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Concerned about my mathematical maturity.

  1. Oct 10, 2009 #1
    I can't read abstract math textbooks/symbols quickly and i find myself skimming over them. I can't do any simple prooves.
    I thought I was good at math, but then it struck...

    With what textbooks/subjects can I get started?
    Level of math: calculus 1,2, and 3 (non rigorous)

    *Textbooks you recommend should be free
    *It must emphasize prooves at some point, and various levels of prooves are a must
    Last edited: Oct 10, 2009
  2. jcsd
  3. Oct 10, 2009 #2


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    One of the main reasons, in my experience, that people have a hard time with abstract mathematics is that they are not familiar with the concrete situations from which the abstract concepts arise. Abstract stuctures like vector spaces, hilbert spaces etc. are always introduced to discuss the common structures in various concrete situations. For example, if you can't think of atleast 5 concrete examples of a vector space, then you are not ready to learn about abstract vector spaces. If you can't think of atleast 5 concrete examples of a group, then you are not ready to discuss abstract groups.
  4. Oct 10, 2009 #3


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    Can you give five (finite) examples of a vector space? Most linear algebra students can only give one, F^n. Examples are not sufficient for understanding. There are numerous mental hurdles. Assume a person has five examples Xi of an X she/he might then say
    -Why should I care about any X but Xi?
    -When I know I am dealing with Xi why should I assume X?
    -Learning about X is hard why not just learn about Xi?
    -I am not conviced there are any X except Xi.
    -I think any X that is not Xi has something wrong with it.
  5. Oct 10, 2009 #4
    Theoretical mathematics texts cannot be "read quickly" by an undergraduate. Even when reading a math paper, you should have spare paper or a blackboard and make sure you understand the implications of every paragraph in the text. Unlike texts at calculus level and below, texts written at analysis level and above require the student to fill in theorems and factoids mentioned in the text themselves before they tackle the exercises. These theorems may not even be labelled "theorem", they will simply be statements the author makes without inline proof. If you try to read the text in a quick fashion like a calculus "recipe book", you will miss all of those little theorems, factoids, and motivations. Also unlike these books, typically the author is past the point of introducing the student to 5 examples of the same theorem, and will either not have an example or motivate only a single example, making these texts very dense. They can contain a great deal more in 5 pages than 5 chapters of most lower level calculus texts, as they expect the student to come up with more examples/applications on their own.
    In short, the text is not meant to be simply read, it must be treated like an extended exercise in itself.
    Last edited: Oct 10, 2009
  6. Oct 10, 2009 #5


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    Where did I say examples are sufficient for understanding?

    Xi have a common structure, which is embodied in X. This structure can be discussed without reference to a particular example. The idea of a function can be discussed without talking about a particular function like cos or sin, and I'm sure you know that this is a very useful thing to do.

    If you can show (not assume) that Xi is a vector space, then various theorems and results from the abstract theory vector spaces become available for application when dealing with Xi.

    Is it easier to learn about the derivatives of each function seperately, or to introduce the notion of derivative in context of general functions and then apply the results (product rule etc.) in each case?

    Then you better convince yourself that there are before you start studying X.

    I have no idea what that means.
  7. Oct 10, 2009 #6


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    Being a physics student, I only deal with real and complex vector spaces, and the theory of vector spaces over finite fields is not useful to me. That's another problem people face: worrying about irrelevant generality.
  8. Oct 10, 2009 #7


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    These simple questions may not be all that make generality difficult to grasp, but the fact that they are often not adressed at all cannot be good.
  9. Oct 10, 2009 #8
    Physics students have it tough: the vector space notation they use is terrible!
  10. Oct 10, 2009 #9
    i condemb your condescending tone.
    you mean only grads can read those texts quickly? that's so unfair.

    what is phisycist's vector space notation?

    also can i get help on where to get started on more abstract math?

