In the papers in this volume, Greg Chaitin has mounted impressive evidence against such a Platonic view of mathematics. Rather than seeing mathematics as an eternally fixed realm, in which "truths" sit like ancient statues covered in the jungle waiting to be rediscovered by intrepid explorers, Chaitin sees mathematics as an evolutionary process in which the mathematician creates these sentinels.
The thread running through these articles is the notion of complexity, defined in a very specific way so as to measure the information content in a mathematical statement. Chaitin shows that mathematics itself has infinite complexity, which among other things implies that it is inexhaustible; the human mind as a finite object is incapable of ever creating that Holy Grail of the physicists, a "theory of everything" containing all possible mathematical truths. This fact of the inexhaustibility of mathematics is the strongest possible testimony to the practice of mathematics being one of creation, not exploration. In short, mathematics is like an empirical science, not simply an exercise in logic.
The ideas uncovered here reside at the very core of the philosophy of mathematics, and deserve the widest possible audience. Greg Chaitin's work will go down next to that of Gödel, Turing, von Neumann and other demigods of the mathematical pantheon, whose ideas changed our view of what is and what isn't. The articles presented in this volume constitute the distilled essence of that work.
----John Casti, Wissenschaftzentrum Wien, Vienna, Austria
Medallion commemorating Leibniz's discovery of binary arithmetic.
In physics, it is normally assumed, there must be a theory of everything that can fit on a T-shirt and that explains the entire universe. Of course, we do not know this TOE yet, but physicists believe that this simple and elegant theory exists. Biology, on the other hand, is the domain of complexity, where everything is diverse, rich and complicated and there are no simple equations that can tell us what is going on.
Is mathematics like physics or like biology? The conventional view is that pure math is like physics, not like biology, because there are simple, self-evident principles that we can all agree on and that in principle permit us to deduce all mathematical truths. Surprisingly enough, however, it turns out that the world of mathematical truth has infinite complexity and therefore pure math is actually a lot more like biology than it is like physics! In fact, in my opinion the right way to think about mathematics is that it is not at all static and eternal, it is much more like a biological organism that is constantly growing and evolving as new concepts and new ideas are created and radically transform our understanding.
To me these ideas suggest a quasi-empirical view of math, in which we are much more willing to add new axioms that help us to organize our mathematical experience even though they are not at all self-evident, for pragmatic reasons, much like a physicist would. I feel forced to believe in this extremely heretical idea, because to me it seems to be an inescapable corollary of this information-theoretic point of view. Of course, I am aware of the fact that at this point in time few others share this view.
A much more tentative corollary of this entire viewpoint is its suggestion that perhaps the physical world is actually discrete and built up out of digital information, out of 0s and 1s, and that continuity is an illusion. Amazingly enough, there happen to be some physicists working on quantum gravity, on the thermodynamics of black holes, who now suspect that any physical system can only contain a finite number of bits of information, which in fact grows as the surface area of the physical system, not as its volume! I'm referring to the Bekenstein bound and the so-called "holographic principle." What a coincidence! Where these ideas will lead, nobody can tell.
Anyway, the four papers in this collection hopefully provide, when read in the order that they are presented here, an understandable introduction to these ideas. They also contain many pointers to the literature and suggestions for further work.
----Gregory Chaitin, June 2006
In the Alps near Turin with Ugo Pagallo, 2006:
