What is meant by learning math historically (NOT learning math history only, but learning math with a historical development perspective)? I've seen some sources that to learn a math topic X, you need to look at the historical development of the topic X and go over the famous questions by yourself to develop a good understanding of the subject.

I also find this method (learning in a historical context) nicer because more often that not traditional books (say a book on Group theory) starts like here are the axioms (eg the group axioms), memorize it and look at the theorems and corollaries which follows from the axioms. Without the historical context it keeps me wondering what was the point of the axioms in the first place (i.e what motivated that).

But on the other hand, regarding learning things historically, how I am supposed to "go over the famous questions by myself" when the problems took tens of years to solve ?

For a concrete example, I'm learning Ring theory now. What should I do to learn it in a historical way ? Am I supposed to work on problems like $x^3+y^3 = z^3$ or UFD over cyclotomic integers and "rediscover" Ring theory ? But how I am supposed to sensibly work on the problems without knowing the theory when they took really long time to solved in the first place ? Or "learning math historically" is done just like reading math from textbook, you just "passively" read the history of how some ideas were developed instead of "actively" working on some historically important questions which took really long time to solve ?

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    $\begingroup$ Consider: Part of the history is why the axiomatic method (axioms, then theorems) was itself revolutionary as a way to organize and present mathematics in Greece ca. 350 BC. "The concept of axiomatic development in mathematics must be ranked as one of the very greatest of the GREAT MOMENTS IN MATHEMATICS" (Eves, Great Moments in Mathematics (Before 1650), Lecture 7). $\endgroup$ – Daniel R. Collins Jul 16 '18 at 14:19
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    $\begingroup$ A good example is “Real Analysis: A Historical Approach” by Saul Stahl. The development of the theory is chronological and quite different than the build-up-from-axioms approach used by usual analysis texts. Alas, I do not know of a similar book for ring theory. $\endgroup$ – Aeryk Jul 16 '18 at 17:48
  • $\begingroup$ @Aeryk: Maybe Classical Algebra. Its Nature, Origins, and Uses by Roger Lee Cooke (2008). However, this is primarily about fields, with rings only entering sporadically (such as the ring of polynomials in one indeterminate over a given field). $\endgroup$ – Dave L Renfro Jul 17 '18 at 18:32

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