Division by zero. Go ahead and laugh.

Discussion in 'Undergraduate Math' started by Lefty, Oct 7, 2004.

  1. You are right. I reject inconsistency root and branch. If tha makes me
    closed minded, then so be it. Logical inconsistency is the death of
    science and mathematics, both of which I care for a great deal.

    I am being both professional and charitable. Serious advocacy of
    inconsistency is either a bad joke (on your part) or a clear symptom of
    inanity. Which is it?

    Bob Kolker
     
    robert j. kolker, Oct 9, 2004
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  2. FOr that reason 0/0 is not permitted either. It is not equal to any
    particular value therefore is not defined. Get it straight. Dividing by
    zero in any circumstance leads to inconsistency which is why we don't
    divide by zero.

    0/0 is not a number.

    Bob Kolker
     
    robert j. kolker, Oct 9, 2004
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  3. It is clearly permitted according to the definition you gave.
    It might be better if YOU got it straight! I was using the definition YOU
    gave.
    No, it doesn't. And I see that you conveniently snipped the end of the
    message in which you attempted to exhibit such an inconsistency, but failed
    to do so.
    Of course, if it did lead to inconsistency, we wouldn't have systems such
    as C* in which division of nonzero quantities by zero is defined.
    What I said above was based on YOUR definition.

    David Cantrell
     
    David W. Cantrell, Oct 9, 2004
  4. O.K. Lets fix the definiton. No division by zero period. That answers
    your objects AND it prevents an inconsistency. Win win all around.

    Bob Kolker
     
    robert j. kolker, Oct 9, 2004
  5. In sci.math, Lefty
    <>
    wrote
    Arithmetic does not "fall apart" if one allows division by the
    arithmetic identity (0); it merely mutates.

    Of course, the classical definition of "real number" precludes 1/0, 0/0,
    and -1/0, but one can easily extend the numbers, although one
    may lose some of the more interesting properties.

    Call this class of numbers the "zero-divisible reals" (not to be
    confused with "the extended reals", which only adds +oo and -oo,
    as I understand it).

    The zero-divisible reals might add three elements.

    1/0, which is greater than any existing real number.
    -1/0, which is less than any existing real number.
    0/0, which is a very weird thing, as you'll see.

    Now one has to define:

    a + 1/0, a - 1/0, a * 1/0, a / (1/0) for any real a.
    1/0 + a, 1/0 - a, 1/0 * a, (1/0) / a, for any real a (*including* 0).
    etc.

    One could get into some interesting if slightly useless territory here.
    For example, is 2/0 > 1/0, leading to a class of "real infinities"?
    Or is 2/0 = 1/0, the more or less standard extension?

    Where did you want to go with this?
    And why would we want to do that?

    Zero is a natural consequence of the definition of subtraction. Briefly
    put, if a = b + c for integers a, b, c > 0, then c = a - b and
    b = a - c. That defines "subtraction" if a > b and a > c; the next
    question is what happens if a < b and a < c, or perhaps defining
    the quantity "a - a". After a little work one gets -- zero, with
    its properties, including the problematic one of x * 0 = 0 for any
    integer, rational, algebraic, real, or complex x. Since d = e * f
    implies e = d/f and d = e/f for any positive integers d, e, f (or,
    for that matter, many other numbers), 1/0 cannot be defined in the
    standard reals -- x * 0 = 1 is unsolvable! -- and 0/0 is meaningless
    in the standard reals, since x * 0 = 0 for all standard real numbers x.

    One can't even be sure whether 0/0 > 0, < 0, or = 0.
    You are welcome of course to define 1/0 and 0/0 as you see fit, bearing
    in mind that the theorems of arithmetic may require some adjustment.
    For example, one can also attempt to define 1/0 * 0 = 0.

    It may depend on how one approaches this problem. For example,

    1/x * x = 1
    1/x^2 * x = 1/x
    1/x * x^2 = x

    for all x != 0.

    So if one takes the limit as x approaches 0, one can either get 1, 1/0,
    or 0, when one multiplies 1/0 by 0. At best, this is rather problematic.
    I would suggest that you define your arithmetic results given the
    three elements 1/0, -1/0, and 0/0, and proceed accordingly (and
    *very* carefully). You may get an inkling of some of the difficulties
    involved.

