I happened to come across a 1993 opinion piece, Theorems for a price: Tomorrow's semi-rigorous mathematical culture by Doron Zeilberger. I think it's a rather fascinating document as it questions the future of mathematical proof. Its basic thesis is that some time in the future of mathematics, the expectation of proof will move to a "semi-rigorous" state where mathematical statements will be given probabilities of being true.

It helps to clarify this with an example even more simple than in Zeilberger's paper. Take the arithmetic-geometric mean inequality for two variables $a,b\geq 0$. It says that

$$\frac{a + b}{2} \geq \sqrt{ab}.$$ This simple identity is just a rearrangement of the inequality $(a – b)^2 \geq 0$. For simplicity, let's say that $a,b\in [0,1]$. Instead of actually proving this inequality, we could generate uniform random numbers in $[0,1]$ and see if this inequality actually works for them. So if I test this inequality 1000 times, of course I will get that it works 1000 times.

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