Hacker News Reader

Comment by soVeryTired | Hacker News Reader

IMO it's not far off how most python or javascript devs don't care about registers or cache misses. Someone's thought deeply about those things so you don't have to.

Mathematicians do care about how much "black magic" they're invoking, and like to use simple constructions where possible (the field of reverse mathematics makes the central object of study). For example, Wiles' initial proof of Fermat's last theorem used quite exotic machinery called "inaccessible cardinals", which lie outside of ZFC. Subsequent work showed they weren't needed.

Another good example of mathematicians caring which 'house of cards' their results are built on is the search for an "elementary" proof of the prime number theorem (i.e. showing it doesn't rely on complex analysis).

Edit: here's a great related discussion on MathOverflow, bringing in analogies from CS: https://mathoverflow.net/questions/90820/set-theories-withou...

View on HN

> Mathematicians do care about how much "black magic" they're invoking, and like to use simple constructions where possible (the field of reverse mathematics makes the central object of study). For example, Wiles' initial proof of Fermat's last theorem used quite exotic machinery called "inaccessible cardinals", which lie outside of ZFC. Subsequent work showed they weren't needed.

In a way mathematicians can afford to do this more readily than people in software development, because if something is actually proven, then you can 100% rely on that. With software not so much. Or rather: Software usually is not proven to be correct, because that's usually expensive. In mathematics they don't have to consider the runtime of an algorithm, when they "merely" need to prove correctness. The time it needs to run is irrelevant for its correctness. And so they can stack and stack and stack, provided that each piece is proven correct, and it won't have negative consequences. Well, almost. There is some negative consequence in that another human being, wanting to understand a proof, needs to know perhaps many concepts and other proofs, in order to be able to do so. But that's probably the only reason to pursue simplicity in mathematics.

> The time it needs to run is irrelevant for its correctness. And so they can stack and stack and stack

This is a very naive take - the very direct translation of what you're saying doesn't happen does in happen in analysis all the time: there are many inequalities which can be "stacked" to prove a bound on something but their factors are too large so you cannot just stack them if you need a fixed bound for your proof to go through. Unsurprisingly this is exactly how actual runtime analysis also works (it's unsurprising because they're both literally math).

I think you are taking it a bit out of context here. Obviously, I assumed in that phrase, that mathematicians are stacking suitable methods and proofs. If some factors are too large to not fit in some bound, then obviously that's not something you would stack. But once you have suitable proofs and proven correct and suitable methods, you can stack, and correctness does not go out of the window. Correctness remaining, you will be able to get to a proven correct result.

Of course proving things in mathematics is also a lot harder, usually, than computer programming, and it is probably still easy to make mistakes.

> Mathematicians do care about how much "black magic" they're invoking, and like to use simple constructions where possible (the field of reverse mathematics makes the central object of study).

I'd be careful about generalizing that to all or most 'mathematicians'. E.g., people working in a lot of fields won't bat an eye at invoking the real numbers when the rational or algebraic numbers would do.

I'm sure some python devs care about cache misses too. I guess my point was that the big results will be picked over again and again to understand _exactly_ which conditions are needed for them to hold.

Reply on news.ycombinator.com