## Archive for September 30th, 2010

### Determination and Definability: Newtonian Mechanics & Statistical Dynamics

September 30, 2010

The last determination-without-reduction example I’ll talk about from Hellman is the one he gives regarding classical particle mechanics and statistical mechanics.   A process is reversible, if it could proceed forward in time as well as backwards (think of playing, in reverse, a video of a model of the orbits of the planets around the sun).  A process is irreversible when it can only proceed in one direction of time and would violate physical law proceeding in the opposite time direction (think of playing, in reverse, a video of a gas escaping a bottle).

Newtonian laws governing motion are time-symmetric and reversible: forward motions of Newtonian systems are on par with motions backwards in time.  Statistical mechanics, on the other hand, attempts to explain irreversible behavior of the higher-level observable phenomena of thermodynamics, like temperature, diffusion, pressure and entropy.  The macroscopic properties of thermodynamics are defined in terms of the phase quantities of Newtonian mechanics with the addition of a measure theoretic probability density function as well as some assumptions a-priori about distribution (e.g., equiprobability of equal volumes of phase space).  For macroscopic properties like entropy, more complex probabilistic concepts come into play: dividing phase space into cells and adding to the mechanical motions the periodical average of the cells, then entropy increases and the distribution density tends uniformly to equilibrium.  This is what the Ehrenfests (Paul and Tanya) called coarse-graining and it’s a method for converting a probability density in phase space into a piece-wise constant function by density averaging in cells in phase space. Coarse-grained densities are needed to avoid paradoxical results concerning how irreversible processes of thermodynamics arise from completely reversible mechanical interactions.

The question Hellman poses is: can the higher-level concepts of thermodynamics be explicitly defined in the language of Newtonian mechanics?  Determination, at least, holds: having fixed any two closed particle systems that are identical at the level of Newtonian mechanics, their higher level behavior (studied by thermodynamics) will be identical.  Each of these systems will be represented by the same trajectory in phase space.  Hellman gives the example of one system entering higher entropy regions at a given time, then the same entropy regions will be entered by the other system.

Definiability on the other hand is more difficult to establish, since the language of classical Newtonian mechanics is not as mathematically robust as that of statistical dynamics.  And ultimately, definability in this case requires a significant change to the language of mechanics: additional vocabulary for measure theory or even set theory to speak of mathematical objects more generally.

We’ve covered some examples of determination without reduction: (1) No explicit definition of the truth predicate $\mathsf{Tr}(x)$ in $\textup{L}$ in spite of  $\textup{L}$-truth and $\textup{L}$-reference determining $\mathsf{Tr}$-truth and $\mathsf{Tr}$-reference, respectively, in $\alpha$-structures; (2) in $\alpha^{\textup{C}}$-structures, $\textup{L}$-reference determines $\textup{DF}arith$-reference, but within this same class of structures, $\textup{L} + \textup{G}(x)$ does not inductively define $\textup{DF}arith$; (3) The mechanical properties of two fixed, closed particle systems determine their macro-level thermodynamic properties without thereby establishing definability, as the language of Newtonian mechanics would have to undergo significant changes incorporating the language, at least, of measure theory.

Next, I’m going to cover an important distinction Hellman makes between the ontological and and ideological status of  properties, relations, attributes, etc. beyond predicates and sets.  After that I’ll go into a detailed, critical, evaluation of some of the anti-materialist claims David Chamers makes in The Conscious Mind: In Search of a Fundamental Theory.