Mathematics, no matter how abstract and symbolic it can be, is founded upon numbers, the number line for those who have taken a geometry or algebra course. Even with the abstractions of group theory and projective geometries, our understanding of mathematics follows our understanding of numbers.
About 15 years ago, Gregory Chaitin of IBM proved that all numbers (in a measure-theoretic sense) are random. This means that the percentage of numbers with patterns is zero in a very strong sense. It suffices to say for this discussion that there are infinities much larger than the ordinary infinity of the integers (1, 2, 3, etc). The infinity which is the count of random numbers is much larger than the infinity of numbers with patterns, including the integers.
This work by Chaitin and others brings a radical implication best captured in a simple claim made by Marc Kac, a prominent mathematician who taught at Cornell: a random number is simply a fact. Professor Kac made this claim in the 1970s, when Chaitin’s work was not yet complete.
Those who wish to see rigorous development of the modern redefinition of randomness (development in a historical sense as well as the sense of the discussion of the final proof) can check out the papers and references to books on Dr.Chaitin’s Home-page.
We should always remember something that even some knowledgeable philosophers and theologians seem to forget, at least when they try to do creative thinking. The metaphysical principles of the Greeks didn’t arise as pure symbolisms or verbal assemblies. They were shaped by the Greek understanding of mathematics and the physical world often seen in terms of mathematics. I don’t wish to deny that heritage from the Greeks, but I do wish to expand our metaphysics beyond that of the Greeks. The best of human thought respects the best of empirical knowledge but that knowledge has grown greatly and forced us to new understandings of mathematics as well as new understandings of physical thing-like being. Right now, our traditional theologies and philosophies are superstitious in the sense that so much no longer valid knowledge is embedded in the speculations about the nature of human thought, the meaning of moral principles, the possibilities of life beyond the grave, the differences between God and Creation, and so forth.
The Greeks assumed that mathematics, hence the world, was built up in an orderly manner from well-defined possibilities. Even stuff that seems messy is understandable as patterns.
I don’t disagree with that, though the patterns have proved to be far more complex than the Greeks could have guessed and also the patterns in our universe tend to develop, and fall apart, rather than being constructed in the way of a theorem or even a human building. Even things such as stars have a ‘biography’, a dynamical nature that goes beyond being which corresponds to axioms or theorems from Euclidean geometry. We have to go beyond the prison that the Greeks constructed for us by teaching us that there is only one possible mathematics and even God must work with a certain body of truths which are directly accessible, at least in principle, by something labeled ‘the human mind’.
I’ve made it clear in my writings that I think we’ve misunderstood the human mind through much of history. I believe it to be the bodily and relational aspects of the human brain, an organ with the amazing ability to encapsulate the messiness of the universe inside itself. The evolutionary biologists are right: our thoughts are not built up from some sort of abstractions drawn from an immaterial world or from necessary truths. I would add there are no such truths which we could understand which are binding upon God. By this, I don’t mean to say that He could have created a universe in which ‘1 + 1 = 3’, but I do mean to say that He could have created a world in which ‘1 + 1 = 2’ is either irrelevant or a higher-level synthetic truth, a theorem of some sorts of higher mathematics.
Why would I make such a claim? Because of the theorem of Chaitin: all numbers are random in a measure-theoretic sense. It turns out that Goedel’s more famous theorem is actually a corollary of this deceptively simple theorem. What we know as mathematics comes from the patterns in a certain set of numbers, patterned numbers, which are a vanishingly small percentage of all numbers.
We now face two choices so far as I can tell. We can see that small set of patterned numbers, the foundation of classical mathematics, as being somehow necessary though now appearing as tiny islands in an absolutely infinite sea of randomness. This would be a heroic stance of sorts, living in a Platonist sub-set of a greater Heraclitian reality. Batten down the hatches, there’s a nasty world out there and we can only make our stand against absurdity and death by standing on these tiny islands which somehow exist in a meaningless world.
The other possible choice is the one I’ve taken. We can see that overwhelmingly infinite sea of random numbers as being an analogy to a more general situation which the human mind can’t directly access. This leaves us in a far worse situation because we lose our ability to see even these tiny islands as holding some meaning which rises above the surrounding chaos.
It’s time to follow Hermann Melville’s advice: have courage and set out into those chaotic seas. The settled life on those islands is an acceptance of a death-like life.
Have courage? No. It’s better to have faith and hope that courage will come, or at least some Christian version of stoic acceptance of one’s hardships. I believe that I might understand. I have faith in God that I might have the courage to face up to the task of understanding the world in which I find myself. I admire Sartre, for all his moral problems, for having the courage to face up to a realistic understanding of the physical universe without even wanting to have faith. But I don’t wish that fate upon any man, certainly not upon myself.
