Comment by midtake
2 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.
> 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.