Pat and Chris, (01)
I agree with the following point: (02)
PH> 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. (03)
But I would claim that the view held by nearly all mathematicians
until the latter part of the 19th century is coherent: Infinity
is a limit, not something that can be attained as a completed
mathematical entity. (04)
In other words, one can accept the point that there is no upper
bound on the size of any integer, but the only sets that are
legitimate objects of mathematical investigation are finite.
That view is quite coherent, and nearly every mathematician
accepted it as dogma in the first half of the 19th century. (05)
The dominant view about points in those days was the approach
advocated by Aristotle and Euclid: a point is a designated
locus on a line, plane, or volume, not a "part" of the line,
plane, or volume. There is no upper bound on the number of
points that a mathematician might designate on a line, plane,
or volume, but it is not permissible to talk about the totality
of all the points that one could designate -- because that is
infinite, and not admissible as an object of mathematical
investigation. (06)
CM> ... which of course means that there are infinitely
> many finite integers, and hence that there is a set that
> contains them... (07)
Wait! A 19th century mathematician would agree with the first
point, but restate it without using the phrases "infinitely many"
or "set of integers". As examples of "correct" 19th century
mathematical English, one could say: (08)
1. There is no largest integer. (09)
2. Any nonempty set of integers has a largest integer. (Note that
this is true because all early-19th-century sets were finite.) (010)
3. There is no largest set of integers. (Follows from #1 and #2.) (011)
CM> ... 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! 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. (012)
No, there is a natural stopping point: only those sets that can be
constructed in a finite number of steps. (013)
I'm not saying that I'm advocating the 19th c. position, but it is
quite coherent, and I can sympathize with people who feel uneasy
about infinite sets. There is no reason why they must accept them
in their ontology if they don't want to. (014)
However, even Euclid would accept statements #1, #2, and #3 above.
It's important to note that Euclid does not imply Cantor. (015)
John (016)
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