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Question about the math degree (for those who have completed it)

  1. Nov 8, 2012 #1
    When you've complete the math degree, did you feel like you got your education's worth and money's worth?

    I have a few goals in mind that i would like to complete in math:

    Start from axioms and build up geometry
    Construct the real numbers
    know what's happening in calculus in a deeper understanding
    do every question in Spivak's Calculus
    Pass real analysis
    take a topology course

    So for the first 4, did you feel like you could do those skills after you finished your degree? Those are my only goals in mathematics to be quite honest. I'm currently in 2nd year and still have no idea how to do most questions in Spivak's calculus. What were you guys able to do at the end of your degree?
  2. jcsd
  3. Nov 8, 2012 #2


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    I've never been too interested in Geometry, so no idea on number 1.

    Did construct the real number line using Dadekind cuts, that was neat.
    I did achieve a better understanding of calc-a-ma-less
    I did do every problem Spivak but I had a year in Afghanistan to do it...
    I did pass a real analysis class!
    I did take one topology class too!

    If those are your math goals, then they are reasonable. After math I ended up Stats, I find it's a happier place in that world.
  4. Nov 9, 2012 #3
    Yeah, I did all that in undergrad.
  5. Nov 9, 2012 #4


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    Hey kramer733.

    I haven't done a great deal of pure stuff at all, but I've done a lot of applied mathematics and statistics and I can say that the experience has been well worth it.

    The pure stuff has its place for sure and all the people that use it trust the pure mathematicians that they have gone through a task that most don't want to, and checked the whole thing out so that it is all consistent and all boxes have been ticked, t's crossed and so on.

    While i value what the pure mathematicians have done, I have found the value of the applied stuff to real life pretty good (and I don't just mean real life as in applications of mathematics to scientific, engineering, or highly intellectual problems).

    A bit of statistics can help you look at a situation and say "something is not right" and give you ways to look at and think about how you decide what is not right about something.

    It's kind of like how knowing what fractions and geometric series mean in the context of reading the news or your loan statement (or credit card statement) and seeing what the real story is: even if the information given isn't false, the real thing is having the ability to make the right interpretation and statistics primarily focuses on systematic ways to approach uncertainty.

    With regard to your goals, I think you'll be surprised at what will happen in a couple of years: you'll see stuff that is hard now not nearly as hard and things will make more sense.

    My advice for deeper understanding would be to consider that mathematics is only about three things regardless of the area (including all the stuff in theoretical computer science) and these are representation, transformation, and constraint.

    Representation asks how do we represent something; transformation asks how can we change something (either keeping all the information for that something or losing it with an approximation or down-wards projection) and finally; constraint asks what particular things are we looking at relative to all the other possibilities out there.

    All of mathematics is based on trying to understand these three both in themselves and also in connection to each other.

    If you are able to understand these three things in any significant depth then you will understand mathematics at a deep level.
  6. Nov 9, 2012 #5
    To add, there is always the possibility more goals appear during your study on some mathematical topic you're not at all aware of as of now. This could be:
    set theory, non-euclidian geometry, etc. Maybe once you've looked at real analysis, you'll want to know what complex analysis is all about. (Usually, this class requires a lot of prereqs but there is always a complex variables course to get a sneak peak).

    Math is really a continuous chain of gaining interests and having the chance to explain them along the way.

    As for applied math, the same process goes on (PDEs is a great example). For statistics, I haven't really "done" any coursework but I'm certain probability theory is full of interesting topics.
  7. Nov 9, 2012 #6
    Most of these can be answered by looking at your curriculum, you should have a pretty good understanding of calculus by the end of calculus 3, and a deeper understanding by the end of real analysis. Assuming you understand real analysis well enough I imagine you having no problem with Spivak.

    Understanding real numbers sounds like you need to maybe take number theory, geometry you would likely have to take a upper division course on geometry, or maybe topology.

    Like anything in life what you put into it will result in what you get out of it. I imagine probably one of the best benefits to doing a math degree would be being able to think logically and problem solve extremely well, regardless of your future career prospects.
  8. Nov 9, 2012 #7
    I'm sorry. but I disagree with this. I have a solid understanding of real analysis, and I have trouble with a non-trivial number of exercises in the Spivak book. I think he will be able to do the Spivak thing, but it will take a while (given that he has other classes to take).

    It seems that the other stuff might be easier to do (in the sense that they are *just* taking classes) with the possible exception of Geometry. At my school, there was no geometry course (well, no euclidean geometry course) and even if there was, it seems highly unlikely that it could "build up geonetry from the axioms" - at least not all of it, simply because it seems like there is waaaay too much stuff to learn.
  9. Nov 9, 2012 #8
    Here, we have euclidean and nonecludean geometry as seperate courses. They are not labeled as strictly proof based but students are given an axiomatic treatment.

    It largely depends on universities because I've not seen identical geometry courses offered in other schools. The requirements is calculus here.
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