alt Hacker NewsPeople think mathematics is about proving theorems.
I think that's just an accident of history.
When we write software, we very seldom write proofs that our algorithms are correct. We just write tests, and we also run the algorithms and when they fail we know we have a bug and then we proceed to debug, fix, and add new tests (if we are disciplined, but most of us are). In time, by usage and testing, we gain confidence that our battle tested software is correct, mostly. And we tell people that we will never be 100% confident that any software is bug free. But that's a slight lie: if we wanted such confidence, we would start using provers, and create bug-free software. That possibility exists, but it's just extraordinarily expensive.
Well, in math that's the only possibility, and we use it. And it is indeed extraordinarily expensive, but it's also the cheapest among the alternatives. The alternatives are 2: be rigorous and do these proofs, or be sloppy and allow bugs to creep in, and allow an entire school of math to collapse like the Italian school of algebraic geometry [1].
There is one more alternative. If a particular math theorem has some applicability, then you write a program and use it in real life. In time you eliminate the bugs as much as you can, and you get to the steady state of "virtually bug free". At that point you don't have a solid proof that the theorem is correct, but in general you don't really care. Because you feel that a formal proof is just a thing one would pursue for getting academic satisfaction only.
[1] https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge...
Replies
It's interesting that mathematics, which is mostly recreational (I received profound disdain at the math department for asking about applications!) has such rigorous standards, but software, which entire civilizations now run on, does not.
This is a bit reductive about what "proof" actually means in mathematics. Even in math, the kind of formal proofs that tools like Coq can automatically verify are an extreme, and lots of accepted and practiced math is not doing that. Proofs are often more abstract and even occasionally hand-wavy (for example not proving "obvious" statements or minor lemmas).
Mathematicians also occasionally build on top of unproven foundations (e.g. all popular asymmetric encryption schemes are built on top the assumption that certain problems such as integer factoring are hard, for which there is no formal proof), or at least explore both possibilities for statements with unknown truth value (e.g. you can find lots of work that explores the consequences of P = NP and/or P != NP).
However, there is a major separation between math and programs that I think mostly invalidates your proposal - most math we're talking about here is simply not applicable directly to the real world in any way. It's only studied for the interest of mathematicians. There is no real world consequence for Fermat's last theorem, for example - it was just a really interesting to prove theorem. In directly applied math, such as engineering, it is in fact much more common to work with unproven but well tested conjectures.
Your example of the Italian school is well taken, although most of their results were later proven to be correct in the right setting. Severi's example is particularly egregious and I think a major reason this became a thing is Severi's refusal to course-correct and accept that some of the results were not correct. It has echoes of Mochizuki, and I fear, once you dig deeper, some issues around the initial declaration of "We've proven the Classification of Finite Simple Groups". There were many genuine gaps, and a lot of lore taken for granted. The sociology around how this happened is interesting - rushing to announce that it was done was the major mistake, it took away almost all incentive to actually write up the proofs and take them through proper peer-review. Genuine mathematical work was falsely reduced to "write up".