    For example, does x = 1/0 solve the equation (1/0) * 0 = 1? If so,
    then one has to define a * 1/0 for real a > 0 carefully, lest one
    misapply the cancellation rule with a * 1/0 = b * 1/0 and get a = b,
    even though a != b initially. Fortunately, there's a precedent; if
    a * c = b * c and c = 0, then there's no drawable conclusion regarding
    a and b; perhaps c = 1/0 could be added to that exclusion set.

    Good luck, but I'm not all that hopeful.
    Sorry, I'm not Catholic. :p
     
    The Ghost In The Machine, Oct 9, 2004
  6. Hmm. I guess that the definition of "normal" in this context must have been
    presented in the same math course in which "nice" and "loose" were defined.
    Sorry I missed that course.
    I wouldn't use the word "blame". Zero's peculiar, but that's just its
    nature. But if something's peculiar, it's abnormal. (Do we have another
    inconsistency here: 0 is both normal and abnormal ?)
    I don't.
    I never have and never will.

    David Cantrell
     
    David W. Cantrell, Oct 9, 2004
  7. Lefty

    Lefty Guest


    That's fine and things work very nicely if you do. I'm very happy that
    algebra is consistent because this special provision that (1/0 DNE) makes it
    so.


    If there were no inconsistency then there would be no debate. The
    inconsistency is traditionally resolved by setting 1/0 as DNE. I think that
    there are other more _rigorous_ ways to resolve this which will also leave
    arithmetic intact, and simultaneously acknowledge the existence of the
    singularity instead of kidding ourselves about it.


    Fine. You can use an arithmetic where 1/0 does not exist and it is
    consistent. Fine. But, if someone else modifies the definition so that
    arithmetic remains consistent and the singularity at 0 can also be handled
    algebraically, then what would you say to that ? One definition is better or
    worse than another - and for what reason ?

    I am saying that the definition of setting 1/0 as DNE creates an arithmetic
    which is consistent, but this definition is rather arbitrary, because you do
    have the option to attempt to assign a symbology which could treat the
    singularity. Why would one be preferable to the other ?
     
    Lefty, Oct 9, 2004
  8. Lefty

    Lefty Guest

    entirety.


    Absolutely not. Sincerely, this would be impossible and undesirable. Yet,
    perhaps this is a popular misconception, and a major motivation to stay away
    from dealing with paradox formally.

    A paradox does not undermine mathematics. It only makes it more fascinating
    in my opinion.


    I am suggesting the exact opposite of what you are saying. I am suggesting
    an operator which is defined in such a way that it leaves arithmetic intact.
    I think that this must be possible.


    Correct. In fact, I dont know what it is except to say that it is a
    contradiction. It exists as such, and there should be a means to handle it
    algebraically within the framework of algebra. It would probably need a
    symbol of some kind, and maybe some other gadgetry. I do not know.

    It is a beautiful thing if only you could see it as such. Math is the only
    science where one can construct absolute truths. Does it seem odd that
    everything should explode at a singularity ? I find it absolutely
    captivating, and I hope you will continue to argue against me because it may
    encourage more contemplation.

    Something is happening - something fundamental and strange, and it does'nt
    seem one bit elegant to amputate zero (for 1/x) and toss it out by stating
    "it does not exist".
     
    Lefty, Oct 9, 2004
  9. Lefty

    Lefty Guest

    You are being very closed minded to this. I am not going to pursue this
    for

    Bob, I'm really sorry if I have offended you. This is not my intention, nor
    do I intend to undermine math. I wish that you could believe me when I say
    that I am trying to enhance the scope of mathematics constructively. I wish
    I could prove this to you that I am not being disingenuous. I have no
    ulterior motives, and again I apologize if it seems that I have seemed to
    make any such overtures.

    I fully respect your feelings in the above. I know that mathematicians are
    _the_hardest_working_ and the most scientifically _honest_ people who have
    ever existed. I know this.

    I'm not yanking your chain.

    Think about it for a moment. Even if you discovered a logical inconsistency
    in the general model somewhere - what do you do with the enormous logical
    structures which have been meticulously erected for centuries ? Throw them
    in the trash ? The works of Gauss ? Euler ? No! I do not think that there is
    ANYTHING which could render these works invalid. This is impossible!

