Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: Andy Smith on 2 Feb 2007 09:37 Andy Smith <Andy(a)phoenixsystems.co.uk> writes infinite volume from a finite surface area should of course have read infinite surface area from a finite volume -- Andy Smith
From: Dave Seaman on 2 Feb 2007 10:23 On Fri, 02 Feb 2007 13:52:54 GMT, Andy Smith wrote: > In message <epvbqo$bjl$1(a)mailhub227.itcs.purdue.edu>, Dave Seaman ><dseaman(a)no.such.host> writes >>Zeno was bothered by this when he talked about an arrow in flight being >>instantaneously "at rest" at each moment. Thanks to calculus, we now >>know that the arrow is never actually "at rest" during flight. > But the positions are reversed? I would be very happy to consider > distance along a line as a continuous function x and rate of change of > position dx/dt etc. But by asserting (or, as I understand it, defining) > the line is an infinite collection of points one reintroduces the > conceptual granularity that bothered Zeno? What is your proposed alternate definition of a line? I don't know what you mean by "granularity". Seems to me there would be granularity only in the discrete case, where each point has an immediate neighbor. Obviously, that doesn't hold on the real line. Hence, no granularity. > You just start with points and define the real line as the set of all of > them. I have an image of a line as a continuous thing of point width, > and it is trying to marry up the perception of continuity with the set > of real points that is difficult for me. I mentioned fractals because > they are a clear example of e.g. how you can get infinite volume from a > finite surface area, etc. so I thought that the essentially recursive > structure of the reals (you can map the real line into the space between > any two points, indefinitely) was some sort of fractal explanation of > how the line could be simultaneously pointlike and continuous. But > doubtless, as with everything else, my problem is getting my head to a > point where I see things in the correct perspective. Continuity is a term that applies to functions, not to lines. A line is connected, but it has the property that the removal of any single point disconnects the line. A plane, for example, does not have that property. -- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. <http://www.mumia2000.org/>
From: Dave Seaman on 2 Feb 2007 10:25 On Fri, 02 Feb 2007 14:37:48 GMT, Andy Smith wrote: > Andy Smith <Andy(a)phoenixsystems.co.uk> writes > infinite volume from a finite surface area > should of course have read > infinite surface area from a finite volume Are you familiar with Gabriel's Horn? That's the curve y = log x for x >= 1, rotated about the x-axis. It has finite volume and infinite surface area, but it is not a fractal. -- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. <http://www.mumia2000.org/>
From: Andy Smith on 2 Feb 2007 11:10 Dave Seaman <dseaman(a)no.such.host> writes >On Fri, 02 Feb 2007 14:37:48 GMT, Andy Smith wrote: >> Andy Smith <Andy(a)phoenixsystems.co.uk> writes > >> infinite volume from a finite surface area > >> should of course have read > >> infinite surface area from a finite volume > >Are you familiar with Gabriel's Horn? That's the curve y = log x for x >>= 1, rotated about the x-axis. It has finite volume and infinite >surface area, but it is not a fractal. > No I wasn't, but that's quite fun. (I guess you meant to write 0<x<=1 for the defining curve) -- Andy Smith
From: Andy Smith on 2 Feb 2007 11:23
Dave Seaman <dseaman(a)no.such.host> writes >On Fri, 02 Feb 2007 13:52:54 GMT, Andy Smith wrote: >> In message <epvbqo$bjl$1(a)mailhub227.itcs.purdue.edu>, Dave Seaman >><dseaman(a)no.such.host> writes > >>>Zeno was bothered by this when he talked about an arrow in flight being >>>instantaneously "at rest" at each moment. Thanks to calculus, we now >>>know that the arrow is never actually "at rest" during flight. > > >> But the positions are reversed? I would be very happy to consider >> distance along a line as a continuous function x and rate of change of >> position dx/dt etc. But by asserting (or, as I understand it, defining) >> the line is an infinite collection of points one reintroduces the >> conceptual granularity that bothered Zeno? > >What is your proposed alternate definition of a line? > Well I am tempted to say the locus of intersection of two planes, but I can see that just defers the question to "what is a plane?". Doubtless you would like me to respond that it is a connected linear sequence of points, and I am sure that you are right. >I don't know what you mean by "granularity". Seems to me there would be >granularity only in the discrete case, where each point has an immediate >neighbor. Obviously, that doesn't hold on the real line. Hence, no >granularity. Yes, I understand, I was speaking figuratively - the whole issue is about the infinite set of real points being connected. > >> You just start with points and define the real line as the set of all of >> them. I have an image of a line as a continuous thing of point width, >> and it is trying to marry up the perception of continuity with the set >> of real points that is difficult for me. I mentioned fractals because >> they are a clear example of e.g. how you can get infinite volume from a >> finite surface area, etc. so I thought that the essentially recursive >> structure of the reals (you can map the real line into the space between >> any two points, indefinitely) was some sort of fractal explanation of >> how the line could be simultaneously pointlike and continuous. But >> doubtless, as with everything else, my problem is getting my head to a >> point where I see things in the correct perspective. > >Continuity is a term that applies to functions, not to lines. A line is >connected, but it has the property that the removal of any single point >disconnects the line. A plane, for example, does not have that property. > Yes, understood, I should have said connected. -- Andy Smith |