Understanding the weird error in thinking

Discussion in 'Recreational Math' started by James Harris, Oct 16, 2004.

  1. James Harris

    James Harris Guest

    Consider

    x^2 + 4x + 3 = (x+3)(x+1)

    where you have the unit 1 paired with 3, which, of course is not a
    unit.

    Now consider

    x^2 + Ax + 3 = (x + 3u_1)(x + u_2)

    where A is some integer other than 4 such that u_1 and u_2 are
    irrational, and u_1 u_2 = 1, but guess what?

    Both u_1 and u_2 cannot be algebraic integers.

    Now that's a specific that's forced upon them based on the definition
    of algebraic integers as roots of monic polynomials with integer
    coefficients.

    With

    x^2 + Ax + 3 = (x + 3u_1)(x + u_2)

    only u_2 can be such a root, so u_1 cannot be an algebraic integer.

    Mathematically that's what follows, but some people take a leap from
    there, and go from the proper conclusion that u_1 cannot be an
    algebraic integer, to the position that the unit case, easily seen
    with

    x^2 + 4x + 3 = (x+3)(x+1)

    has no corollary with irrationals!!!

    It's an assertion of a fundamental difference between rational and
    irrational numbers based on one simple thing: a particular number not
    being the root of a monic polynomial with integer coefficients.

    Once you get started on that path, it's easy to continue, as you can
    find algebraic integers 'b', and 'a', such that b/a = u_1, and find
    that 'a' is a factor within the ring of 3.

    So it seems simple, u_1 is some kind of fraction, right?

    Well, it *seems* simple enough, and people begin working from that
    apparent simplicity to complexity building what are today called the
    modern math tools, including those that Andrew Wiles used in his most
    famous work.

    The trouble is, the idea is easily proven to be wrong with some basic
    algebra.

    That is, the idea that because a particular irrational number cannot
    be an algebraic integer, you know something about the factors of that
    number, is just wrong.

    It makes sense then to not try to create an artificial grab-bag, like
    the ring of algebraic integers, and instead go down to the deeper
    principle, which is to consider a ring which includes integers where
    no integer is a factor of any other integer that it's not a factor of
    in the ring of integers.

    That is managed with two requirements:

    1. No rational unit other than 1 or -1 exists in the ring.

    2. No non-unit member of the ring is a factor of any two integers
    that are coprime in the ring of integers.

    Now then, with those requirements if you go back to

    x^2 + Ax + 3 = (x+3u_1)(x+u_2)

    then u_1 and u_2 may both be in that ring depending on the value of A.

    Notice then they are units and the integer case

    x^2 + 4x + 3 = (x + 3)(x + 1)

    has an irrational corrollary.

    What makes the story here sad is that many people have put a lot of
    time and energy in the wrong idea, but mathematics is a hard
    discipline.

    None of that time and effort matters mathematically.

    So why is it a big deal?

    Well mathematics as a discipline is full of proud people who pride
    themselves on work and ideas they long have thought were correct.
    One of the more potent ideas within the math community itself is the
    belief in their own perfection that the foundations of mathematics are
    so well-worked that there are no errors in thinking.

    You might say, since that idea has been out there for so long that
    it's ironic that lurking within what math people teach is an error in
    thinking older than any of them today, as it has sat there for
    generations.

    Now, for a while, yeah, they'll fight it. But they'll eventually
    lose. Kids growing up today who've yet to learn the wrong way may
    through various means learn of this problem and when the old guard
    tries to teach them the wrong way--reject it and them.

    Intriguingly, there should be a rather steady decline in math students
    worldwide, which I'd think would follow a particular curve, like a
    power curve, until there's something that pushes that society to
    accept the truth.

    Like a precipitating event might be required. I don't know what that
    would be, but my guess is that it would have to be fairly huge.


    James Harris
     
    James Harris, Oct 16, 2004
    #1
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  2. If A is divisible by 3 in the ring of rational integers, then any
    ring containing u_1 also contains u_1(A/3 - u_1) = 1/3?
    So what? In general they can't both be objects.
    Oooh, a new long word!

    But I can't begin to guess what you think it means.
     
    John Roberts-Jones, Oct 16, 2004
    #2
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