Infinity Qs

[sql_lall]

1) Semantically Ambiguous??

3) In this case the number of balls might be indeterminant, or might be zero. - I think we are saying its zero.

However, for those who think its infinite, how about this induction:

n=k=1
Ball 1 is taken out at action #1
=> True for n=k=1

n=k+1
Ball n is taken out 1/n(n-1) minutes after ball (n-1)=ball k is taken out
as ball k is taken out (proved earlier) => ball n is taken out.

=> For all n in the set of integers (which all the ball numbers are), ball ne is taken out, and so is not in the box at the end.

It is an interesting question though, isn't it

[TheAlchemist]

hey sql_lall,
i really didn,t understand what you were trying to say with the ns and ks, maybe you should try put in some commas and all.
shaggy hiker: i think you're approach is very interesting,its certainly given me a new perspective on the topic but
i think GUV is right in stressing the mathematical aspect of this thread and the importance of this in the way we choose to approach the topic. Heres a little thought to hold!

Mathematicians can say whatever they like but physicists must be at least partially sane.
--J Willard Gibbs(professor of mathematics at the yale university a long long time ago)

by the way guv,you're interpretation of the thread was correct, the guy who wrote it intended exactly that.

[sql_lall]

Guy who wrote it being me, or the person I got the problem from? Just for interest sake, i got the problem from a book i got recently, called "Dr. Riemann's Zeros", by Karl Sabbagh. There is a bit at the end (Toolkit 3) about infinite series, and this problem is at the start. In the book it says it comes from "Littlewood's Miscellany", by J.E. Littlewood.

Oh, and incase you are wondering, the book gives the answer as zero. However, on the FIRST PAGE they list the primes INCLUDING 1!!! So i'm not sure how acurate the maths in the book is

BTW: waht i was trying to say that was for every ball that is taken out, the ball with the value 1 greater than it was taken out, => all balls were taken out. (Induction)

[simonm]

However, on the FIRST PAGE they list the primes INCLUDING 1!!!

What's wrong with that?

[Shaggy Hiker]

Quantum physics only works in the quantum world? The last I heard, every partical up to and including a human would demonstrate the same wave/particale duality as light if it were to travel slowly enough. The quantum world is not some esoteric theoretical world divorced from so-called reality. The principles work in the macroscopic as well as atomic level.

So you want to believe that math must come to a conclusion? Must there be an answer? Therefore one of the arguments for infinity or zero is wrong. I disagree.

As for my background, I am 35, and have been a scientist my entire life. I discovered my first mathematical theorem back in high school, and have used original lines of thought in designing a few mathematically oriented programs to locate solutions to chaotic problems. In the course of my career, I have encountered many people who reject certain solutions because they are too complex, and indefinite. Try talking chaos theory with biologists (or economists, if they'll listen) and you'll see what I mean. We have caused more harm than good in this country (USA for those who aren't sure) by looking for a single concrete answer and rejecting the premise that there isn't one.

Guv, this really isn't the forum, due to the time delay, for any kind of discussion on whether there must be an answer to this question, but I urge you to look into this more closely. I get the feeling from your posts that you are another person who wants the world to be black and white. I too, would prefer this be so, but I have found more interesting and practical solutions to problems by considering what would happen if the coin landed on edge.

And if you'd just prefer a flame war, I can do that too, but I'd rather not.

[sql_lall]

Hey, chill, chill.. Lets just talk about the maths.


Anyway, simonm, 1 is NOT a prime!!! it has only 1 distinct positive factor, not the 2 that are needed for primality. Oh, that wasn't the only problem in the book, there were a few more, i'll see if i can find them again

[simonm]

Shaggy Hiker

Quantum physics only works in the quantum world? The last I heard, every partical up to and including a human would demonstrate the same wave/particale duality as light if it were to travel slowly enough. The quantum world is not some esoteric theoretical world divorced from so-called reality. The principles work in the macroscopic as well as atomic level.

It is my understanding also that quantum effects to not manifest themselves on the macro scale (or at least they become extremely unlikely).

The rules that appear to govern the macro world of classical mechanics some how emerge from the micro world of quantum uncertainty.

I think there is a danger of applying quantum results to situations that do not warrent it and I believe you are doing so now.

sql_lall

Anyway, simonm, 1 is NOT a prime!!! it has only 1 distinct positive factor, not the 2 that are needed for primality.

Well, as I understood it, a prime number must only be divisible by one and itself. Since 1 is divisible by 1 and itself (even if they happen to be the same number) it would qualify as a prime number.

[Shaggy Hiker]

Perhaps I have been unclear, and I'm sorry that Guv ticked me off. I don't think that exchange could have occurred except through a faceless medium like this.

