Proving Darwin: a review
In 1944, the quantum physicist Erwin Schrödinger published “What is Life?”, in which he posited a physical basis for all living systems.
“How can the events in space and time which take place within the spatial boundary of a living organism be accounted for by physics and chemistry? … The obvious inability of present-day physics and chemistry to account for such events is no reason at all for doubting that they can be accounted for by those sciences.”
In the 1950s, at least part of his speculations were vindicated when the structure of DNA was discovered, thereby unlocking entire fields dedicated to advancing the idea that life could be understood as natural phenomena.
In 2012, the computer scientist Gregory Chaitin attempted to advance the idea even further in his seminal book, “Proving Darwin: Making Biology Mathematical”, in which he tries to prove that Darwinian evolution is as true a mathematical statement as it is a fact of the natural world. As Chaitin puts it:
“In my opinion, if Darwin’s theory is as simple, fundamental and basic as its adherents believe, then there ought to be an equally fundamental mathematical theory about this, that expresses these ideas with the generality, precision and degree of abstraction that we are accustomed to demand in pure mathematics.”
Chaitin has helped to take more seriously the view that life can be viewed as computational the same way that Schrödinger suggested that life can be viewed as physical. Indeed, the two views are not contradictory: the Church-Turing-Deutsch thesis posits that universal computers are physical systems that can mimic all physical systems, including themselves, which is why we can use computers to try to understand life in the first place. Regardless of whether you think it is true, you should at least try to entertain it in order to be objective, and offer constructive counterarguments, or better explanations.
This is not going to be a comprehensive review (who’s got the time?), but rather a summary of what I think are the most important ideas in each chapter, followed by some thoughts of my own. I hope you will forgive me that I will not attempt to capture every single little detail, especially since the book is full of them. I also cannot explain everything in the limited time that I have, so I insist that you read the book for more details. Any misconception is mine.
Before the preface, Chaitin reprints a nice essay by Gian-Carlo Rota which classifies mathematicians into problem-solvers and theorizers:
“If I were a space engineer looking for a mathematician to help me send a rocket into space, I would choose a problem solver. But if I were looking for a mathematician to give a good education to my child, I would unhesitatingly prefer a theorizer.”
I leave you to read the essay for yourself, which Chaitin seems to use to suggest that he is a theorizer.
The most important thing in the preface is that Chaitin wrote this book as a response to “The Devil’s Delusion” by David Berlinski, “which offers a stinging critique of Darwinism and a withering comparison of biological theory as contrasted with theoretical physics.” Unlike many biologists, Chaitin took Berlinski’s criticism seriously, the latter of whom is often painted as sympathetic to Creationists. This should assuage the skeptical reader that Chaitin is trying to be the good scientist, working with rather than against religion, but who is also not afraid to push back when it is wrong.
Chapter 1 provides a map of things to come, so I shall skip it and go straight ahead.
Chapter 2 begins by asking why we should try to prove Darwinian evolution as a mathematical truth rather than simply accept it as an empirical fact. Yet, recall the quote above: if it is really as fundamental as they say it is, shouldn’t it also be true in a more abstract world?
Specifically, Chaitin is trying to prove that, mathematically, there exists something that meets John Maynard Smith’s definition of life in “Problems of Biology.” Flames have metabolism and self-reproduce, but they do not evolve. Maynard Smith says that life is what happens when there is heredity with mutations (probably still not good enough, because clay crystals might, and certainly viruses, meet that definition), and evolution by natural selection can happen.
But how does Chaitin do this? He proposes using metabiology, which is a new kind of theoretical physics (or, as his friend Stephen Wolfram might say, a new kind of science). Metabiology sees life as evolving software, which cannot be easily modeled using the old methods of theoretical physics. There is natural software, such as what is in ourselves, and artificial software, such as the relatively pitiful ones we have been writing over the last several decades. Nature is the original programmer, and DNA is the original, universal programming language. Chaitin paraphrases La Mettrie, who wrote the book “L’Homme machine” (Man a Machine, Machine Man): it should be “L’Homme software” instead.
But how can we model life using mathematics? Isn’t mathematics closed, static, rigid, inflexible, dead? That is why we must use a new postmodern kind of mathematics called computation, discovered and developed by Gödel, Turing, Church, Post, and friends. Life is plastic, dynamic, flexible, boundless! And software is the closest thing we know that can begin to model it. Chaitin criticizes the traditional approach in population genetics, developed early last century, where it is assumed that that genes are fixed. But this is clearly not what we see in the real world: Nature evolves, genes mutate!
