Space, Time & Stephen Hawking Jive

[FNHinVA]

Actually, one of Stephen Hawking's graduate assistants.

I have a physics question that has baffled me for many years. I posed it to a few physicists and got unsatisfactory answers. So I decided to email Stephen Hawking. (Why fool around with amateurs?)

As expected, he did not answer. But one of his graduate assistants did. But first, the question...

I am going to do a "time trial" over the distance A to B (A|B). I will maintain a constant rate of speed. Obviously, in order to traverse A|B, I must first traverse half of A|B which I will do in half the time. Just as obviously, I must also traverse half of the half of A|B. (You see where this is going...)

Since I have in front of me an infinite number of "halves" I must traverse (and take time doing it), how will I ever pass the B finish line? Obviously, it will take forever. But, because I know I can, in fact, traverse A|B in a finite amount of time, I know it doesn't take forever.

The answer from Mr. Hawking's graduate assistant involved calculus, Planck lengths and the uncertainty principle.

Essentially, what all of this (and he) said was "when things get that small, we can no longer measure them so we don't know what the hell is going on."

Anyone have a better answer?

[kpux]

This is Zeno's paradox. It's an ancient Greek riddle. It's easily solvable with infinite series.

Basically, you can add up an infinite number of positive real numbers and have it come out to a finite sum. For simplicity, let's say the distance A|B = 1. In this example, 1/2 + 1/4 + 1/8 + ... + 1/(2^n) + ... = 1. Even though you are traversing an infinite number of halves, the total distance is still finite. Just because you can express the number 1 as an infinite sum of smaller real numbers, doesn't mean that A|B somehow becomes an infinite distance.

[PairTheBoard]

Just because you measure time in smaller increments doesn't mean you're slowing it down.

PairTheBoard

[drudman]

[ QUOTE ]
This is Zeno's paradox. It's an ancient Greek riddle. It's easily solvable with infinite series.

Basically, you can add up an infinite number of positive real numbers and have it come out to a finite sum. For simplicity, let's say the distance A|B = 1. In this example, 1/2 + 1/4 + 1/8 + ... + 1/(2^n) + ... = 1. Even though you are traversing an infinite number of halves, the total distance is still finite. Just because you can express the number 1 as an infinite sum of smaller real numbers, doesn't mean that A|B somehow becomes an infinite distance.

[/ QUOTE ]

[usmhot]

Absolutely right. Mathematically this is an old problem which is solved exactly as indicated, by a sum over infinite series. It applies to a continuous space - i.e. one in which two points can be an arbitrarily small distance apart.

However, the rest of what Hawking's assistant was talking about is ...
according to Quantum Mechanics space is _not_ continuous - there is a finite distance which is the smallest distance that can be between two points. So in the real Universe there are only a finite number of sub-points (distances) between two points. Thus, in the real Universe you can get from A to B in finite time because there are a finite number of time intervals from sub-point to sub-point.

[FNHinVA]

You are correct. But that is not the issue (or the question). The question is how do I traverse an infinite series of shortening distances in any amount of time. The rate of time has nothing to do with it.

[PairTheBoard]

[ QUOTE ]
You are correct. But that is not the issue (or the question). The question is how do I traverse an infinite series of shortening distances in any amount of time. The rate of time has nothing to do with it.

[/ QUOTE ]

You haven't made the distance longer by dividing it up into infinitely many small increments. Neither have you made the time longer by doing the same with it.

PairTheBoard

[FNHinVA]

OK - now we're getting somewhere (so to speak). This is exactly the answer I was looking for, and didn't really get from Hawking's assistant or the others I asked.

I have a very lame math background so when the answers are in calculus and other languages I don't speak, I get kind of lost.

The concept of a finite measure of time (and distance) is what I have theorized. Obviously, time and space cannot - in reality - be divided into infinity. I don't care about math proofs; reality says it has to stop somewhere or we can never get anywhere.

My theory is that at the ultimate and finite level, motion becomes state changes, rather like the frame changes of a motion picture. These would - I believe - occur at the speed of light.

Now I need to work on the implications of this - if any.

Then, I'm on to World Peace and poverty.

[FNHinVA]

I'm not trying to make anything longer. I'm trying to understand what happens. The math doesn't really explain anything. See my response to usmhot above.

[usmhot]

That argument for space being discrete does not hold - if space were continuous then you would still get from A to B in finite time, as indicated by the post on Zeno's paradox - provable using infinite series. In fact, up until Quantum Mechanics it was assumed that space was continuous and the infinite series proof was totally acceptable as an answer to that original question.

But, there is evidence to suggest that space is not continuous, so the sum over infinite series does not apply. (Not least the energies involved in continuous space becomne infinite.)

As to what 'causes' motion, which is what you're trying to get at ... I think that's a very interesting question. Given that (it seems very likely that) space is discrete and between two points that are a Plank's distance apart (the smallest distance there can be) there is effectively nothing at all, the question of how a particle gets from one point to another is still unknown.

[PairTheBoard]

[ QUOTE ]
I'm not trying to make anything longer. I'm trying to understand what happens. The math doesn't really explain anything. See my response to usmhot above.

[/ QUOTE ]

You seem to think that you're making the travel time longer by dividing the distance into infinitley small pieces. It should be clear to you that you are not making the distance any longer when you do this. Every small distance you imagine takes a correspondingly small interval of time to traverse. Say it takes 1 unit of time to travel the 1 unit of distance from A to B. If you divide the 1 unit of distance into infinitely many smaller distances they still add up to the same 1 unit of distance. As you do so you are dividing the 1 unit of travel time into exactly the same infinitely many small units of travel time which still add up to the original 1 unit of time.

