Re: Hamilton's Rule is Xeno's Paradox ( was Re: Underestimating 'r')
- From: William Morse <wdmorse@xxxxxxxxxxxx>
- Date: Tue, 25 Oct 2005 01:07:19 -0400 (EDT)
"g" <gillawton@xxxxxxxxxxxxx> wrote in
news:dj6jvt$to0$1@xxxxxxxxxxxxxxxxxxx:
>
> <EKurtz99@xxxxxxx> wrote in message
> news:dj4np4$3ej$1@xxxxxxxxxxxxxxxxxxxxxx
>> John Edser wrote:
>> In Xeno's Paradox the algebraic constant "time" has
>>> been deleted as an oversimplification. This reduces the mathematics
>>> to become entirely, just a relative proposition within which the
>>> hare can never
>>> actually catch up to the tortoise.
>>
>> Of course it doesn't, you idiot. Zeno thought he had found a paradox
>> because he did not know how to compute the limit of a convergent sum.
>> Neither do you, of course. However, he has an excuse - he lived 2,500
>> years ago. What's yours?
>>
>> (to compound the problem, you are confusing one of Aesop's fables -
>> The Hare and The Tortoise - with Aristotle's version of Zeno's
>> paradox (or one of them) - Achilles and the Tortoise)
>>
>
> John,
>
> I appreciate your thinking, but not your resort to personal attack.
>
> The issue is this: that Zeno's paradox of the arrow remained a
> paradox even after Newton and Leibniz.
> Newton's realization that we could think of an arrow's trajectory in
> terms of what he termed "fluxions,"
> did not resolve the paradox. It merely found a work-around, by way of
> our manipulating (mathematically) the relationships between functions
> and factors (an over-simplification, but representative). It also
> relied upon 'rounding.' Motion can be expressed only irrationally,
> unless one rounds, just as Pi can only be expressed irrationally.
> Motion and non-motion can not be equated rationally because they are
> mutually exclusive. (For that matter nothing in the cosmos can be
> defined as not-in-motion EXCEPT insofar as it may be arbitrarily
> assigned to be the only non-moving point in the cosmos (or all
> cosmoses, the singleness or plurality of which we cannot be certain).
>
> At the quantum level, the paradox at the Newtonian level is
> potentially resolved in the Newtonian sense, but translates into still
> ANOTHER paradox, since here it is found to move in fits and starts,
> not stopping and starting, exactly, but ceasing to exist at one
> quantum moment and then reappearing at the next one, which is 6.626 x
> 10 to the minus 34 power, perhaps, away (Planck's constant).
>
> For purposes of computation, we can round as much as we like,
> measuring in feet per second, or inches per second, or whatever... but
> these are only rounded approximations until, and unless, we get down
> to the ticks of the clock in the arrow's frame of reference. And EVEN
> THAT varies from one frame of reference to the next (due to the
> expansions and contractions of time, through which photons and arrows
> may pass).
>
> Fortunately, for crude predictions, such as determining for purposes
> of building an aircraft engine -- even to tolerances of a mere
> one-to-five thousandths of an inch, we still have to round... because
> Pi can NEVER be resolved into a real number fraction, nor a terminal
> decimal equivalent). Take a micrometer and hone our engine parts to
> "close enough," and there we settle upon imperfection.
>
> I am not judging whether you have a point worth making here. I am
> only speaking to the issue of whether it is accurate to say a person
> is stupid or wrong for not translating the paradoxes of motion and
> time into a round figure and calling that a resolution of them.
You may want to note that E Kurtz was responding to John Edser's post,
and accusing John of being an idiot. I do agree that while John is
exasperating, and has on occasion made personal attacks himself, ad
hominem arguments have no place on the newsgroup.
I had long thought, as EKurtz has argued, that Zeno's paradox was
adequately resolved by the theory of limits. However, I do remember an
article a number of years ago in Scientific American that challenged this
resolution, as you have done. Unfortunately I am not a good enough
theoretical mathematician to comment more, and in any case I don't think
the discussion belongs on sbe. However, I have enjoyed your further
thought experiments on Achilles-tortoise races.
Yours,
Bill Morse
.
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