Reduction and Elimination in Philosophy and the Sciences

to work this way. Classic examples are the reduction of
Newtonian mechanics to the special theory of relativity in
the case of small velocities (velocity of light c → ∞) and the
reduction of quantum mechanics to classical mechanics in
the case of large quantum numbers (Planck's quantum of
action ħ → 0).
(b) We speak of statistical reductions whenever the higherlevel
theory deals with a large amount of entities of the
lower-level theory, which are treated by means of statistical
methods. The reducing theory is normally concerned
only with the behavior of a single one or at most a small
amount of these entities while the reduced theory deals
with the collective behavior of a huge amount of them. In
much of the literature these types of reduction (a) and (b)
are often lumped together – statistical reduction is considered
a parametric reduction in terms of the number of particles
N. But for several reasons that is itself not a good
reduction of type (b) to type (a).
First, statistical reduction is so abundant, that it
merits to be considered separately. It can be found across
all boundaries in the sciences, whenever large numbers of
similar entities are involved: humans, neurons, atoms,
goods etc. Second, there are important conceptual
aspects in which statistical reduction differs from
parametric reduction. The calculus of probability plays an
essential role. Also, the reduced theory deals with a
peculiar kind of entities and laws: They are statistical in
nature as we will see in the next section. For the rest of
this essay we will concentrate on statistical reductions.
3. Statistical Reduction
Not very surprisingly the decisive property, that distinguishes
statistical reductions from other kinds, is that on
the level of the reduced theory we deal in all aspects with
statistical phenomena: statistical entities, statistical properties
of the entities and statistical laws connecting these
properties. We will illustrate this in the following by means
of the reduction of thermodynamics to statistical mechanics.
Only at the end of this section other examples will be
quickly addressed.
Thermodynamics deals with statistical entities, e.g.
gases, fluids, or solids. These macroscopic entities are
composed of a large amount of the fundamental entities
that mechanics is concerned with – namely a large amount
of point-like masses. The properties of the thermodynamic
entities are also statistical in nature: Quantities like volume
V, pressure p, temperature T, or entropy S require a large
amount of mechanical point masses in order to be
adequately defined. Finally, statistical laws relate these
properties with each other – examples are the ideal gas
law pV ~ T or the second law of thermodynamics describing
the increase of entropy.
Starting from the lower-level theory, which is
mechanics in the considered example, it turns out that
most of the elements of the higher-level theory can only be
defined in terms of continuous and differentiable
probability distributions of the lower-level entities. On the
basis of a frequency view of probability, these continuous
distributions must necessarily refer to an infinite number of
lower-level entities. A finite number of instances, e.g. a
finite number of atoms, can only yield a discrete
distribution function corresponding to a weighted sum of δfunctions.
Thus, in order to establish the concepts of the
higher-level theory the transition to the infinite limit is
essential.
Limiting Frequencies in Scientific Reductions — Wolfgang Pietsch
Microscopically, matter is not continuously
distributed in space – a gas actually consists of many
point-like masses and a lot of empty space. To define
simple properties like volume or pressure on the basis of
the actual discrete distribution function for the atoms turns
out quite difficult: How for example should the boundaries
of the volume along the edges of the gas be determined?
There is just no unambiguous way to decide which parts of
the empty space belong to the gas and which not. The
question is answered on the basis of pragmatic and in
particular symmetry considerations. The actual probability
distribution is smoothed out everywhere and is chosen in
such a way that the edges are as geometrically simple as
possible. In this manner, we are led from the actual
discrete distribution of the atoms to a continuous
distribution which determines the macroscopic concept of
volume.
Similarly, the concept of thermodynamic pressure
involves an infinite limit in order to arrive from the discrete
concept of microscopic collisions to the continuous
macroscopic concept of force per area. Again, symmetry
considerations lead the way from the actual microscopic
events to a continuous probability distribution which
presupposes an infinite number of those microscopic
events.
The laws that connect macroscopic quantities – as
the ideal gas law connects pressure, volume, and
temperature – presuppose the infinite limit as well, simply
because properties like pressure and volume are
otherwise ill-defined. Sometimes, the macroscopic laws
crucially depend on the way the limit is taken. A good
example is the second law of thermodynamics which
describes the increase in entropy during the approach to
equilibrium. Depending on the way the infinite limit is
taken, one either derives from statistical mechanics a
deterministic second law which allows no entropy
decrease at all – as in Ludwig Boltzmann's H-theorem
(Boltzmann 1872). In other cases one may obtain entropy
fluctuations. These results are not contradictory – they just
correspond to different extents of coarse-graining.
When other examples of statistical reductions are
examined one encounters the same need for continuous
distributions and the infinite limit. Statistical reductions can
be found across all boundaries in natural as well as social
sciences, whenever one deals with a great amount of
similar entities: with neurons in neuroscience, goods in
economy, human beings in population science, or errors in
error theory. Considering the abundance of error theoretic
methods in all kinds of scientific enterprises, the last
example once more underlines the ubiquity of statistical
reductions. Whenever Gaussian distributions for
measurement values are assumed, one has implicitly
taken the limit of an infinite number of equally distributed,
miniature errors.
4. Conclusion: The Need for Infinite Frequencies
The indispensable role of limiting frequencies for the macroscopic
concept formation in statistical reductions provides
an interesting test case for the different interpretations
of probability. Lately, limiting frequencies have not
enjoyed a good reputation both among scientists and philosophers
of science, e.g. illustrated in the following quote
by Alan Hájek (2007): “To be sure, science has much interest
in finite frequencies, and indeed working with them
is much of the business of statistics. Whether it has any
interest in highly idealized, hypothetical extensions of ac-
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