
Here’s a hot take i have been sitting on for a while. As AI and formalization converge on mathematics mathematicians are beginning to think and act more and more like computer scientists. See Terry Tao and Ken Buzzard’s talks at the Future of Mathematics Symposium for evidence of what i’m talking about. The more they embrace these methods and principles, the deeper the deeper computer science results will penetrate the field. For example, the notion of Turing complete systems has consequences for many areas of mathematics. Set theory is Turing complete. The constructive reals and the constructive complex numbers are Turing complete. This expressiveness limit is ubiquitous. In this sense research programs like Langlands should not be surprising. In one sense they are building compilers from one programming model to another. Both are Turing complete (or analogs at higher order) so we know beforehand that the compilers exist. Finding and optimizing such compilers can be fun and exciting. But there isn’t a lot of mystery here. It’s completely expected.
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