Why are some mathematics papers seemingly so bad?

Question

I am a PhD student in math beginning to do research. My advisor has given me a paper they want to generalize. It's pretty long with many dense, technical parts, and so, so many typos. The entire paper is about defining/exploring a space, but the definition of the space has obvious typos which make it unclear what the author means. Many definitions are similarly broken. Often the author states something is 'immediate from blah' where blah is a broken definition or a statement that is totally unrelated. I feel like most of my time reading is spent resolving these errors or unhelpful writing.

When I do resolve them and explain to my advisor, they they seem to agree that the paper is gibberish and that what I say sounds right. There is a lot of confusion neither of us can resolve. It's so bad that when I can't understand part of a proof, I start to wonder if it's just wrong.

This paper is not isolated -- the last couple papers I've read have been like this. It's not isolated to my field either -- my peers tell me similar stories about papers with typos that look like key smashing and more. It could be a lack of ability to understand on my part, but from all outside metrics I appear to be understanding what's in front of me. A professor even told a story once in class 'So and so proved this theorem. His PhD thesis was about how blah spaces can't exist, and his first paper was an example of a blah space.'

It's mind blowing to me that published papers/theses can be so broken and impenetrable, or even just false. I would be embarrassed to turn this kind of work in as a thesis. So my question is: how are these papers getting accepted? These are a few explanations I can think of.

1. Doing math is just really hard and explaining it is even harder, and most people who can do math don't put a high value on explaining it well.
2. No one is checking. Because there is a culture of bad writing and everyone is too busy, the peer review process just doesn't catch this stuff.
3. I am just a lowly grad student, still learning how to read papers, and eventually the process will be much smoother and faster so it won't really matter.

Edit: I think a more helpful question would have been this. What can I do as a student to make better progress through papers like this?

Answer 1

There are indeed several reasons for which some papers can look bad and that this is a common feature of math papers but there are a lot of bias.

1. First, well or simply correctly written papers are unconsciously considered as normal and our brain recalls more easy remarkable things, with an emphasis on negative things.

2. Second, even in good papers, whatever it means, you will find things to reproach: too much or not enough details, too much references and not self-contained enough. I also happens to me when looking at papers I wrote ten years ago, to think "oh, this could have been made smoother" or "a transition between two corollaries would have been better".

3. It indeed happens sometimes that bad paper can be accepted. For the reason mentioned in the opening post, that is, that peer-review does not reject bad paper. Sometimes because the referees are busy. But also because, as pointed out by Alexander Woo, when you master well the topic, you read what should have been written and not what is written. This can also happen to a referee. The referees/editors also can be convinced that there is no fundamental flaw in the paper and "only" minor and easy to fix. Moreover, the review delays can favor this phenomenon. It is not seldom to have two or three months to review a 40 pages paper. We can always ask for extensions, but the deadlines can put pressure.

4. A factor explaining why some papers can look like a first draft is the publication pressure. People have to publish, always more and always faster (I consider this as a cause, not an excuse). A consequence of that is that the proofreading and polish can be made in a rush.