Re: [ontolog-forum] Axiomatic ontology

[Pat Hayes <phayes@xxxxxxx>]

At 10:18 AM -0600 2/7/08, Chris Menzel wrote:

> On Wed, 6 Feb 2008, John F. Sowa
> wrote:
> > Avril,
> > ...
> > On the other hand, I would not recommend the following
> approach
> > for number theory:
>
> To say the least.
>
> > > Axiom 9. can be maintained, but the meaning of 'every' has
> to be
> > > interpreted to denote a totality of something around
> 10**120,
> > > or Ackermann(5 5), or something finite that can be written
> down
> > > or understood in some way.
>
> The view isn't even coherent.  If Ackerman(5 5) exists, why
> not
> Ackerman(Ackerman(5 5) 5) -- a massively larger number?  And of
> course
> if *that* number exists, well, you get the idea.
>
> > Many 19th century mathematicians strongly objected to that way
> of
> > talking, and I sympathize with them.  But those
> mathematicians would
> > *never* agree to a fixed upper bound on the integers, such as
> 10**120,
> > Ackermann(5 5), or any other finite integer.
>
> Indeed -- which of course means that there are infinitely many
> finite
> integers, and hence that there is a set that contains them, hence
> a
> power set of that set, and off we go down the Cantorian bunny trail!
> :-)

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
> there is a nonarbitrary point at which you can stop that line of
> reasoning.
>
> -chris
>
>  
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