As I said before, dividing the time into smaller increments doesn't slow it down.

PairTheBoard

[FNHinVA]

First, it is my understanding that Planck's distance is the smallest measurable distance - not necessarily the smallest distance. Second, I have no idea what difference that makes.

As to your comment: "the question of how a particle gets from one point to another is still unknown." Well, hell. That ruins my day. That is the whole (underlying) issue.

I'm going to go with speed-of-light (or instantaneous) state changes. Quantum mechanics must allow for it even if it doesn't explain it.

[FNHinVA]

Explaining why a paradox is a paradox is not really helpful. It is not a solution.

I already know it cannot be true. I am trying to understand what actually happens.

We don't seem to be communicating very well but I appreciate your contribution. Let's end it there.

[Mano]

The fact that you can traverse the distance in a finite amount of time is not reliant on the quantum nature of spacetime. As stated before the sum of the distances 1/2 + 1/4 + 1/8 +... = 1. The time to traverse each of these distances also converges to a finite number - i.e. If your velocity is v, and you travel a distance d, it takes you d/v time units to cover the distance. So it takes 1/2v + 1/4v + 1/8v + ... = (1/2 + 1/4 + 18 + ..)/v = 1/v time units to cover the distance.

[drudman]

I was going to repost a long time ago, but as I got halfway from my chair to my computer, I noticed it was taking me forever to get there.

[PairTheBoard]

[ QUOTE ]
Explaining why a paradox is a paradox is not really helpful. It is not a solution.

I already know it cannot be true. I am trying to understand what actually happens.

We don't seem to be communicating very well but I appreciate your contribution. Let's end it there.

[/ QUOTE ]

Suppose the object is travelling 1 meter per second and the distance from A to B is 1 meter. What "actually happens" is that while you are dividing the 1 meter into imaginary segments of lenths 1/2m, 1/4m, 1/8m, 1/16m ... and the travel time into imaginary periods of 1/2sec, 1/4sec, 1/8sec, 1/16sec ... , One second passes regardless, and 1 meter is traversed. Every 1/2^n meter within the One meter that you imagine has had 1/2^n seconds within the One second amount of time for it to be traversed. There is no paradox.

PairTheBoard

[PairTheBoard]

The real puzzle here is why this so called paradox is psychologically disturbing. I know it disturbed me the first time I pondered it. I think it disturbs our intuition to think about how infinitely Many distances can possibly be travelled in a finite time. Intuition tells us that the Number of Distances matter. Logic tells us that it's only the measure of the Total Distance that matters.

I think this is the kind of thing that makes higher mathematics so difficult for many people. These psychological disturbances happen repeatedly in higher math, where a kind of inner psychological adjustment must be made in order to assimilate new concepts which logic dictates but which seem offensive to intuition and/or previous experience. "This stuff makes my brain hurt" is sometimes more than just an excuse. Those who succeed in higher math are ones who are willing to undergo these disturbing almost painful inner pschological adjustments. It also explains why truly novel and correct ideas are often first met with hostility and ridicule even at the highest levels.

PairTheBoard

[FNHinVA]

Clearly, I did a poor job posing the question.

usmhot is the only respondent who understood where I was trying to go. I refer you to his posts above.

[usmhot]

In our Universe it is reliant on the quantum nature of space-time.
Only in a continuous space does the sum over infinite series actually apply.

[FNHinVA]

Thank you.

I believe the "best answer" I was seeking is that space:time in NOT continuous and can't be.

It is one thing to "theoretically" add up an infinite series of numbers and quite another thing to actually do it. If someone would like to actually do it, please send me your work papers when you are finished (as in: never)

On the other hand, I am quite prepared to "prove" I can go from point A to point B. To me, that means at some point, those "halves" MUST become finite and - as usmhot states - the infinite series sum does not apply.

Once we get to the quantum particle level and Planck lengths and particles behaving like waves and being (apparently) in two places at once and the act of measuring influencing the measured, then, Hawking's assistant is probably right; we don't know what the hell is going on.

I just know that after puzzling over this for many years, I am convinced that space:time can not be a continuum.

I have no idea what difference it makes or why I care.

[PairTheBoard]

FHN --
"It is one thing to "theoretically" add up an infinite series of numbers and quite another thing to actually do it. If someone would like to actually do it, please send me your work papers when you are finished (as in: never)"

We don't need to "actually" add up the infinite series of numbers until you can "actually" produce the infinite division of the unit distance that you theorize.

PairTheBoard

[FNHinVA]

Once again, we have a failure to communicate. I'm sure it is my fault for not properly describing and expressing the problem. (Give me a break; I have an MBA, not a science degree. I barely got past first-year algebra)

My position is - and always has been - that the infinite division of the unit distance is impossible in the real world.... a finite point or distance must be reached.

Quoting usmhot:

"However, the rest of what Hawking's assistant was talking about is ...
according to Quantum Mechanics space is _not_ continuous - there is a finite distance which is the smallest distance that can be between two points. So in the real Universe there are only a finite number of sub-points (distances) between two points. Thus, in the real Universe you can get from A to B in finite time because there are a finite number of time intervals from sub-point to sub-point."

This describes "my theory" exactly.

