Approximation and Idealization John D. Norton Department of History and Philosophy of Science Center for Philosophy of Science University of Pittsburgh

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Approximationand IdealizationJohn D.
NortonDepartment of History and Philosophy of
ScienceCenter for Philosophy of
ScienceUniversity of Pittsburgh
4th Tuebingen Summer School in History and
Philosophy of Science, July 2015

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This Lecture
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Stipulate that Approximations are inexact
descriptions of a target system. Idealizations
are novel systems whose properties provide
inexact descriptions of a target system.

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CharacterizingApproximation and Idealization

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The Proposal
Target system (boiling stew at roughly 100oC )

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A Well-Behaved Idealization
Target Body in free fall
dv/dt g kv
v(t) (g/k)(1 exp(-kt)) gt - gkt2/2
gk2t3/6 -

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Approximation only
Bacteria grow with generations roughly following
an exponential formula.
System of infinitely many bacteria fails to be
an idealization.

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Using infinite Limitsto formidealizations

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Two ways to take the infinite limit
Idealization
The limit system of infinitely many components
analyzed.
Its properties provide inexact descriptions of
the target system.

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Limit Property and Limit System Agree
Infinite cylinder has area/volume 2.

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There is no Limit System
?
There is no such thing as an infinitely big
sphere.

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Limit Property and Limit System Disagree
Infinite cylinder has area/volume 2.

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When Idealization Succeeds and Fails

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Limits in Statistical Physics

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Thermodynamic limit as an idealization

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Two forms of the thermodynamic limit
Number of components
n ?8
V ? 8
Volume
n/V is constant
such that

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Infinite one-dimensional crystal
Problem for strong form.
Spontaneously excites when disturbance propagates
in from infinity.
then
then
then
then

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Strong Form Must Prove Determinism

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Inessential complications??
We emphasize that we are not considering the
theory of infinite systems for its own sake so
much as for the fact that this is the only
precise way of removing inessential complications
due to boundary effects, etc., Lanford, 1975,
p.17

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Continuum limit as an approximation

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Continuum limit
Useful for spatially inhomogeneous systems.
Number of components
n ?8
V fixed
Volume

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Half-tone printing analogy

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Boltzmann-Grad limit as an approximation

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Boltzmann-Grad Limit
Useful for deriving the Boltzmann equation
(H-theorem).
Number of components
n ?8
V fixed
Volume

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Resolving collisions

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Renormalization Group Methods

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Renormalization Group Methods
Best analysis of critical exponents. Zero-field
specific heat CH t-a Correlation length x
t-n for reduced temperature t(T-Tc)/Tc

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The Flow
space of reduced Hamiltonians
Lines corresponding to systems of infinitely many
components (critical points) are added to close
topologically regions of the diagram occupied by
finite systems.

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Finite Systems Control
Necessity of infinite systems
The existence of a phase transition requires an
infinite system. No phase transitions occur in
systems with a finite number of degrees of
freedom. Kadanoff, 2000

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Reduction?Emergence?

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Phase transitions are
Norton, Butterfield
a success of the reduction of thermodynamics by
statistical mechanics.
BOTH!
..and no one is more right.

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Different Senses of Levels
p
Molecular-statistical Description.Phase space of
canonical positions and momenta.Hamiltonian,
canonical distribution, Partition
function.Canonical entropy, free energy.
q

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Where Reduction Succeeds
Level of many component, molecular-statistical
theory
Level of thermodynamic theory
deduce
Renormalization group flow on space of reduced
Hamiltonians.
Critical exponents in vicinity of critical points.

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Where Emergence Happens
Few component molecular-statistical level
A few components by themselves do not manifest
phase transitions in the mean field of the
rest do not manifest the observed phase
transition behavior quantitatively.

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More is Different
P. W. Anderson, Science, 1972.
"The constructionist hypothesis ability to start
from fundamental laws and reconstruct the
universe breaks down when confronted with the
twin difficulties of scale and complexity. The
behavior of large and complex aggregates of
elementary particles, it turns out, is not to be
understood in terms of a simple extrapolation of
the properties of a few particles. Instead, at
each level of complexity, entirely new properties
appear...

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A conjecture
Physicists tend todivide by scale.
Philosophers tend todivide by theory.
Condensed matter physics deals with systems of
many components. Solids, liquids, condensates,
Theory deductive closure of a few apt
propositions.

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Conclusion

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This Talk
1
Stipulate that Approximations are inexact
descriptions of a target system. Idealizations
are novel systems whose properties provide
inexact descriptions of a target system.

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The End

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Appendices

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Recovering thermodynamicsfrom statistical physics
Very many small components interacting.