(Close Window) 
Topic: 2 Math Paradoxes 


Any insights? http://www.youtube.com/watch?v=jUeIjZI32Jg&feature=player_embedded#at=74 


These are certainly intriguing. The flaw in the argument is, I think, the attempt to equate finite and infinite situations. No matter how many steps are drawn, they will always have a finite horizontal and vertical length (which sum to 2 in the example shown). But as long as there is a finite number of steps, they will also each have a diagonal  and it is the sum of these diagonals that equals the diagonal of the square, not the sum of the horizontal and vertical lengths. However, when the limit is reached (i.e., if you have an infinite number of steps) the horizontal and vertical lengths will each be zero (as will the length of the diagonals). So you could use the same argument to say that the length of the diagonal of the square is zero. Monty Hall anyone? Mike 


Another version of this same paradox: [img]http://farm5.static.flickr.com/4147/5221660921_cbfb37c6f1.jpg[/img] All of these paradoxes share the same flaw. It basically boils down to one of "resolution". In the stairsteps (or arcs in the case of the line), there will always be SOME difference between the stairstep approximation and the straight line. Further, this difference will be constant. In the diagonal of the square portion of the video, he starts off by approximating the diagonal with 4 steps with a .25 unit rise and a .25 unit run, giving a total length of 2. As the length of each step decreases, the number of sides increase, and the total remains the same: 4 * .5 = 2 8 * .25 = 2 16 * .125 = 2 And so on. Even when the jagged stairsteps get too small for the eye to see, they're still there. There's always the same amount of difference between the approximation and the actual line (or arc) in question. If you programmed a computer to draw these shapes and their approximations, as well as allow a zoom function, you could always zoom in to a point where you could see that there is a difference. In short, there's no point where the difference magically becomes 0, and the two shapes become the same. This does remind me of the concept of fractals, fractal dimension, and the [url=http://en.wikipedia.org/wiki/How_Long_Is_the_Coast_of_Britain%3F_Statistical_SelfSimilarity_and_Fractional_Dimension]coastline length problem[/url]. Check out the [url=http://www.hulu.com/watch/181084/novahuntingthehiddendimension]NOVA documentary "Hunting the Hidden Dimension"[/url]. 


I agree with Scott. It makes no sense to suddenly jump from (a) an immensely large number of steps (where there will always be a difference between the sides and the diagonal) to (b) an infinite number of steps (where there is no difference between the sides and the diagonal). I am reminded of other paradoxes that are based on flawed extrapolations from the finite to the infinite. For example, Zeno's famous paradox 'proving' that Achilles can never overtake a tortoise that is given a head start because whenever Achilles reaches a point previously occupied by the tortoise, the tortoise will always have moved slightly ahead  ad infinitum. Mike 


These 2 paradoxs demonostrate that the space is not quantized. If it it is then you have this paradox! :) bye Tommy. 


The false statement is the claim that these two cases tend towards being a straight line. Neither does. The do [i]seem[/i] to get closer to a straight line. The only proper way to take a limit is with a formula. The Nth case for the first example has a length L where L = 2^N * (Pi*(R/(2^N))). The first 2^N term signifies the number of halfcircle line segments and the last part of the formula (R/(2^N)) is the radius of each halfcircle. The limit of that formula as N > infinity is Pi*R, not 2R. It's always Pi*R independent of the value of N. The second case doesn't tend towards a line either, even if it appears to. If L is the length of one side of the square and the number of staircase steps is N, the formula for the total length of the diagonal staircase D = N*(L/N + L/N). D = 2L, independent of the value of N. So, as N tends towards infinity, D remains 2L. It never becomes a diagonal line. 


[quote] On 20110527 08:05, Michael Daniels wrote: I am reminded of other paradoxes that are based on flawed extrapolations from the finite to the infinite. For example, Zeno's famous paradox 'proving' that Achilles can never overtake a tortoise that is given a head start because whenever Achilles reaches a point previously occupied by the tortoise, the tortoise will always have moved slightly ahead  ad infinitum. Mike [/quote] I remember talking to my high school math teacher about Zeno's paradox. I told him that if you assume that Achilles shrinks as he runs, the problem is accurate, but it also ceases to be a paradox. ;) 


A similar paradox, explained by my college math teacher C. Stanley Ogilvy in Excursions in Mathematics, has to do with the diagonal of a square. You can draw a step function along the diagonal with steps as small as you want and thus coming as close to the diagonal as you want, but the sum of the lengths of the steps is always equal to 2 times the side of the square. You have to be careful with limits: just because you can get all the points on the step function along the diagonal within epsilon of the diagonal doesn't mean the length of the step function approaches that of the diagonal. 


[quote] On 20110526 13:14, Michael Daniels wrote: Monty Hall anyone? Mike [/quote] Yes. Please! Figures very heavily in bridge, as "The Principle of Restricted Choice." 