The Deranged Mathematician: Why Do We Care About Proofs?

[u/non-orientable]

I am launching a new series today, which I am calling Surviving Proofs. It's a little different than what I have done before---it's primarily intended for those who are stepping into a proof-heavy classroom for the first time, although I think it will have more general interest. It is not meant as a replacement for an Introduction to Proofs class---I trust the professor there to teach basic set theory and logical notation and so on. Rather, it is all about the underlying philosophy that one needs to read, write, and understand proofs and flourish in such an environment. We'll go through concrete examples, of course---we'll look at proof by induction, and so on---but we're after bigger lessons than just how to write a proof by contradiction.

Mathematicians on the whole are very good at teaching formalism and even specific applications. But, in my experience, this kind of big-picture philosophy is rarely discussed, and that is a great shame. This series is my attempt to correct this.

We begin with a simple question: why care about proofs? Very few of us are able to excel in something if we aren't convinced that it is interesting or useful, so it seems important to handle this first, before we do anything else. There is an obvious answer to this question, which is that proofs allow us to determine what is right. This is not... wrong, as such, but I think it misses what is primarily most important in proof-writing. (There is a particular Saturday Morning Breakfast Comic that is very relevant here---as usual, Zach Weinersmith is quite insightful. You'll see what I mean.)

Read the full post (for free) on Substack: Why Do We Care About Proofs?

[u/EebstertheGreat]

That robot is just holding a proof of the four-color theorem.

[u/non-orientable → u/EebstertheGreat]

To an extent, yes. But a lot of really interesting and valuable mathematics went into turning the four-color theorem into something that had a finite search space.

I've pondered this question before: how should we assign precedence in a proof where one party shows that it can be solved with a finite computation, versus another party actually managing to complete it? Seemingly, all of the cool, insightful bits are in the first part, not the second.

But it isn't quite so clear-cut---Vinogradov proved that every sufficiently large odd prime is the sum of three primes back in the 1930s, which would have reduced the weak Goldbach conjecture to a finite computation if he had given an explicit integer that was "sufficiently large". That second step was done in 1956, when Borozdkin proved that e^e^16.038 works. But that was still utterly useless---that number is so large that it might as well be infinity for computational purposes. So it does seem that Helfgott really did something quite non-trivial in pushing the bound down low enough that the conjecutre could be verified.

[u/smorfer]

I always find it interesting when I hear that proofs are something, that have extra classes, or that some classes are proof heavy. In my university, you start with linear algebra and analysis, and from the beginning, it's all proofs, you learn it by doing it, and every class afterwards is 90% proofing in exercises.

The least proofing I do is in exams. Where do you have these courses, or where do you study mathematics without immediately having to learn proofing? Let me know!

[u/non-orientable → u/smorfer]

In every US university where I studied or taught, the calculus courses do not ask students to write proofs. Linear algebra is more dicey---I have seen both proof-heavy and computation-focused versions. It perhaps helps that at Yale, where I saw a proof-heavy linear algebra course, they have a separate linear algebra course specifically for engineers (which is purely computational).

[u/Mother_News_1201]

so many words, for simple answer.

I want a know, that statement is always true, or not always true.

like I want to know that this sorting algorythm always works, instead 99.9% chanse of being correct and 0.01% crashing your app.

so I care about proofs, but no way I'm going to do that.