Comment by midtake
3 hours ago
You have a good point about the human rate of mathematical discovery, but Ayer was an idiot and later Witt contradicted early Witt. For the "already implicit" claim to be true, mathematics would have to be a closed system. But it has already been proven that it is not. You can use math to escape math, hence the need for Zermelo-Frankel and a bunch of other axiomatic pins. The truth is that we don't really understand the full vastness of what would objectively be "math" and that it is possible that our perceived math is terribly wrong and a subset of a greater math. Whether that greater math has the same seemingly closed system properties is not something that can be known.
At this point I think the category theorists hit the foundational idea squarely on the mark:
1. Start with a few simple but non-trivial terms and axioms
2. Define "universal constructions" as procedures for building uniquely identifiable structures on top of that substrate
3. Prove that various assemblages of these universal constructions satisfy the axioms of the substrate itself
4. "Lift" every theorem proven from the substrate alone into the more sophisticated construction
I'm not a mathematician (I just play one at my job) so the language I've used is probably imprecise but close enough.
It may be true that you can't prove the axioms of a system from within the system itself, but that just means that you need to make sure you start from a minimal set of axioms that, in some sense, simply says "this is what it means to exist and to interact with other things that exist". Axioms that merely give you enough to do any kind of mathematics in the first place, that is. If those axioms allow you to cleanly "bootstrap" your way to higher and higher levels up the tower of abstraction by mapping complex things back on to the simple axiomatic things, then you have an "open" or infinitely extensible system.
> Whether that greater math has the same seemingly closed system properties is not something that can be known
negative numbers were invented to solve equations which only used naturals. irrationals were invented to solve equations which could be expressed with rationals. complex numbers were invented to represent solutions to polynomials. so on and so forth. At each point new ideas are invented to complete some un-answerable questions. There is a long history of this. Any closed system has unanswerable questions within itself is a paraphrasing of goedel's incompleteness theorem.
I agree with you all around except it's somewhat up for debate actually that the PI is "contradicting" the Tractatus. That is, there is the so called "resolute reading" of the Tractatus that had some traction for a while.
But note this is more to say that the Tractatus is like PI, not the other way around. And in that, takes like GPs would be considered the "nonsense" we are supposed to "climb over" in the last proposition of Tractatus.