I don't do pure math, but I write the occasional theory paper, and this resonates. So much ends up on the cutting room floor--usually you proved the key result three or four different ways before finding a proof that is actually incisive/aesthetically pleasing/whatever to justify signing your name to it sending it out the door. Also most mathematicians are extremely averse to even saying something that's incorrect, let alone putting in print. There's ton of sanity-checking and trying to break your own result that, again, is rarely if ever mentioned in the final publication.

For what it's worth, this kind of behavior isn't limited to theory papers. Papers presenting analysis and model development also generally only show the "finished" result, omitting most or all of the bad models, training models, development models, etc. that were used to get to the final result. Same thing with observational papers in astronomy and related fields. There's a lot of work done that builds the authors' confidence in the correctness of the result, but that doesn't lend itself to a clean "story" for the paper or perhaps also ends up being supportive but not necessary to demonstrate a result.

I remember being surprised as a student, to learn that papers aren't narrations of the path to a discovery, but rather a narrative to describe the idea in a compelling (and hopefully clear) fashion.

That is what it is, but in the case of theory I think the alternative proofs could actually be really educational. The exact details of a biology technique (and hyperparameter searching type stuff for that matter) don't have the same inherent interest to me.

This was the most frustrating thing for me when I was studying undergraduate maths. Often a result was just stated, with the reasoning behind how it was generated completely omitted. Good professors would go through all of that, but those seemed to be few and far between. The rest just expected you to work it out (despite not ever being exposed to it or pointed to a good source), or rote memorise the result.

This is why there is typically separate Calculus and Real Analysis streams in undergraduate - the former is needed for all the hard sciences and engineering, the latter for math students and fellow travelers who really want to understand what is going on and be able to prove things properly.

Typically the analysis stream will start with a pretty in-depth look at what real numbers actually are (spoiler: weirder than you thought) and the implications of that. You get a much better understanding this way but you don't cover ground nearly as quickly.

I wasn't really talking about proof-based vs. not though - it's true that an analysis course will be more proof based (than Calculus) because that is also a skill you are expected to be developing.

However most calculus graduates, even the ones at the top of their class, still have at best a somewhat superficial understanding of the set of real numbers and functions on it, regardless of whether or not they can regurgitate an epsilon-delta proof.

This isn't a knock on calculus courses, there is an opportunity cost to the time. By the time a continuous math student hits measure theory and understands why they need a(nother) different definition of an integral, most calculus students have been happily chugging along calculating things they need, blissfully ignorant of the "problems" with the integrals they barely remember being defined.

In particular, I found a few papers which claim to provide algorithms which solve some problem. If there is no evidence the algorithm was ever implemented, there is a distribingly good chance it doesn't work -- I used to sometimes give such papers as projects for students, but now I always start on day 1 saying "Be prepared for this to be wrong".

[1] https://www.ias.edu/ideas/2014/voevodsky-origins

I really hope this work trickles down to our programming - state machine problems are basically "solved" (for my needs at least), but more complex programs are really hard to prove.

In a nutshell, the key challenge is taking the core of complexity theory (which is easy to formalize in terms of state machines) and migrating it into the language(s) of category/type theory (which is much more modular/compositional).

A couple interesting recent works in this area:

Categorical Complexity: https://www.cambridge.org/core/services/aop-cambridge-core/c...

Dusko Pavlovic's Monoidal computer series: 1. https://arxiv.org/pdf/1208.5205 2. https://arxiv.org/pdf/1402.5687 3. https://arxiv.org/pdf/1704.04882

I'm a bit of a philistine when it comes to cutting edge CS in this area so I'll leave you to it!

Like I don't mean "this proof has a gap in its justification, but the thing claimed likely is still true". I mean "we thought someone had a proof for X, but actually later someone else showed X isn't true".

The short version is that Gödel claimed in a paper that he had an algorithm to determine satisfiability for a certain logic, but the algorithm actually worked for a slightly different logic instead. People basically took Gödel's word for it for fifty years until a logician called Goldfarb found the mistake.

EDIT: As a disclaimer, I don't have any association with or endorsement of the website I've linked to. I found it just now because I needed a source for the story that I've known for many years (I think I first heard it in person, but it's definitely contained in the book `The Classical Decision Problem' by Börger, Grädel and Gurevich). I skimmed the post and it seemed to cover the relevant details, but that's the extent of my knowledge of the site and its contents.

OTOH, there was the Italian school of algebraic geometry, but that was more than one single flaw...

[0] https://mathoverflow.net/questions/35468/widely-accepted-mat...