    Is there nothing but prove, prove, prove?
    Last edited: Oct 10, 2009
  11. Oct 10, 2009 #10
    Maybe he means "bra" and "ket" notation ... That is [itex] \langle a \vert[/itex] and [itex]\vert b \rangle [/itex] are a bra and a ket, put them together to get a "bracket" [itex] \langle a \vert b \rangle[/itex], which is what mathematicians call an inner product.
  12. Oct 11, 2009 #11


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    Of course you can read faster with practice.

    There is more to math then proofs. There are also examples, definitions, disproofs, applications, computations, and much more. Proofs are important though.

    The first thing to try would be to prove some obvious things you should know like
    -sqrt(2) is irrational
    -There is no largest prime
    -exp(x) is not a polynomial

    Next learn a subject in which simple proofs are use like
    -Elementary Geometry
    -Arithmetic (Number Theory)
    -Linear Algebra

    Though harder one could also use
    -Calculus (though if you know some calculus it would be nice to do a few proofs)

    There also exist 'introduction to math' type books, I think the material they contain is better learned in concert with a particular subject as above, but it would not hurd to browse a few at your desired price of free.

    Proofs and Concepts the fundamentals of abstract mathematics by Dave Witte Morris and Joy Morris

    Sets, relations, functions by Ivo Düntsch
  13. Oct 12, 2009 #12


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    1) space of polynomials with a degree smaller than n
    2) space of nxn Matricies
    3) space of linear transformations from a space to itself (HomV,VV))
    4) space of continuous functions from R to R
    5) Finite fields: http://en.wikipedia.org/wiki/Finite_field

    I'll let you come up with the appropriate fields and operations.
  14. Oct 12, 2009 #13


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    It might help you to realize that "proof" is a synonym for "doing calculations with logic".
  15. Oct 12, 2009 #14
    I'm not sure if that was what was actually meant. I think that most math books from Introductory Analysis and Abstract Algebra and up should be read in a detailed manner with some scrap paper for you to work out any details and examples you may find tricky. Skimming through books and attempting to even get the most basic understanding of the theorems and definitions is not suitable for most science students.
  16. Oct 13, 2009 #15
    This is the impression I get when someone mentions "mathematical maturity". But honestly, if you're not doing that, then you should either put in more effort or the material is too easy for you. I really doubt that "mathematical maturity" really has any more content than simply having the patience to work enough to understand a concept, which is true for other subjects beside math as well.
  17. Oct 13, 2009 #16


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    lolgarithms, What Slider142 said says nothing about graduates, and it is logically incorrect to infer from it some meaning about them. If I were to say that the New York City telephone book cannot be read quickly by an undergraduate, you would probably agree that this is a true statement. However, it would be logically erroneous to infer from this statement that graduates can somehow read this phone book quickly.

    As was mentioned earlier in this thread, math texts typically are dense with notation that requires a lot of time to decode and understand. In addition, there are often steps that are only sketchily described, that usually require some work to verify. The extra needed for comprehension is something that is taken for granted by mathematicians, but probably not many/most math undergrads.

    BTW, the word is "condemn."
  18. Oct 13, 2009 #17

    What topics are covered in the most rigorous undergrad curriculum?
    besides calculus (including complex variables), ordinary and partial diffeqs, linear and abstract algebra, real and complex analysis, set theory/logic...
  19. Oct 14, 2009 #18
    That is essentially what they cover. The analysis becomes more rigorous stepping into the graduate program (there are also other topics in the graduate program, such as topology).
  20. Oct 14, 2009 #19


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    Good response.
  21. Oct 14, 2009 #20
    That really isn't the idea. You have it like this:
    Bra's are row vectors and kets are column vectors. This way you can easily and intuitively define both inner and outer products, and the vector algebra doing multiple vectors and matrixes gets a lot simpler.

    Like [itex] \vert a \rangle \langle a \vert[/itex] for a as a unit vector becomes a matrix representation of a projection on a, since [itex] \vert a \rangle \langle a \vert b \rangle[/itex] will just pick out the a parts of b and give it out again as a's. IF you want projections on more advanced objects just add multiple of these together to form a matrix with higher rank.
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