An article of faith

"Doubt is a pain too lonely
to know faith is his brother."
~ Khalil Gibran

 

 

 

 

 

The Fifth Key of the Tarot, known variously as the Pope or, more commonly, the Hierophant (the word literally means 'teacher of holy things'), is the sixth card along the journey of the Fool. The Law of Fives, that false teacher, suggests it should be significant (not only is it the fifth key, but the prime factors of six sum to... five).

 

We've intimated previously that knowledge is bound by unsurmountable limitations, and that transcendental truth cannot be gained through the application of reason alone. What is left is what necessarily underpins any edifice of reason: faith.

 

Faith is what the Hierophant offers - faith, the "substance of things hoped for, the evidence of things not seen," as the Book of Hebrews relates. Faith is often derided in our materialist culture; but the truth is revealed when we consider the foundation of that culture - for it rests on certain axioms of ontology, of epistemology, which in their nature are not and cannot be proven from earlier principles. Faith is the bedrock of rational consciousness: faith, which appears to admit none of the character of reason, turns out to be essential to reason; just as reason, appearing to ridicule faith, depends upon it. This is an intimate paradox, whose nature I shall leave it to the reader to decide.

 

It is tempting to assert that enlightenment, that cannot be accomplished through Reason alone, can be accomplished through Faith. There are even examples that seem to corroborate this assertion; but, in truth, Faith alone fails too. The reason for this is in fact rather subtle; it has to do with the relative plasticity of Reason.

 

Suppose you hold some view derived logically from certain agreed axioms - as a trivial example, suppose you are of the opinion that there are no black swans, based on the empirical observation that you have never seen anything but white swans and the meta-empirical observation that empirical observations are reliable arbiters of actual fact. Suppose you then encounter a black swan. This new datum contradicts a predicate of your hypothesis, and, as a rational thinker, you revise your hypothesis: you accept the existence of black swans (this possibility is why Hume had a Problem with Induction).

 

Now, suppose your belief that all swans were white stemmed from a pure faith, unsullied by Reason. Suppose you encountered a black swan: your faith would not admit its existence. You would rationalize that it was not a swan, or that it was a white swan painted black, or that you imagined it, or any of a hundred other counterfactuals to avoid having to assail your article of faith.

 

Faith in the transcendent is a precursor to enlightenment; faith in the merely subjective is a barrier to enlightenment. And neither Faith, nor Reason, will enable us to tell the difference...

 

This then, is both the power and the peril of the Heirophant: that he offers a reality more permanent than the one we can apprehend through Reason, yet less certainly true.

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.