Anyway you've already got Apostol - if it's just calculus as such get an older edition. Modern ones have extra goodies like linear algebra but have modern text book pricing (cries softly in $150/volume).

> 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.

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.

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?

> 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.

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.

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

I don't think they are unifiable, the aims and methods one needs to learn are just too different. Limits of covering boxes and scaling your epsilons and so on, stuff from Tao's class on analysis is far away from being able to deal either non-trivial differential equations or stability analysis. You can prove all sorts of things about dense subspaces of Hilbert space and still get totally lost in multiple scale analysis, and vice versa. (Ed: epsilon was spelled espikon)

I'm probably dating myself, but at my college, there was one calculus course for everybody. But also, a lot of the students in those areas had overlapping or double majors. For instance I majored in math and physics.

Perhaps the bigger question is whether it's at the right level of difficulty for the audience.

the Postscript at the end says:

While not every student is expected to read the book sequen- tially cover to cover, it is important to have the details in one place. Calculus is not a subject that can be learned in one pass. Indeed, this book nearly assumes readers have already had a year of calculus, as had the students of MAT 157Y. I hope this book will grow with its readers, remaining both readable and informative over multiple traversals, and that it provides a useful bridge between current calculus texts and more advanced real analysis texts.

My first impression, paging through it, is that it's at a somewhat higher level than the typical college calculus course.