Another question about Math...

Discussion in 'Odds & Ends' started by Fandango, Jul 7, 2012.

  1. Fandango

    Fandango Rookie

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    My question revolves around the fact that math seems to be the only constant in the whole universe- the laws governing of each planet may differ in strength and the forces that go into making the slightest carbon atom may vary as well.

    Since math seems to be the only constant in the universe, do you think it should be considered the most important subject?

    Although I consider myself decent and have a genuine curiosity for mathematics, I am intimidated by anythings past Calculus I and don't really need to learn differential equations or linear algebra for my major (biology). On a related note, is mathematics the purest form of logic?

    Also, if I know the fundamentals of math-multiplication, division, subtraction, variables. Is every process in differential/upper level calculus using an operation like the ones above?

    Are higher level math classes used as applications of those operations as well as linear algebra being the exception by rotating or bringing down terms in matrices.

    I know I don't sound exactly clear but any help is appreciated.

    Thank you for taking the time to answer this if you get a chance!
     
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  2. Talker

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    I always liked math, for me it makes me be more exacting in many other areas.

    There's all kinds of operators besides the basic four, like the modulus operator for example.

    Good math courses that cover most of what you'll ever need.
    Calc.
    linear algebra
    probability and statistics.

    Discrete math is good also as well as advanced calc.

    Enjoying the struggle calmly was what helped me.
     
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  3. Swissv2

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    Everyone will have their opinion about this one. My personal answer is...no, because math isn't the only subject that a person can do to earn a decent wage in life.

    Math is certainly a fundamental part of an education that is complex and difficult to master. That is why a strong foundation of math leads to the most highly skilled jobs in the world.


    A very close yes. But realize that any mathematical formula may be disproved at any given time. String theory is one of those subjects where higgs bosen theoretical equations may be challenged due to the recent discovery of the "god particle" (though they still have to conduct more experiments to confirm their findings.

    These are base fundamentals that get used without thought at higher levels, like breathing or walking.

    You only touch a small sliver of what is to be learned in higher level math.
     
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  4. Claudius

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    It's nice to know that you have genuine interest in math. Let me give you a brief overview of what math is.

    I don't what it means to say that math is the only "constant" thing in the universe. If by this you mean, math is the only "true" thing in the universe, you might want to define "truth." Math is true only with respect to the current accepted axiomatic framework. Some axioms that mathematicians accept are the ZFC axioms of set theory and the Peano axioms. Now, any axiomatic system with arithmetic (of natural numbers), has inherent limitations, in that some mathematical statements about natural numbers can't be proven to be true nor proven to be false. Such statements are said to be undecidable. Here's an example of an undecidable statement:

    Take any natural number n. If n is even, divide it by 2. If n is odd, multiply it by 3 and add 1. Repeat the process and you'll eventually reach 1.


    Let me talk about algebra, and the theory behind when polynomials are solvable by radicals.

    You know that a polynomial with degree 2 is clearly solvable by radicals, since the roots are given by the quadratic formula. The question of when a polynomial of any degree (with rational coefficients) is solvable by radicals can be answered using group theory.

    Definition: A group is a set G with a binary operation * that satisfies the following properties:

    For any x,y,z in G

    - x*y is in G
    - x*(y*z) = (x*y)*z
    -There exists an identity element e in G such that e*x = x*e = x
    -Every element has an inverse element b such that x*b = b*x = e

    Notice that I haven't mentioned commutativity. If the operation satisfies commutativity then our group (G, *) is said to be an abelian group.

    Some example of groups are the integers with addition, the real numbers with addition, the rational numbers with addition, and the rational numbers without 0 with multiplication. Typically they're written as (Z, +), (R, +), (Q,+) and (Q, *) respectively.

    Now we can have functions between groups that satisfy certain properties. Let G, H be groups (let's drop the operation sign for simplicity).
    A function f: G --> H is said to be a homomorphism if for any x,y in G

    f(xy) = f(x)f(y)

    An example of a homomorphism is f: Z --> Z defined by f(x) = x. Typically, functions like these that send every element to itself are called identity functions.

