One idea that comes up a lot in certain technology circles, especially cryptocurrency, is the idea of a “zero-knowledge” proof, or ZKP. Here’s a quick analogy that might be helpful to explain how a ZKP works. Let’s say there is a wall of 1,000 locked safes. I want to verify that you own each safe. You could do that by unlocking all the safes, but that would reveal the combination to me, and you don't want me to know the secret. We seem to be at an impasse. One way out of this dilemma is for me to leave the room. I tell you, “I’m going to leave and come back in 1 hour. Please open all the safes.” If I come back and the safes are open, it's probably because you knew the combinations to each safe. You never had to tell me the secret (the safe combination), and I don’t have to know what the secret is to be satisfied that you know it. That’s the “zero knowledge” part.

Constructing proof is an art, checking them is a science. Making a distinction between the statement of a theorem and the proof is important. It means that if we have a proof of P, we can make a proof of Q. It is a function from the proofs of P to the proofs of Q. It is a function sending an element of P to an element of Q. It is a term of type P → Q. …This is why in the natural number game we use the → symbol to denote implication.

the journal’s acceptance would create a situation that is “historically unparalleled in mathematics: a claim by a well-respected journal that they have vetted the proof of an extremely well-known conjecture, while most experts in the field who have looked into this have been unable to understand the proof”.

Companion files for Logical Verification 2020–2021 at VU Amsterdam - GitHub - blanchette/logical_verification_2020: Companion files for Logical Verification 2020–2021 at VU Amsterdam

Mochizuki had created so many new mathematical tools and brought together so many disparate strands of mathematics that his paper was populated with vocabulary that nobody could understand. It was totally novel, and totally mystifying. “You don’t get to say you’ve proved something if you haven’t explained it,” she says. “A proof is a social construct. If the community doesn’t understand it, you haven’t done your job.”

acceding that temporary ugliness is an essential feature of doing mathematics; you can’t build cathedrals without putting up scaffolds. ​ secrets the erasers and wastebaskets of mathematicians could tell us, if only they could talk!

It is one thing for a proof to be correct; quite another, for a proof to be comprehensible.

Finally, I am concerned about the state of pure mathematics research. More and more, results depend on theorems whose proofs are unpublished or sketchy (or even, in places, incorrect). We rely more and more on unnamed teams of experts who have a sufficiently broad overview of an area to be able to tell us with confidence which papers can be trusted. Fashions change, people desert areas, and I am genuinely scared that we are leaving a mess behind in some areas. There are some theorems whose proofs might be difficult or impossible to reconstruct in 20 years’ time.

I think doing math professionally can be stifling as it can feel like you’re expected to know everything while staying collected. This blog is my outlet to do my best to grow as a mathematician while giving myself and others permission to be wrong, ignorant, and emotional about math!

Unfortunately this sentiment is mirrored among most programmers who claim to be interested in mathematics. Mathematics is fascinating and useful and doing it makes you smarter and better at problem solving. But a lot of programmers think they want to do mathematics, and they either don’t know what “doing mathematics” means, or they don’t really mean they want to do mathematics. ​ Honestly, it sounds ridiculously obvious to say it directly like this, but the fact remains that people feel like they can understand the content of mathematics without being able to write or read proofs. ​ So read on, and welcome to the community. ​ I honestly do believe that the struggle and confusion builds mathematical character, just as the arduous bug-hunt builds programming character. ​ I’m talking, of course, about the four basics: direct implication, proof by contradiction, contrapositive, and induction. These are the loops, if statements, pointers, and structs of rigorous argument, and there is simply no way to understand the mathematics without a native fluency in this language. ​ And so it stands for mathematics: without others doing mathematics with you, its very hard to identify your issues and see how to fix them. ​ And finally, find others who are interested in seriously learning some mathematics, and work on exercises (perhaps a weekly set) with them.