Your comment: "We don't need to "actually" add up the infinite series of numbers until you can "actually" produce the infinite division of the unit distance that you theorize." seems to make a counter-point out of the original point.

Adding up the infinite series of numbers and adding up the infinite division of the unit distance are the same problem. Both are impossible in the real world and lead to usmhot's quantum solution with which I agree.

[drudman]

Adding up the infinite series of numbers is not impossible in the real world.

But I'm sure that's not what you want to hear.

[usmhot]

FNHinVA, let me make one thing clear - we actually _can_ add up infinite series of numbers. Not, of course, by accounting for each term one by one. But, we can prove completely what the value of the sum is for many infinite series, including the one described in Zeno's paradox. In other words, we know, with absolute certainty that if space was continuous in our Universe then getting from point A to B would take finite time.

[carlo]

Reality and Mathematics. This is a case where the logic of the situation and its concomitant mathematical analysis is absolutely correct.

But what is in your head is not a substitute for reality. Square pegs into round holes-you don't need to prove the fallacy by mathematics which is not a substitute for reality. In this case this is tantamount to saying--"I will not be able to digest unless I prove it works".

regards,
carlo

[PairTheBoard]

FNH --
"My position is - and always has been - that the infinite division of the unit distance is impossible in the real world.... a finite point or distance must be reached."

If that was your position you should have just come out and said so to begin with. You don't need Zeno to make that assertion. However, you do need to clarify what you mean by it. "A finite point or distance must be reached"? What does that mean? You started with a finite distance. Every 1/2^n little distance you describe is a finite distance. "A finite point" must be reached? What does that mean? Do you mean there must be some Minimum Distance? In a mathematical continuum model, half that Minimum Distance can still be considered.

If you're saying that the Continuum Model for Space is not accurate at quantum levels then fine. That's quantum theory. It's got nothing to do with Zeno.

PairTheBoard

[FNHinVA]

"Not, of course, by accounting for each term one by one."

Well, that's a hell of an "out."

In my simplistic non-math way of thinking, adding them up one by one is exactly what I see as the requirement in the real world with space:time as a continuum. Then there is no end to the numbers, no end to the adding, no end...

That's how I have to go A to B in the real world; step by step - one (unit) by one (unit). Which is why I fully accept your quantum explanation but can't deal with abstract explanations that require a space:time continuum and an "out."

I'm sure that in the abstract, just as there is a symbol for infinity, there is some symbol for the sum of infinity that can be manipulated in equations but I don't understand (or need) it.

I'm happy that quantum mechanics apparently supports my logically derived theory of finite points of space:time. I didn't know that before I recklessly started this thread.

As I mention earlier in this thread, I am lame with math and untrained in the sciences. The quantum explanation make sense to me; the math-based space:time continuum explanation is beyond my understanding. If there is any controversy between the two, I'm down with the "quantums."

Thanks for your input, usmhot.

From now on, as I go from point A to point B, I will do it with the conviction that I have "only a finite number of sub-points (distances) between two points.." to traverse.

That makes me happy.

[FNHinVA]

"FNH --
"My position is - and always has been - that the infinite division of the unit distance is impossible in the real world.... a finite point or distance must be reached."

If that was your position you should have just come out and said so to begin with. You don't need Zeno to make that assertion."

Perhaps I should have. My intent was to see if anyone came up with an explanation that was consistent with my thinking. I was not trying to mislead or be cute. As to the Zeno reference - I did not make it; I had never heard of it before this thread. It just happens that the way I posed the question apparently was Zeno-like.

"However, you do need to clarify what you mean by it. "A finite point or distance must be reached"? What does that mean?"

The Planck constant - as I understand it. What usmhot described is completely consistent with what I had theorized: "Thus, in the real Universe you can get from A to B in finite time because there are a finite number of time intervals from sub-point to sub-point."

"If you're saying that the Continuum Model for Space is not accurate at quantum levels then fine. That's quantum theory."

Yes, but I couldn't say that because I didn't know before I started this thread that my theory was consistent with quantum theory. Basically, because I know virtually nothing about quantum theory.

As I stated before, I am untrained in the sciences. A number of years ago, I just started thinking about it; how does an object get from A to B if there is always half the distance to go? My logical conclusion was that space:time cannot be a continuum; it has to be some kind of series of points. Whenever I posed the question to someone, the answer was always math-based and assumed a space:time continuum. Those explanation were never satisfactory to me.

So that how this thing got started. So in the past two days, I've learned what the Planck constant is; I learned about Zeno's paradox; I learned quantum theory supports my theory about space:time not being a continuum.

Hell, I've learned a lot. I'm happy.

Thanks for your input...

[PairTheBoard]

You know, mathematics is nearly always working with aproximate models for reality. The reason is that the aproximate models are usually much easier to work with mathematically. We look at what happens "in the limit" even though we know the "limit" situation is not what happens in reality. Brownian motion is only a mathematical construct which is approached by certain phenomenon but not actually achieved. But the Brownian motion Limit is easier to work with.

If the contiuum model for space and time were accurate, I would have no problem with the moving object gobbling up infinitely many imagined little distances right at the end, because they are only imagined distances. The object doesn't know they're being imagined and it doesn't have to pause and catch it's breath at the end of each one. It just gobbles up the final infinitely many all at once - so to speak. Not something I have any experience with and somewhat disturbing to my intuition but so what? It's all happening in the imagination anyway.

