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This question concerns only proven statements. I don't know if research papers do, but most math textbooks don't. Counterarguments:

  1. Space?
    1.1. The increased length from explaining the discovery is justified; if proofs' discoveries aren't explained and revealed, how can they be learned and discovered, especially to solve unsolved questions?
    1.2. The discovery may be more vital than the proof: easier is deducing the latter from the former, rather than searching the latter for the former. So if space is a problem, why not focus on the discovery and leave the proof to the reader, instead of vice versa that is the status quo?

  2. Ignorance of the discovery? Maybe these flashes of genius cannot be described? This seems unlikely for statements proven long ago.

  3. Desire for students to toil, and painstakingly discover the discovery themselves? But this is inefficient; if certain theories expended centuries for a famous mathematician, how can a math student discover within a day?

  4. Suppression of knowledge? This is least likely, and paranoid; maybe authors don't wish students to learn the discoveries easily, because then future students would outperform them?

E.g. Cauchy-Schwarz Inequality is often proven without its discovery explained. Textbooks don't explain how one would divine to consider $ \bf{0} \le \|\bf{u} - \cfrac{\langle u, v \rangle}{\|v\|^2}\bf{v}\|^2$.

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closed as primarily opinion-based by Lord Shark the Unknown, user296602, Morgan Rodgers, JonMark Perry, Claude Leibovici Aug 9 '17 at 8:04

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Re: if proofs' discoveries aren't explained and revealed, how can they be learned and discovered, especially to solve unsolved questions? It seems that you have an underlying assumption that knowing the way that an old fact was discovered (as opposed to proven) is necessary for (or at least a more efficient preparation for) discovering new facts. I don't know if I agree with this. $\endgroup$ – Omnomnomnom Aug 9 '17 at 4:36
  • $\begingroup$ The treatment of the mathematics and the study of its history are typically kept separate; people who are interested in one are not necessarily interested in the other. $\endgroup$ – Morgan Rodgers Aug 9 '17 at 4:54
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There are some textbooks that do take the historical perspective, e.g. Priestley's Calculus: An Historical Approach.

One problem is that history tends to be messy. A new idea often comes out as a by-product of an investigation of something else (often something that we don't care much about any more). The original statement may be quite different from the modern version (often much less general, and lacking a lot of the framework of ideas and terminology that only later grew up around it), and the original proof may be much more complicated than modern proofs. Often different parts of a theorem arise at different times, and are only later put together.

Case in point: you mentioned Cauchy-Schwarz. This was "discovered" by Cauchy (for sums, in 1821), by Bunyakovsky (for integrals, in 1859), and by Schwarz (again for integrals, in 1888, with something like the modern proof). There is a nice discussion here.

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