from obvious assumptions on how<br />
<strong>the</strong>oretical terms can make sense outside <strong>the</strong> <strong>the</strong>ory <strong>the</strong>y<br />
are embedded <strong>in</strong>. None<strong>the</strong>less, Sklar is right when he<br />
po<strong>in</strong>ts out that without fur<strong>the</strong>r specification <strong>the</strong> term<br />
correlation is so vague that it begs <strong>the</strong> question as to how<br />
<strong>the</strong> terms are actually related. The preced<strong>in</strong>g analysis<br />
showed a way out of this dilemma: It is not terms with<strong>in</strong><br />
<strong>the</strong>ories that are mapped <strong>in</strong> one way or ano<strong>the</strong>r, but <strong>in</strong> <strong>the</strong><br />
narrow sett<strong>in</strong>g of concrete models, various descriptive<br />
terms can be identified on (not merely) formal analogies<br />
without <strong>the</strong> metaphysical ballast bo<strong>the</strong>r<strong>in</strong>g us if <strong>the</strong><br />
identification were on <strong>the</strong> level of <strong>the</strong> <strong>in</strong>volved <strong>the</strong>ories.<br />
3. Reduc<strong>in</strong>g structural properties<br />
Even assum<strong>in</strong>g that <strong>the</strong> commonly suggested correspondence<br />
rules successfully reduce models of t to models of<br />
T, this is not yet <strong>the</strong> end of <strong>the</strong> story of reduction to be told<br />
here. Not only <strong>the</strong> observational vocabulary stated <strong>in</strong> <strong>the</strong>oretical<br />
terms like temperature or pressure that take on<br />
specific numerical values needs to be correlated; also<br />
(parts of) <strong>the</strong> structural properties of T have to be mapped<br />
to those of t.<br />
Structural properties refer to those properties of<br />
<strong>the</strong>ories that do not turn on arbitrary choice of units, like<br />
<strong>the</strong> choice of a certa<strong>in</strong> temperature scale, but concern<br />
<strong>in</strong>tr<strong>in</strong>sic features of a system. Consider <strong>the</strong> <strong>the</strong>rmodynamic<br />
concept of quasistatic changes, mean<strong>in</strong>g that <strong>the</strong> system<br />
goes through a sequence of states that are <strong>in</strong>f<strong>in</strong>itesimally<br />
close to (<strong>the</strong>rmodynamic) equilibrium. It is not<br />
straightforward what <strong>the</strong> equivalent of a quasistatic<br />
transformation <strong>in</strong> statistical mechanics might be (cp. Frigg<br />
2008). However, implicit correspondence rules map this<br />
structural property of <strong>the</strong>rmodynamics to that of statistical<br />
mechanics. The claim for quasistatic transformation with<strong>in</strong><br />
<strong>the</strong> <strong>the</strong>rmodynamic framework translates to <strong>the</strong><br />
requirement that on <strong>the</strong> microscopic level <strong>the</strong> relaxation<br />
time tpa of <strong>the</strong> particles is much smaller than <strong>the</strong> typical<br />
time scale tav at which changes occur <strong>in</strong> <strong>the</strong> coarsegra<strong>in</strong>ed,<br />
averaged quantities. Hence <strong>the</strong> microscopic<br />
condition correspond<strong>in</strong>g to <strong>the</strong> <strong>the</strong>rmodynamic<br />
requirement of quasistatic changes is: tpa >> tav, imply<strong>in</strong>g<br />
tpa/tav → 0.<br />
Follow<strong>in</strong>g R. Batterman (2002), this limit<strong>in</strong>g<br />
procedure is a special k<strong>in</strong>d of explanation common with<strong>in</strong><br />
physics, a so-called asymptotic explanation. As C. P<strong>in</strong>cock<br />
(2007) noted <strong>the</strong>se belong to <strong>the</strong> broader class of abstract<br />
explanation appeal<strong>in</strong>g primarily to <strong>the</strong> ``formal relational<br />
features of a physical system'' <strong>and</strong> thus account to what I<br />
have referred to as structural properties.<br />
It is <strong>in</strong>dispensable that reduction accounts also for<br />
(some of) <strong>the</strong> abstract explanations of <strong>the</strong> different<br />
<strong>the</strong>ories <strong>in</strong>volved. One obvious objection aga<strong>in</strong>st this claim<br />
contends that <strong>the</strong> content of a <strong>the</strong>ory is identified with its<br />
empirical content, embedded <strong>in</strong> <strong>the</strong> observables that take<br />
on numerical values. However even <strong>the</strong>n some of <strong>the</strong><br />