I suspect what's really going on is far stranger than we can imagine. It's interesting that quantum theory that was once considered so counter intuitive to our mental assumptions about being able to infinitely break things apart is now with popularization becoming the More Intuitive way to look at things. I wonder what they will come up with next.

Good Topic FNHinVA

PairTheBoard

[Piers]

[ QUOTE ]
Essentially, what all of this (and he) said was "when things get that small, we can no longer measure them so we don't know what the hell is going on."

[/ QUOTE ]

Seems a pretty good answer to me, better than you are likely to get here.

[Piers]

[ QUOTE ]
Essentially, what all of this (and he) said was "when things get that small, we can no longer measure them so we don't know what the hell is going on."

[/ QUOTE ]

Seems a pretty good answer to me, better than you are likely to get here.

[lastchance]

You are asking a math-based question. You have to expect a math-based answer. Limits exist for a reason.

And a better version of Zeno's paradox is that if there are an infinite series of points, how can you get started on the first point?

And "no one understands Quantam Mechanics."

[FNHinVA]

Good point. As I reread my summation of Hawking's assistant's answer, and consider all the back and forth of this thread, his answer looks pretty good. Even if not very "satisfying." Not his fault. I guess there is a reason he is a graduate assistant to Stephen Hawking.,

[FNHinVA]

"You know, mathematics is nearly always working with aproximate models for reality. The reason is that the aproximate models are usually much easier to work with mathematically."

Yes, I understand that. I am sure there is some Greek-looking symbol that represents the sum of an infinite series and that it works very well in equations. I bet you could land a rocket ship on Mars using stuff like that. (Oh, wait. They did that.)

Anyway, it's been fun. I am greatly amused that simple curiosity got me into an issue of quantum theory without having a clue what it is.

I was also rather amused that all the math guys were very helpfully (and insistently) trying to "prove" to me that I could get from A to B.

Hell, I already knew that; I was trying to understand "how." I still don't know, but I remain convinced that space:time cannot be a continuum.

"I suspect what's really going on is far stranger than we can imagine."

Yeah... and Hawking's assistant's answer looks much better now: "When things get that small, we can no longer measure them so we don't know what the hell is going on."

Guess I'll have to settle for that.

Thanks...

[pc in NM]

[ QUOTE ]
Actually, one of Stephen Hawking's graduate assistants.

I have a physics question that has baffled me for many years. I posed it to a few physicists and got unsatisfactory answers. So I decided to email Stephen Hawking. (Why fool around with amateurs?)

As expected, he did not answer. But one of his graduate assistants did. But first, the question...

I am going to do a "time trial" over the distance A to B (A|B). I will maintain a constant rate of speed. Obviously, in order to traverse A|B, I must first traverse half of A|B which I will do in half the time. Just as obviously, I must also traverse half of the half of A|B. (You see where this is going...)

Since I have in front of me an infinite number of "halves" I must traverse (and take time doing it), how will I ever pass the B finish line? Obviously, it will take forever. But, because I know I can, in fact, traverse A|B in a finite amount of time, I know it doesn't take forever.

The answer from Mr. Hawking's graduate assistant involved calculus, Planck lengths and the uncertainty principle.

Essentially, what all of this (and he) said was "when things get that small, we can no longer measure them so we don't know what the hell is going on."

Anyone have a better answer?

[/ QUOTE ]

There is no such thing as a "half"; that is a (useful) figment of your imagination; useful, so long as you don't confuse your imagination with reality....

[usmhot]

FNHinVA,
Maybe I'm not making myself clear ....

Your line of reasoning is not valid in terms of coming to the conclusion you did.

There are other (more complex) reasons why any two points can't be arbitrarily close together, but the argument that you can't see how you could travel from point A to point B in finite time is NOT a valid reason.

I will try to point out step by step why it is a flawed line of reasoning

1. you are trying to prove that space cannot be continuous

2. the definition of 'continuous' is that any two points can be arbitrarily close to each other. Alternatively, between any two points there is at least one other point that is closer to each of them than they are to each other (in fact there are infinitely many points that satisfy this).

3. your proof is based on an argument by reductio ad absurdum - i.e. you start with the opposite of what you are trying to prove and show that it logically leads to a paradox and so you conclude it can't be true.

4. thus, your proof goes something like the following
4.1. Assume space is continuous
4.2. Given two points A and B which are x units distant I wish to travel from A to B.
4.3. To travel x units I must first travel x/2 units which will take a certain amount of time.
4.4. But to travel x/2 units I must first travel (x/2)/2 units which will take a certain amount of time.
4.5. As space is continuous I can divide each half distance by two ad infinitum and find a smaller distance that I must travel.
4.6. Ultimately there are an infinite number of arbitrarily small distances that I must travel
4.7. I (think that I) can't travel an infinite number of distances in finite time therefore space must not be continuous.

5. This argument is wrong, because step 4.7 is invalid and is provably invalid. The proof is a clear and irrefutable mathematical proof. It does not involve the introduction of any weird symbols. It simply and clearly demonstrates WITHOOUT ANY DOUBT that to travel from A to B in a continuous space takes a finite time. There is NO ROOM for question, NO philosophical / logical / reasonable alternate arguments that can refute it.

The reason I'm telling you this is because you seem to want to make some logical reasoning about the real Universe to come to what you consider are valid conclusions. If you are going to do this then you MUST be prepared to accept when some of your steps are simply wrong and find alternate valid steps or live with the irrefutable alternate conclusions. If you are not prepared to do this then you might as well just be writing a little make-believe story, because it doesn't mean a thing.

