Re: infinity

Kirby Cook wrote:
> William Hughes wrote:
> > Kirby Cook wrote:
> >> William Hughes wrote:
<megasnip>
> >>>Yes the problem is set up so intuitively the answer is the sum
> >>>of an infinite series. But the answer to the problem is
> >>>counterintuitive.
> >>>
>
> Thanks to your patience, I think I follow your arguement. The reason I
> don't buy it lies in my reluctance to accept as sense the concurrent
> statements, 1) the balls in the vase increase infinitely as the time
> approaches noon, 2) There is no point at which all the balls vanish (or
> even any discoverable point at which they begin to be reduced), and 3)
> There are no balls in the vase at noon. That seems to me to be as clear
> a contradiction as contradictions get.

I think your reluctance to buy it is understandable. I also think there
is a quite reasonable way to reject the standard Question-and-Answer,
but this is by rejecting the Question. The whole notion that there
could be any sort of ball-and-urn thingy going on is of course quite
absurd, as a physical model, however hypothetical. The problem is that
it starts off sounding quite reasonable, lulling the intuition pumps
into thinking this is going to be easy.

Anyway, there certainly is an unintuitive discontinuity "at noon", but
I think any attempts at different answers will always lead to other
strange discontinuities. For example, it seems that Tony Orlow thinks
that if all the balls are put in the urn well in advance, then indeed
the urn will be empty at five-past-twelve, but if the same [yes, Tony?]
set of balls is added progressively during the emptying process, then
there will be infinitely many left in the urn. Well, consider the
infinitary engineering (ha ha!) implementation: there are ten robot
arms dedicated to putting balls in, and one taking them out. This is
supposed to happen according to the statement of the original problem,
so TO claims that the urn will be full afterwards. BUT! An engineering
failure means that the take-out arm has an operational delay of 1
nanosecond. Aha! This means that *at noon* only a fairly small number
(I think it's about 35) of balls have been removed; the take-out arm is
left working on its owm. I *guess* that in this case, everyone agrees
that the take-out arm will do its job, and all the balls will be gone
by one minute past noon. Well, engineers work on the problem, and
reduce this delay on the take-out arm: but however small, a nonzero
delay means that the pot will be empty, yet get the arm working exactly
on time, and the pot is left full. This surely is a worse absurdity to
deal with? (If absurdities can have degrees of worseness...)

Meanwhile, other engineers have hooked up a chart recorder to monitor
what's going on. This produces marks from an infinite set of pens (ha
ha!), one for each numbered ball, and each pen goes down when the ball
goes in the pot, and up when it comes out. The output is printed on a
nice big *** of white paper, representing the timeline from one
minute before noon to one minute after - noon is right in the middle.
The pen marks are basically superimposed, so we see a grey track
getting blacker and blacker, towards the infinite blackness [ha ha!] of
noon. Right, but what colour is the right hand half of the *** of
paper? Let's be tolerant of smudging, and nanosecond delays and all
that, even let's suppose the pen pressure demolishes a central area of
the paper. How many pens will still be down at the right edge of the
paper?

Brian Chandler
http://imaginatorium.org

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