RE: Mathematics with the potential infinite - not so fast Zeno
dennis.hamilton at acm.org
dennis.hamilton at acm.org
Thu Feb 2 00:07:56 EST 2023
From: FOM <fom-bounces at cs.nyu.edu> On Behalf Of Vaughan Pratt
Sent: Tuesday, January 31, 2023 17:55
To: fom at cs.nyu.edu
Subject: Re: Mathematics with the potential infinite
So far it seems to me that every objection to my proposal (repeated below) concerning gapless geodesics depends on the assumption/axiom/whatever that a nonempty linear order with no greatest (or least) element must be an actual infinity.
To me, this has the flavor of Zeno's paradox, that movement from 1 to 0, say in units of meters, must be impossible because it must pass through the infinitely many points 1/2, 1/4, 1/8, ...
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.
[ … ]
[orcmid]
There’s more interesting meat in Vaughan Pratt’s message, but I want to dwell on Zeno’s paradox versus all we need to know from Newtonian mechanics.
Zeno failed to distinguish that constant velocity is not altered in the pondering about shorter and shorter distances. So if I am moving at rate of 1 meter/second, In one second I will cover 1 meter, no matter how one contends that the second (or the distance) be divided (or increased for that matter). We don’t need to get to infinitesimal traveling at all. And it works under ordinary circumstances.
Concerning potential infinities I suppose one can treat structure N as having a progression of naturals that has no limitation and we can always find more of them given one or more to start from. I suppose we might call that something else to avoid considering N to consist of a set containing all of the naturals with none left out. Does that get to the potential versus actual distinction? (I have a different example that I’m thinking is about a potential infinity and there Is a very rudimentary induction argument .)
* Dennis
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