Re: infinity

Jeroen Boschma wrote:
> cbrown@xxxxxxxxxxxxxxxxx wrote:
> >
> > Jeroen Boschma wrote:
> >
> > <snip>
> > >
> > > So here's what I think about it. It's obvious that the number of balls in
> > > the vase keeps growing as you approach noon.
> >
> > > However, the algorithm which describes the filling of the vase with balls
> > > never reaches noon. Therefore, I think it is not even correct to ask the
> > > original question 'how many balls at noon' (do mathemathicians call that
> > > ill-defined, -posed or something like that?).
> >
> > There is some sense to what you say.
> >
> > There is still some room IMO for claiming that the question itself is
> > "ill-defined". It's certainly not physically possible for such an
> > operation to take place in the real world to start with; so the
> > question of "what would happen at noon?" is really best interpreted as
> > "what would be a logically consistent answer to the question, assuming
> > that it /could/ be done in practice?"
>
> The basic idea I had was that we can make a function B(n) which describes
> the number of balls in the vase from one minute to noon to every time less
> than noon. But the description of the experiment does not allow us to reach
> noon.

It certainly doesn't allow your /function/ as given to be defined at
noon; but that doesn't mean we never "reach" noon. We reach noon, 1
o'clock, etc.

All that says to me is that your function (by itself) doesn't yield
much insight into what a consistent meaning to the question "what balls
are in the vase at noon" might be.

> It does not have a sense of a continuously progressing time which
> is required to reach noon. So I would say that the experiment directly
> implies that 'noon' is not within the domain of B(n), and therefore the
> function, or description of the experiment, cannot give
> an answer to the question.
>

So we must make an appeal to other considerations. Dave Seaman does
this by embedding your function into a different function, which /does/
have a value at noon, and is consistent with the intuitions I gave
below.

> *** For nitpickers :) , as B(n) is a function of variable 'n', statements about the domain should
> be based on 'n' only, I guess. But every 'n' implies a certain time t_n, so I think you get the
> idea... I hope... else beat me again :(
> ***
>
> From the reasoning in this thread, experiment 1 should result in an empty
> vase, and experiment 2 in infinite balls in the vase at noon. Different
> answers while the function B(n) that describes the number of balls in the
> vase at any 'n' is the same for both... The question is a physical one
> (number of balls), and for that reason the answer cannot depend on how the
> balls are numbered or what color they have.
>

What these "experiments" show is that our assumption that it's "just"
about the number of balls isn't consistent with the problem statement;
one needs to consider the ordering of the balls.

> >
> > Suppose instead of your experiment 1, we put a ball into the vase on
> > odd steps, and then take it out the even steps.
> >
> > Then it seems obvious that there is no single well-defined answer to
> > "how many balls are there at noon?". Neither "empty" nor "containing a
> > ball" seems to be correct - there is no answer that is somehow
> > /logically preferable/ to all others; and that is the basic sense of
> > "not well-defined".
>
> For the same reason as I described above: for each n we can find the number
> of balls in the vase, but there is no n which equals 'noon'. Therefore,
> because the number of balls change at every step n, we should not want to
> force an answer. We can just say that no answer is possible because 'noon'
> is not in the domain of the function B(n) which describes the number of
> balls in the vase.
>

We can just say that, but that would be giving up too easily :).

> >
> > On the other hand, if you did /nothing/ at each step, that there would
> > somehow be a ball in the vase "at noon" seems obviously wrong. The vase
> > would "obviously" be empty, because we intuitively require that there
> > can't be a ball in the vase unless it was put there at some finite step
> > n.
>
> Because there are no steps defined where the number of balls in the vase
> changes, we don't need to stick to those discrete moments t_n, but we can
> directly describe the number of balls in 'continuous
> time': nothing happens. So we are not bound by a function B(n) in which
> 'noon' is not element of the domain.
>

How about this: Start with an empty vase. At each step n, if n is odd,
put the ball labelled "1" in the vase; if n is even, remove the ball
labelled "1" from the vase.

Now, is there a ball labelled "2" in the vase at noon? Isn't "no" a
perfectly well-defined answer to this question, regardless of the fact
that there is no well defined answer to "how many balls are in the vase
at noon"?

Similalrly: Start with an empty vase. On step one, place balls "1" and
"2" in the vase. At each step n>1, if n is odd, put the ball labelled
"1" in the vase; if n is even, remove the ball labelled "1" from the
vase.

Is there a ball labelled "2" in the vase at noon?

