At 4:54 PM -0500 2/7/08, John F. Sowa wrote:
>Pat and Chris,
>
>I agree with the following point:
>
>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.
>
>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. (01)
Attained? We don't need to attain it in order to
speak of it. Obviously there is a sense in which
no infinity can be 'attained' in a finite
universe. (02)
>
>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.
>
>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.
>
>CM> ... which of course means that there are infinitely
> > many finite integers, and hence that there is a set that
> > contains them...
>
>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:
>
> 1. There is no largest integer.
>
> 2. Any nonempty set of integers has a largest integer. (Note that
> this is true because all early-19th-century sets were finite.)
>
> 3. There is no largest set of integers. (Follows from #1 and #2.)
>
>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.
>
>No, there is a natural stopping point: only those sets that can be
>constructed in a finite number of steps. (03)
That is circular. Ack(Ack(5,5),5) can be
constructed in a finite number of steps. Numbers
so large that to write out their decimal
expansion would take more than the information
capacity of the known universe can still be
constructed in a finite number of steps. This is
a real problem, even for 19th century
mathematics. In fact, it was from wrestling with
problems like this that 20th century mathematics
emerged. (04)
Finitism seems much more intuitive when dealing
with very large numbers than when dealing with
very small ones, ie their reciprocals. If all our
series have to be finite and we cannot talk of
limits, it becomes impossible to give an adequate
foundation for calculus, for example. On the
whole, I think that the mathematicians have done
a fairly good job and we would all be better off
leaving it to them, and focusing on matters of
more direct importance to our engineering. (05)
>
>I'm not saying that I'm advocating the 19th c. position, but it is
>quite coherent (06)
Its reasonably coherent but it breaks down at the edges. (07)
Pat (08)
>, 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.
>
>However, even Euclid would accept statements #1, #2, and #3 above.
>It's important to note that Euclid does not imply Cantor.
>
>John
>
>
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