Climbing Levels of Complexity II

January 11, 2010

If emergence is not a real, existing phenomenon, but rather a description from a different perspective or a coordinate transformation or a state space transform or … something similar as indicated in my blog post Climbing Levels of Complexity , then it might be invertible.

What does that mean? As an example, starting with quantum theory, statistical mechanics and under technical assumptions like e.g. space being completely filled with matter we can get Hamiltonians describing phases like e.g. crystals. That might be the best understood example of emergence so far. On the other hand, we (considered as human beings) live in a world of solids, liquids and gases. Nevertheless we were able to derive the underlying quantum mechanical laws. From our description in terms of emergent properties we can go back to the ‘fundamental’ equations.

Let’s brainstorm!

Evolution as a notion is not easy to grasp (for mathematicians). Could it be, just as a thought, that this is what evolution means: If we describe a system in terms of emergent laws such that one can get back (some) fundamental laws, then the system is called evolving.

That is bold, I know, but evolutionary adaption might exactly be that: Understanding the environment (which unfortunately evolves itself and thus creates troublesome circularities) in terms of the population and … as a gift … if halves the work we have according to my last post. If ‘life’ can be defined as emergent and evolving and if we have the transform governing emergent systems, then we are done. In the cases the transform is invertible we have ‘created’ life in the other cases not.

A truly random thought as long we we do not have the transform …

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A further glance at ‘Polymath and the Origin of Life’

January 8, 2010

Polymath and the origin of life has finished its second month. Remember, Tim Gowers plans to set up a polymath project to explain abiogenesis. The project should use cellular automata or similar devices to explain the emergence of life. Right at the beginning of his proposal he has posed a couple of questions on what properties these machines or models should have and what exactly should constitute the scope of the project. I quote:

Question 1: Should one design some kind of rudimentary virtual chemistry that would make complicated “molecules” possible in principle?

The alternative is to have some very simple physical rule and hope that the chemistry emerges from it (which would be more like the Game of Life approach).

If the emergence of life does not depend on the details of the underlying chemistry we could choose a ‘simple’ model and proceed. However, that seems to be circular. We do not know enough examples of ‘life’ to know what exactly constitutes a viable approximation to chemistry. We might get lost in arbitrariness.

The other approach uses the one known example of ‘life’and its ‘fundamental’ laws. Approximations to it might still result in the emergence of some sort of chemistry and then ‘life’.

If I had to choose, I would take the second approach. Even if we do not succeed in generating life, finding suitable approximations to Schrödinger’s equation which result in toy chemistries seems to be already a respectable finding.

Question 2: How large and how complicated should we expect “organisms” to be?

If everything turns out to be working we might be able to describe “organisms” in a different frame and size thus may not play an important role.

Added later: I haven’t quite made clear that one aim of such a project would be to come up with theoretical arguments. That is, it would be very nice if one could do more than have a discussion, based on intelligent guesswork, about how to design a simulation, followed (if we were lucky and found collaborators who were good at programming) by attempts to implement the designs, followed by refinements of the designs, etc. Even that could be pretty good, but some kind of theoretical (but probably not rigorous) argument that gave one good reason to expect certain models to work well would be better still. Getting the right balance between theory and experiment could be challenging. The reason I am in favour of theory is that I feel that that is where mathematicians have more chance of making a genuinely new contribution to knowledge.

When I was a teen some twenty-five or thirty years ago I was very impressed by the genetic model in Gödel, Esch, Bach of Douglas Hofstadter. I took my Apple IIe computer and coded a version. (The specification left some elbow room for interpretations to say the least). The microprocessor was a Motorola 6502, 8-bit running at 1 MHz. A month later it became clear: I cannot generate anything even remotely similar to ‘life’. I guess, nobody could that. Today I am writing this blog entry on a dual core laptop Intel P8400 running at 2,26 GHz and I am not trying to code that genetic model again. Why?

It is not only the computing power what distinguishes me from my earlier version. I also do no longer believe that ’emergence’ should be treated as a phenomenon which can be reached in a finite number of steps. I rather think that some sort of ‘limit’ should be involved, like in the definitions of the first infinite ordinal number, velocity or temperature. If that is the case, then the use of computers is limited until the ‘correct’ approximations are known and the question on the ‘size’ of organisms also is answered: they might be huge.

I have distilled a couple of items, which I think have to be addressed in one way or the other to make the project a ‘success’ (whatever that means).

  • Find a suitable definition or concept of life. This definition has to be fairly robust and still open for interpretation. Something like: life is emergent and evolves.  A crystal emerges, but does not evolve and car designs evolve, but do not emerge. If we find something that emerges and evolves we are done.
  • Currently we do not know what emergence and evolution are. Therefore we collect examples of emergent behavior in all branches of science (this is true polymath and emergence seems to be in all concepts proposed so far). Describe these examples in a way accessible to all participants.
  • Use taxonomy or whatever other scientific method to extract the ‘abstract’ information from these examples.
  • Single out or even develop a mathematical theory related to emergence like calculus to mechanics.
  • Explain evolution and its circularity within this framework.
  • Make it happen! This is the more practical part of modelling an emergent and evolving phenomenon.

