Thinking Badly About Infinity
Math1723
want to present it more as a Math lesson, and as a detector for people
to see how critically well they think Mathematically, even when their
intuition goes contrary to logical dictates. (Often, cranks will
abandon logic to embrace the emotion, irrespective of the logic.)
Many of you know this example already as "The Bucket & Balls Problem".
We have an infinite number of Balls, all uniquely labelled #1, #2,
etc. A Clock is counts down from 1 second:
At time t=1 sec, Balls #1 thru #9.
At time t=1/2 sec, Ball #1 is removed, and Balls #10-19 are added.
At time t=1/4 sec, Ball #2 is removed, and Balls #20-29 are added.
At time t=1/8 sec, Ball #3 is removed, and Balls #30-39 are added.
...
At time t=1/2^n, Ball #n is removed, and Balls #10n thru 10n+9 are
added.
The clock and the process all stop at time t=0 seconds.
Note that by rule, no ball is added unless it is labelled with a
finite natural number. Repeat: No ball without a finite natural
number label is allowed to go in to the bucket. (You can pretend
there is a filter checking for the number label, like a bar code scan,
rejecting all balls lacking the appropriate label.)
Question: How many balls are in the bucket when the clock stops at
t=0?
Obviously, the number of balls in the bucket gets larger and larger as
the clock winds down. The number of balls at each time t=1/2^n
increases exponentially. The limit tends to infinity. But remember,
not all functions evaluate at their limit.
As it turns out: there are 0 balls in the bucket. We can know this
since for every Ball #n, we know that it was added at time t=1/2^[n/
10] and removed at time t= 1/2^n. We also know this must be true for
every single finite n (the only n's that were allowed in the bucket).
Put another way, ANY ball that is added at time t=1/2^x MUST BE
removed no later than time t=1/2^10x.
Now the logic is irrefutable. It may be counter-intuitive, defies
imagination, and seem downright wrong. But the logic is absolutely
sound, and if you accept logic, you must accept this. All the balls
are gone.
If you do not, you must ask yourself: If any balls exist in the
bucket, what numbers do they contain? They must be finite numbers (by
rule). Yet how could they have not been removed? Since every non-
empty set of finite natural numbers has a least element, then if any
balls exists in the bucket, there is one with the smallest number.
What is that number?
The purpose of this is to show how easily it is for us to use our
"intuition" and our "common sense" reasoning to think incorrectly
about infinities. Our experience in the finite world gives us
absolutely no handle on how things operate at transfinite levels. We
must rely on the logical rigors of Mathematics to answer our
questions, even if those rigors give us answers we don't like.
Thanks for reading.
David R Tribble
> This is usually given in the form of a puzzle or brain teaser, but I
> want to present it more as a Math lesson, and as a detector for people
> to see how critically well they think Mathematically, even when their
> intuition goes contrary to logical dictates. (Often, cranks will
> abandon logic to embrace the emotion, irrespective of the logic.)
> Many of you know this example already as "The Bucket & Balls Problem".
The Wikipedia page used to be called "Balls and vase problem":
http://en.wikipedia.org/wiki/Balls_and_vase_problem
It's been a year or two since anyone talked about this
on sci.math, so, sure, I'm game for another round...
> Question: How many balls are in the bucket when the clock stops at t=0?
Zero.
You can start counting[*] how many different answers you get to
this puzzle.
[*] Pun intended.
1729 8 my balls
fishfry
<68445693-c076-40c9...@l24g2000vby.googlegroups.com>,
Math1723 <anony...@aol.com> wrote:
<snippage>
This is like the guy who takes ten years to write down the story of his
first year of life. If he lives long enough, he'll eventually be able to
write the story of his entire life. Year n always gets written about in
year 10n (plus whatever year he started writing).
In any event, it's misleading to say what happens "at time zero", since
the function that describes the state of the bucket is not defined at
time 0. All you can say is that every ball added to the bucket is
eventually removed.
This is like the one where the light flips on and off every 1/2^n
seconds, and you're asked if the light is on or off at time t = 1. The
answer is that the state of the light is undefined at t = 1. The
function that describes the state of the light is only defined for 1/2^n
as n ranges over the natural numbers.
Han de Bruijn
Brian Chandler
...
> We have an infinite number of Balls, ...
Oh yes, groan.
> Note that by rule, no ball is added unless it is labelled with a
> finite natural number. Repeat: No ball without a finite natural
> number label is allowed to go in to the bucket. (You can pretend
> there is a filter checking for the number label, like a bar code scan,
> rejecting all balls lacking the appropriate label.)
Ah, but you see, I think you have *no idea* just how slippery balls
can be. (Not to mention their psychological effect in inducing
pleonasms such as "finite natural number" in otherwise sane writers)
Even if you say, twice, that no ball is allowed in the bucket without
a natural number on it, you will not prevent there being balls with
infinite natural numbers on, or pink balls, or even if you say the
balls are white, and painted red on being put in, there will still be
a claim of green balls.
Good luck with it anyway...
Brian Chandler
LudovicoVan
> Many of you know this example already as "The Bucket & Balls Problem".
>
> We have an infinite number of Balls, all uniquely labelled #1, #2,
> etc. A Clock is counts down from 1 second:
>
> At time t=1 sec, Balls #1 thru #9.
> At time t=1/2 sec, Ball #1 is removed, and Balls #10-19 are added.
> At time t=1/4 sec, Ball #2 is removed, and Balls #20-29 are added.
> At time t=1/8 sec, Ball #3 is removed, and Balls #30-39 are added.
> ...
> At time t=1/2^n, Ball #n is removed, and Balls #10n thru 10n+9 are
> added.
>
> The clock and the process all stop at time t=0 seconds.
>
> Note that by rule, no ball is added unless it is labelled with a
> finite natural number.
Then, as already noticed, there is no time t=0.
> The limit tends to infinity. But remember,
> not all functions evaluate at their limit.
Is the limit defined?
> As it turns out: there are 0 balls in the bucket. We can know this
> since for every Ball #n, we know that it was added at time t=1/2^[n/
> 10] and removed at time t= 1/2^n. We also know this must be true for
> every single finite n (the only n's that were allowed in the bucket).
> Put another way, ANY ball that is added at time t=1/2^x MUST BE
> removed no later than time t=1/2^10x.
Sure but, for any ball removed, 10 more get added...
> Now the logic is irrefutable.
I don't see a clear resolution.
> The purpose of this is to show how easily it is for us to use our
> "intuition" and our "common sense" reasoning to think incorrectly
> about infinities.
I think it's just as easy as reasoning incorrectly with math.
-LV
LudovicoVan
For more uniformity, I'll consider that in the bucket at step n=0
there is one ball: ball 0.
We have that at each step n>0, 1 ball gets removed and 10 get added.
If c_n is the number of balls at step n in the bucket, we can write:
c_0 := 1
c_n := c_{n-1} - 1 + 10
That sequence can be rewriten as a direct function of n:
c_n = c'_n := 9*n + 1
Taking the limit for n->oo:
lim_{n->oo} ( 9*n + 1 ) = oo
Ergo: taking the limit over the number of steps, the number of balls
in the bucket tends to infinity.
> > Now the logic is irrefutable.
Is it?
-LV
Han de Bruijn
I'd like to remind you of the fact that, according to _standard_ math-
ematics, the outcome of this problem is: NONE balls left. Denying this
places you in the champ of the "cranks". You're quite welcome with us,
though, because we aren't that picky .. (Needless to say that I agree
with your conclusions, for a change)
Han de Bruijn
Brian Chandler
> I'd like to remind you of the fact that, according to _standard_ math-
> ematics, the outcome of this problem is: NONE balls left.
Correct. (Well, modulo a tiny slip: it's "NO balls left". And you mean
"mathematics".) There does not exist a natural number for which the
value of the "in/out" function is "in" at any time 0 or later. Does
there?
> Denying this
> places you in the [?champ?] of the "cranks".
If this is "camp", then yes, correct. You in particular are an
inveterate crank, since (among other things) you think that limits can
be swapped at will, if it pleases you. Carry on...
Brian Chandler
LudovicoVan
>
> I'd like to remind you of the fact that, according to _standard_ math-
> ematics, the outcome of this problem is: NONE balls left.
Are you sure? I read on Wikipedia that there is no universal agreement
on the solution to this problem.
> Denying this places you in the champ of the "cranks".
I am NOT denying anything: rather you are welcome to check and correct
my argument, or to provide another one. -- Besides, I think 99% of
crankiness is when one uses that word at all.
> You're quite welcome with us
You wish! :)
-LV
Han de Bruijn
NO balls left !! What ?! Do you not universally agree ?!
> Are you sure? I read on Wikipedia that there is no universal agreement
> on the solution to this problem.
>
> > Denying this places you in the champ of the "cranks".
>
> I am NOT denying anything: rather you are welcome to check and correct
> my argument, or to provide another one. -- Besides, I think 99% of
> crankiness is when one uses that word at all.
Your argument is GOOD. The argument of standard mathematics is wrong.
Therefore, again:
> > You're quite welcome with us
>
> You wish! :)
>
> -LV
Han de Bruijn
Han de Bruijn
wrote:
Hahaha .. Achilles and the Tortoise. Your reasoning is outdated, sir.
