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I was wondering if mathematics learning process require the use of textbooks.

When I was a high school student, I read as a preparation for university, Legendre book on Elements of geometry and trigonometry, I notice the power of that style of teaching mathematics: propositions lead to theorems, then I read old books on the foundations of geometry such as Russell's, Hilbert's and Coxeter's. In order to learn some arithmetic, I tried to read Dedekind's Essays on the theory of numbers, off course that was just a big fail, I had to run straight to Niven's book.

I was wondering if a person is capable to substitute textbooks in order to learn directly from the source of Knowledge and creativity, I mean replace a normal HS algebra book with a modern edition of Cardano, Wallis, Vietè, Descartes, etc. Analytical geometry with Descartes, Learn arithmetic with Disquisitiones Arithmeticae; Abstract algebra with Cayley, Dedekind, Galois, Noether, etc; Switch Munkres to Cantor, Poincaré, Hadamard, Borel, etc.

I do not think you can learn Calculus without Spivak's or Hardy's or Rudin's because original calculus was develop for applied purposes and real analysis do require a pedagogical treatment (I think), and those books are really great; but books on Algebra, Topology, Number theory, analytic number theory,categories, representation, even homotopy books are just compiling work and providing useful exercises.

The question is, reading the master, you think yourself capable of learn math without reading the pupils, and instead of that reading the masters, or you think that modern and contemporary math require a depth pedagogical treatment to translate you ideas?, if you think your capable, what articles, books would you read.

I would read Grothendieck, Shannon, Knuth and Mirsky stuff

Note: That is constructional, I want to teach algorithms next term using not textbooks, but old papers, I need an opinion (would it be a waste of time, or inspirational)

Thank you very much

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    $\begingroup$ Of note is that the notation used by the masters is not necessarily the same as, or even compatible with, currently accepted notation. You might want to keep this in mind. $\endgroup$
    – J. M. ain't a mathematician
    Apr 4, 2013 at 6:02
  • $\begingroup$ Yeah, i mentioned modern edition of Cardano,..., etc, but the question is anyway for modern and contemporary math, i don't thing Atiyah notation would differ that much from the one we learned as graduates. $\endgroup$
    – Sebastian Valencia
    Apr 4, 2013 at 6:04
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    $\begingroup$ Original material tends to ne difficult. The exceptions are mainly books written by the masters but intended as textbooks. $\endgroup$
    – André Nicolas
    Apr 4, 2013 at 6:29
  • $\begingroup$ Well, that's basically the only way to do it... once you have your PhD. Before that it's a good idea to read more recently written things. It's not easy to read Principia when you ain't ever done any calculus before. $\endgroup$
    – Alexander Gruber ♦
    Apr 5, 2013 at 5:16
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    $\begingroup$ Obstacles to reading the masters include: papers written in Latin, French, German , Italian, Russian, Japanese, Hugarian or other languages; papers not being readily available - either not online or behind paywalls. $\endgroup$
    – Snor
    Apr 6, 2013 at 12:04

3 Answers 3

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You may find the following website interesting/useful: Teaching with Original Historical Sources in Mathematics

Use of original historical sources in lower and upper division university courses is discussed. Reinhard Laubenbacher and David Pengelley were inspired by William Dunham to cover "mathematical masterpieces from the past, viewed as works of art." However, "Whereas Dunham presents his students with his own modern rendition of these masterpieces, [their] idea was to use the original texts themselves." They have authored at least two books intended for this purpose:

Mathematical Expeditions: Chronicles by the Explorers.
Mathematical Masterpieces: Further Chronicles by the Explorers

As I recall, the books use excerpts from the original sources, liberally augmented with modern explanation/analysis.

Assuming you are a student and not a mathematics researcher, my own personal opinion is that for a first reading it is better to use a (good) contemporary author writing for a student at roughly your level. He or she will have the benefit of history's digestion, simplification, and further development of the subject and, when applicable, will be able to translate outmoded notation into modern notation. If the original source is more recent and has been written at roughly your level then it may be superior. Otherwise, I think original sources are best for second readings and/or supplements.

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André Weil wrote:

... our students of mathematics would profit much more from a study of Euler's Introductio in analysin infinitorum, rather than of the available modern textbooks.

(André Weil, 1979; quoted by J.D. Blanton, 1988, p. xii)

Blanton's translation has made this wonderful 1748 work available to English-speaking audiences. Once you have mastered the Introductio you can go on to Euler's Institutiones of 1755, similarly available in Blanton's translation.

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I'd oppose that idea. What later refiners and lately textbook writers did was to select material, clean it up, simplify and systematize notation, select understandable proofs, add modern extensions and applications. A selection of exercises that challenge and help deepen understanding is a valuable resource. A peek at the history is valuable, but should not be the principal source (unless it is a very new area, but that is another kettle of fish).

Granted, not all textbooks are excellent, but I'd wager most are a step nearer the "average interested student"'s understanding than any random "original source". Besides, some of the towering historical (or contemporary) figures were also accomplished writers (like Euler or Russell), a few were very good teachers too, but most wrote/write opaque gibberish.

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