    We also have what's called a subgroup of a group, which is simply a subset of a group that also forms group under the same operation. If H is a subgroup of G, then we write H < G to indicate this.

    Now back to homomorphisms, we must be aware of a certain class of homomorphisms called isomorphisms. Isomorphisms are homomorphisms that are invertible (equivalently bijective homomorphisms). The identity map mentioned earlier is clearly an isomorphism.

    There are special types of subgroups known as normal subgroups. For a subgroup H < G if a is in G, we let aH ={ ax : x is in H}, that is the set of all elements of H multiplied (on the left side) by a (note: if the operation of the group isn't mentioned, we just call it "multiplication"). We define Ha similarly. If it turns out that aH = Ha, H is said to a normal subgroup of G. What's so interesting these is that they allow us to construction a new group called the quotient group of G by H denoted G/H. We define G/H as {aH : a is in G}. That is the set of cosets of H. The operations is simply aH *bH = abH. This defines a group with the identity element being H itself.


    Now let's tackle the question we asked at the beginning. Given a polynomial p(x) with rational coefficients, we define a splitting field of p(x) to be the smallest extension of Q(the rational numbers), over which p(x) factors as

    a(x - b1)(x - b2).....(x - bn)

    For instance, the polynomial x^2 - 2 has the splitting field Q(√2) which is defined to be

    {q(√2)/r(√2) : q, r are polynomials with rational coeffs and r(√2) =/= 0}


    Let's introduce a couple more concepts from group theory. A group G is said to be solvable if it has subgroups G1,.....Gk such that

    {1} < G1 < .... Gk < G

    and each subgroup is normal in the next and Gk/Gk-1 is an abelian quotient group. Note that here, 1 = e

    Let E be the splitting field of p(x). We define the galois group of E over Q , denoted Gal(E/Q) to be the set of all isomorphisms that fix Q pointwise that is

    {f: E --> E : f is an isomorphism and f(x) = x for every x in Q}

    This forms a group under the operation of function composition.

    Theorem: A polynomial p(x) with rational coefficients is solvable by radicals if and only if Gal(E/Q) is a solvable group.

    The proof is way too long to give here, but here's a nice corollary.

    Corollary(Abel-Ruffini): The general polynomial of degree greater than or equal to 5 with rational coefficients (quintics or higher) is not solvable by radicals.

    That's a taste of what math is.
     
    Last edited: Jul 9, 2012
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  5. mightyrick

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    Most important? Too hard to say. Is science more important than art? Is art more important than language? Who can judge?

    From my perspective, I'd say that language is probably the most important. Because without any kind of language, I really can't communicate thoughts or ideas. Some forms of art might even be considered language. Science (which includes mathematics) is probably least important.

    I also enjoy math. But most need very little of what they learn. If you want a fun math subject-area to explore -- take a look at axioms. Axioms are the lowest-level, machine-language instructions of mathematics. It is really amazing to explore the few very simple rules that all of mathematics derives from -- and then explore more complex derived axioms.
     
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  6. krz

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    Yes :)

    Though I was an applied math and econ major, so I may be a little biased.

    I'm sure I'll offend some people here but, if you suck at math you're probably stupid. Can't just memorize useless facts.
     
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  7. Steady Eddy

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    I sincerely try NOT to believe that, but it's a struggle. At any rate, I really cringe when I hear people brag about being clueless in mathematics.

    But I know alot of people are bad at it b/c they hate it, and they hate it b/c of the poor instruction they received.
     
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  8. mightyrick

    mightyrick Hall of Fame

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    It takes all kinds. I hate cooking. But I'm glad that there are people out there who love to make food. I hate drawing, but I'm glad there are people out there who love to create art. I have no patience for writing, but I'm glad there are people out who love to use the written word to communicate.

    Some of us love mathematics. Others hate it. But those people are also glad there are others in the world who love it.
     
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  9. Mongolmike

    Mongolmike Professional

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    This post says a lot... and not all math related.
     
