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Topic: A Point on Understanding
Replies: 66   Last Post: Jan 6, 2013 11:07 PM

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 Joe Niederberger Posts: 4,391 Registered: 10/12/08
Re: A Point on Understanding
Posted: Dec 30, 2012 7:57 PM

R. Hansen says:
>I was surprised that my son understood that statements involving infinity involved an ongoing process that never completes.

That's the way Gauss understood it. However, prevailing sentiment against actual infinities started to change with Cantor (not that there were not earlier proponents) and its now status quo to accept both kinds.

* http://www-history.mcs.st-and.ac.uk/HistTopics/Infinity.html
* http://www.math.vanderbilt.edu/~schectex/courses/thereals/potential.html
* http://en.wikipedia.org/wiki/Actual_infinity

Now, back to Kirby's triangu-penta-hexa-spheres. I think we can have more fun with it. On one view, you can imagine that with increasing number of vertices, the polygons stay fixed size but the sphere grows large and the surface locally flat. On the other hand, we can equally well imagine the sphere staying fixed size and the polygons shrinking to points. Points have no interior or exterior angles.

So what *really* happens? Why does what looks like the same basic *process* (except for scale) lead to two incompatible results? In this case the "ongoing process" view I think fails, as far as resolving the *conundrum* (a better word than contradiction or paradox?)

Cheers,
Joe N

Date Subject Author
12/15/12 Robert Hansen
12/17/12 kirby urner
12/18/12 Robert Hansen
12/17/12 Haim
12/18/12 Robert Hansen
12/26/12 kirby urner
12/26/12 Robert Hansen
12/26/12 kirby urner
12/27/12 Paul A. Tanner III
12/27/12 kirby urner
12/27/12 Robert Hansen
12/27/12 kirby urner
12/27/12 Robert Hansen
12/27/12 kirby urner
12/27/12 Robert Hansen
12/27/12 kirby urner
12/27/12 Robert Hansen
12/27/12 kirby urner
12/28/12 Paul A. Tanner III
12/28/12 kirby urner
12/28/12 Paul A. Tanner III
12/30/12 kirby urner
12/30/12 Louis Talman
12/30/12 kirby urner
12/30/12 kirby urner
12/30/12 kirby urner
12/30/12 Paul A. Tanner III
12/30/12 kirby urner
12/30/12 Paul A. Tanner III
12/27/12 Domenico Rosa
12/27/12 Robert Hansen
12/27/12 Richard Strausz
12/27/12 Domenico Rosa
12/30/12 Joe Niederberger
12/30/12 Paul A. Tanner III
12/30/12 Robert Hansen
12/30/12 Joe Niederberger
12/31/12 Robert Hansen
1/2/13 kirby urner
12/30/12 Joe Niederberger
1/1/13 Paul A. Tanner III
12/31/12 GS Chandy
12/31/12 Robert Hansen
1/1/13 GS Chandy
12/31/12 GS Chandy
12/31/12 Robert Hansen
12/31/12 Joe Niederberger
1/1/13 Robert Hansen
1/1/13 GS Chandy
1/1/13 GS Chandy
1/1/13 GS Chandy
1/1/13 Robert Hansen
1/1/13 Haim
1/1/13 Joe Niederberger
1/1/13 Joe Niederberger
1/1/13 Paul A. Tanner III
1/1/13 Louis Talman
1/2/13 Paul A. Tanner III
1/1/13 Joe Niederberger
1/2/13 Paul A. Tanner III
1/2/13 GS Chandy
1/2/13 Joe Niederberger
1/4/13 Joe Niederberger
1/5/13 GS Chandy
1/5/13 GS Chandy
1/6/13 Robert Hansen
1/6/13 GS Chandy