When a solution is formalized inside a theorem prover, it is reduced to the level of arithmetic (a theorem prover is an arithmetic-level machine).
So a theory might be a very high-brow math, but a formal derivation is still arithmetic (if one just focuses on the syntax and the formal rules, and not on the presumed semantics).
Yes, the technique of formal proofs, in effect, involves translation of high-level proofs into arithmetic.
So self-reference is fully present (that’s why we have Gödel’s results and other similar results).
What this implies, in particular, is that one can reduce a “real proof” to the arithmetic; this would be ugly, and one should not do it in one’s informal mathematical practice; but your post is not talking about pragmatics, you are referencing “fundamental limit of self-reference”.
And, certainly, there are some interesting fundamental limits of self-reference (that’s why we have algorithmically undecidable problems and such). But this is different from issues of pragmatic math techniques.
What high-level abstraction buys us is a lot of structure and intuition. The constraints related to staying within arithmetic are pragmatic, and not fundamental (without high-level abstractions one loses some very powerful ways to structure things and to guide our intuition, and things stop being comprehensible to a human mind).
When a solution is formalized inside a theorem prover, it is reduced to the level of arithmetic (a theorem prover is an arithmetic-level machine).
So a theory might be a very high-brow math, but a formal derivation is still arithmetic (if one just focuses on the syntax and the formal rules, and not on the presumed semantics).
The theorem prover point is interesting but misses the key distinction:
Yes, theorem provers operate at a mechanical/syntactic level
But they still encode and use higher mathematical concepts
The proofs they verify still require going above pure arithmetic
The mechanical verification is different from staying within arithmetic concepts
In pure arithmetic we can only:
Do basic operations (+,-,×,÷)
Check specific cases
Compute results
Any examples of real conjectures proven while staying at this basic level?
Yes, the technique of formal proofs, in effect, involves translation of high-level proofs into arithmetic.
So self-reference is fully present (that’s why we have Gödel’s results and other similar results).
What this implies, in particular, is that one can reduce a “real proof” to the arithmetic; this would be ugly, and one should not do it in one’s informal mathematical practice; but your post is not talking about pragmatics, you are referencing “fundamental limit of self-reference”.
And, certainly, there are some interesting fundamental limits of self-reference (that’s why we have algorithmically undecidable problems and such). But this is different from issues of pragmatic math techniques.
What high-level abstraction buys us is a lot of structure and intuition. The constraints related to staying within arithmetic are pragmatic, and not fundamental (without high-level abstractions one loses some very powerful ways to structure things and to guide our intuition, and things stop being comprehensible to a human mind).