Unknowability in Mathematics, Biology, and Physics

there are three fields in mathematics that deal with unknowability; in chronological order, they are probability theory, metamathematics, and algorithmic information theory. The theory of probabilities began in a series of letters between Pascal and Fermat. One of the first books in this field is A Philosophical Essay on Probabilities by Pierre-Simon Laplace, published in 1814, which is still very much worth reading.

The theory of probability is about indeterminate events, but it is not about mathematical facts, which are always black or white, not grey, and which are necessary, not contingent, truths. However, the field of meta mathematics is actually about unknowability within mathematics itself—and a fascinating piece of intellectual history associated with the names David Hilbert, Kurt Gödel, and Alan Turing. These three gentlemen are mentioned elsewhere in this volume, so I will not dwell on them.

The newest mathematical field dealing with unknowability is algorithmic information theory. The star of this theory is a real number, the halting probability Omega, which is defined as the probability that a self-contained computer program chosen at random will eventually halt. If all programs halted, the halting probability would be one. If no program halted, the halting probability would be zero. And since some programs eventually halt and others don't, Omega is actually between zero and one.

Omega has a rather straightforward mathematical definition, and it is a single, well-defined real number. Yet its numerical value is maximally unknowable. The base-two bits of the numerical value of the halting probability Omega provide a perfect simulation within pure mathematics of independent tosses of a fair coin. Whether an individual bit is a zero or a one is, to God or to an infinite mind, a single, well-determined individual mathematical truth, but to us down here, it appears maximally unknowable and very much like the result of independent tosses of a fair coin.

Hilbert had hoped that there would be a theory of everything for pure mathematics, but Omega shows that pure math is more like biology, the domain of the irreducibly complex, than it is like fundamental physics, where there is still hope of finding a small set of simple equations that determine the universe.

So that brings us to biology, a very messy, complicated field. I have a wife and a child, so I love messy biology, but from a mathematical point of view, biology looks rather intractable, rather opaque. Will we ever be able to prove mathematically that life must evolve? Does Darwin's theory provably account for the spectacular biological creativity that led to human beings, to you and me? Can one prove that random mutations and natural selection are enough to account for the richness and diversity of our biosphere?

There is a highly developed field called systems biology that deals with computer simulations of complex biological systems, but there are very few mathematical proofs in biology. What can be done about this? Well, in a remarkable 1948 lecture published in 1951 and remarkably forgotten, The General and Logical Theory of Automata, John von Neumann identified the fundamental mathematical idea hidden in biology. That's the idea of software, which accounts both for the plasticity of computer technology and its overwhelming success, and for the plasticity of the biosphere with its spectacular diversity of forms. To put it bluntly, nature invented software millennia before humanity did. Biology deals with extremely ancient software. "Evodevo," or evolutionary developmental biology, is a kind of software archeology.

The new field I call "metabiology" recognizes the impossibility of a fundamental mathematical theory dealing with the effect of random mutations on natural software, on DNA, so instead it proposes studying the effect of random mutations on artificial software, on computer programs. That, in fact, turns out to be tractable mathematically, and it is probably as close as one can get to a theoretical biology as deeply impregnated with mathematics as theoretical physics already is.

When I was a visiting professor at the Federal University of Rio de Janeiro, I sketched out these ideas, and they were published in a little...