Mathematics with the potential or actual infinite or without emptiness
dennis.hamilton at acm.org
dennis.hamilton at acm.org
Sat Feb 11 20:02:33 EST 2023
-----Original Message-----
From: FOM <fom-bounces at cs.nyu.edu> On Behalf Of I.V. Serov
Sent: Saturday, February 11, 2023 08:52
To: fom at cs.nyu.edu
Subject: Mathematics with the potential or actual infinite or without
emptiness
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Dennis Hamilton writes
(https://cs.nyu.edu/pipermail/fom/2023-February/023742.html):
Now it is certainly a fair point that obtaining zero as the limit as n goes to
infinity of 2^-n is the limit of an infinite sequence.
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Antonio Drago replies (in
https://cs.nyu.edu/pipermail/fom/2023-February/023768.html):
Not true. Any approximation does not equate the zero, always a distance
remains; or evenly: a segment, representing an approximation, is defined by
two extreme points; it cannot be reduced to one point only; this is an old
criticism to epsilon-delta technique .
. illusion that leads to believe that the approximations at last crash to
zero. It is just an appeal to actual infinity that justifies this crashing as
effective. Really, it is true in a world of mathematical ideas only according
to the "realist [Platonic] attitude". The entire undergraduated teaching of
mathematics is based on the actual infinity.
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Does zero exist at all, does it not?
Potentially it does not exist as it can never be reached as in the
example of 2^-n.
Limits are always potential unless they are actually achieved.
[orcmid] [ ... ]
Serov
[orcmid] - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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I clearly stepped on quicksand here, and I need to jump for an overhanging
branch.
I will concede that the sequence 2^-n provide an inexhaustible set of
progressive values that gets us as close to 0 as we might desire. To speak of
a limit seems to slip into actual infinities and I will forswear that.
I will also be content to say the same for 1-10^-n providing an inexhaustible
series of values that can be used to get as close to 1 as we desire. It's
basically 0.999...9 with an inexhaustible supply of 9's.
I have no quarrel whatsoever with regard to 0 (and 1) as mathematical entities
and I see no difficulty with the mathematical standing of the empty set (and
the null string) as such. There is no metaphysical claim and I see no need
for concern in that respect. There is significant pragmatic value.
- Dennis
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