> > Denying this
> > places you in the [?champ?] of the "cranks".
>
> If this is "camp", then yes, correct. You in particular are an
> inveterate crank, since (among other things) you think that limits can
> be swapped at will, if it pleases you. Carry on...
>
> Brian Chandler
Indeed. Limits can be swapped at will, even if it doesn't please me.
Han de Bruijn
Han de Bruijn
? Probably you are referring to this:
https://groups.google.com/group/sci.math/msg/66edf2688f0e25c6?hl=nl
Han de Bruijn
Tony Orlow
You may frame the question in terms of what balls remain in the vase
at noon, but perhaps you should ask yourself a different question. If,
at every moment before noon, there are (a growing number of) balls in
the bucket, and if AT noon no balls are added or removed, at what time
does the bucket become empty? Logically, there is no answer to this
question, and therefore no point in your gedanken at which this
magical disappearance can occur. The problem here is an artificial
Zeno Paradox Engine which purports to have completed omega and nothing
else. The fallacy is in assuming any such completion of N can occur
without going beyond N. Since there is no distinct end to N, such
completion cannot occur, and the gedanken is itself invalid.
Tony
Han de Bruijn
Indeed, the fallacy is in _completed_ infinity. As always.
Han de Bruijn
The Qurqirish Dragon
> This is like the one where the light flips on and off every 1/2^n
> seconds, and you're asked if the light is on or off at time t = 1. The
> answer is that the state of the light is undefined at t = 1. The
> function that describes the state of the light is only defined for 1/2^n
> as n ranges over the natural numbers.
No, for this problem the light is off- after turning on and off some
finite number of times, the filament in the bulb will snap from
fatigue, and the light stays off regardless of the position of the
switch (or whatever mechanism you have to turn it on and off) :-)
Tony Orlow
Well, consider it an led light with no filament, which at every
iteration blips for a moment. As noon approaches, the led blips more
and more often, increasing the average brightness perceived, until at
noon, it is blipping infinitely fast and therefore as good as solidly
on, but only for that moment, at which moment the led dies from
overload. Now, how can the led be both on, and dead, at the same
moment? Herein lies the contradiction.
Peace,
:D Tony
LudovicoVan
> On Jan 17, 10:41 pm, Math1723 <anonym1...@aol.com> wrote:
>
>
> You may frame the question in terms of what balls remain in the vase
> at noon, but perhaps you should ask yourself a different question. If,
> at every moment before noon, there are (a growing number of) balls in
> the bucket, and if AT noon no balls are added or removed, at what time
> does the bucket become empty? Logically, there is no answer to this
> question
Logically, the anwer to that question is: never.
> , and therefore no point in your gedanken at which this
> magical disappearance can occur. The problem here is an artificial
> Zeno Paradox Engine which purports to have completed omega and nothing
> else. The fallacy is in assuming any such completion of N can occur
> without going beyond N. Since there is no distinct end to N, such
> completion cannot occur, and the gedanken is itself invalid.
It seems to me that that risks not to be the real problem here, i.e.
assuming that a completion on N is possible. Such completion *is*
indeed possible, for intance by transfinite induction, but my argument
(unless refuted, etc.) shows that you do *not* get an empty bucket by
simply extending via the limit. OTOH, I'd rather think there is a
logical fallacy in how the problem is framed by already assuming the
solution (and that it is an empty bucket!), while a more careful
presentation (I'd think) just leave us with the more mundane
mathematical problem of how to properly do that extension.
-LV
Ray Vickson
If you want to drive yourself crazy, just look at a modified version
(mentioned in outline in the Wiki article):
At time t=1 sec, Balls #1 thru #9 are in the urn.
At time t=1/2 sec, Ball #9 is removed, and Balls #10-19 are added.
At time t=1/4 sec, Ball #19 is removed, and Balls #20-29 are added.
At time t=1/8 sec, Ball #29 is removed, and Balls #30-39 are added.
etc. The same number of balls are added and subtracted, but are just
"re-numbered". Are there balls present at time t = 0? Yes, lots of
them: 1--8 are there, as are 10--18, ... .
R.G. Vickson
Tony Orlow
Yes, and subsequently changing all the numbers on the balls makes them
disappear from the bucket. Crazy!
Tony
Brian Chandler
> On Jan 18, 9:32 am, LudovicoVan <ju...@diegidio.name> wrote:
> > On Jan 18, 3:41 am, Math1723 <anonym1...@aol.com> wrote:
> >
> > > Many of you know this example already as "The Bucket & Balls Problem".
> > > The clock and the process all stop at time t=0 seconds.
(Not quite sure what this means... time stops at noon??)
> > > Note that by rule, no ball is added unless it is labelled with a
> > > finite natural number.
> >
> > Then, as already noticed, there is no time t=0.
(Huh? How can not adding balls labelled with (what? anything that
isn't a natural number?) cause time to stop?)
> > > The limit tends to infinity. But remember,
> > > not all functions evaluate at their limit.
What does this mean? Correctly, that f(lim x: x->n) may not equal lim
x->n (f(x)) ?
Or something more confused? Anyway...
> > Is the limit defined?
> >
> > > As it turns out: there are 0 balls in the bucket. We can know this
> > > since for every Ball #n, we know that it was added at time t=1/2^[n/
> > > 10] and removed at time t= 1/2^n. We also know this must be true for
> > > every single finite n (the only n's that were allowed in the bucket).
> > > Put another way, ANY ball that is added at time t=1/2^x MUST BE
> > > removed no later than time t=1/2^10x.
> >
> > Sure but, for any ball removed, 10 more get added...
>
> For more uniformity, I'll consider that in the bucket at step n=0
> there is one ball: ball 0.
>
> We have that at each step n>0, 1 ball gets removed and 10 get added.
>
> If c_n is the number of balls at step n in the bucket, we can write:
>
> c_0 := 1
> c_n := c_{n-1} - 1 + 10
>
> That sequence can be rewriten as a direct function of n:
>
> c_n = c'_n := 9*n + 1
>
> Taking the limit for n->oo:
>
> lim_{n->oo} ( 9*n + 1 ) = oo
Yes. Absolutely.
The problem is slightly different, though. If you are thinking of
counting steps, and a function from the steps, then there is no end to
the steps. As the step number increases without bound, the number of
balls in the vase at that step also increases without bound. There is
no place called "infinity" which can be reached to view magical things
(like logical contradictions) happening.
But the problem is phrased in terms of this unending sequence of steps
being carried out at particular times, -1/1, -1/2, -1/3, -1/4,
-1/5, ... -1/n, -1/(n+1), ... and so on without and end to the
sequence, but each one of these times being less than zero. We can
think of the process as representing a set of "pulse functions", where
pulse function pf[a,b] (properly a,b might be subscripts) is defined
as follows:
pf[a,b] (x) = 0 if x <= a
pf[a,b] (x) = 1 if a < x < b
pf[a,b] (x) = 0 if x >= b
The presence of ball n in the urn is represented by pf[-1/n, -1/10n],
a function which is zero everywhere outside its particular range, and
a fortiori zero for any n when x>=0. The number of balls in the urn at
any time is represented by the sum of all of these pulse functions.
Call this sigma[pf].
Now you have to be careful what you mean by "limit". There is a very
clear (and not very complicated, totally unmystical) definition in
maths of what is meant (for example) by the limit of sigma[pf] as
(negative) x tends to zero. And it is a mechanical business to work
out that the limit is 'oo'. (This is really not a very good phrasing,
though it is conventional; it is better said that as x tends to 0 from
below, the function sigma[pf] "increases without bound".)
But none of this implies that the *value* of sigma[pf] at zero is
other than zero, as we already noticed.
Now, there are other usages of "limit", under which "limit" means
practically anything, typically starting with "when you get to
infinity". None of these usages are well defined.
Brian Chandler
LudovicoVan
wrote:
> LudovicoVan wrote:
> > On Jan 18, 9:32 am, LudovicoVan <ju...@diegidio.name> wrote:
> > > On Jan 18, 3:41 am, Math1723 <anonym1...@aol.com> wrote:
>
> > > > Many of you know this example already as "The Bucket & Balls Problem".
> > > > The clock and the process all stop at time t=0 seconds.
>
> (Not quite sure what this means... time stops at noon??)
>
> > > > Note that by rule, no ball is added unless it is labelled with a
> > > > finite natural number.
>
> > > Then, as already noticed, there is no time t=0.
>
> (Huh? How can not adding balls labelled with (what? anything that
> isn't a natural number?) cause time to stop?)
I just meant that there is no time t=0 as far as the problem is
concerned -- unless one properly extends to the infinite case, of
course.
> [...]
Well, I just see unneeded extra complication in talking about time,
and the very root of this seeming paradox in it: in fact, I see no
difference in talking about steps n vs. (derived) instants of time
t:=1/n. You cannot get to n=oo (unless by extending), just as you
cannot get to t=1/oo=0 (again, unless...).
