Randomness as a Sign of God’s Presence, Prior Post Updated to 2010

[I’m working on a series of books summarizing my thoughts and writings over the past 3+ years, building upon the contents of my first published book, To See a World in a Grain of Sand. As I review my work to-date, I’ll be republishing some of the articles from my blogs, sometimes pretty much unaltered but sometimes with some updating. This article was first published in September of 2007 on my other website as: Randomness as a Sign of God’s Presence.]

One of the most important, if little noticed, intellectual events of modern times is the development of a rational understanding of randomness to potentially replace an ancient understanding which is surprising mystical for such an important concept in modern mathematics and other fields of modern science. Based on that rational understanding, I made the following claims in my first published book, To See a World in a Grain of Sand:

• Only God can make a truly random number, and

• Only God can act in a truly random way

In this article, I’ll be discussing the reasons for these two tightly related claims and I’ll try to make the discussion accessible to a larger body of readers unacquainted with the work in a field with the forbidding name of algorithmic number theory. This field started to develop in the 1960s. In the middle of that decade, a prominent Russian mathematician, A. N. Kolmogorov, and an American high school student, Gregory Chaitin, both had the idea that randomness was more a matter of algorithmic complexity than of some sort of magic. To cut to the chase, a random number is one which has no patterns which allow it to be described more briefly than simply listing the digits. To be perfectly random, the number can have no patterns at all. It would be essentially a listing of digits which have algorithmic complexity that is absolutely infinite. A number of such an unfathomable nature could be described as the rawest of facts.

Gregory Chaitin continued to work in the field of algorithmic number theory even as he worked as a programmer in an IBM research laboratory. In fact, he ended up proving his major result in this field, to be discussed below, by constructing a compiler for his own dialect of lisp (programming language), the powerful computer language invented by John McCarthy as part of early efforts in the wrongly named field of artificial intelligence. Dr. Chaitin constructed his compiler in such a way that it would ‘work’ only if his theorem is correct. It worked and the theorem was proved.

See Gregory Chaitin’s homepage for both downloadable articles and information on books by Dr. Chaitin. Some of those articles include background information including some comments on those thinkers who are considered by Dr. Chaitin to have had a formative influence on the development of algorithmic number theory. The proof of the main theorem and some background knowledge is presented in his book Algorithmic Information Theory. The first paragraph of the description of Algorithmic Information Theory is:

Chaitin, the inventor of algorithmic information theory, presents in this book the strongest possible version of Gödel’s incompleteness theorem, using an information theoretic approach based on the size of computer programs. One half of the book is concerned with studying the halting probability of a universal computer if its program is chosen by tossing a coin. The other half is concerned with encoding the halting probability as an algebraic equation in integers, a so-called exponential diophantine equation.

Algorithmic information theory, deals with degrees of randomness more than with perfect randomness because we can’t produce a random number. Nor do we have the slightest reason to believe that nature can produce a random number or any movement or change that corresponds to pure randomness — unless God interjects that randomness. It seems to me to be an open question whether God could even do that without violating the integrity of His own Creation. See the ending to the story of Noah in the book of Genesis for an early discussion into God’s promise to honor His Creation. I’d say that promise was inherent in the sort of Creation He chose to bring into being.

In any case, Chaitin’s major result in many ways was a surprisingly simple proof — by the standards of modern mathematics — that every number is random. No number has a pattern. This doesn’t mean that 1.22222… or 1.25 are random nor does it mean that they aren’t numbers. It means that those numbers and similar finitely describable numbers represent a vanishingly small point on the number line. It turns out that all numbers with patterns, all the numbers of our elegant and well-ordered mathematics, add up to a vanishingly small length on the number line. It also means we can’t generate a truly random number yet there are so many random numbers that the infinities of numbers with some patterns are overwhelmed. In the sense of that field of modern mathematics called ‘measure theory’, there are essentially no numbers with patterns in relation to the totality of numbers, ‘all’ of which are true random numbers.

What does this mean? As the mathematician Marc Kac (pronounced ‘cats’) said in the early 1970s when the ideas of Chaitin and Kolmogorov were becoming known:

Now we know what a random number is. It’s a fact.

I quote from memory.

This is the basic insight lying behind my claim that God created the truths of Creation, the truths from which our physical universe is shaped. The number line is a set of facts rather than a construction as Pythagoras and his successors have thought. Elegance in the Pythagorean sense, order in the sense of the theorist of Intelligent design, and randomness in the mystical sense of a typical Darwinist philosopher, play no part in rational mathematics.

But elegant mathematics can arise from this pure factuality. In To See a World in a Grain of Sand, I compared this to the rising of islands of order from an absolutely infinite sea of chaos. Moreover, though I’ve not always been consistent, I’ve tended to at least imply that God created those facts as well. That infinite sea of absolute chaos is also a work of God.

I’ve also denied that God the Creator, He who is His own Act-of-being, works in a manner truly like that of a human workman. Only God exists necessarily and He is a pure act of existence. Mathematics and other truths of Creation are the sorts of truths that have to do with substantial being, the sort of being which is an object (of love) to God who alone is truly a self-contained subject or Person. Even mathematics would be unnecessary in the deepest sense, though perhaps interesting in some sense, to an entity of pure existence. God created mathematical truths and the other truths of Creation as He freely chose to love Creation. His decision to love Creation was the same as His act of creating Creation. To anticipate myself a little, this act was the only one we can (perhaps) know to have been (maybe) purely random. God chose a particular Creation out of possibilities absolutely infinite.

We should also remember that even the absolute truths revealed to us, those which deal with the necessary being of God and not with His free-will acts as Creator, have to be conceived and expressed in creaturely terms. We see the true God through our incomplete and sometimes distorted view of God as Creator and shaper of this universe.

Once God created the basic factual truths of Creation, He could begin to shape it and raise islands of order out of that ocean. As part of His initial Acts-of-being, by which He created from nothingness, He had created the number line and could move on to shape a universe in ways that raised those islands of order that we call Euclidean geometry, algebra, tensor calculus, and logic among many other more concrete laws of substantial being. As part of His acts of shaping this universe, He shaped time and space out of more abstract possibilities. We can imaginatively journey out on those oceans but we really couldn’t live there. In fact, I don’t think that sort of non-thing being could even register on our senses or our minds, though by the imaginative efforts of a disciplined mind, that non-thing being and even non-substantial forms of being can be thought about in a rational but indirect way.

Let’s flip things inside out, to look at matters from inside this universe rather than speaking of truths in the greater Creation. Let’s think in terms of constructing a random number, or executing a random act, inside this universe. To construct a single random number would require an absolutely infinite amount of computer power because each individual digit in the number would have to be checked against its neighbors to assure there are no runs which allow a shortening of the description of that number. Then neighboring groups would have to be checked against all other groups for the same purpose. All sizes of groups would have to be checked against all other groups. Then the differences between neighboring numbers would have to be checked because patterns in differences would allow a shortening of the description of that number. And so forth.

I believe that this understanding of randomness implies the construction of a truly random number would require a single act of thought because the concept of ‘neighbor’ implies a checking of numbers forward and not just backwards as would be true of calculations which seem possible to human beings. Working backward might get you trapped in dead-ends most of the time, though the number of dead-ends is infinitesimal compared to those paths which lead to true randomness.

What’s amazing is that Dr. Chaitin proved that all numbers, in a mathematical sense, pass these unimaginably demanding tests for randomness. I should point out here that Dr. Chaitin might well disagree with much of my use of his ideas in my effort to understand Creation.

I’ll repeat the claims, both mathematical and theological, I made above:

• Only God can make a truly random number, and

• Only God can act in a truly random way