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I use textbook study and am planning on studying Spivak's Calculus, Mathematics It's Content, Methods, and Meaning, How to Prove it by Velleman, etc. However, I'm worried I lack the prerequisite knowledge gained from formal classes in algebra I & II, Trig and beyond. Does Basic Mathematics by Serge Lang, The Zakon Series on Mathematical Analysis, Basic Mathematics, The Kiselev Geometry Books, H.S. Hall's Algebra Series, Euclid's Elements, An Introduction to Logic and the Scientific Method by Cohen and Nagel, and Russel's Mathematical Philosophy cover everything necessary? Does anything need to be omitted as prerequisite or anything added to cover some information?

Thank you.

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  • $\begingroup$ similar: math.stackexchange.com/questions/69848/… $\endgroup$ – Ben Crowell Jul 11 at 0:51
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    $\begingroup$ I would also caution against thinking that you have to do the hardest books that people on MSE always recommend. You may get more bang for your buck mastering (95%+ facility, all hw problems) a standard text and progressing further on to diffyqs, etc. Certainly if going into physics or engineering, it's a better way to spend time. Even if you want to be a research mathematician, I'm not sure the time efficiency of mixing what could be parts of a later topic (real analysis class) into a standard calculus course. You might learn same things, by just progressing. $\endgroup$ – guest Jul 11 at 13:23
  • $\begingroup$ Why the downvote? This seems like a perfectly reasonable question to me. $\endgroup$ – Ben Crowell Jul 12 at 16:00
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Your list seems like overkill to me.

As far as geometry, Kiselev is basically a rehash of Euclid, so I don't see the point in studying both. Just pick one. I don't think you need the solid geometry parts of either.

If using Euclid: -- Euclid contains stuff like number theory done in an ancient style that is now only of historical interest, so if using Euclid, skip that. Euclid's definitions of terms like "line" and "point" are nonsense by modern standards, so skip them as well. (The Russell book will give you a better modern mathematician's intro to how definitions work.) Make sure to work from a well-annotated edition of Euclid. IIRC there's a very good free one online by Kirkpatrick. Some of Euclid's arguments have flaws (including his very first theorem).

The Russell book spends a lot of time developing what we would now call set theory, but it predates the standard modern formulation of ZFC, so I'm not sure I'd recommend using it as an intro to that topic. I don't think any significant amount of the material from this book is needed for Spivak.

I'd avoid making an affectation out of reading books by famous people dating to 100-130 years ago.

For algebra, trig, and geometry, you can get all the preparation you need from any crappy high school books you can find at the library or half.com. The hard part about Spivak is going to be the that it's written for an audience of future mathematicians, so it doesn't dumb things down, and the problems are reputedly hard.

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  • $\begingroup$ what recommendations do you have as an introduction to set theory? Thanks for the tips. $\endgroup$ – Dirac Academy of Self Study Jul 11 at 3:07
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    $\begingroup$ an introduction to set theory --- Given what you've written and your goals, and after looking at the several dozen set theory books on my bookshelf, I'm hard-pressed to recommend any for you at this time. Maybe Introduction to The Theory of Sets (1958 original hardback; 2006 Dover paperback reprint) is sufficiently short and introductory for you now. Probably more than enough is just reading one of the standard introductory set theory chapters in a real analysis or topology text (e.g. Chapter 1 of Munkres). $\endgroup$ – Dave L Renfro Jul 11 at 15:02
  • $\begingroup$ @DiracAcademyofSelfStudy: I don't think you need to know any significant amount of set theory in order to understand Spivak. You could read a brief intro such as the WP article en.wikipedia.org/wiki/Set_(mathematics) $\endgroup$ – Ben Crowell Jul 12 at 0:35
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I definitely recommend a book that carefully explains how to write proofs and various aspects of mathematical reasoning. I haven't read How to Prove It by Velleman, but I have heard good things about it.

I am familiar with Mathematics: Its Content, Methods, and Meaning. It's more of a survey book of various aspects of mathematics, with an emphasis on Soviet mathematics (given the era it was written in). I wouldn't recommend it for learning much actual mathematics, though. I think that a well-written textbook would be more suitable for self-study.

If you know trigonometry and know basic set notation (you can pick this up in any trigonometry textbook), then I think you can go ahead and start Spivak's calculus book.

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