What shall this blog be about? (I)

It seems to me that great mathematical writing is rare, and to be celebrated. It follows that pointing out great mathematical writing, as well as poor mathematical writing, could be a very useful function of this blog. So I propose this as one feature of the blog.

We can review books, blogs, articles, etc. Any mathematics in print is fair game. What should be the criteria by which we judge mathematics to be well-written or not? To some extent, the rules of basic English should apply. We should see punctuated equations, as per N. David Mermin, correct grammar and syntax, and consistent formatting. In addition to these low-level necessities, we should see careful definitions, a concern for the reader, a lively, interesting, engaging style, as well as clarity of expression.

One aspect of mathematical writing not often brought to the fore is the difference between research and scholarship, as mentioned in Morris Kline’s book Why the Professor Can’t Teach, to which I linked above. Research is coming up with new mathematical theorems, procedures, etc. Scholarship is organizing, codifying, and clarifying existing research. One quote from Kline’s book (which I quote loosely) is that “One good scholarly paper is worth a hundred research papers.” Having attempted to read a number of research papers, I can definitely say that the vast majority of them are exceptionally poorly written, tending to be esoteric for the sake of being esoteric, and are generally useless except for the ultra-specialist.

It was V. I. Arnold who wrote the following:

It is almost impossible for me to read contemporary mathematicians who, instead of saying “Petya washed his hands,” write simply: There is a $t_1 <0$ such that the image of $t_1$ under the natural mapping $t_1 \mapsto \x{Petya}(t_1)$ belongs the set of dirty hands, and a $t_{2}, t_{1}<t_{2}\le 0,$ such that the image of $t_2$ under the above-mentioned mapping belongs to the complement of the set defined in the preceding sentence.

This is exactly right. It is this sort of obfuscated “mathematicalese” that I would combat, and I would welcome fellow fighters in this regard.

11 thoughts on “What shall this blog be about? (I)

  1. Working through your example, I finally deduced that at time $t_2$ Petya’s poor hands finally got some overdue attention. The example is not quite correct, however, as it fails to distinguish between the possibility that Petya has already washed his hands, and the case where Petya is washing his hands right NOW. Perhaps we should go further, and partition the set of hands by cleanliness, and show that we have a well-defined mapping to this partition. 😛

    1. Say, Deveno, could you please let me know if you are able to edit your comment? I tried to install a “Preview Comment” feature, but it was much too annoying to use (the preview would move up and down rapidly on the screen!). If users can edit their comments, though, that would more-or-less eliminate the need.


  2. Yes, I was able to edit, and a clock counting down from 30 minutes presumably shows the time I have left to make an edit or request a deletion.

    Once comment though, after making the edit I was left “stranded” on the edit page and had to use my browser’s back button to escape. 😀

  3. I could write an article if only I knew what would the article be about (toungout)

    For me, I think we should write on complicated subjects, especially those whose complexity discourages laymans to even look at it, in an easier (yet not incomplete) perspective.

    I think it would be better to start with insolvability of quintics (my favorite subject, lol). I know, many people might think that galois theory might be too broad of a subject to introduce in it’s fullest, but it’s incorrect that we need galois theory in proving insolvability of quintics. Basic group theory (upto Lagrange) is sufficient to go through Abel-Ruffing-Kronecker line. This so-called pre-Galois _ad-hoc_ proof is very rare and I never saw it described in an openly available source in the internet, i.e., blogs and all. The ones who claim it is Abel-Ruffini just gives a brief idea of Galois’ proof (without FT of Galois theory!) but that’s just ridiculous.

    In particular, we might want to focus on rare mathematical subjects and explain them thread bare and elementarily. I am speaking strictly of expository mathematical writings here, just to note.

    1. Balarka,

      That sounds like a great idea! Just keep in mind that Galois theory is gaduate-level algebra in most places in the US; hence, an explanation of the needed theory would be a good thing. If you’d like to submit your writing sample on the quintic, go ahead.


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