The asymptote of reason

“Science may set limits to knowledge, but should not
set limits to imagination” ~ Bertrand Russell

 

Another important mathematician, who produced another important and paradoxical result, was the Austrian Kurt Godel. Where Mandelbrot considered the objective realm, and applied mathematical models to understanding it, Godel’s focus was on the subjective – he was a logician, and he was investigating how we form an understanding of the world rather than what that world might objectively be like.

We could argue that Godel and Mandelbrot were considering the same substance from opposite sides – that Mandelbrot was actually modeling subjective representations and Godel in truth modeling objective realities; we tend to assume that there is a high degree of concordance between subjective representations we can all intersubjectively communicate and objective entities we can all subjectively observe. This is not necessarily the case, as Descartes found to his dismay, and that is why I draw the distinction here between the logical models of Godel and the applied mathematics of Mandelbrot.

When we talk about a logic, we are talking about a framework in which statements represent either properties or relations of entities – this is why I say a logician deals with the subjective realm, because of course the Cartesian theater is constructed out of these same substrates. A logic consists of axioms, and rules for deriving statements from these axioms, and for assigning these statements some truth value. The most commonly understood forms of logic are bivalent, taking truth values of True or False – and, commonly, what we understand to be true is really just that which we observe to be agreed upon. This is not a particularly logical way of considering truth, as it happens, and we’ll come back to talk some more about the consequences of that later.

For right now, we’re going to follow Godel in the examination of a particular question about logic: how far can a logic go? Is it possible for us to conceive of a logic which allows us to examine every possible statement and assign it a truth value?

Godel was looking at logics that could describe arithmetical properties and relations; Mandelbrot’s work indicates that such properties and relations are sufficient to describe very complex processes in the natural world (indeed, Mandelbrot’s work has applications in many fields including sociology and econometrics). Accordingly, his conclusion that in fact no logical system could completely and consistently describe the natural numbers – and, by extension, the world, although Godel himself didn’t go that far with it – says something very important about the limits of human knowledge.

Godel actually devised a whole new numbering system in pursuing his complex proof of the undecidability of certain formal mathematical propositions, but we don’t need to in order to grasp the central idea. We can use a thought experiment to produce the same result.

Consider the case where we actually can design a thorough logic for assigning a valid truth value to any statement – such a logic is programmable into a computer, and allows us then to build a Universal Truth Machine. Any question we put to the Universal Truth Machine can be parsed by it, analyzed with its powerful logic, and identified as True or False.

Suppose now that we test our Universal Truth Machine, and give it this statement to chew on: “the Universal Truth Machine will find this statement to be false.” The Universal Truth Machine can understand this statement, but it can only find it to be false if the statement is in fact true, and its programming defines True and False as exclusive opposites. By the same token, the Universal Truth Machine cannot find the statement true unless it is in fact false. Therefore, our machine cannot give an answer – cannot prove the statement – and so cannot, in fact, be universal.

Godel’s genius was in formalizing a statement that can be represented in any sufficiently developed logic, and that proves for any logic that it somewhere hits this problem. It must, therefore, be either incomplete – being unable to prove the Godel statement – or inconsistent – being able to prove that it is simultaneously true and false without crashing. This fascinating result leads to the peculiar conclusion that our own brains, which encode sufficiently developed logic to model the world, are themselves subject to Godel’s Incompleteness Theorem. That’s a rabbit-hole for another day, however.