What is the structure and purpose of abstracts in math papers?

[flamengo]

Most published math papers are answers to open questions posed by the authors of the papers, right? So why is this problem that the paper responds to is never explicit in the text by the authors? Would not that be an important thing since it would save time for other mathematicians not to waste time formulating problems that have already been answered? Or is an expert in the field able to identify the open problem that a particular paper responds to even if it is not explicit in the text? Could someone explain to me in detail how this works?

 

[Stephen Tashi]

So why is this problem that the paper responds to is never explicit in the text by the authors?

Why do you think that?

Most published mathematical papers have abstracts and introductions that clearly state the problem they are solving and give background information about previous work on the problem or similar problems.

 

[mfb]

Let's take a random paper as example, the first one I found on arXiv: Irredundant generating sets and dimension-like invariants of the finite group

This is the abstract:

Whiston proved that the maximum size of an irredundant generating sequence in the symmetric group S_n is n−1, and Cameron and Cara characterized all irredundant generating sets of S_n that achieve this size. Our goal is to extend their results. Using properties of transitive subgroups of the symmetric group, we are able to classify all irredundant generating sets with sizes n−2 in both A_n and S_n. Next, based on this classification, we derive other interesting properties for the alternating group A_n. Finally, using Whiston's lemma, we will derive some formulas for calculating dimension-like invariants of some specific classes of wreath products.

Ignore the mathematics behind it, just have a look at the structure: The first sentence presents previous results. The second sentence states the goal (extend these results). The third mentions the methods used and shows what has been achieved (classify some stuff). The following sentences extend that and present more results.

What exactly are you missing?

Here are some more randomly picked abstracts, they all follow a similar structure. Sometimes the abstract doesn't explicitly reference previous work but directly explain the new result. The corresponding question is clear to experts in the field, and even without expert knowledge you can typically figure it out. If the result is "we show number X is Y", then the question is "what is number X".
https://arxiv.org/abs/1712.03247
https://arxiv.org/abs/1712.03224
https://arxiv.org/abs/1712.03226
https://arxiv.org/abs/1712.03861