MathOverflow

What is mathematics about? Is it about numbers? Logic? If we consider it as being about logic we end up with Gôdel type problems. It also cannot be about numbers, and so on. So, what can it be about?
It is usually in physics we develop theory through thought experiments, coming from physics background, I feel the same can be done with mathematics also....

I was exploring the identity
$$1^3+2^3+\cdots+n^3=\left(1+2+\cdots+n\right)^2$$
and I noticed a decomposition pattern that seems different from the standard induction or multiplication-table proofs.

Define the quantity $A(n,k)$ as follows:
$A(n,k)$ denotes the minimum number of $k$-element nonnegative-sum subsets of $n$ arbitrary real numbers $x_1,\dots,x_n$ whose sum is non-negative.
More formally, $A(n,k)$ is given by

I know there are multiple different frameworks for iterating priority arguments as schemes beyond monstrous priority arguments are too complicated to carry out in a classical manner. From what I know the first of its kind was Harrington's workers type...