Pat, (01)
I wasn't advocating that we go back to pre-Cantorian set theory. (02)
But I was trying to distinguish two directions Avril was suggesting:
the indefensible idea of trying to set a finite upper bound to the
integers and the more defensible (but unlikely to be adopted) idea of
going back to 19th c. math. (03)
> 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. (04)
I was summarizing the way a 19th century mathematician would talk,
not advocating it. (05)
PH> Ack(Ack(5,5),5) can be constructed in a finite number of steps. (06)
Yes. A 19th-century mathematician would be quite happy to accept
functions whose values grow very large very quickly. I definitely
do *not* support Avril's finite upper bound, and neither would
any mathematician since the ancient Greeks (or even earlier). (07)
PH> 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. (08)
I agree. But they have developed an abundance of theories, many
of which have not been fully evaluated for possible applications. (09)
My recommendation to Avril would be to avoid trying to stop any
mathematicians from following their own inclinations. A more
useful approach would be to show how some approaches that have
been overlooked have important applications. (010)
As one example, I believe that Peirce's proof procedure has
important advantages for many purposes. Another example is the
application of 19th century geometric algebra to modern physics
by David Hestenes. (011)
John (012)
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