“The biosphere is full of software, every cell is run by software, 3- to 4-billion-year-old software. Our artificial software is only fifty years old or so. But we could not realize that the natural world is full of software, we could not see this, until we invented human computer programming languages. The world was full of software even before we knew what that was! Software is the reason for the plasticity of the biosphere — normal machines are rigid, mechanical, dead. Software is alive!”
It is an inspiring chapter oozing with infectious curiosity (especially if you know how Chaitin speaks) that at least makes the reader want to see where and how far the idea can go.
Chapter 3 presents a revisionist history of biology, which is roughly as follows: Alan Turing’s 1936 paper on computable numbers (remember, no one had systematically discovered universal machines before, and explored what they can and cannot do) had an enormous influence on John Von Neumann, who wrote about how machines can reproduce themselves (also in Appendix 1 of the book), which influenced Sydney Brenner, who in turn influenced Francis Crick, co-discoverer of the DNA.
Chaitin goes into more details, visiting Gödel, Turing, Von Neumann, Brenner, Stanislaw Ulam, and friends. The most interesting comment is what Chaitin observed about Emil Post:
“According to Emil Post — who is not as well known as Gödel and Turing but was at their level (he came up with Turing machines too, and also with an incompleteness theorem that remained unpublished for years)—the axiomatic method, and especially Hilbert’s formal mathematics, was just a terrible mistake, a confused misunderstanding.
According to Post, math cannot provide certainty because it is not closed, mechanical, it is creative, plastic, open! Sound familiar? You bet, we have been talking about biological creativity all through the previous chapter, and now we find something like it in pure math too! So math is creative, not mechanical, math is biological, not a machine!”
It is much too bad that this view is not more widely known. I hope this makes it clearer why Gödel’s incompleteness theorems are not the deadly blow that most have been led to believe. People tend to forget that we got infinitely flexible machines — computers — out of these very limitations! Gödel himself did not think that his theorems were a serious blow to mathematics. Remember: his theorems apply to humans — natural machines — as much as it does to artificial machines.
In Chapter 4, Chaitin first “apologizes” for the toy model of evolution that he is about to present. Some like Jacob Schwartz argue that mathematics is mostly good for describing simple, not complex, phenomena. Others like Maynard Smith and Eörs Szathmáry argue that the point of a mathematical model is to simplify, not to confuse. In any case, Chaitin simplifies things enough so that he can prove that a mathematical life-form exists that meets Maynard Smith’s definition of life.
Here are the rules of the game of his toy model:
- Initial condition: there is a single organism “who” is a mathematician, and whose fitness is 0. No origin of life, no bodies, no metabolism. (And yet it evolves, to paraphrase Galilei, as we will soon see.)
- Mutations: if you have an organism A, then you use an algorithmic mutation M to get to organism B. If M is an K-bit program that transforms A to B, then the probability p of choosing M is 2^-K. In other words, the distance between A and B is -log(p, 2). We do a random walk through the mutation space using this probability.
- Fitness function: if M_K is a K-bit program that transforms A to B, then the fitness F of B is its output. M transforms B in such a way that its output is the first N for which Ω_N ≥ [Ω_(F(A)) + 2^-K], where Ω is the halting probability. Note the heredity in mutations: the fitness of A is used to determine the fitness of B.
- Oracle: the astute reader would have noticed that determining whether B=M(A) halts is an undecidable problem. Therefore, we will assume that we have an oracle, or a Turing machine connected to a fantasy black box that will decide the halting problem for us, so that we can determine whether or not the mutation M_K halts. If it does, the K-th position of Ω is 1; otherwise, 0.
In this manner, the game never ends, because this single organism is continually producing the next largest number. In fact, it is busy solving BB(N), or the Busy Beaver problem, which grows faster than any computable function. There is endless mathematical / biological creativity in this game, because Ω is maximally random, or algorithmically incompressible. Chapter 5 and Appendix 2 discuss all of the technical devil in the details. In particular, knowing the first N bits of Ω effectively gives you BB(N) (the direction discussed above), and vice versa.
Main results for how much time, or equivalently, how many mutations it takes, to get to the fittest organism, where N is the number of first bits of Ω produced by this lone organism:
- “Intelligent design” (trying M_1 first, then M_2, and so on, until M_N): reaches BB(N) in N time steps.
- Cumulative random evolution (the toy model above): reaches BB(N) between N² and N³ time steps.
- Brainless exhaustive search (tries everything at random w/o taking previous organism into account): reaches BB(N) in 2^N time steps.