> We can
> think of the process as representing a set of "pulse functions", where
> pulse function pf[a,b] (properly a,b might be subscripts) is defined
> as follows:
>
> pf[a,b] (x) = 0 if x <= a
> pf[a,b] (x) = 1 if a < x < b
> pf[a,b] (x) = 0 if x >= b
>
> The presence of ball n in the urn is represented by pf[-1/n, -1/10n],
> a function which is zero everywhere outside its particular range, and
> a fortiori zero for any n when x>=0.
That is the point! IMHO, the above formulation does not correctly
model the problem... There is no "outside" of the range (maybe,
domain?) of the function that should bother us.
> The number of balls in the urn at
> any time is represented by the sum of all of these pulse functions.
> Call this sigma[pf].
>
> Now you have to be careful what you mean by "limit". There is a very
> clear (and not very complicated, totally unmystical) definition in
> maths of what is meant (for example) by the limit of sigma[pf] as
> (negative) x tends to zero. And it is a mechanical business to work
> out that the limit is 'oo'. (This is really not a very good phrasing,
> though it is conventional; it is better said that as x tends to 0 from
> below, the function sigma[pf] "increases without bound".)
>
> But none of this implies that the *value* of sigma[pf] at zero is
> other than zero, as we already noticed.
Yep: that is defined to be so in the *paradoxical* formulation! But,
as such, I'm thinking it just doesn't correctly model the problem.
Actually, this seems to me one of the quite common cases in which a
seeming paradox rather betrays a fallacy at the level of the
distinction between the problem space and the problem solver space...
That said, sorry for my improper terminology (I understand that
sometimes hides technical mistakes, though not always), and thanks for
your feedback.
-LV
Virgil
<1301eb16-2db0-412d...@i17g2000vbq.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
As the vase is clearly not empty at any time before noon but is clearly
empty at noon, it is also clear that it "becomes" empty AT noon.
> The problem here is an artificial
> Zeno Paradox Engine which purports to have completed omega and nothing
> else. The fallacy is in assuming any such completion of N can occur
> without going beyond N. Since there is no distinct end to N, such
> completion cannot occur, and the gedanken is itself invalid.
Only in a physical world is it impossible. In a gedanken world it is
quite possible and really rather simple, at least for those who think
straight.
Virgil
<0dcfb03a-77cf-4469...@a28g2000prb.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
If the LED is capable of continuous operation, and and it is designed so
that switching it on or off does not add any stress to it, then nosuch
overload of the LED will occur regardless of switching.
Any problems would occur only in the switch.
LudovicoVan
> In article
> <1301eb16-2db0-412d-8bd8-a106cdd47...@i17g2000vbq.googlegroups.com>,
> Tony Orlow <t...@lightlink.com> wrote:
>
> > You may frame the question in terms of what balls remain in the vase
> > at noon, but perhaps you should ask yourself a different question. If,
> > at every moment before noon, there are (a growing number of) balls in
> > the bucket, and if AT noon no balls are added or removed, at what time
> > does the bucket become empty? Logically, there is no answer to this
> > question, and therefore no point in your gedanken at which this
> > magical disappearance can occur.
>
> As the vase is clearly not empty at any time before noon
> but is clearly empty at noon,
> it is also clear that it "becomes" empty AT noon.
Nice "syllogism". Though, how do you get the second premise?
-LV
Tim Golden BandTech.com
Geeze, I thought you were stronger than this. No, there is such a
thing as a poor construction, and then atop that a poor conclusion,
and this is what I think.
Still, I must leave some content.
The 'clock' is an artifice within this problem, and cannot actually
achieve zero. There is another mapping of this clock which is in the
natural form of the problem which exposes the mapped clock as stepping
continuously toward zero but never hitting zero. This is an old
problem, and we can use the geometry of stepping half way toward zero
from any position on the real line and admit that the result
approaches zero only.
Rather than argue over your conclusion on t=0 I'll just deny that
portion based on this first portion. For instance when t=0 what is n?
This is a motivation to developing aleph type theory right? If we
accept this theory then we conclude that
9 aleph balls
is the correct answer, but this will require the admission of the
aleph as the natural valued infinity, which I am not really that into.
I prefer the continuous and the discrete as fundamental and distinct,
and infinity as a limit that can only be approached. Call me
semiclassical, but at least I have flexed here for this silly problem.
Still, I respect your input here, and wait to see what else you have
to say. Certainly we are getting an education of some sort from you.
- Tim
William Hughes
There is an ordered entity without last element
There is a largest integer.
Choose.
- William Hughes
Virgil
<2fa57d72-68e7-4c5e...@g26g2000vbi.googlegroups.com>,
LudovicoVan <ju...@diegidio.name> wrote:
If it were not empty at noon, one should be able to give the number of
some ball which has been entered but not yet removed.
But for each ball entered before noon, there is also a time before noon
specified for its removal.
So which ball(s) would not yet have been removed at noon?
William Hughes
There is no time *at* which the bucket becomes empty.
There is a time *by* which the bucket becomes empty.
It is possible to complete a process *by* a certain
time but have not time *at* which the process is completed.
This is equivalent to having an ordered entity without last
element.
- William Hughes
William Hughes
Yes, this is Crazy in the sense of wrong. Subsequently changing the
numbers
on the balls does not make them disappear.
- William Hughes
-
LudovicoVan
Your word games add nothing. Your conclusion is equivalent to mine.
Thanks for the feedback.
-LV
LudovicoVan
> > disappear from the bucket. Crazy!
>
> numbers on the balls does not make them disappear.
So you believe that the numbering scheme affects the conclusion?
-LV
LudovicoVan
Logically, there is no noon: the process, as described, is simply
"endless", and the above is properly a *sophism*. There just is no
room for an "observer" and his putative clock, unless one provides for
a much more comprehensive setting.
OTOH, in our basic setting, and thinking again how to extend to the
infinite case:
We had defined the number of balls in the bucket at step n=oo (or,
equivalently, time t=0):
c_{oo} := lim_{n->oo} ( 9*n + 1 ) = oo
So we conclude there is an infinity of balls at (extended) time 0.
Which balls are those?
Again, by stipulation, we could assume those to be balls that are all
labeled with 'oo' -- as just an instance of the simpliest transfinite
ordinal arithmetic I can think of...
-LV
William Hughes
Nope. Your conclusion is that the problem is not well
described. My conclusion is that the problem is well
described. Noting that there is no value largest value
of the form t=0-1/N does not preclude defining a function
on such values, or extending this function to t=0.
- William Hughes
fishfry
<db872cd8-846f-4b9a...@m35g2000vbn.googlegroups.com>,
William Hughes <wpih...@gmail.com> wrote:
The more I think about this problem, the more confused I get. If you
number the balls as in the problem, then its true that every ball is
eventually removed. But if you don't number the balls, and simply remove
a random one, then you can't prove that every ball is eventually
removed. This is troubling.
William Hughes
Yes, the numbering scheme tells you which subset of the natural
numbers to remove. Removing different subsets leads to different
conclusions (and water is wet). However, *subsequently* changing the
numbers
on the balls does not change the cardinality of the set of balls in
the vase.
-William Hughes
fishfry
Virgil <vir...@ligriv.com> wrote:
I don't think it's helpful to try to express this problem in physical
terms. Eventually you'd have to operate the switch so fast you would
violate the laws of physics.
It's better to just model this as a function f:N->{0,1} defined by
f(n) = 0 if n is even, and f(n) = 1 if n is odd. You can then tell
yourself a story that this represents a "light switch" that is operated
at time t = 1 - 1/2^n, but that doesn't add anything to the problem
other than confusion.
It's clear that with the problem as I restated it, asking the value of
the function at infinity is undefined.
The state of the balls in the bucket at t = 0 is of the same nature.
It's undefined.
>
>
> Now, how can the led be both on, and dead, at the same
> > moment? Herein lies the contradiction.
> >
> > Peace,
> >
At what time is it both on and off? At time 1 - 1/2^n, it is either on
or off depending on whether n is odd or even.
LudovicoVan
> described.
Please don't put words in my mouth.
-LV
William Hughes
> In article
> <db872cd8-846f-4b9a-9e01-264fb8e79...@m35g2000vbn.googlegroups.com>,
Not really, The set of balls in the vase at time t=0 is some subset,
A,
of the natural numbers, created by taking the natural numbers and
removing a
subset of cardinality aleph_0. However, to this point there is not
enough
information to determine A. It can have any cardinality from 0 to
aleph_0. The numbering scheme lets you determine A. Different
numbering
schemes lead to different subsets A. If you do not have a numbering
scheme
you cannot determine A.
- William Hughes
LudovicoVan
> numbers to remove. Removing different subsets leads to different
> conclusions (and water is wet).
And you are an idiot.
> However, *subsequently* changing the
> numbers
> on the balls does not change the cardinality of the set of balls in
> the vase.
Sure, that's magic.
-LV
William Hughes
At this point a non-crank would state what his conclusion is.
- William Hughes
H
Math1723
>
> I'd like to remind you of the fact that, according to _standard_ math-
> ematics, the outcome of this problem is: NONE balls left. Denying this
> places you in the champ of the "cranks".
Finally, we agree on something !! :-)
LudovicoVan
Genius no. 2 here!!
