On writing a textbook on algebra

  Throughout my career, I have always sneered upon people who specialize in writing textbooks. After all they are but plagiarizer and are brazen enough to feel smug about results that are not their own. Once I started sitting in a real classroom taught out of lecture notes by the lecturer himself, I started to realize that at least it does not look easy to be in charge of so much material and its internal correlations. But even then I thought it just takes time for the information to sink in and everything becomes spontaneous once you have worked with them for a while in actual research.
  So lately I was inspired by a colleague of mine to start writing a textbook on algebra, partly because it’s one of my weakest areas, and also that it does not take too long to get into some rather nontrivial results. I also convinced myself that Texing things up fosters good memory, especially under the circumstance that I do not look back into notes I have taken in classes.
  So the job started yesterday, with a few words written for the preface and a preliminary structural planning. I based my judgment on how I would choose which textbook to read out of the vast collection in a typical college library. Of course being a greedy person (see my exposition on greed in previous blog entries), I would usually pick the encyclopic ones. Unfortunately my knowledge is quite ramshackled at the present moment, and insufficient to give comprehensive treatment of any subject, much less one that is supposed to kill me all the time, namely algebra. Already in the preface I found my parlance quite affected by the expectation of maturity and pedantry. Nothing of puerile nature should enter the overture of such a promising work. But frivolity and jocularity could easily be an excuse for patent naivety. The sad thing is, however, I am no comedian by trade, nor by predisposition, having resented circuitous thinking earlier on in my life. So maybe honesty would work the best, as I learned from John Kerry’s mother. Thus an ambitious list of topics to be covered concluded this unprepossessing prologue.
  The challenge remained, nevertheless, of finding the right tone to start off the first chapter. Upon second thought, I decided against a formal axiomatic treatment of basic algebraic structure, and replaces such mundane pedagogy with academic catechism. So basically there would be questions interspersed within the text, that serve to motivate new definitions and results. This I believed to be the most critical thing in algebra textbooks. Many formality may seem overly heavy and redundant, only because one does not see its immediate constructive applications. Unfortunately there are no constructivism in things like Galois theory. Instead one tries to disprove something. Much of the same thing happens in virtually any branch of modern mathematics, with the possible exception of analysis. For example topology really is about probing into the nature of spaces. So one must forsake the attitude when he first encountered Newtonian calculus, in which differentiation and integration are taught somewhat like a magic box that produce wonderful things. Just as the age of mathematical formulae were gone with the decease of Ramanujan, so was the presence of such productive and utile magic boxes, perhaps even earlier with Newton and Leibniz. After all, the math done after them was much less straightforward and require case by case techniques.
   One problem with such catechistic style of writing is that one can by no means by comprehensive, unless he awkwardly throws in some random poke-fun questions that whet nobody’s curiosity. On the other hand, however, a good author does not have to stick with some premeditated rules amidst writing his lecture notes. He could switch to different kinds of narrative whenever it behooves to do so. And this is indeed what I may do. Above all, the text is meant to endow the readers with the most useful knowledge in the most efficient manner.
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About aquazorcarson

math PhD at Stanford, studying probability
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