I believe in nothing. It’s hard to have knowledge when you believe in nothing. But it’s hard to believe in anything when you understand that belief is always based on a foundational assumption, which can never be proven from within the presupposed system itself. This is my own recapitulation of what is perhaps a shallow understanding of the German logician Kurt Gödel’s incompleteness theorems. Theorems, which the Stanford Encyclopedia of Philosophy states, “concern the limits of provability in formal axiomatic theories.” These formal axiomatic theories, and the limitations upon which they are necessarily enveloped, are the fundamental basis for all that it is we claim to know. Discovering Gödel’s incompleteness theorems were like discovering a syntax for my madness. It was like finding an interlocutor who, in the midst of philosophical logic and axioms, stumbled upon a hole in the page and for the first time in the history of modern anglo-american philosophy had thought to consider the void as constitutive for/of thought rather than something to be excluded/disavowed by thought. But what exactly do these incompleteness theorems say? Let us return again to the Encyclopedia of Philosophy:
The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F. According to the second incompleteness theorem, such a formal system cannot prove that the system itself is consistent (assuming it is indeed consistent).
In any consistent formal system within which a certain amount of arithmetic can be carried out, there are statements of the language of said system which can never be proved nor disproved in said system. What does this say about systems? What does it elucidate about their character? What does it mean that a system of thought can produce statements which it can neither prove nor disprove? And what does it matter? One can easily see how this line of questioning might bring one towards a bridge with God. What does it mean for this formal world of ours, with its neat geometry, and precise mathematics, data and science, to continue to produce within its formal elaboration a statement which can neither be proved nor disproved? A statement like God. What does this say about systems? What does it elucidate about their character? What does it mean that a system of thought can produce statements which it can either prove nor disprove? There is an atheist somewhere stating, in relation to this question of God and systems of thought, that what all this may mean is that any consistent formal system will always produce an illusion which it knows not how to distinguish from reality. God is such an illusion. And then the biologist might spring forth and state, “Indeed, and this illusion comes with its necessary reproductive and evolutionary benefits and drawbacks.” Gödel, however, went mad at the sight of his own discovery. Indeed, the second incompleteness theorem too is a bit dizzying. A formal system cannot prove that the system it is utilizing is itself consistent. A system of thought cannot prove its own consistency from within its own system. If these theorems are true, then how do we know? How do we know anything at all? How can we trust any of our “consistent” formal theorems and conceptual axioms of knowledge? Perhaps, the question is naïve following its train of thought and if so, I imagine the anglo-americans will be quick to enlighten me on the matter; however, I think the question is a valid one considering philosophical investigations into the nature of what has been so reductively titled “logic” are nothing if they are not tied to epistemological questions as well.
However, the Encyclopedia of Philosophy seems to warn against putting Gödel, God and Epistemology together. As it states:
Sometimes quite fantastic conclusions are drawn from Gödel’s theorems. It has been even suggested that Gödel’s theorems, if not exactly prove, at least give strong support for mysticism or the existence of God. These interpretations seem to assume one or more misunderstandings which have already been discussed above: it is either assumed that Gödel provided an absolutely unprovable sentence, or that Gödel’s theorems imply Platonism, or anti-mechanism, or both.
These “fantastic conclusions” are fantastic only because there is a monolithic regime of truth which grounds the axiological production of the adaptive-truth-for (Wynter) of our current conceptual economy. For what is fantastic in Gödel’s conclusion is not the proof of the existence of God in its Platonic or Anti-Mechanistic sense, but the proof of the incompleteness of our theories about the nature of reality. The proof of the unprovable that always already exist in what is proven today. It is this fact, that haunting fact, that to know is to know that what one knows is necessarily incomplete is what makes this question of Gödel, God and Epistemology so interesting. To paint this picture further I’ll return one last time to the Encyclopedia of Philosophy to read it against itself. The Encyclopedia states:
A common misunderstanding is to interpret Gödel’s first theorem as showing that there are truths that cannot be proved. This is, however, incorrect, for the incompleteness theorem does not deal with provability in any absolute sense, but only concerns derivability in some particular formal system or another. For any statement A unprovable in a particular formal system F, there are, trivially, other formal systems in which A is provable (take A as an axiom). On the other hand, there is the extremely powerful standard axiom system of Zermelo-Fraenkel set theory (denoted as ZF, or, with the axiom of choice, ZFC; see the section on the axioms of ZFC in the entry on set theory), which is more than sufficient for the derivation of all ordinary mathematics. Now there are, by Gödel’s first theorem, arithmetical truths that are not provable even in ZFC. Proving them would thus require a formal system that incorporates methods going beyond ZFC. There is thus a sense in which such truths are not provable using today’s “ordinary” mathematical methods and axioms, nor can they be proved in a way that mathematicians would today regard as unproblematic and conclusive.
