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Thread: Floating glass paradox

  1. #1 Floating glass paradox 
    Forum Junior anticorncob28's Avatar
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    This is similar to my standing pencil paradox. It appears that I cannot post pictures, so I will explain this differently. I will present a two-dimensional version of this paradox. Make a graph of the function y = 1 (a horizontal line with a y-intercept of 1). Then graph the function y = 1 + ln(1 - |1/x|) on the same coordinate grid (we will call this the ground). Got that? Now, imagine the line y = 1 as an infinitely large glass sheet. The glass sheet is hovering above the ground 1 + ln(1 - |1/x|). Why is it hovering above? The glass sheet does not actually touch the ground that it is above. The glass sheet cannot simply sit on top of the ground. If the sheet is lowered even a little bit, it will pass through the surface it is on, because the surface gets arbitrarily close to the glass sheet. Thus, the glass sheet must hover a tiny bit above. What is making it hover? Clearly the ground is keeping the sheet from being lowered, but it isn't touching the sheet. Does mathematics require an invisible force from the ground to make sure the glass sheet hovers above? If you want a three dimensional version of this, the functions z = 1 and z = 1 + ln(1 - |1/(x^2 + y^2)|) should work, but I haven't tested it yet. It might also make the paradox easier to understand.


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  3. #2  
    Malignant Pimple shlunka's Avatar
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    Er... what?


    Strange likes this.
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  4. #3  
    Brassica oleracea Strange's Avatar
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    Quote Originally Posted by anticorncob28 View Post
    Then graph the function y = 1 + ln(1 - |1/x|) on the same coordinate grid (we will call this the ground). Got that?
    plot y = 1 + ln(1 - |1/x|) - Wolfram|Alpha

    That goes to infinity at x = 0 which makes the rest of your text even less comprehensible (I wasn't aware semantic content could go negative before now).

    It might also make the paradox easier to understand.
    Explaining what you are talking about might help. What does a sheet of glass have to do with graphs of functions?
    ei incumbit probatio qui dicit, non qui negat
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    Forum Junior anticorncob28's Avatar
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    I figured out how you got confused. Wolfram alpha was not a good place to go to graph those functions. For -1 < x < 1, the function 1 + ln(1 - |1/x|) gives a complex number for the answer. Try graphing it on a graphing calculator that only does real numbers, and you'll understand better. Here's one good graphing calculator: Function Grapher and Calculator (copy and paste into URL if it doesn't link). Note that for absolute value, you must use abs(x) and not |x|.
    Explaining what you are talking about might help. What does a sheet of glass have to do with graphs of functions?
    The equation y = 1 described a glass sheet, or a piece of paper, or whatever. Then the equation y = 1 + ln(1 - |1/x|) described the floor that the sheet is lying on.
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  6. #5  
    Brassica oleracea Strange's Avatar
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    Quote Originally Posted by anticorncob28 View Post
    Wolfram alpha was not a good place to go to graph those functions.
    Because it does it properly.

    Like this, then: Function Grapher and Calculator

    The equation y = 1 described a glass sheet, or a piece of paper, or whatever. Then the equation y = 1 + ln(1 - |1/x|) described the floor that the sheet is lying on.
    I still don't quite see what point you are trying to make.

    The glass sheet is hovering above the ground 1 + ln(1 - |1/x|). Why is it hovering above?
    If there is a sheet of glass supported by a floor then it isn't hovering. If your imaginary sheet of glass is hovering, then it is up to you to tell us why it is hovering, isn't it?

    The glass sheet cannot simply sit on top of the ground. If the sheet is lowered even a little bit, it will pass through the surface it is on, because the surface gets arbitrarily close to the glass sheet.
    So is this just some confusing version of Zeno's "paradox"? Which is easily resolved, of course.
    ei incumbit probatio qui dicit, non qui negat
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  7. #6  
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    This is actually an interesting question that illustrates the difference between math and physics.

    Your function 1 + ln(1 - |1/x|) has a horizontal asymptote at x = 1. The graph gets arbitrarily close to the line x = 1 but never touches it.

    Now in mathematics, this is perfectly normal and all that we care about. We have the function
    1 + ln(1 - |1/x|) and we can demonstrate that it has the asymptote for mathematical reasons.

    Now if a physicist or biologist or some other scientist had some real world phenomenon they were trying to model, and it turned out that the behavior of their system was modeled by
    1 + ln(1 - |1/x|), then we would still want to try to understand the underlying aspect of reality that is causing the behavior. Maybe the closer the function gets to x = 1, some chemical is being produced that acts to keep the value of the thing being studied strictly less than 1.

    Does that make sense? In math we are only concerned with the mathematical behavior of functions. But if we are applying math to some real world situation, then we would be interested in trying to find out why the situation behaves this way.
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    Brassica oleracea Strange's Avatar
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    Quote Originally Posted by someguy1 View Post
    This is actually an interesting question that illustrates the difference between math and physics.
    Thanks. It makes a bit more sense now. I think...

    But if we are applying math to some real world situation, then we would be interested in trying to find out why the situation behaves this way.
    Or understanding why the real world doesn't behave like a mathematical abstraction. In this case, the electrostatic fields which make the two surfaces "solid" would start to cause a physical interaction between the glass and the floor long before they got infinitesimally close. So the asymptotic relationship isn't really relevant.
    ei incumbit probatio qui dicit, non qui negat
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  9. #8  
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    Quote Originally Posted by Strange View Post
    Or understanding why the real world doesn't behave like a mathematical abstraction. In this case, the electrostatic fields which make the two surfaces "solid" would start to cause a physical interaction between the glass and the floor long before they got infinitesimally close. So the asymptotic relationship isn't really relevant.
    Yes, but you understand that there's no electrostatic interaction between the graphs of the two functions. Any such interaction would be between the real-world things the graphs were modeling in some particular application. You agree, right?
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  10. #9  
    Brassica oleracea Strange's Avatar
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    Quote Originally Posted by someguy1 View Post
    Yes, but you understand that there's no electrostatic interaction between the graphs of the two functions. Any such interaction would be between the real-world things the graphs were modeling in some particular application. You agree, right?
    Absolutely. That is why I was confused by the OP. He is mixing the physical and abstract as if they were the same thing.
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