Being interested in learning rigorous calculus (as opposed to the content taught in AP Calculus and intro calculus courses in university), some textbooks mentioned quite often on the internet include Introduction to Calculus and Analysis, Vol. 1 by Richard Courant, Calculus, Vol. 1 by Tom Apostol, and A Course of Pure Mathematics by GH Hardy. These books are regarded as classics.
My concern, however, is that they are rather old texts, particularly the one by GH Hardy. I do not want to learn from or purchase mathematics books which are inaccurate lest I develop any sort of false beliefs or misconceptions. My question is therefore as follows: are these books, and all relatively older (>5 decades) textbooks in general, sufficiently rigorous by the modern standards of pure mathematics?
Best Answer
Broadly speaking, a book's age does not serve to credit or discredit it. That being said, it's a bit like asking, "Is a Magnavox Odyssey still valid"? If it still plays, then it still plays, but you also have to deal with the fact that old things are meant to do different things than new things, and that even if two things have the same goal, time will still help to refine that thing through innovation.
Will Hardy's analysis be correct? Most likely. But you run into two things, especially when you're talking early twentieth century books. For one, the language will be very unorthodox by today's standards. If you're going particularly early in this period then you might even find dissent among authors as to what to actually call what we would refer to now as a "set" (I believe Russell had used "manifold" at some point for what is now a set, and now manifold has a very particular meaning in geometry). Be prepared to Google what words mean, and be prepared for that Googling to be a nontrivial endeavor. It's also important to note that an important part of mathematics is knowing how to read and communicate it, neither of which will be possible if your lexicon is a century old.
Secondly, their methods and approaches will probably not be what we'd use today. This is the refinement. Over time, mathematicians will look for better ways to do what the older folks did. It's common that a classic theorem's proof when first presented will consist of multiple lemmas, and will be a long and drawn out labor, while later authors will "streamline" and "refine" those methods. So though what folks like Hardy might give you is in all likelihood correct, it's also likely that later authors will have improved upon what the classics present. And when I say "improvement", I don't mean that some stuffy journal has found some generalization of a theorem to the point that it's almost unrecognizable; I mean that your typical undergrad text will do in six lines what an older author did in a page and a half, and probably in a reasonable generalization.
EDIT: This newfound brevity might also sometimes be a mater of saying something very similar to what the older authors were saying, but simply having more precise language to say it with.