Mathematical life forms and really big numbers

I finally got hold of a copy of Gergory Chaitin’s latest book, Proving Darwin: Making Biology Mathematical. The thesis of the book is very appealing to me, since it equates mathematical creativity with biological creativity. And, I would say that Chaitin’s work is a captivating experiment.  He is, as he says, “attempting to find the simplest possible mathematical life-form.”  At the beginning of a tour of this work, he makes these important remarks:

Mathematics isn’t the art of answering mathematical questions – most questions can’t be answered or have ugly, messy, uninteresting answers.  Rather math is the art of asking the right questions, the questions that have beautiful, fertile, suggestive answers.

And mathematics isn’t a practical tool, a way of getting answers.  For that, use a machine, use a computer!  Math is an art form, a way to achieve understanding! The purpose of a proof is not to establish that something is true, but to tell us why it is true, to enable us to understand what is happening, what is going on!

I really enjoyed the point being made here about ‘proof.’  I imagined a proof as a picture of what was going on inside something – like an x-ray or a sonogram – something we can see.

I described some of the points of Chaitin’s work in an earlier post after having listened to him discuss his work online.  The discussion was one that was recorded at a World Science Festival last year.  But, to recap, Chaitin begins his thoughts with two ideas:

1.that DNA is what computer scientists call a “universal programming language,” in other words, a programming language sufficiently powerful to express any algorithm.

2. at the level of abstraction that Chaitin is working “there is no essential difference between mathematical creativity and biological creativity…”(quoting from his talk at the Santa Fe Institute)

Chaitin challenges his mathematical life forms with mathematical problems to force them to keep evolving.  He is studying toy models of evolution, using the simplest life forms he can produce.  They are based on the understanding that life exists when heredity operates, with mutations, and when evolution by natural selection taking place.  Chaitin explains:

So to make things as simple as possible, no metabolism, no bodies, only DNA.  My organisms will be computer programs.

A mutation in this model is an algorithmic mutation, a computer program. The original organism produces the mutated organism as its output.  And in order to ensure that these organisms evolve forever, they are challenged with a mathematical problem that can never be solved perfectly.  According to Chaitin the organisms

are mathematicians that are trying to become better and better, to know more and more mathematics.

I love that.

The problem chosen for these mathematicians is what computer scientists call the Busy Beaver problem – “concisely naming an extremely large positive number, an extremely large unsigned whole number.”  Chaitin points out that to do this effectively one has to be creative, because to do it effectively one needs to invent addition, multiplication, exponentiation, hyper-exponentiation.   If you had the large number N, and you wanted to name a larger number, it will become necessary to consider things like N+N, N*N or N to the nth power, or N to the N to the N….  Successfully finding a larger number, than the one last one found, increases the fitness of the organism/program.  Each of the software organisms calculates a single positive integer and the bigger the number, the fitter the organism.

Chaitin has a way to measure “evolutionary progress,” and “biological creativity,” and he uses these computer science ideas to outline a proof that Darwinian evolution works in his model.   The detail (not obvious here) is provided in the book and it is important if the field of metabiology is going to progress.  I expect that it will.  My confidence, in no small way, lies in the prospect that it isn’t just proving that evolution works, but that it is also revealing some of the aspects of evolution that have been neglected or misunderstood.  In particular, these models highlight what Chaitin sees –

Biology is ceaseless creativity, not stability, not at all.

One last note:

When talking about the Busy Beaver problem, Chaitin referenced an essay by quantum computer complexity theorist, Scott Anderson entitled  “Who Can Name the Biggest Number?” It’s a great piece, and Anderson describes the extraordinarily fast growth of a sequence of Busy Beaver numbers and makes clear what they all mean when they talk about really, really big numbers.  But Anderson also raised an interesting question about whether people are afraid of really big numbers.  To address the question, he referred to studies led by neuroscientist Stanislas Dehaene. These studies suggest that two separate brain systems contribute to mathematical thinking – exact calculations were seen to line up with verbal reasoning while approximations were seen to line up with spatial reasoning.

For approximate reckoning we use a ‘mental number line,’ which evolved long ago and which we likely share with other animals. But for exact computation we use numerical symbols, which evolved recently and which, being language-dependent, are unique to humans…

If Dehaene et al.’s hypothesis is correct, then which representation do we use for big numbers? Surely the symbolic one—for nobody’s mental number line could be long enough to contain , 5 pentated to the 5, or BB(1000). And here, I suspect, is the problem. When thinking about 3, 4, or 7, we’re guided by our spatial intuition, honed over millions of years of perceiving 3 gazelles, 4 mates, 7 members of a hostile clan. But when thinking about BB(1000), we have only language, that evolutionary neophyte, to rely upon. The usual neural pathways for representing numbers lead to dead ends. And this, perhaps, is why people are afraid of big numbers.”  (emphasis my own)

This last statement is pretty interesting.

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