Misinterpretations of Gödel’s Theorem

Philosophers who use Gödel’s incompleteness theorem to make claims about “fundamental limits of human knowledge” have made a category error.

Philosophers who use Gödel’s incompleteness theorem to make claims about “fundamental limits of human knowledge” have made a category error. It’s about axiomatization, not epistemology.

Gödel’s Theorem and Knowledge Limits

Firstly, epistemology is just fancy technical jargon for "what and how we can know." So, knowledge. Questions in the field of epistemology are questions that deal with the nature, sources, and limits of knowledge. The "nature" of this knowledge even includes defining what knowledge is (see Gettier’s problem), but this is beside the point.

On the other hand, Gödel’s (first) incompleteness theorem is regarding what can be proven within a "formal" system. These words are important. "Formal" has a specific meaning, and "within" also has a specific meaning. The claims of Gödel’s theorem are of a completely different domain than claims in epistemology. The confusion between them has led to some quite wild misinterpretations.

Let’s clarify what Gödel actually proved:

Any consistent recursive axiomatization of arithmetic is incomplete.

Again, each word is important.

In simpler terms, if you have a formal system that’s consistent (doesn’t prove contradictions) and can be mechanically checked (recursive axiomatization), there will always be true statements in that system that can’t be proven within the system itself.

Sounds profound. However, unless you already think our knowledge is a formal system (that we’re trapped inside Peano arithmetic), then it’s difficult to see why this would have any implication for our general limitations (or lack thereof) of knowledge.

We use multiple systems for knowledge generation, not just "formal" but also "informal" reasoning like intuition and even empirical observation.

Furthermore, unlike what you may infer from Russell’s 200-page proof of 1+1=2, we don’t use axioms to justify our knowledge, but we use our prior established mathematical knowledge to justify the fittedness of axiomatic systems. It’s the opposite! Russell himself even remarked about this.

In some sense, it’s like building scaffolding to support the ground. It’s impressive, but entirely unnecessary.

Even further, we didn’t need Gödel to tell us there existed some facts about math that will forever be beyond our grasp. You have finite working memory, finite processing speed, and a finite life. You can’t mentally compute a 94,145-digit addition problem, let alone one with googolplex digits, etc.

You can imagine an exam paper with a basic addition problem so large it stretches beyond the boundaries of the observable universe. You’d have no hope of completing it!

Fun Side Note: In high school, we were assigned tedious high-dimensional matrix multiplication for homework. I was so dismayed by this that I called the Toronto District School Board and got my teacher in trouble. From that point on, I was exempt from these glorified numerical dot products and haven’t looked back since...

Finite Minds, Infinite Truths

Our inability to grasp most theorems of Peano arithmetic has nothing to do with Gödel and everything to do with our finite cognitive capacities. Even if we somehow overcame Gödelian limitations, these mundane constraints would still prevent us from knowing many, if not most, mathematical truths.

Also, think about this. You know the Gödel sentence itself is true, despite being unprovable within its system. How? Because you can step outside the formal system and use meta-reasoning! You’re not a formal system. This fact alone dismantles the claim that Gödel’s theorem imposes fundamental epistemological limits.

The Paris-Harrington theorem is another example, as it shows there exist statements that are unprovable in Peano arithmetic yet actually are proven using other mathematical resources. We have knowledge of these truths despite their unprovability in one formal system.

So (as Graham Oppy points out), when theologians like Dennis Turner compare our knowledge of mathematics to our knowledge of God, claiming both are fundamentally limited by Gödelian constraints, they’re making an elementary category error. Our mathematical knowledge is not analogous to our theological knowledge, and Gödel’s theorem doesn’t support any such analogy.

Likewise, when continental philosophers invoke Gödel to defend notions of radical indeterminacy or undecidability of theories, they’re misapplying a precisely defined mathematical result to vaguely articulated philosophical positions.

It turns out that axiomatization ≠ knowledge. To me, that’s the real lesson to take away from Gödel, although you arguably could have concluded that from Löwenheim-Skolem’s theorem a few years earlier.

Even if our minds were formal systems, no one is consistent in their beliefs, and thus Gödel’s theorem doesn’t apply to our knowledge generation anyhow.

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—Curt Jaimungal

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