A mini-conference on Randomness and Constructive Mathematics has been held at Auckland University, New Zealand (under the aegis of the new School of Mathematical and Information Sciences) on Tuesday 16 February 1993.
This conference was organised in honor of G. J. Chaitin, from IBM's Thomas Watson Research Center (New York), founder of algorithmic information theory, who was visiting Auckland University from the 13rd to the 20th of February.
The title of the colloquium Does God Play Dice? is a reference to Einstein's well known comment
questioning the probabilistic nature of quantum theory. This theory, as you no doubt know, pertains to events involving atoms and particles smaller than atoms, events such as collisions or the emission of radiation. In all these situations the theory is able to tell what will probably happen and not what will certainly happen. It is important to note that Einstein was not questioning the use of probabilities in quantum theory (as a measure of temporary ignorance or error), but the implication that the individual microscopic events are themselves indeterminate, unpredictable, random. [It appears that the ancient Greeks and Romans would not object to the idea that the microscopic universe is essentially governed by chance - in fact they made their Gods play dice quite literally, by throwing dice in their temples, to see the will of Gods; the Emperor Claudius even wrote a book on the art of winning at dice. However, from the point of view of Christianity, playing dice with God was definitely a pagan practice -- it violates the first commandment. St. Augustine is reported to say that nothing happens by chance, because everything is controlled by the will of God.]
The classical idea of casuality (i.e. the idea that the present state is the effect of an anterior state and cause of the state which is to follow) implies that in order to predict the future we must know the present, with enough precision. But, for quantum events this is impossible in view of Heisenberg's Uncertainty Principle. It was thought, however, that order reigned at human dimensions, but, in more recent times, with chaos theory and non-linear dynamics, we have got used to the fact that non-predictability is omnipresent in the physical world.
All these facts have undermined our familiar ideas about space, time, matter, casuality. So, it appears that God plays indeed dice with the physical universe.
What about the mathematical universe, our supreme model of solidity and rigor? In 1987, Chaitin showed that even elementary arithmetic has a random structure. Refining work done by Gödel and Turing, but following an original path, Chaitin designed a new mathematical theory of information called algorithmic information theory. Given the central place of arithmetic in mathematics, it seems that God also plays dice with mathematics.
Turing's fundamental result is that the halting problem is unsolvable. Chaitin's fundamental result is that the halting probability is algorithmically random, i.e. the digits of the number Omega expressing the halting probability form a binary random sequence. Omega
embodies an enormous amount of wisdom in a very small space ... inasmuch as its first few thousands digits, which could be written on a small piece of paper, contain the answers to more mathematical questions than could be written down in the entire universe.writes C. Bennett*. And he continues:
Throughout history mystics and philosophers have sought a compact key to universal wisdom, a finite formula or text which, when known and understood, would provide the answer to every question. The use of the Bible, the Koran and the I Ching for divination and the tradition of the secret books of Hermes Trismegistus, and the medieval Jewish Cabala exemplify this belief or hope. Such sources of universal wisdom are traditionally protected from casual use by being hard to find, hard to understand when found, and dangerous to use, tending to answer more questions and deeper ones than the searcher wishes to ask. The esoteric book is, like God, simple yet undescribable. It is omniscient, and transforms all who know it ... Omega is in many senses a cabalistic number. It can be known of, but not known, through human reason. To know it in detail, one would have to accept its uncomputable digit sequence on faith, like words of a sacred text.
We won't go any further into this, but quote a review of Chaitin's work stating that
Chaitin's article makes the world shake just a little.(It is noteworthy that this was written by a Californian reviewer in the Los Angeles Times !)