Why don't textbooks explain proofs' discovery?

Question

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 a student would divine $ \bf{0} \le \|\bf{u} - \cfrac{\langle u, v \rangle}{\|v\|^2}\bf{v}\|^2$.

Answer 1

In many cases, standard theorems in a theory were first proved as special cases before the theory itself was even invented. For example, Lagrange's theorem in group theory predated the invention of group theory. He proved it as a result involving certain sets of permutations (of variables in polynomial equations) which were only much later recognized to form a group in the modern sense. A book in abstract algebra will typically begin with the axioms of group theory -- axioms which were not clearly spelled out in Lagrange's life time. To explain what Lagrange himself actually did you would need a long detour into the theory of polynomial equations, one which would have only an indirect link to the modern result known as Lagrange's theorem (which was only proved decades after Lagrange's original result). This is only a single, somewhat preliminary result in group theory. If you tried to do it for all results in group theory textbooks would blossom into thousand-page tomes that would lose almost all students.

There are some text books that have a somewhat narrow focus and which do have a more historical approach. For example, Harold Edward's Galois Theory does get into what Lagrange actually did. While such books are valuable, and all math students could benefit from at least occasionally working through such things, they aren't a very good way to get a good understanding of the modern theory. Mathematics is hard enough without overburdening its exposition with excessive history.

Answer 2

I'm not exactly sure if I understand what you are asking, but I will try to give at least a partial answer. I think there are a couple reasons why an explanation, of how a proof was first arrived at, is not given:

1. It isn't known

For a lot of proofs not even the author can exactly explain how he arrived at the given proof. "Well I thought about how to prove the problem, thought about what my intuition said, tried to make that rigorous, found the places where I couldn't quite get enough rigor, played around with some other ideas, and finally came up with the proof when I realized that I could just reduce this to a coloring problem which was much easier to solve." And that's one of the more specific examples. Often people spends weeks or months on a problem trying many dead ends before they come up with the correct approach or as happened to me figure out that they are trying to prove the wrong thing and prove the exact opposite.

2. The original proof was quite ugly

This is very often the case for many harder problems. When you first work on a hard problem you might be starting a new direction. The example that comes to my mind is the proof of independence of CH. The original Cohen paper is quite hard to read since when he did it, the technique he invented wasn't formalized and smoothed. He and others went on to refine the technique into something that almost a baby can use (for the easy problems anyway), but this took some time. Explaining how he came up with the original proof would probably be hard and wouldn't really help much.

3. It is time consuming

Explaining how a proof was first arrived at will either be very time consuming (explaining all the dead ends, ideas that didn't work, refinements etc.) or will leave out 3/4 of the important stuff (all the dead ends, ideas that didn't work, refinements etc.). Usually in Mathematics classes (especially at the lower levels) students have more than enough trouble just grasping the refined and cleaned up proofs and ideas, much less being able to follow the complex process that led to their creation.

4. We don't know/ lack of rigor of the original proofs

A lot of especially the original proofs are old. Cauchy lived in the late 18th early 19th century and I doubt he wrote down how he came up with the proof. Further more though quite a bit of older mathematics wasn't proved to current standards. Some of Newton's and Leibniz's results were such and while the results hold in essence the actual founding theory which makes them work nicely wasn't created until much later.

Answer 3

In addition to other good points already made, there is also the mythology that there are "ultimate" proofs, and that the messy human processes that occur along the way to development of "the correct" proof are best ignored... insofar as those human activities are not the "platonic" ideal of mathematics. Timeless, etc.

Similarly, many peoples' affection for "formality" of presentation would not allow them to give examples prior to a definition (which was, in reality, only conceived long after the examples were well understood, and motivated the definition). In my opinion, this is a stylistic conceit that does not help people understand mathematics.

Said in another way, there is an extreme "Whiggishness" in accounts of mathematics, as though (cf. "Whig account of history") people in the past were pathetic and benighted, and everything has been improving toward an ideal, which is represented most closely by the current situation. If we buy that, then there's not much interest in examining the sad, ignorant flailing-about of old-timers like Newton, Leibniz, the Bernoullis, Euler, Lagrange, and so on. :)

More seriously, the sophistication required to understand the complexity of mathematical developments probably is not available to most 18-year-olds, or 22-year-olds (in grad school). So maybe a bad investment of time and effort?

Answer 4

Is there a way to cross-reference MathOverflow? This was asked and answered there.

1. Why is it still common to not motivate results in publications?

2. Examples of Mathematical Papers that Contain a Kind of Research Report

3. Examples of great mathematical writing

Answer 5

I think the history of a subject and its exposition are both incredibly interesting and worthwhile subjects, but combining them in a textbook is often problematic.

The textbook (non-math, but math-related) I know that spends the most time explaining motivations/background/history for discovery is Apologia's chemistry textbook. Students reading the book get really bogged down in the historical explanation. Additionally, it makes the book big and daunting. With all the historical explanation, students get really confused as to what the main point even is. They don't know which part they are supposed to learn and be applying. They get really lost in the details.

Interestingly, another math-oriented science book, Saxon Physics, is actually much more difficult material, but students seem to learn it better, because it is much more direct. The lessons are just a few pages, and very little background is presented.

Again, my point is not that background/motivation is not important, but that, for textbooks, separating it from content helps students because they are better able to see which things directly apply to them.

If someone had an example of a textbook that included the background information but in a way that wasn't distracting to students, I would be interested to see it.