Now, as I said, in this case there are other reasons to suppose that space, in our Universe, is not continuous. As I understand it, this makes you happy because you have other arguments / lines of inquiry that depend on it. But any of your further deductions may be based on fallacious reasoning, as this one was, so be prepared for that.

[hobbsmann]

FNHinVA,

If I recall correctly this paradox is addressed in part of this book...
Elegant Universe (http://www.amazon.com/exec/obidos/tg/detail/-/0375708111/qid=1121094456/sr=8-1/ref=pd_bbs_ur_1/102-4471499-1532927?v=glance&s=books&n=507846)

This book was written 5 or so years ago as popular introduction to string theory and isn't too hard of a read for the non-math people out there. I'd recommend picking it up as you seem pretty interested in this topic.

[FNHinVA]

Gee... I didn't mean to piss you off.

I'm just a simple business major in over my head and trying to understand a few things. And, I'm sure you made yourself perfectly clear to anyone properly trained in the sciences. I'm sure the problem is with me.

Anyway... I'll try to do a better job of making my case.

If I ask you to sum up an infinite series of numbers (say, oh what the hell, .1/2 + 1/4 + 1/8 + ...) and write it down as a real number, you can't do it. I believe you have already acknowledged this.

What you can do is write down some "weird symbol" which represents the sum. You can then stick this symbol into some formula and come up with "...clear and irrefutable mathematical proof."

This is where you lose me... and leave me unconvinced. You use an imaginary number to represent something that - in the real world - can't be... doesn't exist (that is, the sum of infinity). To me, all you've done is prove something about an imaginary world.

I found this with a few searches:

"In math, when you hear people say things like "1 over infinity is zero" what they are usually referring to is something called a limit. They are just using a kind of shorthand, however. They do NOT mean that 1 can actually be divided by infinity. Instead, they mean that, if you divide 1 by successively higher numbers, the result becomes closer and closer to 0. If I divide 1 by a very large number, like a billion, then I get one-billionth, which is a VERY small number, but it isn't 0. Since there is no largest number, I can always divide 1 by a bigger number. But that will just produce an even smaller number, right? It will NEVER produce 0, no matter how high I go. But since the answer to the division is getting closer to and closer to 0, we say that "the limit of the expression is zero." But we have still not divided anything by infinity, since that isn't a number."

Well, hell. That's cheating. Once you operate with a "limit" of course you are going to get a finite result.

I'm sure that this type of "simplifying assumption" (as we call them in business models), works quite well in lots of analytical situations. I just don't buy it in this one.

In the real world, I get to keep dividing 1 by larger and larger number into infinity and there will never be enough time to cover the distances represented.

Thanks very much for your input. I do appreciate it. Even if you do get exasperated with me and a bit snippy at times, it is all very interesting - much more so than income statements and balance sheets.

[FNHinVA]

Thanks much...

I put it on my "wish list" and will add to my next order of poker books.

I need it; I'm tired of usmhot, PairTheBoard, et al, beating me up. They are making my brain hurt.

[drudman]

What makes 0+ any less real than any other integer? The label "real number" is not applied to numbers that are somehow "real" in the sense that they exist in the "real world". Likewise, imaginary numbers are not "imaginary" any more so than integers are.

You can't write down pi as a "real number" either according to your conception of real. Nor can you write down 1/3.

Do you see what I am getting at? You may need to escape the rigid rules you have imposed on mathematics.

[FNHinVA]

I suspect that my use of terms like "real" and "imaginary" are getting me in trouble because they mean something different in math than what I mean by them.

I will try to explain my position without using them.

I understand that I cannot "write down" pi and 1/3 as absolute (finite) values. But when you carry them out enough decimal points they are "good enough" approximation to be useful and accurate (enough) for all kinds of calculations. But they are never absolutely accurate; they are always an approximation.

This is the very problem I have with usmhot's "...clear and irrefutable mathematical proof." It is based on just such an approximation - a limit placed on infinity. By placing a limit on infinity you WILL get a finite time to traverse infinity. And you can ONLY get a finite time by placing the limit. I believe I'm correct in asserting that any math operation involving infinity = infinity.

So, the most fundamental premis of the question has been violated. I am saying "let's imagine infinity." usmhot is saying "I am going to substitue an approximation of infinity and with it prove you wrong."

Not fair.

[drudman]

You are mistaken in your view of infinity.

We are not approximating inifinity. We are not "placing a limit on infinity". The expression 1/(infinity) is not an approximation of a value, it is a bona fide finite value.

[BluffTHIS!]

[ QUOTE ]
The expression 1/(infinity) is not an approximation of a value, it is a bona fide finite value.

[/ QUOTE ]

Informally, 1/infinity=0.

[kpux]

[ QUOTE ]
Anyway... I'll try to do a better job of making my case.

If I ask you to sum up an infinite series of numbers (say, oh what the hell, .1/2 + 1/4 + 1/8 + ...) and write it down as a real number, you can't do it. I believe you have already acknowledged this.

[/ QUOTE ]

1

[ QUOTE ]
What you can do is write down some "weird symbol" which represents the sum. You can then stick this symbol into some formula and come up with "...clear and irrefutable mathematical proof."

This is where you lose me... and leave me unconvinced. You use an imaginary number to represent something that - in the real world - can't be... doesn't exist (that is, the sum of infinity). To me, all you've done is prove something about an imaginary world.

