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Thread: Mathematics

  1. #16
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    Quote Originally Posted by josef k. View Post
    I have put it in a cage... I am now examining it.

    PS - By analyze, I just mean figure out its relationship to other stuff. As in, wobbly lines exist. How do people understand them mathematically? I don't really care about this particular line, although please don't tell it I said that.
    Hm, I can see what you mean, but in a way it's a bit like a kid drawing a totally surreal fantasy animal and the asking an adult what real animals it's related to, if you get me. There's no reason why it should have a relationship to other stuff. In the same way that hfyeuieoffjsgrepl could be considered a 'word', since I can write it using the same letters I use spell real words that have a meaning - but that certainly doesn't endow it with a meaning of its own.
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  2. #17
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    Interesting...

    So, is it that, wobbly lines do not exist? Or that, when they do, it is never arbitrary, as this one is.

    Take the path of a kite through a sky on a rainy day. The path traced by the kite. How do people mathematically analyze this?

  3. #18
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    ps - in the case of the fantasy animal, experts conclude: it is three-headed, lamalian, behorned and tigroid, scale-necked and beastish, amphibious, zebrine, duck-billed and bewinged.

  4. #19
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    Quote Originally Posted by josef k. View Post
    Interesting...

    So, is it that, wobbly lines do not exist? Or that, when they do, it is never arbitrary, as this one is.

    Take the path of a kite through a sky on a rainy day. The path traced by the kite. How do people mathematically analyze this?
    Of course they exist! In the same way the nonsense string of letters typed exists. It's a question of whether it means anything deeper.

    To what extent is anything 'arbitrary'? The path of a kite is determined by gusts of wind and tugs on the string, just as the shape of that line was determined by the movement of the muscles of your hand, in turn produced by electrical signals in your brain and spinal column. These things can all be analysed according to well understood physical laws, which of course have a rigorous mathematical basis. That doesn't mean your kite is going to trace a perfect ellipse in the sky or that you're going to draw a nice neat sine wave; the processes are extremely complex (stochastic, loosely 'chaotic') and cannot, for practical purposes, be predicted - in other words it's non-deterministic.
    Last edited by Mr. Tea; 15-01-2009 at 03:53 PM.
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  5. #20
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    Quote Originally Posted by josef k. View Post
    ps - in the case of the fantasy animal, experts conclude: it is three-headed, lamalian, behorned and tigroid, scale-necked and beastish, amphibious, zebrine, duck-billed and bewinged.
    Is it partly rugose and partly squamous?
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  6. #21

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    There are lots of wiggly lines in the social sciences.

  7. #22
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    Yes, absolutely. It is:

    Monstrine & lamalian,
    Leggaled and armine. Angeloid,
    Green-billed. Cashinal, pervoid.
    Scale-necked, amphextorous
    Olfractic, dreamal & ioned.

    **

    "It's a question of whether it means anything deeper."

    A wobbly line has a relationship to other mathematical properties - area, volume, angle, point, curve. Basically, I'm interested in the relationships between mathematical properties, in figuring out how they fit together.

  8. #23
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    "By analyze, I just mean figure out its relationship to other stuff. As in, wobbly lines exist. How do people understand them mathematically? What mathematical operations are performed on them in order to try and understand them?"
    Well, like Mr Tea said you could do stuff like measure its curvature at a given point or work out its length or whatever and that would be an analysis of a kind I think. You could then compare its curvature to another line or whatever if you're interested in relationships to other things. I don't think that you could come up with a single neat function to describe it but that doesn't mean that it doesn't exist, why would it?

  9. #24
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    Because maths knows only functions?

  10. #25
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    A wiggly line can be described by a continuous function between the set of real numbers occupying the interval 0 <= x <= 1 and a set of points in a Euclidian space. The function takes 0 to the start of the line, 1 to the end of it, and continuously maps all of the points in the interval to points along the wiggly path.

    Some lines wiggle more than others. A Koch curve, for instance, has wiggles on its wiggles, and so on ad infinitum. But that makes it more, rather than less, "mathematical".

  11. #26
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    Quote Originally Posted by josef k. View Post
    Because maths knows only functions?
    Maths knows all sorts of things: functions, functionals, operators, tensors, maps, sets, groups, rings, symmetries, homotopies...it's kind of difficult to know where to start. In part because a lot of the stuff that forms the real conceptual basis of maths is a lot more difficult to get to grips with than the stuff you do at school, much of which has a more fundamental origin that you don't learn about unless you study it to degree level.
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  12. #27
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    "Because maths knows only functions?"
    I don't think that's true. It's certainly not true that it only knows "single neat functions" which is what I said.

    "A wobbly line has a relationship to other mathematical properties - area, volume, angle, point, curve. Basically, I'm interested in the relationships between mathematical properties, in figuring out how they fit together."
    Well, the line in question obviously has no volume or area in itself, you could have a shape of which that line made up one edge and you could ask the same kinds of questions about the shapes that resulted. Again they might not be simply describable (nb you can fairly easily work out the volume of most shapes by seeing how much water they displace but I guess that's not the kind of answer you mean).
    The line has different curvature at different points and its tangents would be at various angles to each other, again it would be hard to sum them up simply.
    I suppose the point is that it is hard to analyse something except in the specific if it is not generated by some kind of rule. It's kind of like in the Borges story where the guy (Funes) develops a new number system that doesn't repeat at all and cannot grasp why it is useless to everyone - except him who has a perfect memory and does not need to have a generating rule to help him comprehend the bigger picture.

    "A wiggly line can be described by a continuous function between the set of real numbers occupying the interval 0 <= x <= 1 and a set of points in a Euclidian space. The function takes 0 to the start of the line, 1 to the end of it, and continuously maps all of the points in the interval to points along the wiggly path."
    Yes of course but the point is that it's hard to say anything more about it than that. That's exactly what I meant by ad hoc.

    "Some lines wiggle more than others. A Koch curve, for instance, has wiggles on its wiggles, and so on ad infinitum. But that makes it more, rather than less, "mathematical"."
    I don't think anyone is saying that that other line is unmathematical. You can have far more complex lines that are nowhere continuous or nowhere differentiable or whatever but I think that if you have a rule for generating it there is some sense in which you have an understanding of it that you don't have with the random wiggle (of course it's perfectly possible that any given apparently random wiggle may be generated by a generalisable rule but I'm not aware of any systematic method to discover what that rule is).

  13. #28
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    "Ad hoc" here seems to mean something a bit like Kolmogorov randomness.

  14. #29
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    I would imagine that most wiggly lines I could draw by hand on the back of an envelope would turn out to be quite compressible - it would be quite a challenge to produce one that couldn't be broken down into a handful of frequencies.

  15. #30
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    Quote Originally Posted by IdleRich View Post
    I don't get what you're saying here? Recurring decimals don't terminate, they recur and any numbers which can't be written as a ratio (ie are irrational) are non-terminating and non-recurring.
    i'm saying there are no terminating numbers

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