Daniel Wqw said:
To me [Spivak] seems in a way like learning geometry using euclids elements. [...] I feel like I might come out from reading it thinking 'why am I learning all this stuff?!'. [...] What will I get out of reading Spivak's Calculus book?
Here is my take; I'll use the example of riding a bicycle. Riding a bicycle certainly seems difficult in the beginning, partly because one doesn't innately know that slower is more hazardous. One might think that one should start off slow and master it before speeding up, but actually it works the other way round. The intuition is wrong and makes it seems much more difficult than it is.
If we extract this wisdom, any new thing will seem difficult at first because our intuition is wrong, but by the simple fact that many other people succeed at it, there must be a way to succeed and we just need to correct our mistaken intuitions.
I'll say the same about Spivak's book, it is difficult because our intuition is wrong. Our intuition is that a person should start at page 1, read everything carefully, answer every question, never turn a page without understanding what it contains, etc, a linear approach. And it is appealing that we can learn every definition and prove every theorem and then know the subject completely, so it seems that this is how one should proceed.
But there is a philosophical problem here. What one is trying to do is learn the subject, and sticking to one particular method of learning can only be a hindrance, it can never be a help. So that's the first point: don't be limited to anyone method. Hopefully this quells your worry about it being like learning from Euclid's Elements. It would still be learning, you wouldn't need to learn in anyone particular way.
Why this book? Because it contains rigorous calculus and is known for having challenging problems.
Why rigorous calculus? I want to defer to Micromass's answer on this point.
What I will say is, I like to think of math as a language. To be philosophical, words are just words, what is important is the things that they refer to. And mathematical symbols are just symbols, what is important is what they refer to, what they are about.
Calculus, rigorous or not, is about a language of symbols to talk about rates of change, slopes or areas and volumes. Rigorous caclulus is careful to define those symbols rigorously, in excruciating detail so as to make the meaning very clear, and that is interesting of course, but really I find the use of the symbols to be more interesting.
One can ask, what is a continuous function? And if one is being precise, one must use the rigorous definition. But why was it defined that way? Because we want it to define what was non-rigorous before. Because if continuity was anything else, it wouldn't be called continuity. If you understand that, that the definition is meant to define something that is intuitively obvious, that it is meant to be somehow simple, and the proofs themselves, although complicated, have simple start and end points because they connect ideas that should be simple, you should start to see that there is a simplicity criterion to the theorems and the complication is buried in the proofs.
But realize that the theorems are chosen so that the proofs have a simplicity criterion as well. Proofs that are too complicated will make use of lemmas to make them easier to understand. So proving any particular lemma or theorem will be to some degree simple or elementary as well. That said, the simplicity criterion for proofs applies only to how the proof looks, not to how easy it was to find the proofs. So any particular proof could be difficult to find.
So hopefully this is a bit of a road map. The theorems themselves, if you don't prove them, are still interesting to know, I believe. And because the proofs have a simplicity criterion, it can be worthwhile to just look them up because they will usually be short and easy to memorize. Why spend hours when you can spend minutes?
As a strange peculiar example, one of the things I wanted to learn from Spivak was how to do limit proofs. I devised a strategy that always worked, a universal procedure to tackle any limit proof (of the type that was in the book). It seems magical but facing a bunch of those questions, I was like, how can I make this more efficient?
Consider this, limit proofs should be similar, very similar; if the function is suitably continuous, why wouldn't the proof be almost identical? So here again, intuition has led us wrong. What seemed magical, that there could be a universal procedure, now perhaps seems even obvious, of course, each problem is the same in a sufficiently small interval (unless the function is strange but no problem was of that sort).
My point is, if you want to read it, it doesn't need to be some kind of pilgrimage to the Mecca of Math, you could read it just to learn about the theorems, to have a more structured knowledge, to reinforce what you know. That said, it is a definition-theorem-proof book so you would need to be able to handle that.
