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Introduction to 'Real' Math (i.e. number theory etc,)

  1. Jun 29, 2009 #1

    I'm basically looking for a book that is very approachable and written for someone who knows very little about math but that goes through actual math, i.e. starts with set theory, then constructs the natural numbers, the integers, yada yada and strongly emphasizes how math is constructed from axiom ->proof -> theorem and so on. It'd be great if it was more layed back and written in a conversational tone and walks a person through such things.
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  3. Jun 29, 2009 #2


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    Aren't books that are "layed back and written in a conversational tone" usually mutually exclusive to books that "strongly emphasize how math is constructed from axiom -> proof -> theorem and so on" (although I suspect you meant axiom -> theorem -> proof)?
  4. Jun 29, 2009 #3
    I hope not. I've seen books that are pretty close, like 'concepts in modern mathematics' by ian stewart. You can emphasize the importance of proofs and construction without be dense, pedantic and unapproachable.
  5. Jun 29, 2009 #4


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    Try 'Foundations of Mathematics' by Stewart and Tall.
  6. Jun 29, 2009 #5


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    learn Discrete Mathematics, I think that fits perfectly with what you described
  7. Jun 29, 2009 #6
    Yes, I'm looking for a number theory book (which is part of discrete math) that is approachable to a layman. Basically I'd like a book to recommend for someone who maybe have some high school math and who in general distrusts math and thinks it's just a bunch of "formulas" that are made up, or determined experimentally or something and I'd like them to see how math is a constuction of logic built up from axioms and i'd like a book that would actually walk them through the basics of number theory.
  8. Jun 29, 2009 #7
    I'd write something myself but you'd think there'd be plenty of books that already do something like this.
  9. Jul 13, 2009 #8
    Probably the best math resource on the internet is the WILLIAM CHEN lecture notes http://www.maths.mq.edu.au/~wchen/ln.html" [Broken]

    From here, you can go from set/number theory to more advanced topics like complex analysis. Fairly rigorous.
    Last edited by a moderator: May 4, 2017
  10. Jul 14, 2009 #9
    Wow. That's awesome
    Last edited by a moderator: May 4, 2017
  11. Jan 9, 2010 #10
    “What is the mathematics” by Richard Courant is the best I think!
  12. Jan 31, 2010 #11
    Thanks for the thread, guys. I was wondering something similar, and I think I may have my answer- I have been looking for a while for an introduction to rigorous mathematical thought, and what higher math is really like, for someone with just a high school/ some calculus background to get a glimpse of the real thing. So I'm giving this thread a bump, and any more thoughts would be great! thanks.
  13. Feb 13, 2010 #12
    MIT's openware is quite entertaining and it is free.
  14. Feb 14, 2010 #13
    https://www.amazon.com/Foundations-...dp_top_cm_cr_acr_txt?ie=UTF8&showViewpoints=1 by Landau is a book that completely develops the natural numbers, the rationals, the reals, and the complex numbers from the Peano Axioms. It's one of the most terse math books you'll ever read and it's typeset horribly, but if you want rigor for all things arithmetic that are always taken for granted and you want it done in an axiom->lemma->proof format, this book absolutely delivers.

    If you had a lot of calculus background I would recommend Baby Rudin (Principles of Mathematical Analysis) so you could learn something more useful, but I wouldn't advise going into it until you've completed an entire university calculus sequence (through Stokes' Theorem) and a maybe a linear algebra class that had a little bit of theory in it.

    Landau won't require any specific knowledge to complete though; you just have to be a very attentive reader. I personally think intuition and visualization are a hell of a lot more important than rigor when it comes to math, but rigor has its place when you're checking your ideas and communicating them.
    Last edited by a moderator: May 4, 2017
  15. Mar 10, 2010 #14
  16. Mar 12, 2010 #15
    "Fundamentals of Mathematics." By Moses Richardson. 1960's ed., Newer edition (70's, I think) in collaboration with Leonard Richardson (Moses' son, I presume.) Long out of print, available used through Amazon $14.95 or thereabouts. This book is described exactly by your request, and, as an extra, is a pleasure to read.
  17. May 27, 2010 #16
    I didn't know about these. I'll have to check them out. I've been looking for "Fundamentals of Mathematics" in my local library but haven't been able to find it.

    Last edited by a moderator: May 27, 2010
  18. Jun 22, 2010 #17
    This set of notes is absolutely fantastic. What a gem you've shared. Thank you very much for this recommendation.It's quite difficult to find pre-calculus material presented in an intelligible and fairly rigorous manner, but these lectures seem to do just that. I only wish there were more material.
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  19. Jun 22, 2010 #18
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  20. Aug 1, 2010 #19

    Gib Z

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    Wow, I've only looked through the Complex Analysis notes, but from what I see, William Chen's notes are extremely well written, succint, and includes plenty of problems at the end! Excellent resource!
  21. Aug 2, 2010 #20
    There's one book that I know of that's laid out exactly how you have said in your OP. I'm sure you know about it. "The Principles of Mathematics by Allendoerfer and Oakley. At the beginning it definitely starts out conversational as it introduces basic mathematical logic and set theory but gets into the nitty gritty of proofs later on.

    From what I can tell, a lot of the same titles pop up whenever someone asks a question like this (I have most the books that have been mentioned in this thread for example). So I'd like to know what else it out there as well.
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