-LV
LudovicoVan
> On Jan 18, 7:47 pm, LudovicoVan <ju...@diegidio.name> wrote:
> > On Jan 18, 11:34 pm, William Hughes <wpihug...@gmail.com> wrote:
>
> > > Your conclusion is that the problem is not well
> > > described.
>
> > Please don't put words in my mouth.
>
> At this point a non-crank would state what his conclusion is.
At this point you can safely get lost, moron.
-LV
William Hughes
Nope. The fact that there are aleph_0 changes of state of the
balls in the vase before noon, does not mean there is no noon.
Or are you playing Zeno "it is not possible to perform an infinite
number of actions in a finite time".
- William Hughes
fishfry
<ff81b3c1-aec4-42e2...@e4g2000vbg.googlegroups.com>,
William Hughes <wpih...@gmail.com> wrote:
At any time before noon there have only been finitely many changes of
state. That's key.
William Hughes
Indeed. Let s be any time *at* which there is a change of state.
Then at times up to and including s, there have been finitely
many changes of state. There is no time *at* which there is a change
of state such that there are an infinite number of changes
of state before that time.
To complete the task you need an infinite number of changes
of state. So there is no time *at* which the process is completed.
There are however times *by* which the process is completed (the
smallest
such time is noon). There is no last change of state.
- William Hughes
Math1723
wrote:
> Math1723 wrote:
>
> Even if you say, twice, that no ball is allowed in the bucket without
> a natural number on it, you will not prevent there being balls with
> infinite natural numbers on, or pink balls, or even if you say the
> balls are white, and painted red on being put in, there will still be
> a claim of green balls.
Okay, I admit, when I posted this, I expected to see some outlandishly
cranky justifications for the wrong answer, but THIS has got to take
the cake! I must admit, it is pretty funny. Mathematics definitely
has some mystical attributes in your mind. If Underwood Dudley ever
updates his "Mathematical Cranks" book, I hope he finds this post.
Classic!
Math1723
>
> Then, as already noticed, there is no time t=0.
Of course there is time t=0. That's when everything stops.
> > The limit tends to infinity. But remember,
> > not all functions evaluate at their limit.
>
> Is the limit defined?
Sure. You can calculate the exact figure yourself, but essentially
the number of balls grows nearly tenfold with each t=1/2^n. So the
limit tends toward infinity.
> Sure but, for any ball removed, 10 more get added...
Agreed.
> > Now the logic is irrefutable.
>
> I don't see a clear resolution.
There's nothing to resolve. There is a proof that the bucket is
empty. There is no proof to the contrary.
Math1723
>
> For more uniformity, I'll consider that in the bucket at step n=0
> there is one ball: ball 0.
>
> We have that at each step n>0, 1 ball gets removed and 10 get added.
>
> If c_n is the number of balls at step n in the bucket, we can write:
>
> c_0 := 1
> c_n := c_{n-1} - 1 + 10
>
> That sequence can be rewriten as a direct function of n:
>
> c_n = c'_n := 9*n + 1
>
> Taking the limit for n->oo:
>
> lim_{n->oo} ( 9*n + 1 ) = oo
>
> in the bucket tends to infinity.
Excellent. Well done!
> > > Now the logic is irrefutable.
>
> Is it?
Certainly.
Math1723
wrote:
>
> What does this mean? Correctly, that f(lim x: x->n) may not equal lim
> x->n (f(x)) ?
Essentially yes, although badly phrased. More rigorously:
Let Sn be the set of balls in the bucket at time t=1/2^n, and Card(x)
is the cardinality of set x.
Then what I am saying is that you are confusing
lim Card(Sn)
with
Card(lim Sn)
The first is infinite, the second is zero. There in lies the
confusion.
Math1723
> > not all functions evaluate at their limit.
>
> > since for every Ball #n, we know that it was added at time t=1/2^[n/
> > 10] and removed at time t= 1/2^n. We also know this must be true for
> > every single finite n (the only n's that were allowed in the bucket).
> > Put another way, ANY ball that is added at time t=1/2^x MUST BE
> > removed no later than time t=1/2^10x.
>
> > Now the logic is irrefutable. It may be counter-intuitive, defies
> > imagination, and seem downright wrong. But the logic is absolutely
> > sound, and if you accept logic, you must accept this. All the balls
> > are gone.
>
> > If you do not, you must ask yourself: If any balls exist in the
> > bucket, what numbers do they contain? They must be finite numbers (by
> > rule). Yet how could they have not been removed? Since every non-
> > empty set of finite natural numbers has a least element, then if any
> > balls exists in the bucket, there is one with the smallest number.
> > What is that number?
>
> > The purpose of this is to show how easily it is for us to use our
> > "intuition" and our "common sense" reasoning to think incorrectly
> > about infinities. Our experience in the finite world gives us
> > absolutely no handle on how things operate at transfinite levels. We
> > must rely on the logical rigors of Mathematics to answer our
> > questions, even if those rigors give us answers we don't like.
>
> > Thanks for reading.
>
> at noon, but perhaps you should ask yourself a different question. If,
> at every moment before noon, there are (a growing number of) balls in
This is not a different question, really, but speaks to the heart of
the problem.
> ... at what time does the bucket become empty?
There is hidden ambiguity in your phrase "becomes empty"? Certainly
the buck is empty at t=0, and at no time prior to t=0 is the bucket
empty. So this is the earliest time at which the bucket is empty. So
by "become empty" do you simply mean this earliest time at which it is
empty? If so, then the answer is clearly t=0.
However, at time t=0, there are no balls added or removed, since all
the balls have been added and removed by this point. So if you mean
by "becomes empty" a time at which the last ball is removed, then
there is no such time, since(obviously) there is no last ball.
> Logically, there is no answer to this question ...
Actually there is, I just answered it. What you need to do is define
what you meant by "becomes empty". Once you do, the answer is clear.
> The fallacy is in assuming any such completion of N can occur
> without going beyond N. Since there is no distinct end to N,
> such completion cannot occur, and the gedanken is itself
> invalid.
This is not a fallacy. I'm sorry to say, the fallacy is actually at
your end: your assumption that completion cannot occur without "a
distinct end to N" is fallacious. This is merely an assumption (an
incorrect one at that) on your part. You have offered no proof of
this (which you cannot because it is untrue). You are relying upon an
intuition that does not hold up to logic.
Have you considered the possibility that your intuition could be wrong
and Mathematics is right?
Math1723
>
> There is no time *at* which the bucket becomes empty.
> There is a time *by* which the bucket becomes empty.
>
> It is possible to complete a process *by* a certain
> time but have not time *at* which the process is completed.
> This is equivalent to having an ordered entity without last
> element.
Well stated.
Math1723
>
> So you believe that the numbering scheme affects the conclusion?
If by this you mean changing which ball is removed, then Yes! And
provably so! Just change the problem so that at time t=1/2^n Ball #2n
is removed instead of Ball #n. Then indeed you DO have an infinite
number of balls left. Specifically: Balls #1, 3, 5, 7, ... etc!
Math1723
>
> The more I think about this problem, the more confused I get. If you
> number the balls as in the problem, then its true that every ball is
> eventually removed. But if you don't number the balls, and simply remove
> a random one, then you can't prove that every ball is eventually
> removed. This is troubling.
Not troubling, but just a different problem. If at time t=1/2^n, you
remove Ball #2n instead of Ball #n, you have an infinite number of
balls left, specifically Balls #1, 3, 5, 7, etc.
You can modify the problem to leave zero, a predefined finite number,
or an infinite number of balls left.
Choosing one at random can leave any number, but I believe the
possibilities leading to a finite or zero answer have measure zero, so
the probability is 1 that you have an infinite number.
Math1723
wrote:
>
> Geeze, I thought you were stronger than this.
Sorry to disappoint.
> Rather than argue over your conclusion on t=0 I'll just deny that
> portion based on this first portion. For instance when t=0 what is n?
There is no n at t=0 (just as there is no n at t=1/3, t=+17 or
t=-200).
> This is a motivation to developing aleph type theory right? If we
> accept this theory then we conclude that
> 9 aleph balls
> is the correct answer, but this will require the admission of the
> aleph as the natural valued infinity, which I am not really that into.
No, none of those. My intention is to show that people tend to
gravitate toward emotional embraces to intuition, despite the cold
hard logic.
I had an experience in college with another counter-intuitive result.
I gave this puzzle to a friend of mine: Suppose you fold a piece of
paper in half. Then in half again. And again. No assuming a
sufficiently large enough sheet of paper, suppose you fold it in half
25 times. How thick would that wad of paper be? How about 50 times?
Well, that thickness can be (under)estimated by simply taking the
thickness of a sheet of paper and multiply by 2^25 and 2^50 (since
folding always doubles the thickness). To the surprise of many
people, after 25 times, the wad would be a mile thick. At 50 times,
the thickness exceeds the distance between the Earth and the sun.
My friend's response: "Well, this is just proof of how Mathematics
breaks down." He refused to accept it. I even printed an Excel
spreadsheet showing him from fold #1's thickness (that of a piece of
paper) and double each step. He refused to believe it. "Math is just
wrong at those numbers."
