Date: Dec 30, 2012 7:57 PM
Author: Joe Niederberger
Subject: Re: A Point on Understanding

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.

Here are some links for reading should anyone desire:

* 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