I feel that both sides, zero and infinity, have made compelling arguments for their interpretation. However, I have seen no mathematical argument against either side. The arguments against each side have been logical ones along the lines of "This shows that the answer must be x, and therefore, other answers can't be right." However, both sides are making that argument, and doing so in a manner sufficient to prevent invalidation on a mathematical basis. I say that because there hasn't been such an invalidation yet.

In the early days of math, or so we have been told, there was no 0. That was added, then negative numbers, fractions, decimals, etc. At some point people found that there were non-repeating decimals (e.g. pi) for which there is a name that I forget. However there was still a gap.

At some point, people began accepting calculations using unreal numbers (i) for which there really isn't any logical real-world representation. Surely there were those who felt that using the square root of a negative number in your calculations was absurd.

During the Age of Reason, mathematicians were looking for a "complete" system of mathematics. A complete system was one with the power to define and identify all valid theorems within the system. Godel proved that wasn't possible. No system can be created that can derive all theorems, and once you get close to such a system you can derive at least one theorem that is both true and not true, a paradox.

In math, as in physics, we can find paradoxical results once the system becomes sufficiently close to completeness. The current thread is a similar case, since solid arguments can be made to "prove" two mutually exclusive results.

I am disturbed by this incompleteness we are seeing in some systems. Usually, when we find flaws in the patterns coming from theoretical science, the reason is that our comprehension is inadequate. Being a biologist, I don't think humans are particularly intelligent, since our actions aren't substantially different from other animals, but in this one area, our intelligence is our asset. We must keep moving forward in knowledge. A series of problems of this sort, arising as they do in different fields, suggests that current models can be improved upon. Numbers are only a model of the world around us. We have already expanded that model to inculde unreal numbers, perhaps we need to, again, expand the model.

We know of equations (quadratics) which have two algebraic solutions that are both valid, but are not at all contradictory. Perhaps we will eventually come up with a further set of math to cover equations that have two contradictory solutions. This thread is an example of such a problem. As the number of iterations approaches infinity, the number of balls in the box approaches infinity, but at infinity itself, the number of balls in the box can be either infinite or zero, as many have demonstrated. Why can't there be a further set of numbers beyond unreal.

Perhaps the inconceivable numbers.

[NotLKH]

Originally posted by sql_lall
Glad to see someone has replied.

You see, the common sense reply would be infinity, cos surely it is the same as putting in 9 balls each time? Or perhaps the time never gets to noon?

Some things to think about:

1) The time surely has to get to Noon. Otherwise, time will stop, and that doesn't happen too regularly.

Obviously, the closer you get to noon, the faster you have to move, in order to get the ever increasing number of balls into this infinitely voluminous box. So, Naturally, you end up approaching the speed of light, where you will start to observe some relativistic effects on time. Therefore, can you actually guarantee that, for the person adding and subtracting the balls, that he will ever get to noon?

Sounds like a greek tragedy to me.

[Guv]

I wish those who are discussing this issue would pick from the following alternatives or propose some fourth possibility.

• The original post is semantically ambiguous. In this case there can be no further analysis. I hope you understand the implications of the term semantically ambiguous.
• The post describes the series (10 -1) + (10 -1) + (10 -1) . . . In this case the number of balls grows without bound.
• The original post describes the difference between two infinite series. I think this alternative leads to zero.

To me it is very clear that the answer depends on which of the above interpretations one chooses.

I would not argue with any valid analysis which starts by mentioning which of the above interpretations is considered correct. A valid analysis can only be refuted by claiming that it misinterprets the problem description.

The finite time frame seems to be obfuscation. The process describes an unbounded number of operations, although the time schedule was not precisely described in the original post. Sql_Lall clarified in a later post, resulting in a schedule with an unbounded number of operations. Briefly, that post said the following (using numbered balls).

Put in #1- #10 & take out #1
Put in #11 - #20 & take out #2
Put in #21 - #30 & take out # 3
. . . .

Simonm: Please note below, the difference of two transfinites with same cardinality need not be zero. You interpret the second part of the operation as removing balls #1 to #n, which is clearly the removal of an unbounded number of balls. After reconsidering a bit, I think this interpretation is correct or at least describes the intent of the author. I change my vote to agree with Simonm: Zero balls are in the box when the process is complete (if ever). The result is the difference of two transfinites with the same cardinality.

It is strange, but consider the following change in the description.

Put in #1 - #10 & take out #1
Put in #11 - #20 & take out #11
Put in #21 - #30 & take out # 21
. . . .

The latter describes putting in ten and taking out one, or adding nine at each step. Clearly, balls #2 - #9, #12 - #20, et cetera are never removed. The result is a lot of balls in the box after a lot of steps. A transfinite numbers of balls are removed, but a transfinite number remain.

Both descriptions specify putting in ten and taking out one at each step, but are subtly different. Transfinite arithmetic is counterintuitive.