This is similar: people refuse to let go of their own intuitions,
irrespective of the irrefutable logic. The Bucket & Balls problem is
very plain: Only balls with finite natural numbers on them can get
in. I can specify for each n when the ball goes in and when it goes
out. None others can get in except by this rule. EVERY ONE LEAVES AT
A SPECIFIC TIME. The logical is completely flawless. Yet, people
will throw away logic because of their feelings.
I am hoping that illustrating this will help some people come to grips
with placing truth (and not their emotions) as the primary role in
Mathematics. But I know, most people will be like my friend: "Math
fails."
Transfer Principle
wrote:
> Rather than argue over your conclusion on t=0 I'll just deny that
> portion based on this first portion. For instance when t=0 what is n?
Out of curiosity, what is "aleph type theory"? By "aleph
type theory," do you mean ZFC? If so, then let's see what
ZFC does prove.
Let's define a function V : (0,1] -> P(N) as follows:
V(t) = {aeN | floor(1-lg(t)) <= a < 10floor(1-lg(t))}
where lg is the base 2 logarithm. Then V(t) is the set
of all labels of balls in the vase at time t. (I chose
the letter V for _V_ase.)
Now let's extend this function V to the domain [0,1]
rather than (0,1]. To do so, we need to define the
concept of a limit of a sequence sets. This thread
hinges on the notion of such a limit -- consider how
many times such "limits" have already been mentioned in
this thread. The posters in most of these threads are
divided into two "camps" (as HdB puts it) on this issue:
1) If x_n is a sequence of sets, then the limit x is a
set such that mex if mex_n for all but finitely many n
and ~mex if ~mex_n for all but finitely many n, if such
a set exists. (This is often called the "cofinite
topology", and we say that x_n converges to x in the
cofinite topology if such an x as described exists.)
2) If x_n is a sequence of sets, then the limit x has
the property that if card(x_n) >= k for all but finitely
many n, then card(x) >= k, and if card(x) <= k for all
but finitely many m, then card(x) <= k.
In either case, x_n is the set V(2^-n) of all labels in
the vase at time 2^-n, and the limit x is the set V(0)
of all labels in the vase at time 0.
Now in neither case is a limit guaranteed to exist. But
in case 1), if a limit exists, then it is unique. This
is because in ZFC, "e" is the primitive, and case 1)
tells us whether mex or ~mex for each natural m. But
case 2) only determines what card(x) is, but unless k
is 0, there are many sets with of cardinality k. So
case 2) almost never provides a unique limit. Because
of this, most ZFC users use method 1) to find the limit
of a sequence of sets, if it exists.
So now we use method 1), the cofinite topology, to find
the limit of the x_n. We notice that for all n >= m,
the ball labeled m is not in the vase -- and this is
true for _all_ m. Thus, most ZFC users conclude that
for all m, ~mex -- i.e., ~meV(0). Therefore, they
conclude that the vase is _empty_ at time 0.
This conclusion is unpalatable to those who favor
method 2) to find V(0). They find the result of method
1) to be counterintuitive. But those who use method 1)
strongly criticize those who use method 2). After all,
2) cannot determine whether meV(0) for any natural
number m, method 1) users love to ask method 2) users
to provide a specific element of V(0), and insult them
when they are unable to provide it.
> If we
> accept this theory then we conclude that
> 9 aleph balls
> is the correct answer, but this will require the admission of the
> aleph as the natural valued infinity, which I am not really that into.
But what is aleph? Standard ZFC states that there are
many alephs. Indeed, for any ordinal alpha, aleph_alpha
is a cardinal. But the only ordinal I see that you
mention is 9 -- unless you mean aleph_9.
Since card(V(1)) is 9, many method 2) users do feel
that "9" should appear in the correct answer. But note
that in ZFC, 9aleph_0 = aleph_0. Thus, this "aleph type
theory" is starting to sound less and less like ZFC.
> I prefer the continuous and the discrete as fundamental and distinct,
> and infinity as a limit that can only be approached. Call me
> semiclassical, but at least I have flexed here for this silly problem.
You say that you prefer thinking about "infinity as a
limit" to infinite cardinals. By this, does this mean
that you accept the "oo" infinity symbol? OK, I see
nothing wrong with accepting "oo" while rejecting
infinite cardinals and ordinals -- but other posters
aren't always as openminded.
Han de Bruijn
> In article
> <1301eb16-2db0-412d-8bd8-a106cdd47...@i17g2000vbq.googlegroups.com>,
> > does the bucket become empty? Logically, there is no answer to this
> > question, and therefore no point in your gedanken at which this
> > magical disappearance can occur.
>
> As the vase is clearly not empty at any time before noon but is clearly
> empty at noon, it is also clear that it "becomes" empty AT noon.
>
> > Zeno Paradox Engine which purports to have completed omega and nothing
> > else. The fallacy is in assuming any such completion of N can occur
> > without going beyond N. Since there is no distinct end to N, such
> > completion cannot occur, and the gedanken is itself invalid.
>
> quite possible and really rather simple, at least for those who think
> straight.
Straight?
Han de Bruijn
Han de Bruijn
>
> LudovicoVan <ju...@diegidio.name> wrote:
> > > In article
> > > <1301eb16-2db0-412d-8bd8-a106cdd47...@i17g2000vbq.googlegroups.com>,
> > > Tony Orlow <t...@lightlink.com> wrote:
>
> > > > You may frame the question in terms of what balls remain in the vase
> > > > at noon, but perhaps you should ask yourself a different question. If,
> > > > at every moment before noon, there are (a growing number of) balls in
> > > > the bucket, and if AT noon no balls are added or removed, at what time
> > > > does the bucket become empty? Logically, there is no answer to this
> > > > question, and therefore no point in your gedanken at which this
> > > > magical disappearance can occur.
>
> > > As the vase is clearly not empty at any time before noon
> > > but is clearly empty at noon,
> > > it is also clear that it "becomes" empty AT noon.
>
>
> If it were not empty at noon, one should be able to give the number of
> some ball which has been entered but not yet removed.
>
> But for each ball entered before noon, there is also a time before noon
> specified for its removal.
>
> So which ball(s) would not yet have been removed at noon?
Sure, keep thinking that pure reasoning helps against reality. Mental
hospitals are full of those people.
Han de Bruijn
Han de Bruijn
> On Jan 18, 7:47 pm, LudovicoVan <ju...@diegidio.name> wrote:
>
> > On Jan 18, 11:34 pm, William Hughes <wpihug...@gmail.com> wrote:
>
> > > Your conclusion is that the problem is not well
> > > described.
>
> > Please don't put words in my mouth.
>
> At this point a non-crank would state what his conclusion is.
See, LV ?? Welcome to the club !!
> > > My conclusion is that the problem is well
> > > described. Noting that there is no value largest value
> > > of the form t=0-1/N does not preclude defining a function
> > > on such values, or extending this function to t=0.
Han de Bruijn
Han de Bruijn
Yeah, but .. MIND THE QUOTES !!
Han de Bruijn
Han de Bruijn
Simply admit that your conclusions do NOT meet the requirements of the
other club, LV. So you're no longer a member of the common mathematics
community. Pity for you .. (?)
Han de Bruijn
Han de Bruijn
Sure. That's why _such_ mathematical reasoning is deemed unreliable.
Han de Bruijn
Han de Bruijn
Confusion? Iterated limits are always commutative XOR nonsense. Thus:
lim Card(Sn) = Card(lim Sn)
XOR nonsense, I said.
Han de Bruijn
Han de Bruijn
> On Jan 18, 5:59 pm, "Tim Golden BandTech.com" <tttppp...@yahoo.com>
> wrote:
>
> > Geeze, I thought you were stronger than this.
>
> Sorry to disappoint.
>
> > Rather than argue over your conclusion on t=0 I'll just deny that
> > portion based on this first portion. For instance when t=0 what is n?
>
> There is no n at t=0 (just as there is no n at t=1/3, t=+17 or
> t=-200).
>
> > This is a motivation to developing aleph type theory right? If we
> > accept this theory then we conclude that
> > 9 aleph balls
> > is the correct answer, but this will require the admission of the
> > aleph as the natural valued infinity, which I am not really that into.
>
> No, none of those. My intention is to show that people tend to
> gravitate toward emotional embraces to intuition, despite the cold
> hard logic.
Your "emotional embraces to intuition" is called "experience" in some
circles.
> I had an experience in college with another counter-intuitive result.
> I gave this puzzle to a friend of mine: Suppose you fold a piece of
> paper in half. Then in half again. And again. No assuming a
> sufficiently large enough sheet of paper, suppose you fold it in half
> 25 times. How thick would that wad of paper be? How about 50 times?
>
> Well, that thickness can be (under)estimated by simply taking the
> thickness of a sheet of paper and multiply by 2^25 and 2^50 (since
> folding always doubles the thickness). To the surprise of many
> people, after 25 times, the wad would be a mile thick. At 50 times,
> the thickness exceeds the distance between the Earth and the sun.
>
> My friend's response: "Well, this is just proof of how Mathematics
> breaks down." He refused to accept it. I even printed an Excel
> spreadsheet showing him from fold #1's thickness (that of a piece of
> paper) and double each step. He refused to believe it. "Math is just
> wrong at those numbers."