These items do not have a natural order. Currently most work was done on developing foundations for the practical part (the last item). Gowers gave a list of 7-8 desirable properties and discussed momentum- and energy conservation.

Let me just note that energy conservation seems problematic. While fundamental physical laws exhibit time translation symmetry, it is not obvious whether and how the same holds for e.g. evolutionary adaption. What does that mean? The following could happen: If we switch from the description of the system on a fundamental level (with energy conservation) to the description of the system on the ‘life’ level by say some ‘limit’ procedure we might get emergent laws depending on time. Such an effect might be necessary or even desirable to explain concepts like adaption, learning and free will. Energy conservation (aka time shift symmetry) might play the same negligible role for ‘life’ as quantum tunneling for cannon balls.

In the project, the emphasis so far seemed to be on understanding how one has to code the problem. However, also definitions were given, toy chemistries were proposed, examples of emergent behavior were given and so on. My items do not seem to be too far off and if that is true there seems to be much work to be done in 2010.


Program

January 5, 2010

Currently I cannot keep my biweekly rhythm. Year end means too much work for me. However, I can see the light at the end of the tunnel and plan to continue soon.


Climbing Levels of Complexity

December 16, 2009

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.


Origin of Life – Revisited

December 11, 2009

Roughly one month ago I have reported on Tim Gowers’s polymath project to model the origin of life. Now it is time to go back there and see what has happened.

It is in the nature of polymath, which is simply put ‘more of the same’, that progress on such a fundamental question is slow. Organizers and participants should simmer that at low heat. After all, it is not just a math Olympiad problem with a solution known to exist.

What have we got so far (according to my frail understanding)?

A first observation is, that the players seem to be interested in implementation and representation rather than the underlying symmetries or rules of the problem. They have in mind a cellular automaton like a sand-pile model or Conway’s game of life. This is interesting in so far as the best mathematicians tell us first to learn about the problem, state the laws governing it and then, as a last step, search for representations as known objects. Is it obvious that the problem has a solution among Turing machines? Not to me, at least. Why should we then restrict ourselves to Turing machines? One reason might be, that they are well understood. A second might be that if you believe in the Church-Turing thesis, Turing machines are all you need.

This  summarizes the first observation: The players apparently presuppose the Church-Turing thesis (albeit without mentioning it).

But, what else could they do instead of looking for Turing machines?

They could go for the laws and they did. It was rapidly recognized that Gowers’s item 5 or even the one before play a central role.

4. It should have a tendency to produce identifiable macroscopic structures.

What does that mean? In his comment Tim Gowers explains:

My impression is that there are several levels of complexity and that each level provides a sort of primeval soup for the next level up, if that makes sense.

Anyhow, a sort of answer is that from any model I would look for the potential to produce objects that could serve as the basic building blocks for a higher-level model.

This last sentence seems to be a recurring motive among players. There seems to be a consensus that such levels should be part of a model. If you now start with some bottom level, then how do you proceed from one level to another? That is indeed a problem.

Some believe, for example, that if you put electrons and other elementary particles together in the right combination you will end up with a human. Like a watchmaker making a watch. The watch obeys the same laws as do their parts. In this spirit a human obeys the same laws as say an electron. That line of thinking nearly inevitably leads to the ‘free will vs. determinism‘ controversy.

Others believe that there is no evidence for the above to hold. There is essentially only one known process to ‘produce’ humans. The natural one. This process is radically different from ‘putting pieces together’ and it is not obvious that the laws governing the pieces hold for the whole. There is, let’s call it, a ‘barrier’ separating the level of humans from the level of electrons. This is hard to imagine for a reductionist.

If we take the existence of ‘levels of complexity’ as a given, then the decisive question currently seems to be:

  • What are the rules to go from one ‘level of complexity’ to the next?

I am very excited on what answers they come up with …


An Uncertainty Principle for Markets

December 9, 2009

Today our goal is to derive an exact formulation of an uncertainty principle in markets. To that purpose we have established in earlier posts a commutation relation between demand {d_i} and price {p_i} of a good {i} in a market. I state it again:

Prices {p_i} and demands {d_j} interact according to

\displaystyle  		 		[p_i,d_j]=i \mu_i p_i \delta_{i,j} \ \ \ \ \ (1)

for a fixed real {\mu_i\in\mathbb{R}}.

