More Really is Different is one of the more philisophical topics that I’ve dealt with. The basic idea is to provide some analytic evidence towards the idea of emergence – that not all macroscopic physical laws can be derived from a theory of everything of fundamental particles.
Understandable, the research has generated a good deal of interest amoung the general public, including an article in New Scientist. However, due to its nature, its results can also be easily misunderstood. In this article, I’m hoping to clear up some of it!
When we look at the physical universe around us,we often observe some sort of ‘macroscopic order’. When we analyze the flow of water, or the dynamics of a glacier, we do not need to compute the exact motion of every atom. The trick here is that when we observe the macroscopic world, we generally neglect the microscopic details. We see and measure macroscopic features, such as pressure, density and stress. The earliest scientists like Galileo and Archimedes observed relations between such quantities, without a care for the existence of atoms or quantum mechanics.
When scientists write down equations relating such quantities, one should note that the quantities themselves implicitly assume a continuity limit. Pressure gradient, stress etc., are only formally defined in the limit where the medium is continuous, i.e., where it contains an infinite number of infinitesimal particles. We call these equations macroscopic laws. A reductionist view is that all macroscopic laws are logically implied by microscopic laws. Since a metal bar is made out of atoms, then the microscopic laws governing the dynamics of these atoms would allow you to systematically derive the laws that govern how the bar behaves under stress. Here ‘systematically derive’ implies formal, mathematical implication. That is, feed the microscopic laws that governed each atom and their interactions into a computer, along with formal definitions of the macroscopic observables, and it would eventually be able to output any law that governs those macroscopic observables. The derivation of macroscopic laws requires no additional ‘insight’ or ‘creativity’
This reasoning would seem at first very reasonable. For example, if at the level of macroscopic objects, a macroscopic law required some additional assumption, or axiom if you like to call it, then it would seem to suggests that this additional law would add additional constraints to the microscopic world. This `backwards causation’ seems deeply unsettling, given that the microscopic laws have already fully specified on microscopic objects behave. There is really no place for any additional assumptions.
In the paper `More is Different’, what our goal was to show that macroscopic laws can require additional axioms, without backwards causation. The idea really stems from Godel’s incompleteness theorem. We know that any mathematical system of sufficient complexity can be fully defined by a finite set of axioms, and yet the truth of many propositions on the system cannot be formally proven. We essentially show that this phenomena also occurs in mathematical models of physical systems, as soon as you make assumption that the system has a infinite number of infinitismal particles. Since virtually every macroscopic law makes this assumption y the virtue of its macroscopic variables (such as resistance, pressure gradient, etc.), it presents the possibility that such laws cannot be derived from fundamental principles.
The presentation I discussed during the Quantum Biologic Workshop in Singapore during January 2008. The talk focuses particularly on the implications of of this research, in particular to question of whether mathematical relations that govern biological observables can be derived, in principle, from fundamental physics.
New Scientis Article:
- Why nature cannot be reduced to mathematical laws
The article published on New Scientist in regards to this research. While it is not very accurate, it gives a quick idea of what the research means.
More Really is Different
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.