> >
> > Similarly, it seems intuitive that if at step 1, you put ball "1" in
> > the vase, and then at no later step do you remove it, then that ball
> > "must be" in the vase "at noon".
> >
> > These two intuitions are the assumptions we are making when we ask
> > "assuming it is possible in the real world, what balls are in the vase
> > at noon?"; and a well-defined answer would be one which is consistent
> > with these intuitions, and also be the /only/ one consistent with these
> > intutions.
>
> Again, IMHO, forgetting that the answer must be given by a function which
> describes the number of balls in the vase, and that 'noon' is not in the
> domain of that function.

You are confusing your function B(n), which doesn't have noon in its
domain, with a counting function (such as that given by Dave Seaman)
which does have noon in its domain.

> We don't _need_ to give a number as an answer.
>

Sometimes we can't. But sometimes we can.

> In this experiment, we try to connect some logical reasoning (about putting
> balls with number in the vase and getting some out) with a physical entity
> (number of balls in the vase). That connection cannot be made because we have
> no description or function which can make the connection!
>

Sure we do. The connection is that if a particular labelled ball is put
in the vase and never taken out, then it is still in the vase. If a
particular labelled ball is removed from the vase and never put back in
(or was never put in to start with), then it cannot be in the vase.

If the process of putting labelled balls in and out of the vase results
in every ball obeying one of the two above conditions, then it is
possible to determine the number of balls in the vase at noon.

In particular, if the process of describing how the balls are put in
the vase tells us that every ball is not in the vase at noon, then the
vase is empty.

> >
> > Thus, in experiment 1, we can agree that, if at some definite step t_n,
> > we remove the ball labelled "n", and then there is no later step that
> > we put the ball back in, then it is perfectly reasonable to insist
> > that, /whatever/ we mean by "the balls in the vase at noon", it can't
> > mean that ball "n" is in the vase at noon.
> >
> > Since, in your construction, the above is true for every n, the only
> > /possible/ answer consistent with our assumptions is that there are no
> > balls in the vase at noon.
>
> So your explanation is that for each ball n in the vase, there is a step t_n
> where that ball is taken out.

AND /never/ put back in. If we happen to stumble on the experiment in
progress at a time after t_n, we see a situation where the ball
labelled "n" is not in the vase, and is never put into the vase. Why
should we assume that it somehow will be in the vase at noon?

> So each ball is taken out eventually which results in an empty vase at noon.
> This is based on _physical_ observations: every ball put in the vase is
> taken out at some time.

Certainly, it's based on a very simple premise: If at some time t_n <
noon, the ball "n" is not in the vase, and for all t > t_n, the ball is
not returned to the vase, then it cannot be in the vase at noon.

But it's not a "physical observation", it follows from the statement of
the problem. If a ball is in the vase at time t0, then there must be
some definite step in our progession that put it there. If it is not
there at some later time t1, then there must be some "later" step in
the progression that removes it. That's all that's being assumed.

> Another _physical_ observation I made in my original reply
> is: at every 'n' the number of balls in the vase
> increases by 9. So there is no interval between one minute to noon and noon
> where the number of balls in the vase decreases. So how can the vase be empty
> at noon?

Yes, it boggles the mind; but you are assuming something here without
justifiction. The vase can be empty at noon, because there is no
requirement (as Dave Seaman demonstrates) that your function B be
continuous at noon.

>
> IMO, both reasonings are OK, but they _seem_ to end up with different
> answers.

Both reasonings _seem_ OK, but they _definitely_ end up with different
answers. So one reasoning, at least, must be wrong.

> Reason for this contradiction is that we're not allowed to
> formulate a numeric answer to the original question,
> because noon is not in the domain of the experiment.
>

As you can see, in certain situations it is entirely allowed.

Cheers - Chas

.

Relevant Pages

  • Re: infinity
    ... Physically: Infinity is undefined physically. ... >> to show that each ball is removed from the vase before noon and is not ... >> the vase is empty before noon. ... The operation of adding or removal of balls is undefined at noon. ...
    (sci.math)
  • Re: An uncountable countable set
    ... To become empty means there is a change of state in the vase, from having balls to not having balls. ... There are always a specific number of balls, if additions and removals occur instantaneously. ...
    (sci.math)
  • Re: An uncountable countable set
    ... Since the vase was empty to start with, it cannot later "become" empty after once having been empty, at least according to that definition. ... Noon does not exist in the experiment, or else you have infinitely numbered balls. ... insertion or removal or location of balls is a function of time. ...
    (sci.math)
  • Re: infinity
    ... >>> Physically or mathematical it is not difficult to prove that the vase ... # of balls in the vase can be ... this statement alone is not sufficient to claim ... >>> the vase is empty before noon. ...
    (sci.math)
  • Re: infinity
    ... Number of balls in the vase at noon is f= OO. ... Unfortunately, if infinity gets involved, this statement alone is not sufficient to claim the vase is empty before noon. ... then the sum becomes 0. ...
    (sci.math)
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