Considering for example group theory or complex analysis one quickly realizes that successful mathematical theories have plenty of examples. Therefore, if one takes a step into uncharted domain it is never bad to start such an enterprise with an example.

In my last post on the ‘origin of life’ – polymath project I have observed that people expect models explaining the emergence of life having ‘levels of complexity’. Of course, I was very vague about what ‘levels of complexity’ actually are. Anybody is vague about that.

The goal of this post is to remember and discuss a well-known example of emergence and to shed some light on the problem.

In the beginning of the last century physicists recognized classical mechanics as emergent. For centuries Newton’s laws built a solid fundament for science and then, in (essentially) a sudden, everything became different. Classical mechanics was realized to be a ‘limit’ of quantum mechanics. This is commonly known as the correspondence principle and its formulation is remarkably vague. In what sense has this limit to be taken?

A theorem of Ehrenfest tells us that if we consider means of observables we get back Newton’s laws quite easily. Classical momentum (position) is interpreted as the mean of the momentum (position) observable of quantum mechanics. Since the mean is just some (well-defined) limit everything seems fine. However, that is not the problem. Let me just ask:

What is it, that has classical momentum?

Is there a cannonball (and not just some quantum state)? I would certainly approve this if one is flying towards me and I won’t start to calculate the probability that it is tunneling through me. In a way the cannonball has become independent of the underlying quantum laws and now just obeys the new classical ‘in the mean’-laws.

For me this is the essence in climbing ‘levels of complexity’. We do not just get new laws, which are suitable interpreted, means of fundamental laws. We also get new objects or states governed by the new laws and no longer by the fundamental laws. There is no tunneling of cannonballs however I will get shot to pieces, something quantum theory does not cover adequately. Quantum theory simply is not supposed to tell me how to stop the bleeding.

Forget about my clouded comments. Is there something that we can learn from this example? I think so. The lesson we learn is that there is at least one important example of ‘climbing the levels of complexity’ in which the state space changes with the level. Sure, a reductionist could argue that in principle we can describe the cannonball by a quantum state. For all practical purposes however the state space changes. This observation is not sufficiently appreciated.

For my taste this was not enough mathematics. Next time I return to set theory. There are still two paradoxes left to talk about, namely Cantor’s paradox and Burali-Forti.