To: |
"[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx> |
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From: |
Pat Hayes <phayes@xxxxxxx> |

Date: |
Thu, 7 Feb 2008 11:06:02 -0600 |

Message-id: |
<p06230903c3d0e1b6e969@[10.100.0.43]> |

At 10:18 AM -0600 2/7/08, Chris Menzel wrote:
On Wed, 6 Feb 2008, John F. Sowa wrote: It is kind of fun to see if there is a coherent alternative,
though. How could one make sense of the thesis that there are only
finitely many integers? The counterproof seems very simple and is
probably one of the oldest mathematical proofs ever devised.
1 Suppose there were a largest integer.
2 Call it N.
3 Consider N+1.
4 N+1 is larger than N: contradiction.
5 Ergo, there is no largest integer.
Where can one object to this? There are several
possibilities.
a. Maybe the largest integer can't be named, so step 2 is
illegitimate (but why not?)
b. Maybe N+1 isn't larger than N for that N (how? What happens to
arithmetic up at the large end?)
c. Maybe N+1 doesn't exist at all, its impossible. That
seems more intuitive. But what would it mean to say that N exists but
N+1 doesn't? After all, if N exists than the set {1, ... ,N} seems to
exist, and the power set of this set has more than N elements. Its
very hard to cleave to the strict finitist intuition and still do any
kind of set theory.
d. This is an argument by contradiction, and so is inherently
suspicious. I find this quite unconvincing, myself, even though its
popular. The actual
logic of this argument seems quite sound to
me.But my main point is that in order to maintain a coherent
strict-finitist position one does need to consider arguments/debates
like this and to think hard about the consequences of various
positions. Its not enough to just announce as an obvious doctrine that
infinity is wrong; still less to seem to link conventional mathematics
to some kind of dark political conspiracy. Mathematicians tend to be
Platonists because they are driven to it by following chains of
thought which seem to be inevitable and conclusive. If you want to
announce an alternative, you have to tell us where the less travelled
paths branch off the mathematical highway.
Pat
You may not like where that leads, but it is very hard to argue that -- ---------------------------------------------------------------------
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