Exactly. Take a _real_ piece of paper and SHOW us (your friend and I).
> This is similar: people refuse to let go of their own intuitions,
> irrespective of the irrefutable logic. The Bucket & Balls problem is
> very plain: Only balls with finite natural numbers on them can get
> in. I can specify for each n when the ball goes in and when it goes
> out. None others can get in except by this rule. EVERY ONE LEAVES AT
> A SPECIFIC TIME. The logical is completely flawless. Yet, people
> will throw away logic because of their feelings.
Nope. According to physics, the bucket will turn into a Black Hole.
Thus time will slow down so much that it is impossible to reach noon.
(Not that I truly believe in black holes, but ..)
> I am hoping that illustrating this will help some people come to grips
> with placing truth (and not their emotions) as the primary role in
> Mathematics. But I know, most people will be like my friend: "Math
> fails."
It's analogous to wrong extrapolation. You're deeming math irrelevant,
while it is highly relevant, once a decent problem is properly posed.
Han de Bruijn
LudovicoVan
> On Jan 19, 1:21 am, LudovicoVan <ju...@diegidio.name> wrote:
> > On Jan 18, 11:55 pm, William Hughes <wpihug...@gmail.com> wrote:
> > > On Jan 18, 7:47 pm, LudovicoVan <ju...@diegidio.name> wrote:
> > > > On Jan 18, 11:34 pm, William Hughes <wpihug...@gmail.com> wrote:
>
> > > > > Your conclusion is that the problem is not well
> > > > > described.
>
> > > > Please don't put words in my mouth.
>
> > > At this point a non-crank would state what his conclusion is.
>
> > At this point you can safely get lost, moron.
>
> other club, LV. So you're no longer a member of the common mathematics
> community. Pity for you .. (?)
Mate, don't get fooled: that just is not the mathematical community...
In Naples, bad people have a saying: Pity for me, pity for everybody.
I can't save the world, but for the sake of some fun I'll kick this
bunch of paid spammers and liars in the ass until I have feet to kick.
-LV
LudovicoVan
You are a dishonest cheat whose theorems are based on snipping:
<< Logically, there is no noon: the process, as described, is simply
"endless", and the above is properly a *sophism*. >>
Have you got any math reasoning, informal or otherwise, to support
your senseless statements?
Let us know in case,
-LV
Jesse F. Hughes
> My friend's response: "Well, this is just proof of how Mathematics
> breaks down." He refused to accept it. I even printed an Excel
> spreadsheet showing him from fold #1's thickness (that of a piece of
> paper) and double each step. He refused to believe it. "Math is just
> wrong at those numbers."
That's a pretty exceptional response.
Most of the doubt around here occurs with infinite sets, not finite
values.
--
Jesse F. Hughes
Mama: "I had a very good steak when I was in Bonn."
Quincy (Age 4): "A stick? I wish you brought it home. Was it very
big and did it look like a gun?"
Jesse F. Hughes
>> My friend's response: "Well, this is just proof of how Mathematics
>> breaks down." He refused to accept it. I even printed an Excel
>> spreadsheet showing him from fold #1's thickness (that of a piece of
>> paper) and double each step. He refused to believe it. "Math is just
>> wrong at those numbers."
>
> Exactly. Take a _real_ piece of paper and SHOW us (your friend and I).
Er, another remarkable response. Do you honestly doubt that the width
of the paper at least doubles at each fold?
You do also realize that he *can't* fold a piece of paper twenty-five
times, right? You honestly expect him to have the tools to fold paper
until it's over a mile thick?
But you can try yourself. Fold a piece of paper seven times. Measure
it. Fold it one more time. Measure that, too. It should be double the
thickness it was. Extrapolate.
--
Jesse F. Hughes
"Love songs suck and losing you ain't worth a damn."
-- The poetry of Bad Livers
Jesse F. Hughes
> This conclusion is unpalatable to those who favor
> method 2) to find V(0). They find the result of method
> 1) to be counterintuitive. But those who use method 1)
> strongly criticize those who use method 2). After all,
> 2) cannot determine whether meV(0) for any natural
> number m, method 1) users love to ask method 2) users
> to provide a specific element of V(0), and insult them
> when they are unable to provide it.
Yes, that is indeed what I, as a user of method (1) love to do. I look
forward to the insult bit in particular.
But I wonder why we users of method (1) don't just point out that the
reasoning employed in method (2) is not valid in ZFC. That seems
easier. Here it is again:
> 2) If x_n is a sequence of sets, then the limit x has
> the property that if card(x_n) >= k for all but finitely
> many n, then card(x) >= k, and if card(x) <= k for all
> but finitely many m, then card(x) <= k.
These here users of method (2) are just choosing a different method, I
guess. We all march to the beat of different drums, right? That's what
I suppose you want to imply, but there really *is* a difference here.
The claim you describe as (2) above is simply *not* a theorem of ZFC.
It is invalid.
Doesn't that make a difference?
--
Jesse F. Hughes
"Really, I'm not out to destroy Microsoft. That will just be a
completely unintentional side effect." -- Linus Torvalds
Jesse F. Hughes
You badly misunderstood Brian. He is parodying the reactions of others,
not his own beliefs in this case.
--
Quincy (age 5): Baba, play some [computer games].
Mama: Quincy, if you want [Baba] to live, don't make those
suggestions.
Quincy: Make those suggestions. Got it.
LudovicoVan
That is senseless, amounting to taking into account the balls you
remove and forgetting that you are actually adding more than you
remove.
The very fact that, in your setting (yeh, I mean the common one),
changing the numbering scheme leads to different conclusions should be
enough as a signal that your reasoning is going adrift.
(Can you fix your argument so that the balls added are taken into
account?)
-LV
Jesse F. Hughes
Right! We know the answer beforehand, so if math says something else,
it's wrong! What could be simpler?
In fact, since our intuitions are infallible, I'm not sure why we study
math in the first place.
--
Jesse F. Hughes
"All Chinese are Confucianists when successful, and Taoists when they
are failures."
-- Lin Yutang, /My Country and My People/
LudovicoVan
> Han de Bruijn <umum...@gmail.com> writes:
> > On Jan 19, 4:00 am, Math1723 <anonym1...@aol.com> wrote:
> >> On Jan 18, 4:32 am, LudovicoVan <ju...@diegidio.name> wrote:
>
> >> > Then, as already noticed, there is no time t=0.
>
> >> Of course there is time t=0. That's when everything stops.
>
> >> > > The limit tends to infinity. But remember,
> >> > > not all functions evaluate at their limit.
>
> >> > Is the limit defined?
>
> >> Sure. You can calculate the exact figure yourself, but essentially
> >> the number of balls grows nearly tenfold with each t=1/2^n. So the
> >> limit tends toward infinity.
>
> >> > Sure but, for any ball removed, 10 more get added...
>
> >> Agreed.
>
> >> > > Now the logic is irrefutable.
>
> >> > I don't see a clear resolution.
>
> >> There's nothing to resolve. There is a proof that the bucket is
> >> empty. There is no proof to the contrary.
>
> > Sure. That's why _such_ mathematical reasoning is deemed unreliable.
>
> Right! We know the answer beforehand, so if math says something else,
> it's wrong! What could be simpler?
>
> In fact, since our intuitions are infallible, I'm not sure why we study
> math in the first place.
I have given argument and proof to the contrary. The problem is not
mathematics, it's these guys that just don't play by the rules.
-LV
Jesse F. Hughes
>> If it were not empty at noon, one should be able to give the number of
>> some ball which has been entered but not yet removed.
>>
>> But for each ball entered before noon, there is also a time before noon
>> specified for its removal.
>>
>> So which ball(s) would not yet have been removed at noon?
>
> Sure, keep thinking that pure reasoning helps against reality. Mental
> hospitals are full of those people.
If you think this experiment involving hypertasks has much to do with
reality, perhaps you should consider the hospital as well.
--
Jesse F. Hughes
"All information is subject to change without notice."
-- California Alternative High School
Han de Bruijn
> Han de Bruijn <umum...@gmail.com> writes:
>
> >> My friend's response: "Well, this is just proof of how Mathematics
> >> breaks down." He refused to accept it. I even printed an Excel
> >> spreadsheet showing him from fold #1's thickness (that of a piece of
> >> paper) and double each step. He refused to believe it. "Math is just
> >> wrong at those numbers."
>
> > Exactly. Take a _real_ piece of paper and SHOW us (your friend and I).
>
> Er, another remarkable response. Do you honestly doubt that the width
> of the paper at least doubles at each fold?
>
> You do also realize that he *can't* fold a piece of paper twenty-five
> times, right? You honestly expect him to have the tools to fold paper
> until it's over a mile thick?
>
> But you can try yourself. Fold a piece of paper seven times. Measure
> it. Fold it one more time. Measure that, too. It should be double the
> thickness it was. Extrapolate.
Here you say it: EXTRAPOLATE. That's exactly the trouble. Everywhere.
Han de Bruijn
LudovicoVan
> Han de Bruijn <umum...@gmail.com> writes:
>
> >> If it were not empty at noon, one should be able to give the number of
> >> some ball which has been entered but not yet removed.
