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Testing Accessible Math Greats?

  1. Jun 7, 2012 #1

    I'm currently a junior undergraduate, with a relatively weak background in mathematics. I've done linear algebra, differential equations, and up to multivariable calculus. But no statistics, discrete mathematics, analysis, geometry etc.

    Usually when I begin to explore a new field (or world, in this case), I find it very helpful to read seminal works with strong historical impact. I was wondering if anyone could recommend to me some important books or papers that might be accessible to someone of my limited background. For example, I read Neumann's Theory of Self-Reproducing Automata, and loved what I understood, which I actually felt like was a good deal. The time period does not particularly matter, and I'm really just looking to become better at reading mathematics and developing a strong intuition for the subject. So maybe Euler or Euclid would be good to read. I realize this is probably too-broad a request, but I really am open to reading from any area of mathematics, but simply don't know what to begin with. If anyone could recommend 3 or 4 of their favorite works, which are also accessible, I would appreciate it a lot. (I emailed my professor with the same question, and he didn't even reply..lol).

  2. jcsd
  3. Jun 7, 2012 #2
    Gauss' disquisitiones spring to mind
  4. Jun 7, 2012 #3
    Try Euclid's Elements. Thirteen volumes of awesomeness!!

    Diophantus' Arithmetica had a great impact during its time too. Probably the kind of book you're looking for?

    A more recent one(and a better read) is Principia mathematica by Bertrand Russel.

    All of them are freely available online, so they have great accessibility :smile:
  5. Jun 7, 2012 #4
    The bourbaki books are pretty significant I believe
    For the most part they're pretty nice books even if they are a little outdated now a days

    Although I'd argue that 'historical impact' doesn't really make a paper or a textbook good. Most of the best (imo) textbooks I have read haven't really been the most widely used or the most historically significant.

    The first bourbaki book, the theory of sets is one of my favourites and it's pretty good at developing that 'mathematical maturity' stuff :biggrin:
  6. Jun 7, 2012 #5
    These are all excellent recommendations, thank you so much everyone. I doubt I would have stumbled upon any of them by myself, with the exception of Elements.

    Also genericusrnme, I agree on how petty history can often be, and impact was perhaps the wrong word to use. I'm just interested in gaining the type of foundations necessary to approach mathematics confidently.
  7. Jun 7, 2012 #6
    If you're looking for good foundation books then I'd recommend these bad boys;
    1. Vellerman - How to Prove it: A Structured Approach (good for learning how to read proofs)
    2. Bourbaki - Theory of Sets
    3. Hoffman - Linear Algebra
    4. Spivak - Calculus
    5. Rudin - Principles of Mathematical Analysis (although it pains me to say this I think it's an ok text, it's just incredibly painful work(
    6. Roman - Advanced Linear Algebra
    7. Paolo - Algebra: Chapter 0
    8. Kolmogorov - Elements of the theory of functions and functional analysis

    I'd go on but with those books under your belt I'm pretty sure you'd do just fine reading whatever other texts you feel you may need
    I'm sure other people will probably disaprove of my list too, they are just some books I quite enjoyed - they may or may not work for you
    Good luck! o:)
  8. Jun 7, 2012 #7
    I'd like to read more old stuff. A lot of good material can be found in old papers, rather than in books. My favorite reasonably old book is by Hilbert and Cohn Vassen: Geometry and the Imagination. Another fantastic book that I have checked out from the library is "On Riemann's Theory of Algebraic Functions and their integrals, by Felix Klein. Another one by Klein that I want to read is Development of Mathematics in the 19th Century. Vladimir Arnold said that he learned half the math he knows from that book.
  9. Jun 7, 2012 #8
    For the OP, historical material is excellent for giving context to mathematics, but we have often worked out much more effective approaches to teaching and notation since things were originally discovered. I think finding a modern book that also conveys some historical context is the best option until you have the maturity to tackle historical sources directly. There are certainly exceptions to this, such as some of the books already mentioned.

    FYI, the author's name is Paolo Aluffi. This is an excellent book, but perhaps not the best first textbook for Abstract Algebra. It is more aimed at grad students, even though it is written very well and a well-prepared undergrad could tackle it.
  10. Jun 8, 2012 #9
    Yeah, I was gonna add the bourbaki group's book algebra 1 before that but I didn't think I should reccomend two boubaki books :biggrin:
  11. Jun 8, 2012 #10
    A more contemporary classic, "What is Mathematics? An Elementary Approach to Ideas and Methods" by Courant and Robbins provides an overview of the scope of the field presented with rigor, problems, and proofs.

    "It is a work of high perfection, whether judged by aesthetic, pedagogical or scientific standards. It is astonishing to what extent What is Mathematics? has succeeded in making clear by means of the simplest examples all the fundamental ideas and methods which we mathematicians consider the life blood of our science."--Herman Weyl
  12. Jun 8, 2012 #11


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    Euclid: Elements;
    Euler: Elements of Algebra;
    Gauss: Disquisitiones Arithmeticae;
    Hilbert and Cohn-Vossen: Geometry and the imagination.

    that should hold you for a few years.
  13. Jun 8, 2012 #12


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    Some suggestions:

    Hamiltons original treatise on Quaternions
    Grassmanns original work on geometri tc algebra (could also include Clifford, Cayley, and others)
    Riemanns paper on the stuff relating to the Zeta function
    Von Neumanns original works and papers

    So much stuff that is out there like mathwonk said.

    Can't say you won't be busy!
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