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I bought Spivak's classic Calculus a month or so ago, and after doing a few problems from the first chapter, it's become apparent that some type of foundational knowledge in formal maths and proofs are necessary.

My Question to those of the stackexchange community that have worked through this book: What some of the things you studied prior to Spivak, and what were some of the books you used? As I said before, it feels that a foundation in formal maths will be necessary, so I've purchased Velleman's How To Prove It, but I'm not particularly sure if this book will lead me into the right direction to be able to tackle an introduction to elementary analysis book.

Thanks in advance.

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What specifically are you having trouble with? Maybe the first few chapters of this: math.wustl.edu/~sk/newtrans.pdf –  Potato Feb 5 '12 at 6:45
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Maybe it would be better to present specific problems and your attempts? In theory, I don't think you're supposed to need a lot of preparation for Spivak's book, since it is usually assigned to first-year students, and you want to avoid reading a whole book on proofs and logic if you can help it; of course, that doesn't mean that adjusting to a higher standard of rigor isn't difficult. My friend Mitya wrote a short handout on writing proofs for an introductory class using Spivak that you might enjoy. –  Dylan Moreland Feb 5 '12 at 7:04
    
Thank you for encouragement and motivation to keep at it. I will do so, and certainly take a look at the papers provided to supplement. –  Miles Stevenson Feb 5 '12 at 11:01
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Just realized, possible duplicate of (which also contains some good answers for you) math.stackexchange.com/questions/69848/preparing-for-spivak –  user23784 Feb 5 '12 at 22:24

4 Answers 4

up vote 12 down vote accepted

What you need to read a book like Spivak for the first time is what can loosely be termed "mathematical maturity." Or at least the beginnings of it (the definition keeps getting more demanding the further your studies take you). The only way to obtain mathematical maturity is by reading books like Spivak. I do not think there is another book you need to prepare you; Spivak is one of the canonical choices for a first "rigorous" maths book.

Mostly, you just have to press ahead, keep putting in effort, and ask questions. If you put in real work on a problem and post it here you are likely to get some outstanding answers. Even if you don't understand some things completely, I would argue to keep moving forward. Many of the things you don't understand in Chapter 1 will start to solidify in your mind as you move forward. (That is not to say just skim over things... really, really work at them, but don't be afraid to continue covering new material even while the earlier chapters are still solidifying in your brain. Just don't get bogged down on one problem or section.)

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Thank you for the wonderful answer. I will try not to be so discouraged, and keep pushing. –  Miles Stevenson Feb 5 '12 at 10:49
    
What are other 'canonical' books if you don't mind me asking? –  seeker Jan 2 at 13:02

I think persistence and patience is the only requirement for Spivak's Calculus. But, if you are still not sure, maybe give Ross a chance. The book is intended to be an introduction to Calculus without any proof based knowledge.

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+1 for Ross. He's the Boss atleast at this stage for the OP! –  user21436 Feb 5 '12 at 8:08
    
Thank you for the wonderful answer. –  Miles Stevenson Feb 5 '12 at 10:48

Not sure if this post is still active. I typed in "mathematics Proofs" into amazon books and got 5-7 decent looking books. Rotman is a good author I think …

I sympathize with your situation. I think a book like Velleman's could potentially be helpful. Ultimately it is just a hard road learning about proofs. But it is fun as you get the hang of it. Get a tutor once in a while for feedback.

I made my transition to advanced math using Spivak's book. I took a class using it in my first year after high school, and it took lots of time to get used to and help from friends over beers. Then I had a great Summer working through Spivak a second time slowly doing all the problem sets form the more advanced class. It was a formative experience for me. (I eventually got my Ph D and was in academics for a while, but will probably move on soon.)

So advice on Spivak: Yes, give it time … keep working at problems, go over it again and again. Get some help, talk to people if you know any. Learning math is very osmotic I think. It is very helpful to have someone who has it in their bones give you a feel for it.

For limits, I found it helpful to just focus on polynomials at first. Spivak does a detailed example. You can follow the same pattern for any polynomial, like an algorithm for how to write the proof. Once I had that under my belt, I gained some confidence with limit proofs.

For properties of real numbers it is a bit subtle and you will gain more and more appreciation of their subtle aspects as you go on. Don't get hung up on it.


I looked at Velleman's book you mentioned. Seems like it should help a bit. But don't get stuck there, jump into proofs and problems that really interest you, there are lots of cool ones in Spivak.

If Velleman doesn't help try some of the ones you get after typing "mathematics proofs" into amazon.

Here are some. Unfortunately I cannot recommend any one in particular (I believe Rotman is a good author). Good luck.

Mathematical Proofs: A Transition to Advanced Mathematics (3rd Edition) (Featured Titles for Transition to Advanced... by Gary Chartrand, Albert D. Polimeni and Ping Zhang (Sep 27, 2012)

Proof in Mathematics: An Introduction by Albert Daoud and James Franklin (May 17, 2012) Formats

Proofs and Fundamentals: A First Course in Abstract Mathematics by Ethan D. Bloch (Jun 1, 2000)

Journey into Mathematics: An Introduction to Proofs (Dover Books on Mathematics) by Joseph J. Rotman (Dec 21, 2012)

A Transition to Mathematics with Proofs (International Series in Mathematics) Hardcover – December 30, 2011 by Michael J Cullinane

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I'm a new posted that was drawn to post by this thread. Was looking for Spivak's Calculus myself on the recommendation of a math grad student who was helping me with an analysis problem today.

Here is my $.02, FWIW. I am a hard-working math student (older) who is not a natural. I get by on elbow grease and depressing amounts of time invested. I am suffering in real analysis right now, but to place myself in the spectrum of math students, did very well at intro proofs, linear algebra, reasonably well in the calc series.

SO...we are using Ross in my analysis course. It is decidedly not for beginners with no proof-based knowledge in my humble opinion. It is basically a book full of proofs with a modest handful of examples interspersed throughout it. I'm looking for alternatives to it right now, or at least complements to it.

I would really, really recommend checking out something like Zorn's Understanding Real Analysis. It has a lot of discussion about the mechanics of the proofs one at a time, the language used to write proofs, graphical depictions of differing sorts of convergence, just a lot of different ways to try and make things clear. Definitely a better way to tiptoe into analysis.

cheers!

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