    There is no way to shake the structures which exist. Even if mathematics
    were succeptible somehow to space/time, the structures which have been
    observed and proven are all still valid.

    My aim is to demonstrate somehow that math and physics reflect each other in
    yet another way - the existence of paradoxes. Singularities. Whatever they
    are, whatever you call them, I think that logic can break down "locally" in
    math and also in physics. The universe wont collapse, and neither shall
    math.



    I only wish we could formally axiomatize physics. Actually, what I mean by
    this is that the physical universe itself be axiomatizable to create a
    "non-abstract mathematics", a math of real objects which is non-abstract. I
    dont think that I'll ever succeed at this, and perhaps this will only
    illustrate how truly insane I might be. But I'll make you a promise. I
    promise to not harrass people to try to force them to believe me. Nobody
    will ever notice this material, and no-one will ever care.

    Please consider the following statement:
     
    Lefty, Oct 9, 2004
  10. Lefty

    Lefty Guest

    If you find having consistency a burden

    Imagine for a moment that there is some strange thing out in space
    somewhere. Could be a black hole, - whatever. Imagine for one moment that
    this object has the very real physical attribute that logic breaks down or
    is altered when things go near it.

    Do mathematicians have any tools to describe this thing ? Can you model it
    mathematically ? Or, would you have to conclude that it does not exist
    because there are no algebraic tools available to model what you are
    observing in the physical world ?
     
    Lefty, Oct 9, 2004
  11. Good. Now do the following. Produce an algebraic system with the
    following properties.

    1. The elements form a ring with respect to addition.

    2. The elements form a multiplicative group (that means 0 has a
    multiplicative inverse).

    3. The distributive low of multiplication over addition holds.

    4. The system is consistent. No logical contradictions.

    5. The system is non-trivial, i.e. it has more than one element.

    6. You can use an instantiation of the system to do your income tax.

    Get back to us when you figure out how to do it.

    Bob Kolker
     
    robert j. kolker, Oct 9, 2004
  12. Forbidding division by 0 is precisely admiting the singularity.
    You think so? Fill the room with your brilliance. Do it.

    Bob Kolker
     
    robert j. kolker, Oct 9, 2004
  13. A contradiction i.e. a proposition of the form A and not-A destroys
    mathematics. It enables -any- propostion to be proved. This is a well
    know excercise in any logic 101 course. A and not-A has the value false
    and a truth table excercise show that FALSE implies P for -any-
    proposition P. If FALSE (i.e. A and not-A) were provable any proposition
    follows. Such a calamity renders mathematics unusable except for finite
    systems for which models exist. Good bye calculus. Goodbye real and
    complex variables. Goodbye number theory. Since physics is based
    squarely on the theory of real and complex variables, goodby physics.
    Do it then. Let us see it and show that it is consistent. No
    consistency, no mathematics.

    Bob Kolker
     
    robert j. kolker, Oct 9, 2004
  14. You mean by no two, that the objects are distinct. Hence any pair of
    distinct objects are distinct. With which I agree wholeheartedly.

    The contrary to the existence of two (or more) distinct objects as that
    there is either one object or no objects. Which would you prefer?

    Do you believe there is only one object and we are deluded in seeing
    distinct object? If so, our two eyes, which are really one eye and our
    ass and our stomach is deluding us badly.

    Bob Kolker
     
    robert j. kolker, Oct 9, 2004
  15. Thanks for the suggestion, but I'll pass.
    What you request is clearly impossible, as we are both well aware.

    David Cantrell
     
    David W. Cantrell, Oct 9, 2004
  16. That there thing may be physical but our math cannot touch it. Physical
    existence does not depend on our being able to describe whatever it is
    mathematically. Our mathematics is limited by logical consistency. It is
    entirely possible that a thing can exist such that we cannot comprehend
    it by a logically consistent system of concepts.

    Just because we have nimble three pound brains and we can build rocket
    ships does not mean we are able to comprehend everything. There are more
    things in heaven and earth than are dream't of in our philosophy.

    Bob Kolker
     
    robert j. kolker, Oct 9, 2004
  17. Lefty

    Lefty Guest


    I dont want to see any of these things going away, except perhaps the
    physics. The models of physics I have seen are all essentially mathematical
    contraptions which do not exctly match the physical world. Physics should be
    a mathematics in it's own right, where the number system and operands match
    the physical world _exactly_ . I dont know if this is possible, but I do
    know that I have never seen anything like that.