>
> >> But for each ball entered before noon, there is also a time before noon
> >> specified for its removal.
>
> >> So which ball(s) would not yet have been removed at noon?
>
> > Sure, keep thinking that pure reasoning helps against reality. Mental
> > hospitals are full of those people.
>
> If you think this experiment involving hypertasks has much to do with
> reality, perhaps you should consider the hospital as well.
Hypertasks don't justify wrong conclusions: the above is a sophism at
noon.
(And Han de Bruijn is right about derangement.)
-LoVe
Han de Bruijn
> Han de Bruijn <umum...@gmail.com> writes:
>
>
>
>
>
> > On Jan 19, 4:00 am, Math1723 <anonym1...@aol.com> wrote:
> >> On Jan 18, 4:32 am, LudovicoVan <ju...@diegidio.name> wrote:
>
> >> > Then, as already noticed, there is no time t=0.
>
> >> Of course there is time t=0. That's when everything stops.
>
> >> > > The limit tends to infinity. But remember,
> >> > > not all functions evaluate at their limit.
>
> >> > Is the limit defined?
>
> >> Sure. You can calculate the exact figure yourself, but essentially
> >> the number of balls grows nearly tenfold with each t=1/2^n. So the
> >> limit tends toward infinity.
>
> >> > Sure but, for any ball removed, 10 more get added...
>
> >> Agreed.
>
> >> > > Now the logic is irrefutable.
>
> >> > I don't see a clear resolution.
>
> >> There's nothing to resolve. There is a proof that the bucket is
> >> empty. There is no proof to the contrary.
>
> > Sure. That's why _such_ mathematical reasoning is deemed unreliable.
>
> Right! We know the answer beforehand, so if math says something else,
> it's wrong! What could be simpler?
>
> In fact, since our intuitions are infallible, I'm not sure why we study
> math in the first place.
Our "intuitions" are called "experience" in some circles. Mathematics
_can_ be counter-intuitive, but it should _not_ be counter-experience
all the time. Instead, it should help us to acquire better experience
than before.
Han de Bruijn
Han de Bruijn
Who's rules ? Say farewell to the club, buddy: you're now one of us.
Ah, you still can't believe that standard mathematics plays the game
with _such_ rules ? So pity for you ..
Han de Bruijn
Dan Cass
To conclude "that's the trouble Everywhere" is a bit of
an extrapolation, don't you think?
Han de Bruijn
> Han de Bruijn <umum...@gmail.com> writes:
>
> >> If it were not empty at noon, one should be able to give the number of
> >> some ball which has been entered but not yet removed.
>
> >> But for each ball entered before noon, there is also a time before noon
> >> specified for its removal.
>
> >> So which ball(s) would not yet have been removed at noon?
>
> > Sure, keep thinking that pure reasoning helps against reality. Mental
> > hospitals are full of those people.
>
> If you think this experiment involving hypertasks has much to do with
> reality, perhaps you should consider the hospital as well.
Even hypertasks can be carried out with _some_ sense of reality. Apart
from this, it would be a definite improvement if LIMITS, indeed, were
applied according to the proper rules (and LV has shown how to do it).
http://hdebruijn.soo.dto.tudelft.nl/jaar2006/ballen.jpg
Han de Bruijn
Brian Chandler
With some posters, it's possible to believe that they are basically
confused. With others, such as the owner of approximately one brain*
being yourself, it's difficult to escape the conclusion of simple
dishonesty.
* Note that I'm not allowed to swap quantifiers at will. You perhaps
are.
Anyway, "intuition" is the word we use relating to mathematics,
*precisely* because it it not about "experience". No-one has ever
"experienced" a pile of balls labelled with all the natural numbers,
so the "thought experiment" is not about experience, and, rather
obviously, experience will be no guide to what is happening. Of
course, one honest response to questions like this is simply to deny
that it makes sense to ask the question, because the experiment is
physically impossible, even in principle (unlike the paper folding). A
confused response is to assume that since from physical experience all
functions are continuous this means that somehow all functions in
maths are continuous. Then there's your response, which as usual
doesn't actually make any sense at all.
Brian Chandler
>
> Han de Bruijn
LudovicoVan
> Ah, you still can't believe that standard mathematics plays the game
> with _such_ rules ? So pity for you ..
The rules of mathematics in this case. Weak thinking is not for me,
pal, so save your pity and the clubs for yourself.
-LV
Han de Bruijn
You're quite right.
Han de Bruijn
William Hughes
For any natural number n define
state_n(t) = 1 if 1-([n/10] + 1) <= t <= 1-1/n, 0 otherwise
Define balls_in_vase(t) to be the set of all natural numbers such that
state_n(t)=1. (Note that this set is finite for all t and we can
determine
the set in a finite number of steps)
Define number_of_balls(t) = cardinality of balls_in_vase(t).
Note that number_of_balls(t) is defined for all t, in particular
number_of_balls(0) is defined.
We have lim t-> 0 number_of_balls(t) = inf
number_of_balls(0) = 0
Number of balls(0) is defined, nor is it necessary to
do an infinite number of operations to calculate it.
- William Hughes
Han de Bruijn
wrote:
All functions, Brian, _are_ continuous as well as discrete. Both ways
of looking at the universe are in the eye of the beholder. If you took
the effort to really _read_ what I've written, then you would probably
agree with this stand. But I understand that is a hypertask for you ..
> Then there's your response, which as usual
> doesn't actually make any sense at all.
Then there's your response, which _could_ have been honest if you just
had left out the "as usual". But now it's too late.
Han de Bruijn
Brian Chandler
> On Jan 18, 10:22 am, Tony Orlow <t...@lightlink.com> wrote:
<snip>
> > Logically, there is no answer to this question ...
>
> Actually there is, I just answered it. What you need to do is define
> what you meant by "becomes empty". Once you do, the answer is clear.
>
> > The fallacy is in assuming any such completion of N can occur
> > without going beyond N. Since there is no distinct end to N,
> > such completion cannot occur, and the gedanken is itself
> > invalid.
>
> This is not a fallacy. I'm sorry to say, the fallacy is actually at
> your end: your assumption that completion cannot occur without "a
> distinct end to N" is fallacious. This is merely an assumption (an
> incorrect one at that) on your part. You have offered no proof of
> this (which you cannot because it is untrue). You are relying upon an
> intuition that does not hold up to logic.
Hmm, I think you seriously estimate Tony's intuition. It got him all
the way through school, and a finite combinatorics [or similar] course
at university. He can see the answer others have to work out; he has
also been to Infinity (which I bet you haven't) and seen the tunnel of
love, the largest pofnat, resplendent in its mantle of nonexistence,
and a whole lot more.
Where was I?
> Have you considered the possibility that your intuition could be wrong
> and Mathematics is right?
No, not really, I think.
Look, the talk of urns and balls is provocative, even misleading to
our weaker brethren. Why not ask questions about functions? Represent
the (we're not really mentioning it, but) presence of ball n by a
function pf[a,b] (as defined above somewhere: values 0 and 1) where a
is the value on the real line where the ball is put in, and b is the
value where it is taken out. Given a clutch of these pulse functions,
ask them (the cranks, I mean) if they accept adding the functions?
Obviously, unless there are only a finite number of the pfs with value
1, then we cannot work out the sum. But in your example, (if we
rearrange the clock to run from -1 to 0 instead of 1 down to 0) for
any real value we can calculate (tediously!) exactly how many of the
pfs are 1, and it is always finite. So we can always give the sum as a
finite value. Only the cranks ever have an infinite number of balls in
the urn, and that's actually when it's empty -- that's the power of
intuition for you.
You could also look at the example where the pulses are all of length
1. This gives indeterminate (what the cranks call "oo") values for the
sum from 0 to 1, followed by zero. I bet there are some cranks that
will get this one right.
If pulse n goes from -1/n to 1/n, then the sum is only indeterminate
at 0, and has finite values everywhere else. Quite possibly almost all
of the cranks will get this one right.
But now (the "exploding balls" model): go back to the original
example. But let each ball have an automatic fuse, so that if one
minute [ er, units? ... after time interval 1] elapses without the
ball being removed, it just goes "poof!" and vanishes. (Cue for some
drivel about explosives from HdB). The cranks have to claim that if
the robot doing the ball removal is operating perfectly, this is the
original example, and an infinite number remain. But if the robot
breaks, and the ball removal fails in part or whole, mysteriously
there are fewer balls left in the urn.
Well, you can make up examples, but it doesn't work. Cranks are immune
to reason. Good luck!
Brian Chandler
Marshall
> Math1723 <anonym1...@aol.com> writes:
> > My friend's response: "Well, this is just proof of how Mathematics
> > breaks down." He refused to accept it. I even printed an Excel
> > spreadsheet showing him from fold #1's thickness (that of a piece of
> > paper) and double each step. He refused to believe it. "Math is just
> > wrong at those numbers."
>
> That's a pretty exceptional response.
I think you're understating the situation. "Exceptional" isn't
strong enough a word to describe someone that fucking
stupid.
