Conversations with a Mathematician is a collection of three lectures and six interviews originally given between 1989 and 2001. What emerges from this somewhat disparate set of transcripts is a surprisingly coherent picture of Chaitin and his work. While there is the inevitable repetition of ideas, stories, and examples, in the end that very repetition helps to leave the reader with the impression of having become acquainted with Chaitin himself. One way we know our friends is by knowing their important stories.
Of course, it is no accident that Chaitin's work on algorithmic information theory is at the center of these conversations. One way he describes his work is as "discovering randomness in pure mathematics." His technique is to use the size complexity of computer programs to define randomness. This is quite different from run-time complexity, which has been the more common way to measure program complexity. The rough idea is that randomness is equivalent to incompressibility: if the shortest program to reproduce a given set of data is the same size as the data itself, then the data must have no patterns and should be considered random.
This notion of randomness leads to what Chaitin sees as the real reason for Gödel's incompleteness theorem. To fully understand Chaitin's argument here, one would probably need to read his technical works. It is too easy to be led astray in the landscape near the incompleteness theorem. Chaitin's own critique of Penrose's use of Gödel's theorem provides a good example of the difficulty.
However, it is still possible to appreciate Chaitin's claim that his approach makes incompleteness appear natural. When reading Gödel's proof, it is easy to feel that the statement he constructed is just a paradoxical trick. It works, but how many mathematical statements are really like that? Chaitin sees his own proof as being in line with Turing's, making incompleteness appear less surprising and more motivated. In his words, "when you start looking at program size, well, incompleteness, the limits of mathematics, it just hits you in the face!"
Throughout these lectures and interviews, Chaitin explains that his ideas for studying randomness in mathematics came from physics, particularly the work of Ludwig Boltzmann on entropy. Probably the most controversial claim Chaitin makes is that, because of the prevalence of incompleteness, mathematics should adopt a more experimental methodology, closer to that used in the other sciences. His argument is that, given the fact that incompleteness is everywhere and that axioms can only capture so much truth, we should no longer expect to find self-evident axioms. Rather, we should adopt the more pragmatic approach of physicists: if a theory works, use it! For example, if the Riemann Hypothesis (RH) is validated by all experiments so far and leads to other fruitful results, then why not assume it? If later work shows it to be false, then we look for a better theory at that time.
Although this argument has some appeal, it is not persuasive. First, it is not clear what we gain by assuming statements like RH. Chaitin acknowledges that people already prove theorems on the implications of RH — would they prove more theorems if we assumed it? Would the new theorems be better? Secondly, Chaitin's own assertion that axioms are no longer "self-evident" argues against our ability to agree that RH, for example, should be adopted as an axiom. People will disagree. As mathematics currently operates, everyone agrees that implication theorems are valid (although they may disagree about their value); if a part of the community begins assuming different axioms, then questions of validity arise. But again, to what end? It is not clear what we gain.
Given this claim of his, though, it should not be surprising that physicists tend to be more comfortable with Chaitin than mathematicians. And this is not the only provocative statement Chaitin makes in the course of this book. In one television interview, he claims to no longer believe in either the real numbers or the natural numbers. What I found interesting in reading this collection, though, is that in one of the last pieces, the interviewer, an undergraduate from Bard College, asks Chaitin directly about many of these claims. It is almost exactly what I wanted to do as a reader at that point — just to probe a little bit more to see if I understood what he was really saying. Another interesting aspect of this particular interview is that the student asked him what he would recommend to a humanities student who wanted to read some of his work. Chaitin suggested some lecture transcripts and an interview on his web site, saying that they were "more understandable" than his books.
That is, until he compiled this one. Conversations with a Mathematician fills the gap left by Chaitin's other works by providing a highly engaging and readable account of his explorations into randomness in the foundations of mathematics. Those who want details are given enough pointers to find them.
Mark Johnson (firstname.lastname@example.org) is Associate Professor of Computer Science and Mathematics at Central College in Pella, IA.