Cum Deus calculat, fit mundus.
As God calculates, so the world is made.
---Leibniz
[Inaugural Académie Internationale de Philosophie des Sciences lecture by Gregory Chaitin, Tenerife, September 2005.]
I am a mathematician and my field is algorithmic information theory (AIT). AIT deals with program-size complexity or algorithmic information content, which I regard more or less as the complexity of ideas. I think that this has much greater philosophical significance than the much more popular complexity concepts based on time or other measures of computational effort or work.
Thank you very much for making me a member of the Académie Internationale de Philosophie des Sciences. And thanks for squeezing me into the program and making space for me to give a short talk. Thank you very much for asking me to give a talk even though I was not scheduled to be a speaker at this meeting.
I've brought with me, hot off the press, a copy of my new book Meta Math!. This book has been 40 years in the making. I've been working on these questions for that long. In this brief talk, I'll merely touch on topics that are developed at much greater length in my book.
To start the ball rolling, let's consider physics versus biology. Is mathematics more like physics or is it more like biology? Well, in physics we have simple equations, whereas biology is the domain of complexity. So normally people think that math is much closer to physics than it is to biology. After all, mathematics and physics have co-evolved, and not much mathematics is used in biology. However, as I'll explain in this talk, mathematics contains infinite complexity and is therefore, in a fundamental sense, much closer to biology than it is to physics!
How does AIT manage to show this surprising and unexpected connection between mathematics and biology? AIT is at this point in time a fully developed elegant mathematical theory of program-size complexity. But for the purposes of this discussion, we, as philosophers of science, do not really need to know the mathematical details of AIT. Instead it suffices to understand the basic concepts, which amazingly enough can be traced back to Leibniz. Here are three texts by Leibniz that caught my eye as a mathematician. They are, as far as I'm aware, his key texts on the concept of complexity:
Having used Leibniz as an introduction, let me now leap into the heart of AIT. One of the central topics in AIT is a number that I've discovered that I like to call Ω. Briefly, Ω is the halting probability of a computer, it's equal to 2 raised to the power −K summed over the size in bits K of every program that ever halts:
Ω is important because it's an oracle for Turing's halting problem, it's the most compact, the most concise way of summarizing all the possible answers to questions asking
as originally discussed by Turing in 1936. Ω is irredundant, the infinite stream of base-two bits in its binary expansion are irreducible mathematical facts. In other words, whether each bit in the base-two binary expansion of Ω is a 0 or a 1 is a mathematical fact that is true for no reason, no reason simpler than just directly knowing the bits themselves. More precisely:
Please note that this information-theoretic limitative meta-theorem contradicts Leibniz's principle of sufficient of reason, which says that if something is true then it has to be true for a reason. (Of course this applies only to necessary not to contingent truths.) Those reasons as Leibniz points out in The Monadology would necessarily have to be simpler than the bits of Ω in order to be able to count as the reasons determining their individual 0/1 values. But in the case of Ω, which is irreducible, no simpler reasons are possible. In other words, the bits of Ω are logically as well as computationally irreducible, that is why they refute the principle of sufficient reason. Essentially the only way to establish what these bits are is to add that information directly to your mathematical theory as a new axiom. But anything can be established by adding it as a new axiom. That's not using reasoning, that's not much of a proof, it's a new assumption.
Furthermore, in toto the bits of Ω are infinitely complex, which establishes the promised link between mathematics and biology.
To conclude, I would like to thank you all again for making me a member of this Academy. I know I've rushed through this material very, very quickly. But if you want to know more about all of this, please take a look at my new book. In fact, it's actually my système du monde, it's my attempt to formulate a complete speculative metaphysics. Meta Math! is a serious book in spite of the frivolous-sounding title. For example, let me mention three important topics in my book that I haven't had time to discuss here:
Thank you.