structural properties need to be bridged. Any model or<br />
<strong>the</strong>ory makes predictions only with<strong>in</strong> some range of<br />
applicability. Beyond <strong>the</strong>ory-external conditions, <strong>the</strong>re are<br />
always specifications of <strong>the</strong> range of applicability <strong>in</strong>ternal<br />
to <strong>the</strong> <strong>the</strong>ory or <strong>the</strong> model under consideration. In<br />
specify<strong>in</strong>g this range of applicability, we fall back on <strong>the</strong><br />
structural properties of <strong>the</strong> <strong>the</strong>ory. Thermodynamics, for<br />
example, makes predictions about <strong>the</strong> state of a system if<br />
<strong>the</strong> undergone changes are quasistatic. A successful<br />
reduction requires that at least those structural properties<br />
of <strong>the</strong> reduced <strong>the</strong>ory required to specify <strong>the</strong> range of<br />
applicability are connected to <strong>the</strong> reduc<strong>in</strong>g <strong>the</strong>ory, <strong>and</strong> vice<br />
versa.<br />
Conclud<strong>in</strong>g this section, it is worth not<strong>in</strong>g that <strong>the</strong>re<br />
is a genu<strong>in</strong>e difference <strong>in</strong> how <strong>the</strong> connection of <strong>the</strong><br />
descriptive vocabulary <strong>and</strong> <strong>the</strong> structural properties of two<br />
<strong>the</strong>ories describ<strong>in</strong>g <strong>the</strong> same physical system are treated<br />
with<strong>in</strong> <strong>the</strong> sciences. While for <strong>the</strong> former explicit<br />
correspondence or bridge rules are stated – as for<br />
example <strong>in</strong> relat<strong>in</strong>g <strong>the</strong> microscopic <strong>and</strong> <strong>the</strong><br />
<strong>the</strong>rmodynamic temperature discussed above – <strong>the</strong><br />
relation between <strong>the</strong> formal relational features of two<br />
ma<strong>the</strong>matical descriptions is mostly given implicitly <strong>and</strong><br />
often rema<strong>in</strong>s among <strong>the</strong> `tacit knowledge' of <strong>the</strong><br />
scientists, shared by <strong>the</strong> practice of do<strong>in</strong>g a specific<br />
scientific research.<br />
4. Conclusion<br />
This paper argued that even when well established <strong>the</strong>ories<br />
exist, <strong>the</strong> reduction might not be an <strong>in</strong>ter<strong>the</strong>oretic one.<br />
Ra<strong>the</strong>r a concrete model of a <strong>the</strong>ory T is correlated with a<br />
model of <strong>the</strong>ory t <strong>in</strong> a way that qualifies as reductionist.<br />
Our discussion of <strong>the</strong> alleged reduction of <strong>the</strong>rmodynamic<br />
to statistical mechanics thus explicitly showed how some<br />
of Kitcher's classical criticism of Nagelian reduction with<strong>in</strong><br />
biology translate <strong>in</strong>to <strong>the</strong> more formal sciences. With a<br />
view to <strong>the</strong> debate with<strong>in</strong> philosophy of physics, <strong>the</strong> central<br />
role of models as regards <strong>in</strong>ter<strong>the</strong>oretic reduction can be<br />
taken as a h<strong>in</strong>t to not only ``take <strong>the</strong>rmodynamics less<br />
seriously'' (Callender 2001), but also take statistical mechanics<br />
as a <strong>the</strong>ory less serious.<br />
Turn<strong>in</strong>g to more general debates with<strong>in</strong> philosophy<br />
of science, both po<strong>in</strong>ts made <strong>in</strong> this paper – <strong>the</strong> central<br />
role of models <strong>in</strong> <strong>the</strong> process of reduction <strong>and</strong> <strong>the</strong><br />
necessity to also connect (some of <strong>the</strong>) structural<br />
properties of T <strong>and</strong> t – reveal a more complex picture of<br />
scientific progress as commonly recognized with<strong>in</strong><br />
philosophy of science. Although <strong>the</strong> raised po<strong>in</strong>ts do not<br />
refute <strong>the</strong> hopes <strong>and</strong> fears advanced <strong>in</strong> <strong>the</strong> microscopic,<br />
reduc<strong>in</strong>g <strong>the</strong>ories like molecular biology or<br />
nanotechnology, <strong>the</strong>y do raise serious doubts as regards<br />
<strong>the</strong> common view that `microscopic' <strong>the</strong>ories are generally<br />
more embrac<strong>in</strong>g.