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I am interested in published articles, and also more informal writing (blog posts, talk slides etc.) which discuss the importance of textbooks (where this word encompasses research monographs etc.) in the long-term health of a sub-discipline in Mathematics.

Motivation: I have been thinking of late about how large Mathematics is getting (compared to, say, 50-60 years ago) with many more mathematicians and many more published papers. It also seems to me that at least some areas are becoming increasingly technical.

Furthermore, it can be very hard to follow a field by just reading the original papers: there can be false steps, or incomplete results, which only reach final form after some attempts. Often the pressures of space mean that motivation, or background material, is omitted in articles.

A good textbook can solve all of these problems. It seems to me that especially graduate students, or more established mathematicians seeking to move field, or use results of a different field to their own, face these sorts of problems in the extreme. By contrast, people working in the field probably carry around a lot of the "missing content" in their heads. This then makes me wonder if the lack of textbooks might lead to an ever increasing barrier to entry, and perhaps to sub-disciplines dying out as younger/newer mathematicians do not take up the study.

Hence my question of whether these thoughts have been stated in a longer, more thought out way before.

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    $\begingroup$ I hope this is suitable here: I guess also it might belong at academia.stackexchange but I felt that the issues involved are really quite specific to Mathematics (a slowly changing, technically deep field which it's not unusual to look at papers published many, many years ago, for example) where "curating knowledge" in the form of textbooks etc. might be rather different to other subjects. I have asked for this to be made community wiki. $\endgroup$
    – Matthew Daws
    Commented Feb 4, 2020 at 10:53
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    $\begingroup$ much of the literature on math text books addresses elementary mathematics, as taught in high school; for some thoughts on the use of text books in advanced mathematics, see mathoverflow.net/q/13089/11260 $\endgroup$
    – Carlo Beenakker
    Commented Feb 4, 2020 at 12:09
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    $\begingroup$ Wish there were a textbook for complexification. I'm fairly certain for each correct claim I make here or here, at least 1 mathematician has already proved it. $\endgroup$
    – BCLC
    Commented Dec 4, 2020 at 0:17
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    $\begingroup$ I have rolled back to Matt's original question. The use of CW status on MO usually serves a different role from on other sites, and I think Matt's original wording should be left to stand. $\endgroup$
    – Yemon Choi
    Commented Dec 5, 2020 at 0:36
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    $\begingroup$ @JohnSmithKyon FWIW: my impression from my own conversations with Matthew is that the level of mathematics which he has in mind is much higher than the level of your questions about complexification, where it seems that most of what you have been working out and pasting on this website is routine calculation that follows logically from the definitions. (I seem to recall that some of this is outlined in a section of the Kostrikin--Manin book that you refer to.) $\endgroup$
    – Yemon Choi
    Commented Dec 5, 2020 at 0:39

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The following is purely my opinion, but the specific question that you posed seems to invite such answers.

Strictly speaking, I'd say that the answer to your question is "Yes", a lack of textbooks in a new area is a barrier to entry. However, my experience over the past 30+ years is that areas don't stay "new" for long, and as things become more solidified, people write introductory (graduate level) textbooks. This may be to promote their vision, or it may simply be because they find the subject beautiful and want to share that beauty with others. To take an older example, Grothendieck's revolution in algebraic geometry created a large barrier for entry, but Hartshorne's book appeared when I was in graduate school, and it provided a way in. Was it perfect? No. Have other books, possibly better introductions, appeared since. Sure. But it was there, and I think it's fair to say that it helped train a generation (or more) of algebraic geometers and those in allied fields. (I'm in the latter group.)

So "yes", lack of graduate level texts in a area is a barrier to entry. But is it a long-term problem. I'd suggest that the answer is "No", because a new area of mathematics that's thriving tends to acquire such textbooks.

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