This extremely powerful standard axiom of Zermelo-Fraenkel set theory still cannot overpower this axiom of incompleteness. It is this axiom of incompleteness and its ever penetrating consistency in relationship to the “non-provability” of certain arithmetical truths (which even ZFC cannot prove) that makes this question of Gödel, God and Epistemology one worth considering. For, what is it that maintains the formal system of incompleteness? Why incompleteness at all? Whence and where does incompleteness arise? In addition to this, the fact that “today’s ‘ordinary’ mathematical methods and axioms,” do not provide us with an intelligible matrix which “mathematicians would today regard as unproblematic and conclusive” makes one wonder again: why is it that even the extremely powerful standard axiom systems like ZFC still possess incompleteness at the core of its systematic elaboration?
Control F, “Cantor,” on the Standford Encyclopedia page dedicated to Gödel’s incompleteness theorem and you’ll see that Georg Cantor, the mathematician that founded set theory, only appears as part of the bibliography as reference from within certain academic papers and yet, not in the body of the page at all. This is fine but I think it is important to note here that Cantor thought that God could be understood as the set of all infinite sets – the absolute infinity. Thus, once again, counter-acting the idea that this question of Gödel, God and Epistemology are simply “fantastic conclusions.” They are as fantastic of conclusions as the founder of set theory itself are willing to draw. Admittedly, Cantor lost his mind too but, what separates the mathematicians and the philosophy of mathematics from myself is that the mathematician’s believe in math and I believe in nothing.
In other words, I believe in incompleteness as the void through which every constructible (Gödel once again) formal system elaborates itself, only for that system to never be able to be completely consistent within itself, as well as for this same system to produce unsolvable problems that therefore require additional systems which are oftentimes regarded by mathematicians today as problematic and/or inconclusive. While, this is about math, I want to suggest that if Alain Badiou and a host of other philosophers of mathematics, physics, science, etc. are right in thinking that mathematics is ontology (which I’m not quite sure they are) then the metaphysical question being considered here are metaphysical questions with implications for every sphere of ontic activity period. But, I believe we must start with the void, with nothing and then, proceed to understand that each and every understanding is necessarily a constructible system that always already must disavow its inconsistencies and its unprovable statements in order to proliferate forward towards “Knowledge,” and “Truth.”
Here is where faith comes in. Here faith is understood not as the evidence of things not seen, but the evidence of things which are always already incomplete and the procedure through which we continue to act and press forward even with full cognizance of the necessarily incomplete formation of our cognitive maps, schemas and theories (one of the most powerful of which is ZFC). Once it is understood and accepted that incompleteness is a necessary constituent element of our formal conceptual mapping and theoretical language then it must necessarily follow that we are always already in the midst of not-knowing. That the first statement of truth we can utter, following Gödel, is that our statements of truth are always already preconditioned by a kind of ignorance and inconsistency. Then, we can go forward with Faith as a supplement to what we consider “Knowledge” in order to elaborate formal systems that are constructed, imagined, enacted, performed, materialized, discovered and instrumentalized.
It is the history of modernity, of Anglo-American, Anglo-Saxon, Euro-Modernity, to turn Faith into Ignorance or Blindness. But, it is this moment in post-modernity with its discontents and disruptions that require us to reconsider our disdain for faith. For what we know to be “Fake News” and the “Information Bomb” is far more difficult than what news pundits have been willing to articulate because rather than a clear-cut distinction between the Real and the Fake, what we are seeing is a war of formal systems of enunciation and the perpetual unwillingness to sit with the inconsistences and the ignorance inherent to our formal systems of enunciation. Faith is not an absence, but the presence upon which our artificial intelligence has been built and the entire world is known through an artificial intelligence masquerading as all-knowing while at the same time failing to know what Gödel has made known to us. Namely, that to know is to know that knowing is incomplete.
Now, while this known incompleteness does not give us God, it is understandable why one might want to situate a deity within this indeterminate field of existence. But, I believe in nothing. I believe in incompleteness. I believe in indeterminacy. Perhaps, this places me within the other category of scorn for the Encyclopedia of Philosophy namely, what they call “the mystics.” While, I place no shame nor issue in the mystics and I find many a mystic to have stumbled upon Gödel’s idea (or the general Idea behind Gödel’s idea) through different systems of illumination, I much rather remain nameless and disaffiliated. I am what I am not. I am not what I am. Or am I?
Some things (not all of them) I looked at to help me understand Gödel and Cantor
I also took a course on the concept of Infinity in Grad School and would be willing to share some papers for very interested folks.