[/ QUOTE ]

There are no weird symbols involved here at all. Infinite series are completely valid mathematical objects. The one we need to use in this example sums to 1. I think your problem is with the fact that we are summing an infinite number of rational numbers. But this is perfectly valid.

[ QUOTE ]
I found this with a few searches:

"In math, when you hear people say things like "1 over infinity is zero" what they are usually referring to is something called a limit. They are just using a kind of shorthand, however. They do NOT mean that 1 can actually be divided by infinity. Instead, they mean that, if you divide 1 by successively higher numbers, the result becomes closer and closer to 0. If I divide 1 by a very large number, like a billion, then I get one-billionth, which is a VERY small number, but it isn't 0. Since there is no largest number, I can always divide 1 by a bigger number. But that will just produce an even smaller number, right? It will NEVER produce 0, no matter how high I go. But since the answer to the division is getting closer to and closer to 0, we say that "the limit of the expression is zero." But we have still not divided anything by infinity, since that isn't a number."

Well, hell. That's cheating. Once you operate with a "limit" of course you are going to get a finite result.

I'm sure that this type of "simplifying assumption" (as we call them in business models), works quite well in lots of analytical situations. I just don't buy it in this one.

[/ QUOTE ]

It doesn't matter whether you buy into it. Limits are concrete mathematical tools and must be accepted as so. They aren't cheating, they are a way to solve exactly the kind of problem that you posed.

Limits aren't a simplifying assumption. They exist. Without the development of limit theory there would be no calculus.

[ QUOTE ]
In the real world, I get to keep dividing 1 by larger and larger number into infinity and there will never be enough time to cover the distances represented.

[/ QUOTE ]

The distance represented is the same. Just because you can partition the number 1 into an infinite sum of rational numbers doesn't change the nature of the number 1.

[FNHinVA]

The infinite series of 1/2 + 1/4 + 1/8 + ... approaches (but can never reach) a limit of 1. So you are either going to use 1 or something very, very close to 1. It may indeed be a "bona fide finite value" but it is still an approximation of the actual (non-existent) value.

This article (http://mathforum.org/library/drmath/view/62486.html) at mathforum.org supports my view of infinity and how it is represented in calculations. If it is not satisfactory, I can come up with others. I saw many that said essentially the same thing.

There is simply no way any value can be anything other than an approximation of infinity.

[kpux]

[ QUOTE ]
The infinite series of 1/2 + 1/4 + 1/8 + ... approaches (but can never reach) a limit of 1. So you are either going to use 1 or something very, very close to 1. It may indeed be a "bona fide finite value" but it is still an approximation of the actual (non-existent) value.

[/ QUOTE ]

The sum of this series is equal to 1 and is defined as 1. It's not like the numbers are adding themselves up and they just can never get to 1 because there isn't enough time.

[ QUOTE ]
There is simply no way any value can be anything other than an approximation of infinity.

[/ QUOTE ]

The article you posted, which is pretty good, clearly states that infinity is a concept and not a number. If infinity (I could use a sideways 8 key right now) isn't a number how could it have an approximation? I think you are confusing using infinity in calculations, which is nonsense, and performing an operation an infinite number of times, which makes perfect sense.

[BluffTHIS!]

FNH, you need to study up on what kpux is talking about regarding limit theory to understand these concepts better. And if you really want to boggle your mind, then study transfinite numbers as well.

[FNHinVA]

It's been entertaining but I am making a strategic withdrawal. Note that I am not surrendering.

I have spent way too much time on this and I need to get back to work. I also need to do some more reading.

We are clearly at an impasse. While I don't question the math being thrown at me, I don't understand its applicability to the issue at hand. I understand how limits work in calculus and other applications but I don't understand them in this context. I'll work on that.

I shall return (after some finite duration of time which cannot be divided into infinity no matter what you math geeks say).

It's been fun - gotta go....

[drudman]

Unfortunately, you've withdrawn, but in the hope that you will check back once more and read this, thing of it this way:

.9(repeating) = 1

There is no number that exists between those two numbers on the number line. It's a weird coexistence, but a valid one. It is similar in our infinite series.

[PairTheBoard]

I'm a little hurt that you feel I've been beating on you FNHinVA. I think you should be grateful that we've given you so much of our attention here.

You said,
"In the real world, I get to keep dividing 1 by larger and larger number into infinity and there will never be enough time to cover the distances represented."

This is the point I tried to make before. You say that you get to do this in the real world but then after you have done this division of the distance "into infinity" we don't "get to" add them up.

FNHinVA --
"If I ask you to sum up an infinite series of numbers (say, oh what the hell, .1/2 + 1/4 + 1/8 + ...) and write it down as a real number, you can't do it. I believe you have already acknowledged this."

The thing is they were already PreAdded before you gave yourself permision to create them "into infinity" and the number you would write down is the number "1".

When I pointed this out to you before you declared that what you really meant was that you really can't divide 1 up into those smaller and smaller pieces "into infinity", whereas above you say you do "get to" do this. This kind of thing is frustrating.

The thing is, this all depends on the kind of math you want to do. Some people object to the Axiom of Choice and do Math without it. At one time there were people who objected to the Real Numbers and insisted on using only the Rationals. You might be more comfortable with the Mathematics of Whole Numbers, although you might restrict yourself to finite sets of whole numbers since All the integers can't really be produced one by one.

usmhot really Can't prove that 1/2 + 1/4 + 1/8 + ... add up to 1 unless we assume it's ok to add up the series to begin with. Most Modern Measure Theory and therefore Probability Theory is based on the property of "Countable Additivity" ie. The Measure of the countable Union of Disjoint Measurable Sets is the Infinite Sum of the Measures of Each Set. But there are some Probabalists who insist on working with the more restrictive property of Finite Additivity ie. The Measure of the Finite Union of Disjoint Measurable sets is the Finite Sum of their Measures. Clearly you would be in this camp.

