Reduction and Elimination in Philosophy and the Sciences
Reduction and Elimination in Philosophy and the Sciences
Reduction and Elimination in Philosophy and the Sciences
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All <strong>in</strong> all, we have seen that reduction via deduction<br />
has no <strong>in</strong>terest<strong>in</strong>g examples because <strong>in</strong>ter<strong>the</strong>oretic<br />
relations typically are no deductions but comparisons<br />
between both ma<strong>the</strong>matical structure <strong>and</strong> terms differ<strong>in</strong>g<br />
completely <strong>in</strong> its usage. So, if we want to def<strong>in</strong>e a relation<br />
of reduction we cannot rely on deduction.<br />
Hav<strong>in</strong>g <strong>in</strong> m<strong>in</strong>d that <strong>in</strong> our <strong>in</strong>vestigation we are<br />
look<strong>in</strong>g for a concept of reduction that is able to support<br />
claims of reductionism, one of <strong>the</strong> most <strong>in</strong>terest<strong>in</strong>g<br />
answers to that problem is that of (Schaffner 1967). For<br />
him a <strong>the</strong>ory is not reduced to ano<strong>the</strong>r if <strong>the</strong>ir laws are<br />
logically deduced, but if it is possible to deduce a new<br />
corrected <strong>the</strong>ory from <strong>the</strong> reduc<strong>in</strong>g one which is formulated<br />
<strong>in</strong> <strong>the</strong> latter’s vocabulary <strong>and</strong> strongly analogous to <strong>the</strong><br />
orig<strong>in</strong>al one to be reduced. In a similar manner, reduction<br />
is def<strong>in</strong>ed <strong>in</strong> (Hooker 1981), <strong>and</strong> recently <strong>in</strong> (Bickle 1998)<br />
as <strong>the</strong> so called new wave reduction, which is a result of<br />
merg<strong>in</strong>g Schaffner’s <strong>and</strong> Hooker’s concept of analogy with<br />
<strong>the</strong> structuralistic approach to physical <strong>the</strong>ories, as e.g.<br />
(Endicott 2001) po<strong>in</strong>ts out.<br />
Let us have a closer look at <strong>the</strong>se concepts. At first,<br />
Schaffner’s def<strong>in</strong>ition rests on a very vague <strong>and</strong> not clearly<br />
def<strong>in</strong>ed “strong analogue”-relation between <strong>the</strong> orig<strong>in</strong>al<br />
<strong>the</strong>ory to be reduced <strong>and</strong> a “corrected <strong>the</strong>ory” deduced<br />
from <strong>the</strong> reduc<strong>in</strong>g one. This relation can surely be made<br />
more precise with<strong>in</strong> <strong>the</strong> structuralistic approach: As stated<br />
<strong>in</strong> (Moul<strong>in</strong>es 1984), a reduction <strong>in</strong> structuralistic terms<br />
yields a “ma<strong>the</strong>matical relationship between two sets of<br />
structures” with<strong>in</strong> a “scheme of reduction” which “does not<br />
require semantic predicate-by-predicate connections nor<br />
deducibility of statements” (l. c., p. 54-55). So this<br />
approach delivers <strong>in</strong>deed a very sophisticated concept of<br />
compar<strong>in</strong>g <strong>the</strong>ories that avoids <strong>the</strong> difficulties of a concept<br />
based on deduction, but can on <strong>the</strong> o<strong>the</strong>r h<strong>and</strong> be noth<strong>in</strong>g<br />
but a compar<strong>in</strong>g relation between both concepts <strong>and</strong> laws.<br />
Moreover, this account yields a comparison between two<br />
<strong>in</strong>dependent <strong>the</strong>ories <strong>in</strong> such a way, that “we could have a<br />
reductive relationship between two <strong>the</strong>ories that are<br />
completely alien to each o<strong>the</strong>r” (ibid.).<br />
Such a compar<strong>in</strong>g relation now substantiates only<br />
reductionistic claims, if it is a comparison between<br />
concrete explanations as <strong>in</strong> our simple case above, but<br />
that seems not to be <strong>the</strong> case if <strong>the</strong> <strong>the</strong>ories <strong>in</strong>volved are<br />
more complex as we will see below consider<strong>in</strong>g general<br />
relativity. A topological comparison <strong>in</strong> <strong>the</strong> sense of<br />
(Scheibe 1999), which is possible also between <strong>the</strong>ories<br />
“that are completely alien to each o<strong>the</strong>r”, doesn’t make any<br />
<strong>the</strong>ory superfluous <strong>and</strong> hence cannot on its own support<br />
reductionistic claims. It is much easier to establish a<br />
ma<strong>the</strong>matical-conceptual compar<strong>in</strong>g relation between two<br />
<strong>the</strong>ories than to show that <strong>the</strong> one can expla<strong>in</strong> <strong>the</strong><br />
phenomena which are typically expla<strong>in</strong>ed by <strong>the</strong> o<strong>the</strong>r.<br />
Now, if it is possible to deduce a “corrected <strong>the</strong>ory” from a<br />
reduc<strong>in</strong>g one, this would show that <strong>the</strong> latter is able to<br />
cope with <strong>the</strong> phenomena described by <strong>the</strong> <strong>the</strong>ory to be<br />
reduced. However, compar<strong>in</strong>g relations can be established<br />
between <strong>the</strong>ories without deduc<strong>in</strong>g a corrected <strong>the</strong>ory or<br />
expla<strong>in</strong><strong>in</strong>g phenomena, <strong>and</strong> <strong>the</strong>re are <strong>in</strong>deed <strong>the</strong>ories (as<br />
general relativity) not permitt<strong>in</strong>g such a deduction or<br />