The curious reader might be realizing something at this point. Whatever judgments are ultimately made upon the quality of my thought, I’m one of the very few truly creative thinkers in modern times because I’ve broken out of the ruts in which modern thought trapped us with the politically useful but wrong-headed claim that theology and philosophy and history and physics and literature can be split up into these streams called ‘fields of knowledge’. The structure of modern knowledge has been shaped to the needs of modern liberalism, a way of life and thought which has proven best at producing nice people who are morally spineless, people who can continue to teach physics or push paper or build railroad lines when the Jews and Gypsies are being rounded up a quarter-mile away. As a consequence of my understanding of human knowledge and the possibilities of understanding our environments, our physical universe, and the morally ordered universe I call a world, I will sometimes burst out into discussions of moral or literary issues in the midst of discussing mathematics or cosmological physics.
Now, I return to mathematics to repeat three claims I made in To See a World in a Grain of Sand:
1. Only God can make a random number.
Under the more modern understanding developed by Chaitin, anticipated by Chaitin as a high-school student and also by Kolmogorov around 1965, a random number is nothing more than a number with no patterns at all which allow a shorter description than the actual listing of the number. Amazingly enough, Chaitin proved — in a book published as an undergraduate computer science text — that the measure of non-random numbers ‘divided’ by the measure of random numbers is 0. In a measure-theoretic sense: all numbers are random. I can imagine a comic saying that mathematicians have proven the integers don’t exist.
2. Randomness, as the term is commonly used, is a type of factuality viewed through superstitious eyes.
As I said above, the prominent measure theorist Mark Kac said a random number is simply a fact. He made that statement more than 15 years before Chaitin completed his work.
3. Any patterns in the world are small islands of order in the midst of tumultuous oceans of randomness or factuality which threaten to overwhelm us, physically and also spiritually.
If mathematical descriptions of the universe point to deeper truths and are not just coincidences, then we can say that there is a 0 probability that anything should exist. The sorts of patterns that support thing-like being have a zero probability of existing in a mathematical sense.
I’m not a big fan of the idea that we can prove, in the modern sense of ‘prove’, the existence of God — I discuss some of the reasons for this position in To See a World in a Grain of Sand. On the other hand, we need to make sense of this universe, also our local environments and human communities. Systems of mathematics don’t suffice — narratives are needed. Cosmological physicists have found themselves in the position of writing narrative descriptions of the evolution of this universe which aren’t so much different from the creation ‘myths’, actually narratives, of the so-called primitive peoples. Because of our mule-headed insistence on separating fields of knowledge, there are two important differences which make our modern creation narratives less adequate than the traditional narratives:
1. There is no recognition of the need for necessary being to ‘explain’ how contingent things could have come into existence. Various thinkers throughout history have had the instinct that this sort of a world has what I call a zero-probability of existing. Prior to Plato’s “Timaeus” and then Philo’s understanding of the book of “Genesis” in light of “Timaeus”, there was no real understanding of the concept of “creation from nothing”, but earlier thinkers, including those who wrote the early chapters of “Genesis”, knew that some being of a greater inherent coherence was necessary as a shaper of chaotic being.
2. There is no moral structure in our modern scientific narratives and those are the dominant narratives of our days. In terms I’ve developed on my two blogs, narratives become truly such and make a deeper impression on the human brain when they are morally purposeful. During the first stage of making sense of this claim, it’s sufficient to remember that human beings are social mammals, rational and dependent animals as Alasdair MacIntyre pointed out. Our moral natures are not particularly strong compared to those of some other social mammals, but our ability to think abstractly means we can make sense of the universe and can also adapt ourselves to new conditions. Unfortunately, we can also learn to create and justify various sorts of hells on earth. St. Francis of Assisi and Clara Barton are possible but so are Atilla and Hitler.
The traditional narratives which seek to understand our world are inadequate because they are superstitious in the sense of being built upon older understandings of empirical reality which are known to be inadequate. We have some strong hints in the bloody wars of the modern world, as well as the horrors of Auschwitz and the Maoist murders of millions of peasants, that maybe moral structure is more important than scientific ‘truth’. With the fragmented nature of knowledge and beliefs in modern times, we inhabit not a world but rather some sort of battleground in the midst of chaos.
However, I’ve taken the position that we can move on and achieve a new understanding of the world which respects revelations, the moral needs of our race, modern empirical knowledge, and the human need to creatively speculate on how this all fits together. In To See a World in a Grain of Sand, I constructed a worldview which tries to pull together our modern empirical knowledge and the revealed truths of Christianity in a creative, even playful, way. My speculations were formed and informed by a radical version of Thomistic existentialism, a way of thought which can respect both God’s revelations and also man’s knowledge gained by exploration of God’s Creation. Moreover, this way of thought teaches us we can’t even understand God’s most blunt revelations without a reasonable understanding of Creation. Most certainly, we can’t understand Creation without some understanding of God’s revelations.
I believe that I might understand, but my understanding enriches my beliefs.