Your balking at infinite sums is not without merit because it is only countable sums which give us a powerful workable useful mathematics. If we try to go to sums of uncountably infinite many numbers we get nowhere.

So there are restricted forms of Mathematics that would work for you but they would not be nearly as powerful as the mathematics commonly used today.

PairTheBoard

[FNHinVA]

I' m baaack... and I've been reading stuff.

Check this out...

"Mathematicians thought they had done away with Zeno's paradoxes with the invention of the calculus and methods of handling infinite sequences by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century, and then again when certain problems with their methods were resolved by the reformulation of the calculus and infinite series methods in the 19th century. Many philosophers, and certainly engineers, generally went along with the mathematical results.

Nevertheless, Zeno's paradoxes are still hotly debated by philosophers in academic circles. Infinite processes have remained theoretically troublesome. L. E. J. Brouwer, a Dutch mathematician of the 19th and 20th century, and founder of the Intuitionist school, was the most prominent of those who rejected arguments, including proofs, involving infinities. In this he followed Leopold Kronecker, an earlier 19th century mathematician. It would be incorrect to say that a rigorous formulation of the calculus (as the epsilon-delta version of Weierstrass and Cauchy in the 19th century or the equivalent and equally rigorous differential/infinitesimal version by Abraham Robinson in the 20th) has resolved forever all problems involving infinities, including Zeno's.

[http://en.wikipedia.org/wiki/Zeno's_paradoxes#Status_of_the_paradoxes_today]

A more detailed treatment - with essentially the same observations and conclusions - can be found at:

http://www.mathpages.com/rr/s3-07/3-07.htm.

These articles are only two of many I found that made the same basic points.

Stuff I learned....

We can put aside the debatable and controversial issue of continuous vs. non-continuous space [the verdict is undecided; I will accept a draw on that... at least for the time being].

I was wrong in my conjecture that the sum to infinity was some "weird symbol." I now understand that it is an actual number. But I also understand it is a number that the actual sum to infinity approaches but never reaches. So in my mind it is still a symbol (or substitute) for something that doesn't exist. Just not weird looking.

[Aside: Please remember; business-trained... not science trained. I barely remember the intro calculus course they made me take before business school. And I hated it. I've been winging it here. Just thinking; no tools. Hell, when I put up the OP, I had never heard of Zeno's paradoxes. It was just an interesting problem that had occurred to me. I didn't know I was 2500 years late. If I had known about it, I would have just looked it up on the Web and never bothered you guys.]

Nonetheless, it turns out that it was not so unreasonable for me to challenge the applicability of calculus (and, specifically, the limit) to the problem. Even though I didn't know enough about it to properly express myself, I knew there was something troublesome going on there.

That is the issue I got pounded on - not one poster supported me - but now I discover there are eminent philosophers and mathematicians down with my position. Since I arrived at that position on my own and had to defend it by myself, I was - understandably I believe - very happy to find some company.

I have to wonder what the aforementioned eminent philosophers and mathematicians might have to say about this:

"This argument is wrong, because step 4.7 is invalid and is provably invalid. The proof is a clear and irrefutable mathematical proof. It does not involve the introduction of any weird symbols. It simply and clearly demonstrates WITHOOUT ANY DOUBT that to travel from A to B in a continuous space takes a finite time. There is NO ROOM for question, NO philosophical / logical / reasonable alternate arguments that can refute it."

Oh, really? Go tell that to my posse.

(I know... argumentum ad verecundiam - but against that statement, it's all I need.)

Sorry, usmhot, I couldn't resist.

I just had to make the point that I was not so unequivocally wrong as you made me out to be; I was simply on the other side of a controversial issue - with company. I'm sure you still disagree with me but at least I know I am not alone and now I know my position is... well... somewhat respectable. Cool.

So where does all of this leave us? Continuous vs. non-continuous space [the core issue] remains controversial and unresolved, due in no small part to the questionable application of the calculus limit. Blah...

Ironically, it appears we have come full circle. The answer by Hawking's assistant:

"...When things get that small, we can no longer measure them so we don't know what the hell is going on."

may still be the best answer.

Well, it's been interesting, some fun and very challenging. And I've learned a lot of stuff I really didn't need to know. Just shows what brute force analysis combined with ignorance can get you.

A damn headache, for one thing - I'm going to check the latest ME chip counts and go to bed.

Thanks, all...

Later.....

[FNHinVA]

It was certainly not my intent to offend. I was being a bit facetious. I had several guys coming at me from different directions all at the same time and all of them taking shots at something or another and I - in the discipline at hand - don't know what the hell I am talking about. So I definitely felt under siege. Yeah, I got lot's of attention... more than I could handle.

I will say this: nobody was mean-spirited or nasty and I do appreciate that. Some guys got (understandably) frustrated with me and showed it a little but, hell, nothing compared to some stuff you see on the forums.

Several of your points in this posting are addressed in my last posting ("RESTART...") which was done before I read this post.

