User:Yto

Hi, i've been an apeirologist for approximately 4 years. I thought about writing articles on this wiki, but realized that i'm bad at finding sources, so i'll just use this page for unsourced explanations and philosophy related to apeirology. Hopefully that's ok.

Social and formal proofs

There are two kinds of proofs in math. Formal proofs start with specified axioms and use well-defined rules to make inferences. Meanwhile, social proofs don't necessarily start with any axioms, and inferences can be made intuitively. The goal of a social proof is to convince the reader that something is true, while the goal of a formal proof is to show that if the axioms are true, then the theorems are also true, therefore a social proof of the axioms would lead to a social proof of the theorems.

I'll edit this to provide examples soon.

Platonism

Platonism is the philosophy of Plato. It asserts that abstract objects such as beauty and injustice exist. In the context of mathematics (mainly set theory), i think "platonism" commonly refers to a view asserting that something has a value independent of our thoughts. For example, a platonist may believe in an independent universe of sets, or only in an independent set of natural numbers. Due to this variety, platonism is not a single belief, but a way to compare beliefs - one belief is "more platonic" than another if it asserts that more things are independent of thoughts. A view with absolutely no platonism is pointless, because that would mean nothing can be certain, not even statements such as 1+1=2, or the existence of anything at all.

For example, if we abandon the idea that there is no integer between 0 and 1, despite this being true in our physical world, there are suddenly many things we cannot be certain about. What if 7 is the largest prime number, and all the larger numbers that we call "primes" are actually divisible by this hidden integer n? How would a function with n inputs work? Is 2+2 still 4, or could it now be 3? Maybe 2+2 is an integer between 3 and 4. Being too unplatonic leads to confusing (but entertaining) questions like this, and if we want to get any advanced results, it's more useful to simply assume that some things are the way we expect them to be.

This is closely related to social proofs. A more platonic view allows more social proofs. For example, the belief that the axioms of Peano arithmetic are true independently of our thoughts basically directly assigns social proofs to all formal proofs in Peano arithmetic. However, a belief in an independent set of natural numbers can lead to social proofs of much more, probably including things like the consistency of Peano arithmetic. It also implies that there are independent truth values of the twin prime conjecture and the goldbach conjecture, but we don't have any social proofs of those truth values yet, and i don't know whether it's possible that social proofs of these would require a more platonic view, or even not exist at all.

I'll add more to this eventually.