R: ?Mathematics with the potential infinite or not
Antonino Drago
drago at unina.it
Thu Feb 9 17:17:14 EST 2023
Dennis Hamilton writes:
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.
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 (see Kogbetlianz E.G. (1968). Fundamentals of Mathematics from an Advanced Point of View, App. 2) It is a parallelism with an optical 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.
Best
Antonino Drago
Da: FOM [mailto:fom-bounces at cs.nyu.edu] Per conto di martdowd at aol.com
Inviato: domenica 5 febbraio 2023 17:38
A: fom at cs.nyu.edu
Oggetto: Re: ?Mathematics with the potential infinite - some inexhaustible?
Dennis Hamilton writes:
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. What is bothering me is the idea that because the sequence is infinite, it is therefore not something we can experience.
This is true of mathematical objects in general. An integer n can be "experienced" in everyday life in various ways. But what about Z_n, the ring of integers mod n? Also, infinity can be experienced in some ways. A line segment in 3-dimensional Euclidean space has uncountably many points. Everyday life example of countably infinite sets seem more involved.
Martin Dowd
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