Mathematics can be written in many ways. One approach, very popular with professional pure mathematicians, is to write as little as possible. Often the best proof of a mathematical theorem is the shortest and most elegant. This fact, combined with some of the history and culture of mathematics, leads to the classic terse mathematical style: theorem-proof, theorem-proof, lemma-theorem-proof, definition-prposition-theorem-proof, and so on.
(The fact that most mathematicians dislike writing may also have something to do with this!)
I think that, on every mathematical subject, there ought to be texts which are written in this way: short, crisp, elegant, minimalist. But there should also be others.
The standard terse style is, however, imperfect for learning mathematics — especially anyone below PhD level. Perhaps this style is tolerable for learning how the proofs go. It’s useful for understanding the exact steps in rigorous proofs of theorems. And it often works well with a highly motivated or sophisticated reader — one who understands that reading such books is not actually about reading, but about knowing when and how to ask oneself questions, filling in the details which have been omitted. The standard style is hard, both in the sense of “not easy” to read, and in the sense of “not soft”, with no surrounding story or context.
Such an approach is fine for insiders: those who already understand the culture and the conventions of mathematical literature. But for learners — particularly those with a weak background, as is increasingly the case — it is a different matter.
What tends to be left out in the standard terse style? Everything that makes mathematics human: history and context; motivation; commentary; connections within and beyond mathematics. And even a mathematician may not appreciate every book being so hard to read.
Other mathematicians may disagree, but given a choice between terse text, and a gentle version which is twice as long but twice as easy to read — and full of interesting details and tidbits — my preference is clear: genial is better than brutal and terse.
Why are we interested in the topic we are talking about? What are its implications and connections? Why do we cover the parts of the subject that we do? Why do we use the arguments we do, and why not others? Where did this proof come from? How could we use similar ideas to prove other results? These questions are often as important as the mathematical content itself.
More generally, contemporary curriculum and culture, at least in Australia, leads to the situation that students may know little about the background of their subject — even when they are studying at an advanced level.
Further, the classic terse approach can descend into a combination of intimidation and disrespect. Proofs and arguments are routinely omitted as “obvious” or “trivial”. Steps are skipped. Some readers may be fine with one or two gaps to fill in themselves, though no harm would have been done had the author included it; but every skipped step is a potential hazard, and a successful reader must navigate them all.
In extreme cases, authors leave a trail of breadcrumbs which the reader may be able to pick up and follow along, if they have enough knowledge or curiosity or insight or gumption or tenacity or luck. Mathematical writing then becomes a set of puzzles, where every sentence must be solved by the reader to progress to the next. Mathematicians in certain fields will know certain “classic” texts in the mathematical literature are precisely of this type. All this in the supposed pursuit of communicating mathematics as fast and efficiently as possible!
Such an approach makes reading mathematics, in its terse classic style, a completely different affair from reading almost any other subject.
Why erect walls of unexplained argumentation, and browbeat those who cannot scale them with cries of “obvious” and “trivial”?
For students first arriving upon the abstract world of pure mathematics, it can seem a harsh, even brutal subject. That is because it is a harsh, brutal subject. Mathematics does not forgive your one mistaken observation: your proof will come crashing down despite your pleadings. Most of your thoughts on mathematics will be wrong — to the extent they are even precise enough to be wrong. To do mathematics is to work through all the wrong thoughts to make them right.
Mathematical arguments are true independent of what humans think of them: in this sense, the truths of mathematics live in their own world, a world that has no feelings and is not human. The independence of mathematics from the human world is the source of an austere beauty, but it can also make the subject seem cold and desolate.
It is a cold world, it is a harsh world, but it is a beautiful world, and its statements are pure, honest, and beautiful. And while it is not human, it is a world inhabited by humans. It also provides the language of science and the universe.
Some can brave entry to this world themselves. But why should we not provide some guidance as to the nature of this world, as we enter it?