So, for instance, maybe Baby Rudin is the paradigm of "how to do it right" which includes proving stuff like Baire's Theorem. By "rigorous", I don't necessarily mean "Baby Rudin". On the other hand, Thomas and Finney does, in fact, have the formal definition of a limit, for instance, on p70 of my edition. But, that's after a couple sections covering limits and even theorems about limits, done as sort of a digression and a computational one at that, if you look at the exercises. Clearly, students aren't really expected to be able to do epsilon-delta proofs, and the material of the book doesn't appear to entail that level of sophistication. So, by "rigorous", I also don't mean that sort of thing, either -- where it is technically "in there" somewhere but not really taught. Apostol is not quite Real Analysis, yet he definitely has an impressive table of contents for a calculus book. For instance, if you look at the way he approaches integration, he starts out with simple finite sums of step functions and you get exercises on that. Then, he extends that using the Archimedean Property. If you go back to the section where he covers the Archimedean Property, he's got all kinds of nice exercises in there asking the student to prove stuff like that the rationals are dense in the reals. On the other hand, on that section for integration, those exercises are pretty much all computational. I don't think there is a single problem in that exercise set that asks the student to mimic his proof that a monotonic function on a closed interval is integrable or that otherwise requires the use of the Archimdean Property to figure out or do. Of course I don't want to judge a book on two sections -- I'm just looking for other people's opinion on the book. How rigorous is it? Is this really just Thomas and Finney on steroids? Is Courant, for instance (which I have just ordered a copy of but have not really been able to really look at), qualitatively different from this? Is this just calculus for you -- if I really want to teach rigor in this venue then I will just have to wait until real analysis to do it?
With apostol? Well, that's the question. How much do I have? For instance, just because the author is rigorous in their presentation, that doesn't mean the student is actually expected to understand it. To what extent is the student expected to be rigorous (as opposed to something else that is perhaps very hard but nevertheless not rigorous).
You go on and learn other things and it will come with time. Now, you're also assuming that someone's understanding of mathematics is through their rigorous treatment of the topic at hand. Well, that's false.
The truth of a belief will never make the argument you used to arrive at it is any less spurious. And, knowledge isn't knowledge without correct justification. Rigor doesn't come with time and it doesn't suddenly emerge from years of heuristic rationalization. It is a completely separate skill that must be taught independently of any of that, independently even of the facts, themselves. And, by the way, it can be taught to a student the moment they are capable of virtually any abstraction at all. It is just a question of choosing it over telling them about more facts that they are incapable of truly understanding.
I disagree. Lots of areas of mathematics are connected and work together. When you learn about other areas, you learn of its applications elsewhere and how they mingle together. Then you'll go back to stuff you've studied earlier and it's really not that bad at all. (When you're in 4th year and look back at 1st and 2nd, you'll be like... wow that wasn't so bad after all.)
The assumption, I suppose, is that Apostol is not rigorous enough to adequately prepare for a course in Real Analysis. Having used mainstream Thomas and Finney like texts and then Apostol, I can personally say that his presentation is a large step up in terms of rigor from the former example. Apostol helps develop a better understanding of the development of calculus and its applications than most other calculus books. His exercise sections ususually contain both computations and proofs so I'm not so sure about your claim that they are solely computational. Even in his preface, Apostol states he tried to extablish a balance between theory and application and in my opinion he largely succeeded.
Clearly the issue is the exercises and not the presentation. So, are you saying then that a typical student out of Apostol can do stuff like use the Archimedean Property to prove that a monotone function is integrable? Can they do epsilon-delta proofs? I really am just curious? It doesn't look like it to me from the exercises.
I'm definitely not asking about how good of a calculus text it is. I certainly am not asking if this will prepare you for yet some other course. I'm asking about the nature of the material covered in it. For instance, "A student who made an A in his calculus class where they used Apostol probably can do...," what? A meaningful response to the question will bring up some sections, talk about what's in them and how it is rigorous. For instance, you could say "Aha! You are looking at the section where he first introduces the integral of a monotone function, but if you skip to three sections beyond that you will see that all of those problems do challenge the student on...." Maybe you would say something like "An alternative way to rigorously do such and such is to blah blah blah, and if you look at the exercises in sections yadda, yadda, and yadda, you can see that a student can definitely blah blah blah by the end of yadda yadda yadda." Now, whether or not I'm some kind of a big ******* for asking this question in the first place is another matter. Answers like "Apostol is rigorous enough," don't answer my question. Above all, by "rigorous", I am talking about the math, itself, (that the student does or is expected to do) and not the difficulty of the undertaking or something like that. "Apostol represents a very rigorous treatment of freshman calculus," is not what I am talking about. I'm sure it is. Yes, I know that Apostol is much better than Thomas and Finney. I'm asking about the mathematical rigor. Now that we know that the book is the best one out there, how rigorous of an understanding of the mathematics does a student actually walk away with? Not how much math do they know or how good of a problem solver they are or how great of a preparation for further study in mathematics this is -- those are all other questions.
You're answer is here. The reason why I said it is enough is because mathematics is not about rigour! You can learn mathematics in its highest rigour, but are you really walking away knowing mathematics? In my opinion, no it doesn't now imply you know mathematics at all. Rigour is a part of mathematics but mathematics is clearly not about rigour. Mathematics has this beauty and if you do not see it, feel it, sense it, and so on, then in my opinion, you don't know mathematics at all. You can not explore mathematics with rigour because constant rigour is a constraint and it should only be consulted after you let your intuition ride free of it and feel you're way around mathematics. Of course you may argue that by exploring mathematics without rigour, you'll get to all kinds of nonsense. Well, that is also NOT true. If you have a feeling for mathematics, like I said a few lines ago, when exploring things you will just know it doesn't make sense. It's that feeling that I'm talking about that you should worry about. The rigour will come later. Note: I ain't a professor, but I know enough to know that it's not about rigour so I would suggest not to worry about it right now. Explore the things that are interesting about mathematics (you can do this without the utmost rigour). If you're interested in how rigorous mathematics is, then you should go into the foundations of mathematics. Something completely different.
adrian I cannot cite any examples from Apostol because I no longer have the text but my previous statements are from experience. I assume this is your main question. And I think I already part of it answered when I said "Apostol helps develop a better understanding of the development of calculus and its applications ..." From your questions about rigor, it seems like you're looking for analysis material in a calculus book and I'm not sure why. If you want a rigorous understanding of the theorems used in Calculus get an Analysis book such as Rudin or Apostol's Mathematical Analysis. Apostol's Calculus however does give you a fair base before you tackle these higher level books. I understand "fair base" is very vague but I think the whole issue about "rigorous understanding" is vague. How do you expect to gauge your understanding of a topic? Exams, I suppose, have this purpose but then again the level of difficulty varies from school to school. If by rigor you mean the ability to prove certain things then as I said before, get an analysis book.
well only a weak student restricts himself to working the exercises. a good student tries to understand the proofs, and even to provide some of them himself, without looking at the ones in the book. and tries to extend the results himself. i have not looked at the exercises in apostol, as that is my view. a student should not expect a teacher to tell him everything, not only explain the material but tell him how to practice it as well, and what to do on his own. just take every other theorem in apostol as an exercise for****'s sake. get some initiative!
also in foundations, intuition is needed, you can't research there without having some intuition, which is gained by practice and understanding what the definitions are all about.