    I'm not saying that physics is wrong.

    What I mean by this is that mathematics is pure abstraction. But physical
    processes exist completely in the physical universe, with the exception of
    the mind and maybe some other oddities somewhere which intersect both
    worlds, the abstract world and the world of space/time. Physics as we know
    it utilizes abstractions to model real things. This is a pulling together of
    two separate worlds. And typical physics models _never_ describe things
    completely, they always leave out information. I think that physics can be
    axiomatized so that it need not borrow from the abstract world when building
    models.

    Maybe.

    I will try to put something on paper and will post here if I can come up
    with such a thing, trying hard not to sound too stupid, I know I've already
    blown my credibility, but I certainly dont mind being criticized for being
    wrong, it's all just experimentation.

    It's probably folly.

    But where would man be today if he were'nt such a fool ?


    (a paradox?)
     
    Lefty, Oct 9, 2004
  18. Lefty

    Lefty Guest

    -------------------

    Actually, I am trying to explain that the naturals do not _exactly_ describe
    physical objects. If you count 5 imaginary marbles in your mind, then you
    have counted 5 identical marbles. No problem. But if you count 5 real
    marbles which you bought from the marble store, then your counting process
    is inexact. No two marbles in space/time can be identical. You cannot count
    to 5. You must count to 1, 5 separate times. Each marble is unique.

    You have either 1 marble, or 0 marbles. Marble exists, or it does not. This
    line of reasoning is _exact_ . It matches the universe with a precise
    exactness which you dont ordinarily see in most models of real events.

    It reminds me alot of some existentialist stuff, more philosophical garbage
    you may say - but I think that the axiom I have given is a statement of a
    number-theoretic nature. I come to this group because others know much more
    about these things than myself.

    The bulk of reality does exist as a single object. It is also composed of a
    plurality of subunits known as "things".

    I think that separate things do exist. There is 1 of each. For some things,
    there are zero pieces available, these things are non-existent in the
    universe of space/time.
     
    Lefty, Oct 9, 2004
  19. Exact matches are a physical impossibility. If for no other reason than
    the uncertainty principle.
    You can't get an -exact- match between experiment and theory. The
    instruments used in the experiments are susceptable to thermal
    perturbation and other environmental disturbances not to say anything
    about defects in the workman ship and just plain instrument reading
    errors. Exact measurements of complementary observables with non zero
    commutators is not possible courtesy of the uncertainty principle. All
    experments come with error bars. If the prediction falls inside the
    error bars the experiment supports the prediction.

    Physics is NOT mathematics although mathematics is necessary to express
    the quantitative relations of physical theory. Physics is -empirical-. A
    physical theory is precisely as good as the predictions it makes.
    Expermental fact trumps physical theory each time, every time. Theories
    serve. Facts rule.
    You just did.
    Carnap and the logical positivists tried to axiomatize physics for three
    decades. They failed. Reality is just too big to be fit into a
    formalism. Physical theories will always be incomplete because we are
    always learning New Stuff. When light from far reaches of the kosmos
    reaches us it tells us about Newt Stuff we have literally never seen
    before. There is not way we are ever going to be able to deduce physical
    reality a priori.
    Much better off. Paradoxes are the stigmata of sloppy workmanship and
    fuzzy thinking. If our theories can not catch up to facts, that is NOT a
    paradox. It is a deficiency of our thinking and understanding. We are
    Mark 3 apes with three pound brains, mediocre senses and the gift of
    gab. We are the smartest baddest apes in The Monkey House. That is what
    we are.

    Bob Kolker

    God ever geometrizes. The rest of us pay taxes and commute to work -
    Plato Rabinovitz
     
    robert j. kolker, Oct 9, 2004
  20. Lefty

    N. Silver Guest

    If the models exactly matched the "physical world," then they would
    be the "physical world." The point of having a model is that it is a
    a stripped-down-to-essentials rendition of a physical process that
    allows us to analyze and make mathematical predictions about the
    process.
    We want to construct a simple system that mirrors a more
    complicated one. Then we can modify the model as we
    use it to more accurately reflect the process. If we are
    successful, our sequence of models will converge.
    You don't have the authority to even think that.
     
    N. Silver, Oct 9, 2004
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