The number of degrees of freedom in a system is the number of numerical parameters that need to be set to fully determine the system. Scientists have an epistemic preference for theories that posit systems with fewer degrees of freedom.
But any system with n real-valued degrees of freedom can be redescribed as a system with only one real-valued degree of freedom, where n is finite or countable. For instance, consider a three-dimensional system which is fully described at any given time by a position (x, y, z) in three-dimensional space. We can redescribe x, y and z by real-valued variable X, Y and Z in the interval from 0 and 1, for instance by letting X = 1/2 + π−1arctan x and so on. Now write out these new variables in decimal:
- X = 0.X1X2X3...
- Y = 0.Y1Y2Y3...
- Z = 0.Z1Z2Z3...
Finally, let:
- W = 0.X1Y1Z1X2Y2Z2X3Y3Z3....
Then W encodes all the information about X, Y and Z, which in turn encode all the information about (x, y, z) and hence about our system at a given time. (This obviously generalizes to any finite number of degrees of freedom. For a countably infinite one, things are slightly more complicated, but can still be done.)
There is a lesson here, even if not a particularly deep one. The epistemic preference for theories that have fewer degrees of freedom cannot be separated from the the epistemic preference for simpler theories. For of course rewriting a theory that made use of (x, y, z) in terms of W is in practice going to make for a significantly messier theory. So we cannot replace a simplicity preference by a preference for a low number of degrees of freedom.
Objection: Instead of a simplicity preference, we may a priori specify that laws of nature be given by differential equations in terms of the variables involved. But when, say, x, y and z vary smoothly over time, it is very unlikely that W will do so as well.
Response: But one can find a replacement for W that is smoothly related to x, y and z up to any desired degree of precision, and hence we can give a differential-equation based theory that fits the experimental data pretty much equally well but has only one degree of freedom.
