These are some reflections on what chaos means in the context of mathematics and physics. Chaos must be understood as the absence, or privation, of order, not as the opposite of order in any active sense.

To explore what chaos may mean from the mathematical point of view, we can consider an irrational real number between zero and one in its decimal expansion to try to represent a number representing true lack of order, that is, the ultimate random number. A rational number is not random, since its digits will repeat after a certain point. The ultimate random number will have two properties:

1. Not Computable: There is no algorithm to determine the sequence of digits
2. Normal: Every digit, and sequence of digits, is evenly represented in the decimal expansion.

Numbers such as √2 and π fail the first condition since there are mathematical formulas to compute the decimal expansion to any desired precision.

In the decimal (or any base) expansion of normal numbers, each digit will appear with the same frequency, i.e., 10%. Furthermore, each sequence of digits will also appear with its corresponding frequency. That is, 10 through 99, 100 through 999, and so on, will also appear 1% and 0.1% of the time respectively. √2 and π are believed to be normal, but there is no proof. The mathematician Gregory Chaitin defined a number Ω, Chaitin’s constant, that has both properties.

Condition 1 is necessary because an algorithm is the hidden order behind a seemingly random sequence of numbers. Condition 2 is necessary because complete chaos would have to include all possibilities.

In our random number, then, each digit is independent in the sense that its value does not depend on the digit, or sequence of digits, that precede it. Investigating this number, we will observe at some point the sequence of numbers from 1 to 1 million. This poses a dilemma. Do we conclude, then, that we have discovered a region of order within that alleged chaos, or are we convinced that it is truly random? This is not merely speculative as it has real world applications.

• In Intelligent Design theory, recognized patterns are presumed to result from a designer rather than random variation.
• In the search for extra-terrestrial intelligence, cosmic radio waves are scanned to look for patterns hidden within the white noise.
• In the multiverse theory of physics, all possible worlds exist and we just “happen” to live in one of them that exhibits the necessary degree of order to support human life.
• Then, there is the Infinite Monkey Theorem that claims that enough monkeys with typewriters, given enough time, will produce the works of Shakespeare.

We have previously asked this question about whether each moment in time is independent of any previous moment or if there is a deep connection between the past and the present. The first is chaos; the second may be either order or chaos deceptively ordered. There is no way, apparently, within a chaotic situation to determine if order is accidental or real.

Two of the conditions of manifestation are form and matter. Chaos can only be associated with prime matter, Prakriti, without property and infinitely receptive to any form. Matter, in this sense, can never “exist”, that is, it can never be a possibility of manifestation. Hence, Chaitin’s constant, is only a thought construction and not a reality. That is why Guenon says that at the heart of the Kali Yuga, where chaos reigns, the state of total disorder can never be reached.

Thus, form is a necessary condition of manifestation; it is transcendent to matter and we know it through our will and intelligence. I design, hence I recognize design; I communicate, hence I recognize a communication; I create, hence I recognize an artistic creation.