I once watched a lecture on YouTube about Wittgenstein’s philosophy of mathematics.

The speaker said something very interesting: Wittgenstein did not simply dismiss G?del’s incompleteness theorems as nonsense (though he famously said that the liar’s paradox does not interest him). Rather, he thought that G?del himself did not fully understand the meaning of his own discovery.

The issue, according to this interpretation, is not that a formal system is incomplete per se. Instead, the point is that there are things within a system of integers that are not, strictly speaking, integers. For example, the famous self-referential G?del sentence (“I cannot be proven”) can be encoded numerically using G?del numbering (e.g., via primes raised to powers), yet what it expresses is not purely numerical. This suggests that something non-numerical is embedded within the formal system of arithmetic.

If this interpretation is correct, what could that mysterious non-numerical thing be?