>> the author’s wish to present ... mathematics, as intuitively and informally as possible, without compromising logical rigor

The books in the West in general kept getting less rigorous, with time. I don't see Asian or Russian books doing this. The audience getting less receptive to rigor and wishing for more visuals and informal talk. When they get to higher studies and research, would they be able to cope with steep curve of more formalism and rigor?

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nabla9 2 hours ago | parent | next [-]

> Russian books doing this.

Mathematics: Its Content, Methods and Meaning by A.D. Aleksandrov, A.N. Kolomogorov, M.A. Lavrent’ev,.. https://www.goodreads.com/book/show/405880.Mathematics

It's still a masterpiece. Originally published in 1962 in 3 volumes. The English translation has all in one.

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zozbot234 39 minutes ago | parent | prev | next [-]

If you care about getting all the nitty gritty details of a "rigorous" proof, maybe the quicker approach is to install Lean on your computer and step through a machine-checked proof from Mathlib. What you get from even the most heavyweight math books is still quite far from showing you all the steps involved.

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actinium226 an hour ago | parent | prev | next [-]

> The books in the West in general kept getting less rigorous, with time.

I wonder if it's because more people are going to college who would have otherwise gone to a vocational or trade school? If the audience expands to include people who might not have studied calculus had they not chosen to go to college, I feel like textbooks have to change to accommodate that.

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elcapitan 2 hours ago | parent | prev | next [-]

This may be a stupid question, but what do people usually mean when they refer to a mathematical text as being "rigorous"? Does it mean that everything is strictly proof-based rather than application-oriented?

	▲	actinium226 an hour ago \| parent [-]
		Generally that's what it means. And also when proofs are presented, a rigorous book will go through it fully, whereas a less rigorous one might just sketch out the main ideas of the proof and leave out some of the nitty gritty details (i.e. it's less rigorous to talk about "continuity" as "you can draw it without lifting the pen" as compared to the epsilon-delta definition, but epsilon-delta is pretty detailed and for intro calculus for non-mathematicians you don't really need it).

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kardianos 2 hours ago | parent | prev | next [-]

I agree with this. But I don't see the students rejecting this, but the education degreed peoples who choose texts and the publishers want to make all learning for all people. This is foolish. Most people don't need to know calculus. And if you do learn it, do so with rigor so you actually learn it and not just the appearance of it, which is much much worse.

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mjburgess 2 hours ago | parent | prev [-]

Nope, but mathematics research is one of the most rarefied fields being extremely difficlt to get into, hard to get money, etc. -- (this is my understanding, at least). Progress is made here by people who, aged 10 are already showing signs of capability.

There's not much need for a large amount of PhD places, and funding, for pure mathematics research.

Likewise, on the applied side, "calculus" now as a pure thing has been dead alone time. Gradients are computed with algorithms and numerical approximations, that are better taught -- with the formal stuff maintained via intuition.

I'm much more open to the idea that the west has this wrong, and we should be more focused on developing the applied side after spending the last century overly focused on the pure