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  10. Claudius

    Claudius Professional

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    What do you mean by "not all math related"?
     
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  11. bad_call

    bad_call Legend

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    not getting that one.

    anyway no discussion about P NP? might be a little "heavy" for most...based on this peeps observation.
     
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  12. Eph

    Eph Professional

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    Indeed. It is so much easier to be "good" at humanities than the be good at something objective, like mathematics.
     
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  13. r2473

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    "Shakespeare was a dumbsh!t" - Eph
     
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  14. Eph

    Eph Professional

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    And a plagiarist.
     
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  15. krz

    krz Professional

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    What's more sad is that here in the US at least it's socially acceptable to be bad at math.

    In fact, people wear it like a badge of honor.

    Off topic kinda but, the other thing that gets me is people who consider themselves "bad test takers." So you struggle with the part where we find out what you know? (Tosh reference)
     
    Last edited: Jul 10, 2012
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  16. Eph

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    Really? People display being bad at math like a badge of honor? Can you give an example? I've never experienced that-- the people I know in the humanities wish they could do math, and are not proud that they cannot.
     
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  17. bad_call

    bad_call Legend

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    possible indication why an individual's personal budget isn't kept? or a lack of reasoning/comprehension when reading/communicating with others?
     
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  18. Swissv2

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    He is probably talking about those guys that really don't give a rip about math, and are not shy about saying they will "never use it in their life" and "math is for nerds". Then, they complain about the drama that is happening at the local fast food restaurant they have been working at for the past few months.
     
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  19. ninman

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    What is math? I've heard of maths but not math, is that some kind of weird new subject?
     
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  20. Claudius

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    My response to, "I was never good at math" is "But you can read, right?"

    :)
     
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  21. krz

    krz Professional

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    Or that annoying girl "teehee I'm so bad at math teehee"

    From across the pond are we?
     
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  22. millicurie999

    millicurie999 Semi-Pro

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    #22
  23. OriginalHockeytowner

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    Is that the Collatz conjecture?

    There are several great XKCD comic strips related to this thread...but I won't post them :)
     
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  24. mightyrick

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    Here's another, the MR conjecture:

    Take any natural number n. If n is even subtract 1. If n is odd, subtract 1. Repeat the process and you'll eventually reach 1.
     
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  25. Claudius

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    There's actually more to that statement than meets the eye. What you said is true, precisely because the integers under addition form a cyclic group (there exists one element, x, such that every element is a power of x). Here, x = 1.
     
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  26. ninman

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    No I just know the difference between plural and singular.
     
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  27. Timbo's hopeless slice

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    Math people always make me smile.

    Often have difficulty in understanding the distinction between the math and the reality. I think it is why mathematicians and scientists often think they have much in common with musicians but the inverse is NEVER true. The notes on a page are NOT the music, but math people have trouble understanding that.

    For the record, I have taught Math at College level, I just don't value it very much as a subject. Useful, just not very interesting.
     
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  28. ninman

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    As someone who works in a university researching mathematics I think about 99.9% or more of what is done research wise is the most pointless tripe ever.

    The problem is that we are judged on how many publications we have, how good the journals that we publish are and so on. This encourages people to aim for the middle ground, just take some old problem change it slightly then get a paper published.

    The whole system has made the subject very stale and boring. I personally try not to play these pointless games and go for quality over quantity, but take one look through arxiv on any given day and you'll find that there are literally no papers worth reading.
     
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  29. krz

    krz Professional

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    I've had this debate a million times, neither is wrong. I'm sure you say "maths are my best subject too" right?
     
    Last edited: Jul 12, 2012
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  30. ninman

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    There are exceptions to just about every rule in the English language, e.g. "the sheep is eating over there", "the sheep are eating over there".

    Just because it's plural doesn't mean it necessarily follows all the rules of a plural.
     
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  31. krz

    krz Professional

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    Now we are talking about English and way off topic.

    Your sheep example is not exactly the same as math or maths. Both "the sheep is eating over there" and "the sheep are eating over there" make perfect sense, but mean different things. The first example is used to refer to a single sheep and the second to more than one.