What I didn’t tell you so far is how measurement of market observables is supposed to work. Let me just close this gap. Measurement of an observable, e.g. the price of good {i}, in a market in state {\xi} (e.g. in this case selling a small quantity of good {i}) will result in a jump of the market into a new state {\zeta} being an eigenvector of the observable. The outcome of the measurement will be a real number {\zeta_i} (e.g. the price), the eigenvalue of the observable corresponding to {\zeta} with probability

\displaystyle \textnormal{prob }(\zeta_i)=\frac{\left\langle \xi|\zeta \right\rangle \left\langle \zeta|\xi\right\rangle}{\|\xi\|^2}.

For an observable {a} on {X} one can show that its mean value at state {\xi\in X} is given as

\displaystyle  	\overline{a_\xi}=\frac{\left\langle a \xi|\xi\right\rangle}{\|\xi\|^2}.

The dispersion of an observable {a} on {X} is given as

\displaystyle \overline{\left(\triangle a \right)^2_\xi}= \frac{\left\langle\left(a-\overline{a_\xi} \text{id}_X \right)^2 \xi|\xi\right\rangle}{\|\xi\|^2 }.

Now we are in the shape to state the uncertainty principle in markets. In essence it claims that prices and demands of a good cannot be measured with arbitrary precision. Moreover, an explicit lower bound on the maximal simultaneous precision is given. Its proof is essentially a straight forward application of Cauchy-Schwarz inequality.

Proposition. For a market in state {\xi} the dispersions of {p_i} and {d_i} satisfy

\displaystyle \overline{\left(\triangle p_i \right)^2_\xi} \, \overline{\left(\triangle d_i \right)^2_\xi} \geq \frac{\mu_i^2}{4} \| \sqrt{p_i} \xi\|^4.

In the asymmetric case {\mu_i\neq0}, the right-hand side is strictly larger than zero.

Proof. Since dispersion and mean do not depend on the norm of a state we can, without loss of generality, assume that {\|\xi\|=1} and obtain

\displaystyle \overline{\left(\triangle p_i \right)^2_\xi} \, \overline{\left(\triangle d_i \right)^2_\xi} = \left\langle\left(p_i - \overline{p_i} \text{id}_X \right)^2\xi|\xi\right\rangle \left\langle\left(d_i - \overline{d_i} \text{id}_X \right)^2\xi|\xi\right\rangle.

Now Cauchy-Schwarz inequality implies

\displaystyle \begin{array}{rcl} \overline{\left(\triangle p_i \right)^2_\xi}\, \overline{\left(\triangle d_i \right)^2_\xi} & \geq & \left\langle\left(p_i - \overline{p_i} \text{id}_X \right) \left(d_i - \overline{d_i} \text{id}_X \right) \xi|\xi\right\rangle \\ & & \qquad \times \left\langle\left(d_i - \overline{d_i} \text{id}_X \right) \left(p_i - \overline{p_i} \text{id}_X \right) \xi|\xi\right\rangle. \end{array}

Since {ab = \frac{1}{2}[a,b]_+ + \frac{1}{2i}i[a,b]} with {[a,b]_+=ab+ba} we obtain

\displaystyle \begin{array}{rcl} \overline{\left(\triangle p_i \right)^2_\xi} \, \overline{\left(\triangle d_i \right)^2_\xi} & \geq & \left\langle\frac{1}{2} [d_i - \overline{d_i} \text{id}_X, p_i - \overline{p_i} \text{id}_X]_+\xi|\xi\right\rangle^2 \\ & & \qquad +\left\langle\frac{1}{2i} [d_i - \overline{d_i} \text{id}_X, p_i - \overline{p_i} \text{id}_X]\xi|\xi\right\rangle^2 \end{array}

and since the first term is positive

\displaystyle \begin{array}{rcl} \overline{\left(\triangle p_i \right)^2_\xi} \, \overline{\left(\triangle d_i \right)^2_\xi} & \geq & \left\langle\frac{1}{2i} [d_i - \overline{d_i} \text{id}_X, p_i - \overline{p_i} \text{id}_X]\xi|\xi\right\rangle^2 \\ & \geq & \left\langle\frac{1}{2i} [d_i,p_i]\xi|\xi\right\rangle^2. \end{array}

Now (1) and the fact that positive observables have a square root yields the final inequality

\displaystyle \begin{array}{rcl} \overline{\left(\triangle p_i \right)^2_\xi} \, \overline{\left(\triangle d_i \right)^2_\xi} & \geq & \frac{\mu_i^2}{4} \|\sqrt{p_i} \xi\|^4. \end{array}

Since {\mu_i \|\sqrt{p_i} \xi\|^4} can only be zero if {\mu_i} is zero the proposition is proved.

Admittedly, that was a bit dry, but it does the job and that is sometimes all that is necessary in mathematics. Now the pace increases and we are heading with giant leaps towards the time evolution equations for markets …


A New Look

December 4, 2009

This blog got overhauled.

If the content is not clear, then at least the presentation should be.