Reduction and Elimination in Philosophy and the Sciences

4. Property language and theoretical
reduction
Physics offers many examples of theoretical reduction. We
will consider one from classical mechanics. It has several
equivalent formulations of which we will discuss two, the
Newtonian and Hamiltonian mechanics.
The structure of Newtonian mechanics is defined by
a set of axioms covering
Euclidean space and time (abstract)
Action of the Galilei group
Operational definitions of velocity, length and time
measures determining coordinatisations
Calculus
Newton’s second and third laws
The set of axioms supplemented with terminological definitions
constitute an ontology for the property language of
Newtonian mechanics.
A model is defined by the specification of a set of
equations, the equations of motion. The equations of motion
implement Newton’s second law and include quantities
representing the identification properties of the system
modelled and empirical constants, i.e. the masses of the
objects and the gravitational constant. The solutions,
moreover, depend on another set of empirical quantities
defining initial conditions.
Hamiltonian mechanics is a formulation of classical
mechanics that is a more restrictive way of looking at classical
mechanics. It is based on the following elements
Phase space and time as a differential manifold
Action of Galilei group
Operational definitions of momentum, length and
time determining coordinatisations
Hamilton’s principle of least action
The set of axioms supplemented with terminological definitions
constitute an ontology for the property language of
Hamiltonian mechanics.
Formal Mechanisms for Reduction in Science — Terje Aaberge
A model of a system is defined by a function on
phase space, the Hamiltonian, which includes reference to
identification properties of the system modelled. Given the
Hamiltonian, the equations of motion are derived from the
hypothesis that the dynamics satisfies Hamilton’s principle.
The passage from Newtonian mechanics to Hamiltonian
mechanics is a theoretical reduction; the axioms of
Hamiltonian mechanics impose more structure than those
of Newtonian mechanics but at the same time they define
a more restrictive theory. The definition of a model is thus
more compressed in Hamiltonian mechanics than in Newtonian
mechanics. In fact, while the definition of a model of
a simple system needs the specification of three functions,
the force, in Newtonian mechanics, it is defined by only
one function, the energy, in Hamiltonian mechanics. The
domain of application of Hamiltonian mechanics is however,
smaller than that of Newtonian mechanics. In fact,
while Newtonian mechanics can model dissipative systems,
Hamiltonian mechanics can only handle conservative
systems.
It should be noticed that the terms reduction is also
used to denote the limit of physical theories for parameters
going to zero.
Literature
Blanché, Robert 1999: L’axiomatique. Paris: Presses Universitaires
de France
Gion, Emmanuel 1989 Invitation à la theorie de l’informatique,
Paris: Éditions du Seuil
Smith, Barry and Casati, Roberto 1994 Naive Physics: An Essay in
Ontology, Philosophical Psychology, 7/2, pp. 225-244.
Tarski, Alfred 1985 Logic, Semantic, Metamatematics (second
edition), Indianapolis: Hackett Publishing Company
Tarski, Alfred 1944 The Semantic Conception of Truth and the
Foundations of Semantic. Philosophy and Phenomenological Research
4, pp. 341-375
Wittgenstein, Ludwig 1961: Tractatus logico-philosophicus, London:
Routledge and Kegan Paul
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