Encefalus

Stephen Wolfram

Lately I’ve been reading Stephen Wolfram’s A New Kind of Science. As you can guess from the title, Wolfram makes a few great remarks about the future and the methods of science.

Traditional science uses mathematics. What mathematics do, is to quantify problems, find regularities and then use equations to describe these regularities. What this means, is that mathematics lie in a continuum, upon which we can jump from one point to the next.

However, this approach has met various difficulties. For example, mathematics never had any real chance in social sciences, or in biology. Mathematics have worked wonders for physics, where systems are characterized by stability to a very large extent, but for other systems, where we are faced with a high degree of randomness or complexity, mathematics have failed to us to a great extent.

Encefalus, for example, has written some articles on the problems that mathematics had to deal with in economics and what this has to do with the recent economic recession: A different view on economics: maybe all we really need, Behavioral economics revisited in the face of the recent economic crisis

What Stephen Wolfram proposes as a complement (or replacement) to mathematics, is computation.

Wolfram studied a set of systems called cellular automata, and other similar systems. What these systems have in common is that they are compromised by a set of discrete steps and a few simple rules.

What Wolfram proposes is that every system, no matter how complex, can be represented by a set of simple rules, which through the evolution of the system, can create complex and emergent phenomena, like certain shapes or the phenomenon of continuity.

Of course, what Wolfram proposes, while it may seem very interesting, it could also be false. After all, computational models are just that, models, and they could be completely wrong and oversimplifying. However, somewhere in the middle of the book I found something that intrigued me. In chapter 9, in section 3 (http://www.wolframscience.com/nksonline/section-9.3), Wolfram shows a computational proof concerning the violation of the second law of thermodynamics. What Wolfram says, is that, throughout the evolution of a system, there can be trends towards increased order and towards increased randomness other times, while the second law implies that all systems move towards entropy.

Example of cellular automata

Well, that may seem interesting, but still, cellular automata might not agree with reality. A New Kind of Science was written in 2002. In November 2008, this article came out in Scientific American: Does Nature Break the Second Law of Thermodynamics? In the key points we read

1. Evolution leads to a homogeneous state.

2. Evolution leads to a set of separated simple stable or periodic structures.

3. Evolution leads to a chaotic pattern.

4. Evolution leads to complex localized structures, sometimes long-lived.

It is class 4 that Wolfram considers the most interesting. In this class, simple rules create complex structures, and any more complexity in the rules does not lead to more complex behavior.

Note: Wolfram’s classification scheme is believed to face some problems which Epstein tried to correct in his own classification scheme (see here: Classification).

It was originally thought that systems capable of universal computation would be found among Class IV automata – since only they exhibited interesting behaviour and signal propagation mechanisms such as gliders.

However, as Eppstein pointed out rules in all of the classes actually support gliders, and some non-class IV rules also look as though they exhibit universal computation e.g. see here. Since they contain gliders, universal computation may well show up among the other classes as well.

Eppstein’s scheme is as follows

1. No pattern expands: If no pattern can never expand, no gliders exist, and the rule is not universal. A similar phenomenon occurs with rules which remain within a finite bounding box – though they may compute functions which only require bounded resources to calculate;

2. No pattern contracts: If no pattern can ever shrink, no gliders exist. However universal computation could still occur in other ways; for instance the boundary of an expanding pattern could simulate the behavior of a 1d universal automaton.

3. Both contraction and expansion possible: Only in the remaining cases can gliders and universality exist. Our investigations show that a large fraction of the remaining cases do indeed support gliders; much more work would be required to show that they are universal.

Well, certainly there are many things we didn’t cover in this article. What I wanted to show, however, was that maybe computation can indeed be proven to be that new kind of science that Wolfram dreams of. Only time (and science) will tell.

This entry was posted on Tuesday, December 30th, 2008 at 9:05 pm and is filed under General, Philosophical. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.