More and Less

Bee and Lubos are now blogging on a fairly recent paper by Mile Gu, Christian Weedbrook, Alvaro Perales, Michael A. Nielsen
entitled "More Really is Different." This title is borrowed from Phillip Anderson's paper of the similar name ("More is Different") arguing for the importance of emergent phenomena, such as the collective excitations of the solid state and plasmas. As might be expected, Bee produces a didactic post with little editorial embellishment, while Lubos skitters all over the place, says a lot of interesting things, but mainly just misses the point.

The abstract:

In 1972, P.W.Anderson suggested that `More is Different', meaning that complex physical systems may exhibit behavior that cannot be understood only in terms of the laws governing their microscopic constituents. We strengthen this claim by proving that many macroscopic observable properties of a simple class of physical systems (the infinite periodic Ising lattice) cannot in general be derived from a microscopic description. This provides evidence that emergent behavior occurs in such systems, and indicates that even if a `theory of everything' governing all microscopic interactions were discovered, the understanding of macroscopic order is likely to require additional insights.

I don't want to discuss the paper, except to say that their point is to map the idealized physical system on to a logical system for which it has been proven that no finite and complete system of axioms exists.

From a practical standpoint, Anderson's point cannot be challenged. There is no foreseeable future in which the properties of complex molecules, much less living systems, could be deduced directly from (say)quantum field theory and the standard model. Neither does Anderson or anybody else suggest that the physical laws that apply to complex molecules and living systems don't conform to and ultimately reduce to fundamental physics.

Lumo's argument covers a lot of ground, and makes some sensible points, but he is confused on a central point:

My countrymate Kurt Gödel has shown that every axiomatic system that is at least as powerful as set theory including integers admits a proposition that can be neither proved nor disproved within the system itself. But it seems to me that what many people don't understand is the fact that at the very same moment when the proposition is shown to be unprovable, it is also proved to be correct - within a slightly extended system of logical methods.

Actually, since the statement's negative satisfies the same criterion, whether the statement is true or false is an arbitrary choice independent of the original system. If a crucial intial assumption of the proof is valid (that the original axiom system is consistent), each such Goedelian statement allows one to extend the axiom system by adding such additional assumptions. The point is that no finite number of such additions can constitute a complete description of the system, and if the paper's logic is correct, no reduction to a finite set of laws for the idealized sytem is possible.

Lumo goes on to add that rules are not enough to describe a physical system, one also needs some sort of initial conditions or "history." Perhaps it is possible in this case to consign those undecidable elements of the system to initial conditions - but I don't know.