If you describe two independent processes.

• The former could be described as put in ten balls at each step for an unbounded number of steps, followed by the removal of all the balls one at a time.
• The latter could be described as put in ten balls at each step for an unbounded number of steps, followed by removal of every tenth ball starting with #1. This is the removal of an infinite subset, but not all the balls. Clearly, an infinite subset remains. Following are never removed: #2 - #10, #12 - #20, et cetera.

I would like to see a description that leads to both zero and an infinite number simultaneously as the answer to the above, which I still claim is a silly and/or meaningless statement.

Shaggy Hiker: I do not view everything as black or white, although the world of mathematics seems to have definite answers to many questions. I certainly do not view Quantum objects as well defined until measured.

However, I do not think it is proper to say that light is both a wave and a particle until some experiment results in measuring some wave or particle property. I seem to remember descriptions using words like Limbo State, implying neither wave nor particle until measurements are made. Similarly for the properties of other quantum objects.

Perhaps your claim for simultaneously existing paradoxical values is not even valid in the Quantum world. Do you have any good citations for such simultaneity? I seem to remember incorrect conclusions resulting from making assumptions about the state of an unmeasured quantum property.

[Shaggy Hiker]

Sorry Guv, I don't have any citations handy, I might be able to dig some up in a day or so, but I guess it doesn't really matter.

I prefered your earlier stance that the answer was not 0, but agree that there is a good argument for 0. I'd rather not vote on this, but would prefer that there were an answer. I suppose I choose the semantically ambiguous option, though semantically ambivalent seems more correct. I don't think the ambiguity lies in the semantics, but that may just be more antics with semantics.

[sql_lall]

Firstly, i am not sure what part of the Questin is ambiguous.

However, i can say that you never reach the speed of light. NotLKH, we have assumed that putting balls in/out of the box takes no time at all.

I think the problem comes with treating the balls as objects, not every ball as '1'.

You see, if we treat every ball as '1', and adding/subtracting to box as +/-, then sure, the sum ends up as (10-1)+(10-1)+(10-1) etc. = 9+9+9... = infinity

However, the balls are objects, you can't take any 1, you have to take a certain 1.
I.e. the sum then becomes:
(1₁+1₂+1₃+1₄+1₅+1₆+1₇+1₈+1₉+1₁₀ - 1₁)
+ (1₁₁+1₁₂+1₁₃+1₁₄+1₁₅+1₁₆+1₁₇+1₁₈+1₁₉+1₂₀ - 1₂)

This is a telescoping sum, which cancels out to 0.
I.e. if the balls ar treated as a value, "1" then you get infinity. If you treat them as objects (i.e. strings: "ball1", "ball2" etc) then they cancel out to 0.

This is what i don't agree with in Guv's (10-1)+(10-1)+... case, as you are really taking an arbitrary "1", from a 10 which the 1 isn't in. Sounds weird, i know.

[Guv]

Sql_Lall: Check my most recent post, which agrees with you and SimonM.

In particular note that the finite time schedule is unnecessary confusion.

[teriaki]

ive browsed this site of and on ever since i started teaching myself VB, but it wasnt till i saw the truely in-depth discussion of this thread that i felt i had something to contribute.

DISCLAIMER: im not a mathematician, i just like playing around with it. im not a physicist, i just find it interesting

as i understand it, any hypothetical universe actually DOES exist, even if it is only a mathematical construct. what i mean is: any hypothetical set laws is just as valid as the set under which we live (so long as it does not violate itself). so this experiment could actually take place once a set of laws is in place. the true outcome (or measurement) really depends on these laws. so in effect Shaggy Hiker was right in saying, "We can't know the exact state until we observe it..." but not because of the uncertanty princible or because of a physicist's cat. we will know the outcome simply because we set it to be true by the predefined laws.

this i think is the only instance of symantical ambiguity, but only because we cannot seem to agree apon what the laws ARE. the only constant it seems is time. since the problem statement suggested that we could *expand* one minute to perform an infinite number of operations while maintaining that it is a finite amount time, i put forth my (tenative) conclusion:

the answer seems to 0. since each ball that ever enters the box eventually leaves it, it shouldnt matter how quickly the box is filled or emptied, so long as the entire process is completed before noon.

[sql_lall]

or because of a physicist's cat

hehe

If it were a dog, many more people would have liked Shroedinger (sp?)
However, as it is a cat, no-one really cares. Even if it were dead, it would still come back as the ghost of uncertainty past.

[teriaki]

good point,

Actually, I once read a book about quatum theory in which the protagonist meets Shroediger's cat. The cat relates that as the experiment was taking place, he escaped the box while the physists weren't looking. when they opened the box to discover that there was no cat inside, they were unable to make the necessary measurement. thus, the cat was instantly doomed to wander the earth in constant flux between life and death.