What do you mean by "doesn't have proofs" here? It's demonstrably possible to develop a formalised philosophy: see Spinoza's Ethics for an example. It's impossible to prove that one formalisation is more "correct", but the same applies to mathematics, via Gödel's second incompleteness theorem.
>The regress argument has nothing to do with ethics in particular. It applies just as much to any non-proof area of study, including all of science, history, and even ordinary observations and recollections.
It also applies to maths in the form that a system cannot prove its own consistency. Yet that doesn't stop us formally analysing, comparing and making use of different mathematical systems.
This couldn’t be further from the truth. There is a wealth of logics that have been developed and studied within philosophy, that aim to formalize various philosophical concepts. See modal logics, deontic logics, logics about knowledge and belief, and so on.
Logic, mathematics, computer science, proof theory... same thing. https://en.wikipedia.org/wiki/Proof_theory
We are eternally stuck interpreting all of our theories in the context of their meta-theoretic assumptions.
Subject to all the constraints of formal language theory/linguistics.
It fails to address the symbol-grounding problem.
It turns out you can't formalize anything that doesn't have proofs -- in other words, it basically only works for what we'd call "mathematics" or "pure logic". You can't formalize literally anything else in philosophy, at all.
The regress argument has nothing to do with ethics in particular. It applies just as much to any non-proof area of study, including all of science, history, and even ordinary observations and recollections.