Posted by Jason Polak on 02. September 2018 · Write a comment · Categories: opinion · Tags: ,

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.
More »

Posted by Jason Polak on 23. August 2018 · 2 comments · Categories: book · Tags: , ,

When it comes to the philosophy of science, not many publications are relevant to modern practice. Let's take math. The current literature still talks about platonism. Look harder and you might find the rise of non-Euclidean geometry or other breakthroughs like cardinality. In short, the bulk of mathematical philosophy still consists of math that's hundreds of years old. While these topics are still important, I find it much more interesting to look at the new philosophical issues present in modern mathematics and science.

That's why I was delighted to find Lost in Math by Sabine Hossenfelder, who is also the author of a popular physics blog called Backreaction.
More »