I have to admit that I was, at first, perplexed when I first heard that Chaitin assumed the existence of oracles. Recall Chaitin’s “apology” for using this toy model. He has to be able to prove that evolution happens here, and there is no way to prove the results above without assuming the existence of these oracles that, thus far, exist only in fantasy. He is the first to admit that future generations should try to improve his results by using more realistic models where, for example, the running time is always bounded.
Speaking of which, how does Nature never seem to run into the halting problem? (The typical treatment of the halting problem, by the way, considers noninteractive computers, which is unrealistic.) She clearly doesn’t know which mutations will work or not: witness the tragic, broken phenotypes of some organisms born due to no fault of their own. However, She does seem to spontaneously abort some fetuses which don’t appear viable, possibly using some unknown heuristics.
Lest this toy model look simple, Chaitin himself took a few tries to get there. Consider, for example, the probability of producing a mutation M that flips all the bits of an organism A. Without algorithmic mutations, this probability would be infinitesimal.
Chapter 6 is about Chaitin’s theological views that color his metabiology. (Anyone who pretends not to have some sort of theology, even atheism, is lying.) He insists that metabiology is not an aesthetic theory. He thinks that we can use Reason to understand Leibniz’s God, a God that does not miraculously intervene with life, for that would make the world a lot less perfect. (I am sympathetic to this view.) He admits that he doesn’t have a fully coherent view of all of this. He says that he does not believe that we live in an imperfect realization of the Platonic realm of perfect ideas. Rather, he believes in Tegmark’s Level IV Multiverse: that we live in a tiny sliver of all mathematically / physically possible universes. With all due respect, I disagree: I think mathematics is a consequence of physics, not the other way around. Anyway, Chaitin concludes that metabiology does not make Nature mechanical or reductionist like a closed Hilbertian formal axiomatic system (see Post’s view above). Rather, Nature, by creating software, becomes forever open, plastic, and creative. Darwin replaces God with randomness, but this is not a bad thing: it is much better than living in a dull and predictable universe. Creativity is, by definition, unpredictable, a view also expounded by David Deutsch.
Chapter 7 is about Chaitin’s political views regarding metabiology. Chaitin begins by discussing the mysterious creativity of Bach, Mozart, Euler, Cantor (who wanted to understand God by reasoning about infinity), and Ramanujan (who claimed that the Tamil goddess Namakkal wrote equations on his tongue as he slept). Chaitin uses Ramanujan to make fun of reductionists. I cannot emphasize this enough: computation makes the world more, not less, creative. Computation teaches us that we must be “Against Method” à la Feyerabend: sometimes, we must make progress through bold speculations, not pure Reason (my words, not Chaitin’s). Chaitin says that we should measure the wealth of a nation not through its GDP, but through its creativity. That is why Chaitin is for decentralized city-states, not centralized empires. He laments that we are being micromanaged by gigantic bureaucracies. He wants more of the Talmudic and Hindu culture where there is no centralized authority. (Does all of this sound familiar to the Nassim Nicholas Taleb tribe?) He ends by quoting G.H. Hardy in “A Mathematician’s Apology”:
“It is never worth a first class man’s time to express majority opinion. By definition, there are plenty of others to do that.”
Finally, Chapter 8 is about Chaitin’s epistemological views about metabiology. Largely, he talks about how others can advance metabiology from what he has done. He offers the extended Copernican principle, which says that there is no reason to think that our scientific knowledge is anywhere near complete.
Overall, this is a fantastic, little pocket book. There are so many delicious quotes and details in this book that I simply cannot capture them all. The book also has many great references to papers and other books littered all over the place. It is also sometimes redundant and repetitive, but this is easy to forgive, especially for such an interesting and thought-provoking book. I highly recommend that everyone read it.
So what is the point of this book? I don’t think anyone is convinced that this is truly a living organism. No, I think the point is that, while we know from empirical fact that evolution by natural selection is true (at least until we find a better theory that explains the gaps), it is also good to know that evolution is mathematically true, which suggests that life is inevitable, in a sense. We are “at home in the universe,” as Stuart Kauffman might say.
I want to end this review with a call to arms. While I subscribe to Chaitin’s view that software is open, plastic, and creative, and alive, the problem is that our current state of software is closed, rigid, mechanical, and dead. Why do we think that we can create an artificial general intelligence worthy of its name by designing software top-down? Nature certainly didn’t do that, and I don’t think we can, either. We can hardly understand ourselves, let alone other people. No, I think that we will somehow need to find out how to make evolving hardware that can reprogram their software in multiple, cascading layers of emergence — in other words, complex technology. It will be simultaneously powerful and dangerous, which is why I think we should take it seriously, if it is even remotely possible. Maybe it is ultimately infeasible, who knows, but what good is it to presuppose a Tower of Babel, and not even try?
