Categories: communal human being, Freedom and Structure in Human Life, Magical ways of thought, Mathematical models, Narratives and truth
In my last two essays I posted, Mathematical Models of Human Communities: Parts and Wholes and Mathematical Models of Human Communities: We Live in Narratives, Not in Models, I acknowledged the usefulness and potential truthfulness of mathematical models but claimed we need to consider wider aspects of this world and of all of Creation. In particular, I discussed in the first of those essays, very briefly, the ways in which many complex systems, those of physical spacetime and—with near certainty—those of human social relationships, have global properties which don’t fully come from summing up local properties. In the second essay, I discussed, with equal brevity, the nature of one of those global aspects of complex systems and especially human communities—they are stories or narratives with the properties which we expect in novels or tales and which don’t come from mathematical models as such.
At the same time, the reader should keep in mind that I believe our `mathematical’ understanding of this world can be expanded greatly by further use of the proper mathematical fields and ways of thought such as those fields of geometry which deal in qualities rather than quantities. There are also the fields of geometry, such as differential geometry, which have the power to deal separately with local and global properties.
I’ll continue to respond to specific quotes from a book, Sociodynamics: A Systematic Approach to Mathematical Modelling in the Social Sciences (Wolfgang Weidlich, Dover Publications, 2006), which does a good job of dealing with the power and limitations of mathematical modeling in the `social sciences’.
On page 155, the author writes:
Historical and/or social phase-transitions are by definition revolutionary events in which the macrovariables of the system change their whole dynamical mode. A necessary concomitant circumstance of such a phase-transition is the appearance of critical fluctuations. These critical fluctuations are crucial for deciding the question which direction the path of the system will take at the cross-roads. In our case they are decisive for the question whether the political system will remain a liberal democratic one or whether it tumbles into the new totalitarian phase.
However—and this is the essential argument—the critical fluctuations are of random nature and are neither predictable by the research of historians, nor by the macroequations of any mathematical model! At best the full set of macrovariables and (not predictable) fluctuating microvariables which both together are causative for the concrete course of historical events at a phase-transition can be recognized by historians only retrospectively.
Therefor the general conclusion must be: In the rare cases of historical phase-transitions fluctuations become decisive (in contrast to smoothly and continuously evolving situations). These fluctuations consist of thoughts, decisions and activities of one or a few persons in key-positions in a global situation on the verge of a possible phase-transition.
This does of course not mean, that the continuous—to a high degree “calculable” and therefore predictable—macrovariables would be unimportant. In the contrary! They lead to the “revolutionary situation”, i.e. into the vicinity of a destabilizable situation where “everything can happen“. However, at the phase-transition these macrovariables are insufficient to make the further course of events predictable!
Weidlich tells us, “the critical fluctuations are of random nature.” True enough, but what is random nature? I’ve dealt with this issue before and have claimed in various ways that the usual definition of such terms seems to smell a bit of the occult, even of outright superstition.
In February of 2010, I published a slightly updated post I had first published on my other blog, Randomness as a Sign of God’s Presence, in September of 2007. In the updated post, Randomness as a Sign of God’s Presence, Prior Post Updated to 2010, I wrote:
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
What is this all about? The short story is:
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.
One of my college professors put it in a slightly different way. He told us that all of probability theory can be enfolded into a fully deterministic Measure Theory without losing any content. Still another way to express this insight is: probability theory is useful mostly as an introduction to measure theory, though many don’t really go beyond the simple applications which can be taught using decks of cards or pairs of dice or bins of colored balls. A naive and pseudo-rational version of mystical randomness remains valid as a teaching tool. What is remarkable is the number of people who learn their probability and statistics from this viewpoint, never move on from the mystical viewpoint, and yet advocate a fully deterministic understanding of our complex world.
From facts come—sometimes—patterns. We’ve become somewhat accustomed, by way of terribly vulgarized mathematics and biology and other sciences, to the idea that patterns come from `randomness’ or `chaos’. Something of an overview can be communicated to those who have not heard of Poincare or Hadamard or Duhem, Ruelle or Smale or Prigogine and to those who don’t know what a nonlinear equation is; we should wonder what sense these people make of it. We are at a more complex transition point than the one noted by Oystein Ore, prominent number theorist and teacher (see Number Theory and Its History republished by Dover Publications in 1988): in the 14th century or so, long division was coming into use and was considered to be a topic for mathematical geniuses, well beyond those even of more normal high intelligence. Nowadays, we start learning long division in mass education elementary schools, though many still have trouble with it and some can never master it even to the point of figuring how much per pound a roast costs if 4.5 pounds costs $25.
The main point is that a shift from a `mystical’ or `irrational’ understanding, or misunderstanding, of probability theory to a more rational understanding of measure theory changes little except to clear our minds of rubbish and to allow us to move on. The famous distributions of probability theory (Poisson and binomial and so on) remain as does the remarkable tendency for disorder, mystical randomness or factuality, to produce patterns. Those who see a Creation and those who see a Universe barren of divine presence can continue their debates, perhaps on a somewhat more rational level. The various arguments remain equally strong or weak.
Moreover, most scientists including physicists such as Weidlich and many evolutionary biologists and certainly most geneticists use the term `random’ without qualification but seem to be using that term in the more modern sense—that of algorithmic complexity theory. And, to be quite fair, I think many philosophers and historians and scientists and engineers have always interpreted `randomness’ in terms of factuality or even some sort of complexity. After all, there is nothing non-deterministic about those standards teaching tools in probability theory, cards and dice and bins of colored balls.
We’ve allowed our thinking to be constrained and distorted by popular misunderstandings of such terms as `random’ and `deterministic’ and `non-deterministic’. To a certain extent, this deep confusion has even spread into our understandings of `factuality’ and `causality’.