Monday, June 13, 2005

Evolutionary Mathematics

Chaitin, as he often does, has got me thinking. He writes:

Von Neumann also said that we ought to have a general mathematical theory of the evolution of life... But we want it to be a very general theory, we don't want to get involved in low-level questions like biochemistry or geology... He insisted that we should do things in a more general way, because von Neumann believed, and I guess I do too, that if Darwin is right, then it's probably a very general thing.
For example, there is the idea of genetic programming, that's a computer version of this. Instead of writing a program to do something, you sort of evolve it by trial and error. And it seems to work remarkably well, but can you prove that this has got to be the case? Or take a look at Tom Ray's Tierra... Some of these computer models of biology almost seem to work too well---the problem is that there's no theoretical understanding why they work so well. If you run Ray's model on the computer you get these parasites and hyperparasites, you get a whole ecology. That's just terrific, but as a pure mathematician I'm looking for theoretical understanding, I'm looking for a general theory that starts by defining what an organism is and how you measure its complexity, and that proves that organisms have to evolve and increase in complexity. That's what I want, wouldn't that be nice?

And if you could do that, it might shed some light on how general the phenomenon of evolution is, and whether there's likely to be life elsewhere in the universe. Of course, even if mathematicians never come up with such a theory, we'll probably find out by visiting other places and seeing if there's life there... But anyway, von Neumann had proposed this as an interesting question, and at one point in my deluded youth I thought that maybe program-size complexity had something to do with evolution... But I don't think so anymore, because I was never able to get anywhere with this idea...

Tons of interesting stuff to chew on, but I'll limit myself to this: Imagine a simulation where you have two entities: organisms and resources. The organisms are just data structures which reproduce when they have been getting enough resources. The resources are re-generable and are of various types.

Now let's add on a few complexities: Assume that an organism 'eats' only certain types of resources. So Organism 42 can only live on Resource 118 for example. Further assume that the quantity of Resources stays relatively stable...with exceptions of rare time units of plenty and others (also rare) of drought. Also assume that there can be more than one type of Organism that consumes a certain type of Resource, and also that there are Resources that are not consumed by any organism when the simulation starts.

An Organism will then have the following data elements: Its type [corresponds to the species it belongs to]; its number [i.e. its name]; the Resource number(s) it consumes; its wellness number - a measure of how well fed the organism is - if the wellness number goes over a limit the organism will reproduce; an organism competitive index which will measure how well the individual competes within his species; and a species competitive number that measures how well the species competes with other species vying for the same resource. Reproduction passes on the competitive indices to the progeny. When the wellness index falls below a certain level, the organism dies.

Now also imagine that you have random mutations. A random mutation could change the type of resources an individual consumes and/or its competitive indices (either up or down).

These are only the barest details...but I hope you believe that it is possible to capture the main points of Darwin theory in a reasonable simulation.

Hit start and run the simulation: You will probably see organisms dying and being born; species will be created by the right mutations - they will also thrive or struggle - but eventually all will die out. The world itself may reach some kind of stable equilibrium, but more likely than some point we'd hit zero organisms or zero resources.

All this is worth doing in its own right (in fact I'd be shocked if someone hasn't already done it), but now, just for fun, imagine one last externality: Say that organisms of a certain complexity level can perceive a proportional complexity of mathematical truths. So for example an organism of complexity index 1088 could really 'get' that there can be no largest prime (but other, more difficult theorems are beyond it), and an organism of complexity index 4063 could 'get' the prime number theorem ('get' = a deep understandig that does not allow for the result not be true. Similar, but not equal to proof).

It seems to me then that there will always be mathematical statements that we humans couldn't get, no matter what.

This is far from air tight, but there may be something to chew on here.

--Gaurav Suri

Tuesday, June 07, 2005

The Voynich Manuscript

The Voynich manuscript is a very old 230+ page manuscript written in a code that no one has been able to crack. Here's the Wikipedia entry:

The Voynich manuscript is a mysterious illustrated book of unknown contents, written some 600 years ago by an anonymous author in an unidentified alphabet and unintelligible language.
Over its recorded existence, the Voynich manuscript has been the object of intense study by many professional and amateur cryptographers — including some top American and British codebreakers of World War II fame — who all failed to decipher a single word. This string of egregious failures has turned the Voynich manuscript into the Holy Grail of historical cryptology; but it has also given weight to the theory that the book is nothing but an elaborate hoax — a meaningless sequence of random symbols.
The book is named after the Russian-American book dealer Wilfrid M. Voynich, who acquired it in 1912. It is presently item MS 408 in the Beinecke Rare Book Library of Yale University.

The book has strange drawings of flowers, alien looking plants and naked women. Its history is utterly fascinating. Find out more here. Almost as fascinating as the book itself is the history of the men who have attempted to decipher the symbols. In many cases they have bolted on semi-plausible theories even though they were few supporting facts to be had.

All this forces us me to ask - where does the meaning of anything lie? Surely it is not in the symbols we use to communicate the ideas. No, meaning must lie in the mind of the humans deciphering the language/symbol. The human mind has the power to give anything meaning; it also has the power to force meaning where there is none to be had. Be it the notion of the color red, the undecidability of the Continuum Hypothesis or the rhythms of the Voynich manuscript.