Consider this problem: Are there are an infinite number of 9s in the digits of pi? The answer is obviously yes. The only way it could be no is if pi interacts in some strange way with base-10 representations, and pi has nothing to do with base-10.
But how do you prove that no such interaction exists? You have to rule out an endless number of possible interactions, and even then there could be some grand conspiracy between pi and base-10, hiding in the places you haven’t yet looked.
Proving the absense of an interaction between two areas of math is much harder than proving its presence. If you want to prove presence, you can just find the interaction and explain it. But you can’t “find an absence.”
Most of the hard math problems turn out to have this issue at their core. If you dig into Collatz, you find that it’s very likely to be true. The only way it could be false is if there’s an undiscovered conspiracy between parities of integers and the collatz map. How to prove there is no conspiracy?
Consider this problem: Are there are an infinite number of 9s in the digits of pi? The answer is obviously yes. The only way it could be no is if pi interacts in some strange way with base-10 representations, and pi has nothing to do with base-10.
But how do you prove that no such interaction exists? You have to rule out an endless number of possible interactions, and even then there could be some grand conspiracy between pi and base-10, hiding in the places you haven’t yet looked.
Proving the absense of an interaction between two areas of math is much harder than proving its presence. If you want to prove presence, you can just find the interaction and explain it. But you can’t “find an absence.”
Most of the hard math problems turn out to have this issue at their core. If you dig into Collatz, you find that it’s very likely to be true. The only way it could be false is if there’s an undiscovered conspiracy between parities of integers and the collatz map. How to prove there is no conspiracy?
Aren’t proofs by contradiction the standard technique for proving the absence of a property.
You can prove ¬P by proving that P ⇒ FALSE?