    No one says "maths are my best subject."

    Like I said it probably depends on where you are from. Most people who grew up in the US tend say math. Over yonder people sway towards maths, torques and horsepowers.

    But, hey what do I know I'm not an English major, nor a maths grad student. You should probably go ahead and contact the Mathematics departments over at Columbia and Harvard to let them know to stop using the word "math."
     
    Last edited: Jul 13, 2012
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  32. Eph

    Eph Professional

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    Very good post! I completely agree (I'm in theoretical physics at a major university)
     
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  33. Claudius

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    Depends on what you mean by "worth reading." Can you give an example of a paper that is (worth reading) and isn't?
     
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  34. vkartikv

    vkartikv Hall of Fame

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    One of the problems with the education system (universally) is that they teach you calculus but never tell you why you need to know it or where it is applied in real life. if they had told me differential calculus is the core of groundwater flow processes (which is my specialty now) i would have paid a lot more attention in school. math doesnt suck, its the teachers.
     
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  35. jaggy

    jaggy G.O.A.T.

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    Learn the President's English
     
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  36. ninman

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    #36
  37. Eph

    Eph Professional

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    really? are you an undergrad?
     
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  38. ninman

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    ah, pretend mathematics, got you. I tried reading a book about Quantum mechanics once but gave up because it didn't make any sense at all, mathematically and I literally could not see the point of it.

    We have this equation, then we have this equation, then we have this equation.

    Whatever happened to Definition, Theorem, proof? Also they never at any point say what they are trying to work out, what they are trying to achieve, i.e. this is our goal and this is how we're going to get there and now we've reached it.
     
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  39. Eph

    Eph Professional

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    Simply, mathematics is the language we use to prove what ideas we are working on. Surely you know this.
     
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  40. ninman

    ninman Hall of Fame

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    Yes, but you don't prove anything.
     
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  41. Eph

    Eph Professional

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    Prove, in what sense?
     
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  42. Claudius

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    It's interesting what you consider to be "worth reading." I'm not particularly well versed in functional analysis (which that second paper is about), but I know for a fact that the theory of isometries between Hilbert spaces has been indispensable for quantum mechanics.

    Is your approach to math "if it's beyond my level of understanding, it can't be useful?"
     
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  43. Claudius

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    Yes, in math, you prove everything.
     
    Last edited: Jul 13, 2012
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  44. ninman

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    Ok, what I do is basically applied maths, although it's not what people who applied maths/theoretical physics would call applied maths.

    When I read about theoretical physics they don't use rigorous mathematics at all to do anything. i.e.

    Definition: A particle is...

    Definition: An electron is...

    Theorem: When electrons do...this happens always.

    Proof:

    If you do that way you really get proper rigorous mathematical results, not this kind of hand waving, easy to pick apart, unproven tripe. That's why I hated physics when I was an undergraduate.

    They don't do anything or go anywhere, just write a load of equations down and say "see we proved that".

    Go to my website, and read some of my papers www.combinatorialgametheory.com, that is what applied mathematics should be like.
     
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  45. ninman

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    No my approach is, if it's derivative drivel it can't be useful.
     
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  46. r2473

    r2473 Legend

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    "math" is what uneducated fools talk about on the "internet"

    "maths" is what you and i (and other equally educated persons) discuss on the "internets"
     
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  47. Avles

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    Sorry, you're just wrong here. Merriam-Webster says that "maths" is just fine:

    http://www.merriam-webster.com/dictionary/maths

    The BBC uses "maths" http://www.bbc.co.uk/programmes/p00t6kkg (See paragraph 3)

    The Cambridge math department also uses "maths": http://www.maths.cam.ac.uk/. (See upper right, and the web address)

    Why don't you write the Cambridge math department and tell them that they're using the wrong abbreviation for their subject? I bet they'll be impressed by your assertiveness.
     
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  48. ninman

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    I'm sorry at what point did I say it should be "math"?
     
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