I'd be very surprised if there aren't huge areas of undiscovered math that can't be explained with either geometric or algebraic views.
Math is entirely subjective. "Proof" essentially means "Other educated practitioners have the same experience when trying to understand this."
The logical steps that proofs are built on all have that common foundation. Our concept of logic based on our subjective experience of "truth." We've built machines that reproduce our subjective processes mechanically, but there is no sense in which this idea of "true" is truly objective. It happens to be computationally convenient, and it has some relationship to experience, but that doesn't make it an independent reality that all possible observers, human and otherwise, would agree on.
We're really just mapping our own minds through our own experiences.
Animal brains can't abstract like (some of) our brains can. What are the odds our brains are limitless and don't have some similarly crippling limitations from a couple of levels up?
One of the tells for ASI is that it will start reasoning at those levels, using cognitive techniques that are completely incomprehensible - not just because of brute volume, but because our brains won't have the wiring to get a foothold on them.
Some of the products will be reducible to human cognition, in a distorted and simplified form, but many won't.
So - I disagree with Egan. I don't think there's going to be a universal proof library, and even if there were we'd only ever get the Cliff Notes version.
Math is entirely subjective. "Proof" essentially means "Other educated practitioners have the same experience when trying to understand this."
The logical steps that proofs are built on all have that common foundation. Our concept of logic based on our subjective experience of "truth." We've built machines that reproduce our subjective processes mechanically, but there is no sense in which this idea of "true" is truly objective. It happens to be computationally convenient, and it has some relationship to experience, but that doesn't make it an independent reality that all possible observers, human and otherwise, would agree on.
We're really just mapping our own minds through our own experiences.
Animal brains can't abstract like (some of) our brains can. What are the odds our brains are limitless and don't have some similarly crippling limitations from a couple of levels up?
One of the tells for ASI is that it will start reasoning at those levels, using cognitive techniques that are completely incomprehensible - not just because of brute volume, but because our brains won't have the wiring to get a foothold on them.
Some of the products will be reducible to human cognition, in a distorted and simplified form, but many won't.
So - I disagree with Egan. I don't think there's going to be a universal proof library, and even if there were we'd only ever get the Cliff Notes version.