explanation despite be<strong>in</strong>g comparable to ano<strong>the</strong>r one (for<br />
<strong>in</strong>stance to Newton’s <strong>the</strong>ory of gravitation) – hence <strong>the</strong><br />
Schaffner-Hooker-Bickle account of reduction seems not to<br />
be adequate.<br />
<strong>Reduction</strong> <strong>and</strong> <strong>Reduction</strong>ism <strong>in</strong> Physics — Rico Gutschmidt<br />
Guided from <strong>the</strong>se observations, I’d like to propose<br />
<strong>the</strong> follow<strong>in</strong>g two def<strong>in</strong>itions. First it seems to be<br />
appropriate to call most of <strong>the</strong> <strong>in</strong>ter<strong>the</strong>oretic relations <strong>in</strong><br />
physics a relation of compatibility: One actually can<br />
compare two <strong>in</strong>dependent <strong>the</strong>ories with each o<strong>the</strong>r <strong>and</strong><br />
mostly such comparisons show that <strong>the</strong> <strong>in</strong>volved <strong>the</strong>ories<br />
are via approximate derivation <strong>and</strong> related concepts<br />
compatible to each o<strong>the</strong>r. This term doesn’t evoke any<br />
reductionistic claim <strong>and</strong> isn’t meant to do so. If we want to<br />
f<strong>in</strong>d, secondly, a def<strong>in</strong>ition of a relation of reduction <strong>in</strong> such<br />
a way that a reduced <strong>the</strong>ory is <strong>in</strong> pr<strong>in</strong>ciple superfluous, it<br />
seems that we have to refresh an idea of (Kemeny <strong>and</strong><br />
Oppenheim 1956). Their def<strong>in</strong>ition of reduction is based on<br />
<strong>the</strong> explanation of phenomena (“observable data” <strong>in</strong> <strong>the</strong>ir<br />
terms, cf. p.13): Any phenomenon expla<strong>in</strong>able by means<br />
of <strong>the</strong> <strong>the</strong>ory to be reduced must be expla<strong>in</strong>able by <strong>the</strong><br />
reduc<strong>in</strong>g <strong>the</strong>ory. If, fur<strong>the</strong>rmore, <strong>the</strong> explanations of <strong>the</strong><br />
reduc<strong>in</strong>g <strong>the</strong>ory are <strong>in</strong> a sense better <strong>and</strong> if <strong>the</strong> <strong>the</strong>ories<br />
<strong>in</strong>volved are compatible <strong>and</strong> <strong>the</strong>refore <strong>in</strong> a way related to<br />
each o<strong>the</strong>r, it seems legitimate to say that <strong>the</strong> one is<br />
reduced to <strong>the</strong> o<strong>the</strong>r. A <strong>the</strong>ory reduced <strong>in</strong> this sense is<br />
<strong>in</strong>deed superfluous: “Their” phenomena are expla<strong>in</strong>ed<br />
better by ano<strong>the</strong>r <strong>the</strong>ory, to which it is compatible (<strong>and</strong><br />
<strong>the</strong>re is no need to go a long way round via “corrected<br />
<strong>the</strong>ories”). This is now surely <strong>the</strong> case e.g. for Galileo’s law<br />
of fall<strong>in</strong>g bodies or Kepler’s laws of planetary motion but<br />
not for Newton’s <strong>the</strong>ory of gravitation: We will now see that<br />
we have “only” compatibility here.<br />
The reason is that <strong>in</strong> spite of hav<strong>in</strong>g comparable<br />
laws as shown e.g. <strong>in</strong> (Misner et al. 1973) or (Scheibe<br />
1999), <strong>the</strong>re are many phenomena expla<strong>in</strong>ed by<br />
Newtonian physics but not by general relativity, because<br />
no one solved <strong>the</strong> field equations for <strong>the</strong>m. While <strong>the</strong> twobody<br />
problem is directly solved by Newton’s law, it has<br />
(<strong>and</strong> as a matter of fact can have) only numerical solutions<br />
with<strong>in</strong> general relativity. And while <strong>the</strong> orbits of <strong>the</strong> planets<br />
can be described as geodesics with<strong>in</strong> <strong>the</strong> Schwarzschild<br />
solution, <strong>the</strong>ir <strong>in</strong>teraction as described by Newtonian<br />
physics is not yet expla<strong>in</strong>ed by general relativity for <strong>the</strong>re<br />
are no solutions of <strong>the</strong> field equations for mov<strong>in</strong>g<br />
gravitational sources. Similarly, <strong>the</strong>re are no general<br />
relativistic explanations for complex formations as star<br />
clusters or spiral galaxies ei<strong>the</strong>r, while <strong>the</strong>y can too be<br />
h<strong>and</strong>led with Newtonian physics. It is surely possible to<br />
claim that one will f<strong>in</strong>d relativistic explanations of <strong>the</strong>se<br />
phenomena one day, but because of <strong>the</strong> difficult <strong>and</strong><br />
abstract character of <strong>the</strong> field equations <strong>in</strong> contrast to <strong>the</strong><br />
high applicability of Newton’s <strong>the</strong>ory we can also put <strong>the</strong><br />
possibility of such explanations <strong>in</strong> question. And <strong>in</strong>deed, as<br />
a matter of fact numerical simulations of such phenomena<br />
on <strong>the</strong> basis of <strong>the</strong> field equations depend due to <strong>the</strong>ir<br />
complex structure on <strong>the</strong> heuristic help of Newton’s law of<br />
gravitation (with<strong>in</strong> <strong>the</strong> so-called post-Newtonian<br />
approximation). Therefore, Newton’s law of gravitation can<br />
be improved by general relativity, but is not superfluous – it<br />
is <strong>in</strong> our terms not reduced to general relativity despite<br />
compatibility.<br />
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