3 ms·
It's both. The axioms are invented, the corpus of theorems is discovered. As once the axioms are chosen the provable theorems are already fixed. But the axioms
by pa7x1 3y ago
It's both. The axioms are invented, the corpus of theorems is discovered. As once the axioms are chosen the provable theorems are already fixed.
But the axioms are a choice, and we can pick different ones. The common choice of axioms is utilitarian, they lead to interesting math that helps us describe the universe.
- random3 3y agoAre the theories beyond axiom fundamentally different if axioms are changed, though? And if not, aren't then axioms merely props or placeholders for invariants?
- sweezyjeezy 3y agoYes they are different - example: https://en.m.wikipedia.org/wiki/Parallel_postulate https://en.m.wikipedia.org/wiki/Parallel_postulate
- sweezyjeezy 3y agoI'm not sure I see why the axioms were not also discovered though? Choice between irreducible assumptions does not seem to make them any more 'invented'.
- pa7x1 3y agoWithout entering into an endless debate about semantics and metaphysics I would simply say that if you want to use the word discovered for the axioms then you must acknowledge that the theorems are not the same kind of discovered.
- Mimmy 3y agoI would agree the axioms are chosen, but what’s the connection between choosing something and inventing it? Choosing to study molecular biology doesn’t mean cells are a human invention.
- mensetmanusman 3y agoThat the universe chose axioms is indeed the mystery.
- hurryer 3y agoWhat about natural numbers? I've read that Godel's result and diagonalization procedure shows that they exist (not invented).
- openquery 3y agoProving a theorem given a set of axioms is a search problem. Given a set of axioms you can apply rules of inference to generate the graph of all provable theorems. Proving a theorem is about finding a path from the axioms to the vertex which is your theorem. But you can make the same case for axioms - that they are not invented but discovered through a process of search in the space of axioms.