Marshall
Marshall
> Han de Bruijn <umum...@gmail.com> writes:
>
>
>
> > On Jan 19, 4:00 am, Math1723 <anonym1...@aol.com> wrote:
> >> On Jan 18, 4:32 am, LudovicoVan <ju...@diegidio.name> wrote:
>
> >> > Then, as already noticed, there is no time t=0.
>
> >> Of course there is time t=0. That's when everything stops.
>
> >> > > The limit tends to infinity. But remember,
> >> > > not all functions evaluate at their limit.
>
> >> > Is the limit defined?
>
> >> Sure. You can calculate the exact figure yourself, but essentially
> >> the number of balls grows nearly tenfold with each t=1/2^n. So the
> >> limit tends toward infinity.
>
> >> > Sure but, for any ball removed, 10 more get added...
>
> >> Agreed.
>
> >> > > Now the logic is irrefutable.
>
> >> > I don't see a clear resolution.
>
> >> There's nothing to resolve. There is a proof that the bucket is
> >> empty. There is no proof to the contrary.
>
> > Sure. That's why _such_ mathematical reasoning is deemed unreliable.
>
> Right! We know the answer beforehand, so if math says something else,
> it's wrong! What could be simpler?
>
> In fact, since our intuitions are infallible, I'm not sure why we study
> math in the first place.
Perfect! This is just perfect! Now if I ever want to know what
Han thinks about something, I need only remember this brief
post. Bravo!
Marshall
PS. Although I'm not sure why I'd want to know what Han
thinks about something.
Marshall
wrote:
> Han de Bruijn wrote:
> > On Jan 19, 1:26 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> > > In fact, since our intuitions are infallible, I'm not sure why we study
> > > math in the first place.
>
> > Our "intuitions" are called "experience" in some circles. Mathematics
> > _can_ be counter-intuitive, but it should _not_ be counter-experience
> > all the time. Instead, it should help us to acquire better experience
> > than before.
>
> With some posters, it's possible to believe that they are basically
> confused. With others, such as the owner of approximately one brain*
> being yourself, it's difficult to escape the conclusion of simple
> dishonesty.
Dishonesty is present, yes. However it seems to me that the
central motivating characteristic of the crank is arrogance; it
is the arrogance that motivates the intellectual dishonesty. Just
a dollop of humility is enough to cause anyone to pause in the
face of universal disagreement; the arrogant instead reach the
conclusion that they are the first ones ever to see clearly. (Or
the first ones to see through the conspiracy, or whatever conclusion
they use to avoid recognizing that they made a mistake.)
Marshall
Han de Bruijn
Which rules of which mathematics? Certainly not standard mathematics.
Han de Bruijn
LudovicoVan
> > room for an "observer" and his putative clock, unless one provides
> > for a much more comprehensive setting.] >>
>
> > Have you got any math reasoning, informal or otherwise, to support
> > your senseless statements?
>
> For any natural number n define
>
> state_n(t) = 1 if 1-([n/10] + 1) <= t <= 1-1/n, 0 otherwise
That is:
state_n(t) := 1 iff ball n is in the bucket at time t,
where t := 1/k, k in N.
(The countdown is a convolution of a natural counting.)
[That is by our intended interpretation, i.e. how we model the
problem: then, based on our intended interpretation, we can say if a
statement is true or false.]
> Define balls_in_vase(t) to be the set of all natural numbers such that
> state_n(t)=1. (Note that this set is finite for all t and we can
> determine the set in a finite number of steps)
So: balls_in_vase(t) := { n in N | state_n(t) = 1 } is the set of ball
labels in the bucket at time t.
Such set is surely finite for any given t and, yes, we can determine
it in a finite number of steps.
> Define number_of_balls(t) = cardinality of balls_in_vase(t).
> Note that number_of_balls(t) is defined for all t, in particular
> number_of_balls(0) is defined.
Not yet: there is no k such that t=0, IOW there is no end to the
countdown, unless by extension...
> We have lim t-> 0 number_of_balls(t) = inf
> number_of_balls(0) = 0
That's a rather weird stipulation: insensical I would call it.
> Number of balls(0) is defined, nor is it necessary to
> do an infinite number of operations to calculate it.
Sure, if you say so!
-LV
William Hughes
We do not define number_of_balls(0) by finding a k such
that t=0, The fact that there is no such k does not mean
that number_of_balls(0) is not defined.
number_of_balls(0) is defined to be the cardinality
of the set of natural numbers n such that state_n(0)=1.
This is easily seen to be 0.
>
> > We have lim t-> 0 number_of_balls(t) = inf
> > number_of_balls(0) = 0
>
> That's a rather weird stipulation: insensical I would call it.
Why? Both statements follow directly and simply from the definitions.
- William Hughes
Math1723
>
> Confusion? Iterated limits are always commutative XOR nonsense. Thus:
>
> lim Card(Sn) = Card(lim Sn)
>
> XOR nonsense, I said.
If you are correct, then the bucket will have an infinite number of
balls with labels displaying finite natural numbers. Please name any
such number. (Remember, by the rule, it must be a finite natural
number.)
But of course you cannot. So your assumption that lim Card(Sn) =
Card(lim Sn) is false, at least in this instance.
Have you stopped to consider why you continue to hold onto your
assumption that lim Card(Sn) = Card(lim Sn) when there is evidence
that it is not always true?
LudovicoVan
Yes it does, until you provide an extended definition -- and that
should take into account what t is and how it ticks.
> number_of_balls(0) is defined to be the cardinality
> of the set of natural numbers n such that state_n(0)=1.
> This is easily seen to be 0.
Maybe easy for you, still an unsupported and insensical stipulation
for what I can see.
> > > We have lim t-> 0 number_of_balls(t) = inf
> > > number_of_balls(0) = 0
>
> > That's a rather weird stipulation: insensical I would call it.
>
> Why? Both statements follow directly and simply from the definitions.
You say so, but you have shown no such derivation for the extended
case.
-LV
Math1723
>
> Your "emotional embraces to intuition" is called "experience" in some
> circles.
This would of course the same justification used by those who reject
relativity, quantum mechanics, and members of the Flat Earth Society.
Sad to say, our experience can sometimes lead us to false
conclusions. It tends to work very well for ranges that we experience
every day. But for ranges that far outstrip our ordinary experience,
what makes you think that our experience is a could indicator then?
> > I had an experience in college with another counter-intuitive result.
> > I gave this puzzle to a friend of mine: Suppose you fold a piece of
> > paper in half. Then in half again. And again. No assuming a
> > sufficiently large enough sheet of paper, suppose you fold it in half
> > 25 times. How thick would that wad of paper be? How about 50 times?
>
> > Well, that thickness can be (under)estimated by simply taking the
> > thickness of a sheet of paper and multiply by 2^25 and 2^50 (since
> > folding always doubles the thickness). To the surprise of many
> > people, after 25 times, the wad would be a mile thick. At 50 times,
> > the thickness exceeds the distance between the Earth and the sun.
>
> > My friend's response: "Well, this is just proof of how Mathematics
> > breaks down." He refused to accept it. I even printed an Excel
> > spreadsheet showing him from fold #1's thickness (that of a piece of
> > paper) and double each step. He refused to believe it. "Math is just
> > wrong at those numbers."
>
> Exactly. Take a _real_ piece of paper and SHOW us (your friend and I).
Obviously the difficulty would be in obtaining a sufficient large
piece of paper. However, do you actually deny that double the
thickness of a piece of paper 25 times would yield a thickness greater
than a mile? If so, how thick do *you* think it would be?
LudovicoVan
> On Jan 19, 4:41 am, Han de Bruijn <umum...@gmail.com> wrote:
>
>
>
> > Confusion? Iterated limits are always commutative XOR nonsense. Thus:
>
> > lim Card(Sn) = Card(lim Sn)
>
> > XOR nonsense, I said.
>
> If you are correct, then the bucket will have an infinite number of
> balls with labels displaying finite natural numbers.
No, that would be a "transfinite mistake": in the simpliest case, the
balls that remain are consistently defined to be all labeled by 'oo'.
> (Remember, by the rule, it must be a finite natural
> number.)
No, not in the extended case -- if you mean to get there to begin
with.
> But of course you cannot. So your assumption that lim Card(Sn) =
> Card(lim Sn) is false, at least in this instance.
Unwarranted conclusion.
-LV
Math1723
> Math1723 <anonym1...@aol.com> writes:
> > My friend's response: "Well, this is just proof of how Mathematics
> > breaks down." He refused to accept it. I even printed an Excel
> > spreadsheet showing him from fold #1's thickness (that of a piece of
> > paper) and double each step. He refused to believe it. "Math is just
> > wrong at those numbers."
>
>
> values.
Agreed. Yet to my surprise, even Han de Brujin (in this very thread)
questions this result. Some people are so trapped within the walls of
their fantasy world, that nothing can break it down, even something as
seemingly concrete as this.
Aatu Koskensilta
> Agreed. Yet to my surprise, even Han de Brujin (in this very thread)
> questions this result. Some people are so trapped within the walls of
> their fantasy world, that nothing can break it down, even something as
> seemingly concrete as this.
There are also highly intelligent and sensible people who simply don't
have any aptitude or inclination for mathematics. Usually of course
these people don't post on sci.math.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus