# Learning math historically

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 mention 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 (to quote exactly the source: "read the author, reproduce the results in your own way, and think about it to internalize, and repeat for the entire history. Also, don't get stuck, if you are stuck, move on to something else and come back. Time is limited").

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 the definitions in the first place).

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 ?

I also want to know whether this is a good approach.

EDIT: My main focus is not on the books which approaches mathematics historically (though book recommendations are very much welcome ! But they should be in comments section rather than as a stand alone answer) but on the method itself.

• When I studied math history (admittedly, only a single semester course), it seemed to me that it was teaching math with a historical perspective. The advantage you have over the mathematicians who originally worked on these problems, however, is that you can peek ahead to the results that they eventually found, and moreover you can benefit from notation and other helpful devices that were only developed hundreds of years later and can condense ten years of work into a few hours. – David K Jul 16 '18 at 16:14
• I don't know who said "go over the famous questions by yourself" or what precisely they meant by it, but what I would expect from a historical approach to math education is mostly a guided tour and not like "throw you in the water and see if you learn how to swim." Stillwell was mentioned in another comment; he also wrote the math history textbook that I studied from. Almost any chapter of that book (together with its exercises) would be an example of what I meant. – David K Jul 16 '18 at 17:35
• Why this is getting heavily downvoted ? What did I write wrong ? – cdt Jul 19 '18 at 13:12
• It was a mistake to set up a bounty. Since you are asking for opinions there is no "detailed canonical answer". – Christian Blatter Jul 20 '18 at 20:02
• The trouble with studying math historically is that mathematics exhibits a "trickle down" phenomenon. An abstract concept is first developed to solve very advanced problems (where the motivation for the concept is clear and natural), but after a few decades it's realized the more elementary theory can be reformulated with the new ideas, and so mathematicians do so in the interest of uniformity and beauty. The trouble is that now undergraduates have to study the abstract notions in a context where the motivation is not so obvious anymore. – Jack M Jul 25 '18 at 16:06

Here I'd like to address three books with a dedicated historical development perspective. Since OP's question is highly opinion based I think it might be helpful to look at some statements from the authors themselves, in order to get an impression how this subject should be treated.

A Radical Approach to Real Analysis by David Bressoud

• (From the Preface): The task of the educator is to make the child's spirit pass again where its forefathers have gone, moving rapidly through certain stages but suppressing none of them. In this regard, the history of science must be our guide. (Henry Poincaré).

This course of analysis is radical; it returns to the roots of the subject. It is not a history of analysis. It is rather an attempt to follow the injunction of Henry Poincaré to let history inform pedagogy. It is designed to be a first encounter with real analysis, laying out its context and motivation in terms of the transition from power series to those that are less predictable, especially Fourier Series, and marking some of the traps into which even great mathematicions have fallen. ...

... Careful definitions mean nothing until the drawbacks of the geometric and intuitive understandings of continuity, limits, and series are fully exposed. For this reason, the first part of this book follows the historical progression and moves backwards. It starts with infinite series, illustrating the successes that led the early pioneers onward, as well as the obstacles that stymied even such luniaries as Euler and Lagrange.

• There is an intentional emphasis on the mistakes that have been made. These highlight difficult conceptual points. That Cauchy had so much trouble proving the mean value theorem or coming to terms with the notion of uniform convergence should alert us to the fact that these ideas are not easily assimilated. The student needs time with them.

The highly refined proofs that we know today leave the mistaken impression that the road of discovery is straight and sure. It is not. Experimentation and misunderstanding have been essential components in the growth of mathematics. Exploration is an essential component of this course.

In the same spirit the author has also written A Radical Approach to Lebesgue's Theory of Integration.

The next one is a classic from a famous mathematician which is already an essential part of mathematical historical development.

• (From the Preface): ... Until rather recently, number theory, or arithmetic as some prefer to call it, has been conspicuous for the quality rather than for the number of its devotees; at the same time it is perhaps unique in the enthusiasm eloquently expressed in many utterances of such men as Euler, Gauss, Eisenstein, Hilbert.

Thus, while this book covers some thirty-six centuries of arithmetical work, its bulk consists in nothing more than a detailed study and exposition of the achievements of four mathematicians: Fermat, Euler Lagrange, Legendre. These are the founders of modern number theory The greatness of Gauss lies in his having brought to completion what his predecessors had initiated, no less than in his inaugurating a new era in the history of the subject.

• Our main task will be to take the reader, so far as practicable, into the workshops of our authors, watch them at work, share their successes and perceive their failures.

... The method to be followed here is historical throughout; no specific knowledge is expected of the reader, and it is the author's fond hope that some readers at least will find it possible to get their initiation into number theory by following the itinerary retraced in this volume.

The last one is from an author which was (and is) important for me when I was a student.

Theory of Complex Functions by Reinhold Remmert

• (From the Preface): ... The present book contains many historical explanations and original quotations from the classics. These may entice the reader to at least page through some of the original works.

"Notes about personalities" are sprinkled in "in order to lend some human and personal dimension to the science" (in the words of F. KLEIN on p. 274 of his Vorlesungen uber die Entwicklung der Mathematik im 19. Jahrhundert - see [Hs]).

• But the book is not a history of function theory; the historical remarks almost always reflect the contemporary viewpoint. Mathematics remains the primary concern. What is treated is the material of a 4 hour/week, one-semester course of lectures, centering around Cauchy's integral theorem. Besides the usual themes which no text on function theory can omit, the reader will find here ...

In the same spirit the author has also written Classical Topics in Complex Function Theory, which can be seen as volume II of the former book.

OP's quote: "... read the author, reproduce the results in your own way, and think about it to internalize, and repeat for the entire history ..." is not a challenge to reinvent some theorems, mathematical structures or concepts.

It is an invitation to

• read the author's text

try to understand it in some depth

try to extract the essential ideas and internalize them

... and then check if you really got some understanding by

• trying to reproduce it, but now without some additional means.

You could imagine a friend which is curious about the stuff and you try to write down and verbally explain the single steps. This way you can test if you really got a proper understanding.

You could also reflect about the historical approach and think about how the current approach differs (in terms of definitions, methods, ...).

The books referred in this answer provide many examples in appropriate historical context which can be formidably used to study and train this way the own abilities by grasping the essential ideas and trying to reproduce them independently.

The development of mathematical ideas is by far not only a single thread connecting a point $$A$$ in the past and another point $$B$$ in later times.

A nice characterization of mathematical development is given by Will Dunham in The Calculus Gallery: Masterpieces from Newton to Lebesgue:

• In a continuing ebb and flow, mathematicians develop grand theories and then find pertinent counterexamples to reveal the boundaries of their ideas. This juxtaposition of theory and counterexample is the logical engine by which mathematics progresses, for it is only by knowing how properties fail that we can understand how they work. And it is only by seeing how intuition misleads that we can truly appreciate the power of reason.

... and even this is a simplified description, since there are many superpositions of ebb and flow with respect to different ideas in mathematics.

• @alxchen: Thanks a lot for granting the bounty, although you are looking for slightly different information. I think I could add some thoughts within the next days. I also encourage you to go mindfully through the passages in my answer. They contain important information regarding your question and reflection about the content should help you to find a good way. Especially the statements of D. Bressoud could help you to decide whether an approach is convenient or rather not advisable. – Markus Scheuer Jul 26 '18 at 21:20
• @alxchen: I've added some thoughts. – Markus Scheuer Jul 27 '18 at 16:31