Re: Marble problem--put in 10 marbles, then remove 1

> On Sun, 23 Oct 2005 03:29:16 +0000 (UTC),
> magidin@xxxxxxxxxxxxxxxxx
> (Arturo Magidin) wrote:
>
> >In article
> <djevjl$7i9$2@xxxxxxxxxxxxxxxxxxxxxxxxxx>,
> >Dave Seaman <dseaman@xxxxxxxxxxxx> wrote:
> >>On Sat, 22 Oct 2005 22:39:04 +0000 (UTC), Arturo
> Magidin wrote:
> >>> In article
> <39kkl11tcdhigqin42holgebcq5oldhcop@xxxxxxx>,
> >>> David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx>
> wrote:
> >>>>On Fri, 21 Oct 2005 16:32:03 +0000 (UTC),
> magidin@xxxxxxxxxxxxxxxxx
> >>>>(Arturo Magidin) wrote:
> >>>>
> >>>>>In article <tz86f.10494$cg.789@xxxxxxxxxxxxx>,
> >>>>>Steven <sgottlieb60@xxxxxxxxxxx> wrote:
> >>>>>>
> >>>>>> My friend and I have have different answers to
> the following problem. Who
> >>>>>>is correct and why?
> >>>>>>
> >>>>>>You have a countable number of marbles and a
> really big bag.
> >>>>>>The marbles are labeled 1, 2, 3, ... and so on.
> >>>>>>You put the first ten marbles into the
> aforementioned bag and then you take
> >>>>>>marble number 1
> >>>>>>out of the bag and discard it. Then you put
> marbles 11 through 20 into the
> >>>>>>bag and then you
> >>>>>>take marble number 2 out of the bag and discard
> it, and so on.
> >>>>>>In the end how many marbles are in the bag?
> >>>>>>
> >>>>>>
> >>>>>>Friend's Answer: This process does not have a
> well defined limit.
> >>>>>
> >>>>>Depends on your definition of limit.
> >>>>
> >>>>I don't see what "limits" in whatever sense have
> to do with it.
> >>
> >>> I agree... up to a point.
> >>
> >>>>Every marble that's added is also removed, so
> there are no marbles
> >>>>remaining.
> >>
> >>> Yes: if you go on long enough, you will reach a
> point where any
> >>> specific marble is taken out.
> >>
> >>> But if you want to talk about what happens "in
> the end", "and so on", with
> >>> an infinite process, surely some sort of limiting
> process is needed to
> >>> discuss this "end state".
> >>
> >>That only makes things more difficult. To solve
> the problem directly
> >>(without limits), you only need to answer one
> question: which balls are
> >>left "in the end"?
> >
> >What does "in the end" mean in a process that has no
> end?
>
David Ullrich replied:

> It means "after all the infinitely many steps have
> been carried
> out".<<

There are no infinitely many steps for a bag that
was defined to be finite but unbounded -- in this
context, an inexhaustible space.

However much this problem has been discussed and debated,
I find Arturo Magidin's explanation is still the most
rational. It relies on a necessary context.

One cannot speak of states that involve the counting of
discrete objects without reference to the limit of the
counting process, which is itself a discrete state.
We would otherwise be compelled to introduce mystical
concepts in which infinity is operational, instead of
admitting (as in this context)"no well defined limit."

So the discrete states that are tractable to definition
in this problem reduce to "empty" or "not well defined."
In fact, we know that in the physical world, quantum
states analagously relate to a counting process where
the energy level is measurably known ("empty" state) or
unknown ("not well defined"). One would get in trouble
confusing the unbounded bag (quantum phase space) with
the quantum number (measured energy).

One of the most marvelous aspects of mathematics, IMO,
is that while we creatively use the concepts "limit" and
"function," we still don't know much about them, or
even how or if "time" discretely separates defined
states in the limit. This is the problem that Einstein
ran up against in trying to reconcile continuous functions
with quantum uncertainty.

>
> Which of course is impossible in the real world. But
> saying
> something about limits does not change the fact that
> in
> the real world we can only do finitely many things
> before
> we die - the problem _is_ an abstract mathematical
> thing
> regardless.
>
> The state of some system "after infinitely many
> steps"
> need not always be well-defined. For example, you
> toggle
> a light switch on and off infinitely many times;
> there
> is no answer to the question of whether it's on or
> off at the end. But that's not really because a
> certain
> limit does not exist:
>
> Q: Start with x = 1. At each step replace the value
> of x with the current value of x divided by 2.
> What is the value of x after infinitely many
> steps?
>
> If someone said the answer was 0 because
> the limit of x was 0 I wouldn't feel inclined
> to argue with that. But if someone said that
> for _this_ problem "the value of x after infinitely
> many steps" was undefined I wouldn't be inclined
> to argue with _that_ either! Because the limit
> of x_n is _not_ the same thing as "the value
> of x after saying n = n+1, x = x_n infinitely
> many times"...
>
> In the current problem one could say that a certain
> sequence of sets has an empty limit. But that
> doesn't seem to me to be exactly what the problem
> is asking, and the answer to exactly what the
> problem is asking _is_ clear, without mentioning
> limits: Every marble that is added is also removed
> at some later stage, so after infinitely many
> steps the jar is empty.
>
> >How does the
> >question you ask me to answer even begin to make
> sense without
> >addressing that point? To solve the problem
> "directly" as you suggest
> >requires you to interpret what "the end" means for a
> process with NO
> >end. You may be hiding it, but you are still
> considering limits of
> >some kind at the very instant you start talking
> about "in the end". <<

That seems mystical me. "Hiding" it? There is quite some
semantic difference between hiding a fact and knowing
-- i.e., being able to define -- what it is.

> >
> >That said, since for most people it is intuitively
> clear what "in the
> >end" means for the purpose of this mental exercise,
> then for most
> >people it is indeed completely unnecessary (and only
> more confusing)
> >to introduce limits. Simply pointing out that ball n
> will be removed
> >in step n is enough to convince most people that
> there will be no ball
> >"in the end" (whatever that means).
> >
> >>If you introduce limits, you have to answer at
> least 3 questions:
> >>
> >> (1) What sort of limit should we consider?
> >> (2) What sort of value do we get for this limit?
> >> (3) Does this limit agree with the answer that we
> would have gotten
> >> without applying limits? That is, is the
> function in some sense
> >> continuous "at the end"?<<



"In some sense?" It is either in a discrete definable
state, or it is continuous -- no?


Tom


> >>
> >>It seems to me that in order to answer question
> (3), you have to answer
> >>the very same question as before, namely, what
> answer do you get without
> >>applying limits at all? And if you fail to
> consider question (3), then
> >>you simply have not answered the question that was
> asked.
> >
> >I did not bring up limits to answer the question, in
> any case. The
> >original poster said his friend answered that there
> was "no answer"
> >because "the" (singular definitive article) limit of
> the sets in
> >question did not exist. I pointed out that even if
> you want to try to
> >interpret the question by invoking limits
> explicitly, his friend's
> >analysis was flawed because the kind of limits he
> was considering was
> >far too limited for the task at hand. If ->he<-
> really wants to
> >consider limits, he has to address your question
> (1), at which point I
> >pointed out a reasonable extension of ->his<- notion
> of limit that
> >would render ->his<- objection moot.
>
>
> ************************
>
> David C. Ullrich
.

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