"When I pointed this out to you before you declared that what you really meant was that you really can't divide 1 up into those smaller and smaller pieces "into infinity", whereas above you say you do "get to" do this. This kind of thing is frustrating."

Sloppy on my part; I get to try... I can't do it.

"Your balking at infinite sums is not without merit because it is only countable sums which give us a powerful workable useful mathematics. If we try to go to sums of uncountably infinite many numbers we get nowhere."

And, I was pleased to discover, I am not alone here. You will see how pleased I was with myself on this issue in my last posting.

"So there are restricted forms of Mathematics that would work for you but they would not be nearly as powerful as the mathematics commonly used today."

I understand this and do not dispute it in the least. But it is not really the point. I got accused of attempting to undermine calculus but that is just silly. Its usefulness in both everyday stuff and extraordinary stuff is beyond obvious. I just take exception to its applicability in this case. And - again - I am not alone.

Thanks much for your contributions. I do appreciate it.

Seeya down the road...

[FNHinVA]

WARNING: Math Dolt in the house.

If you read my last, last post, you will find that I am going with an appeal to authority argument (argumentum ad verecundiam), declaring a small, pathetic victory and getting the hell out of Dodge.

I am over-matched and out-numbered so it's all I've got.

I know it doesn't prove anything other than this: my intuitive sense that it is wrong to use the "limit" to measure infinite distances has support among better minds than are likely to be found around here. The opposite is also true; so there is an ongoing controversy.

After being told repeatedly and emphatically that I was wrong... wrong... WRONG, I see that as at least a partial vindication. That would be my small, pathetic victory.

I remain convinced that space:time is non-continuous; I can't prove it but I believe it. And nobody can prove I'm wrong. Another small, pathetic victory.

So... on my way out of Dodge, please... don't anybody shoot me in the back. It's over.

Thanks for your contributions. It's been... well... infinitely amusing.

seeyadowntheroad....

[malorum]

[ QUOTE ]
If you read my last, last post, you will find that I am going with an appeal to authority argument (argumentum ad verecundiam), declaring a small, pathetic victory and getting the hell out of Dodge.

[/ QUOTE ]

In addition to the latin terminology could u cite an appropriate bible qoute plz /images/graemlins/confused.gif

[FNHinVA]

Not a quote. I believe that is properly called a citation.

And usmhot started it... just following his lead.

If he starts quoting the bible in a logic & math context, I'll start quoting Britney Spears.

Should work...

[usmhot]

/images/graemlins/laugh.gif I would never quote the Bible in any context that demanded rational scientific discussion!!!

Incidentally, I have always supported your claim that in our Universe space is not continuous - but on the basis of our understanding of Quantum Mechanics.

"I know it doesn't prove anything other than this: my intuitive sense that it is wrong to use the "limit" to measure infinite distances has support among better minds than are likely to be found around here. "
I don't think you should be so quick to dismiss real live posters here over people who have written in other forums. There is no reason to assume that some of the posters here (potentially _even_ including yourself) aren't the intellectual equal of those you call 'better minds'.
I read the material you referenced (indeed one of the references is part of a book on relativity and I'm continuing to read the whole book), and I have to say it doesn't really add anything to your argument.

The point I would make about the mathematics of infinite series is this - as you rightly point out, and as is succinctly discussed in the material you referenced, the sum of an infinite series is never actually reached. However, what the mathematics does prove is that the sum CAN NEVER BE MORE THAN the limit. I.e. if you prove that the limit of an infinite series is, e.g., 1 then you prove that adding the terms for infinite time will never give a value greater than 1.
In other words, the proof shows that you don't actually have to add all the terms for ever because you know that the answer can't be larger than a specific value.
In the case of the motion from A to B using half distances, the maths proves that the time taken can never be larger than some specific value - which, coincidentally enough, happens to be the distance divided by the velocity.

Undoubtedly, there are some very useful mathematical systems which limit themselves to finite sets, but a 'feeling' or 'intuition' about something is irrelevant in any formal system. And, just because some professional mathematicians work within finite systems doesn't mean that their systems are any more 'real' than infinite systems. Mathematics, in all its forms, is an abstraction, the results of which are often applied to the Universe as we see it to enable us to understand and manipulate it better. And, incidentally, mathematics of finite sets is merely a subset of mathematics of infinite sets.

What the mathematics behind the infinite series says is that 'in a Universe in which space is continuous, motion from one point to another is possible'. This means that you can't use the paradox you started with to deduce that space in our Universe isn't continuous.

Be content that there is other evidence to suggest that space in our Universe isn't continuous.

I accept your (unconditional) withdrawal (/ surrender) /images/graemlins/grin.gif

[Darryl_P]

[ QUOTE ]
I remain convinced that space:time is non-continuous; I can't prove it but I believe it. And nobody can prove I'm wrong.

[/ QUOTE ]

FWIW I agree with you. There is a level of smallness beyond which we can never acquire reliable information because we are limited by the size and speed of the photons which transmit the information to us. We also know that our attempting to observe it actually changes it so we can never be just observers -- like it or not we are participants in the event also.

The quantum mechanics part of the explanation was to remind you that this type of stuff happens when the measurements get small enough.

Besides, infinity is a man-made concept, so to ask the question in the way you did you are bringing in elements which are not part of the observable, physical world. If you bring them in, you must do so with a definition, so to get out of it you need to refer to the same definition.

[FNHinVA]

"I would never quote the Bible in any context that demanded rational scientific discussion!!!"


I didn’t really think so. Nor would I. Hence the reference to the airhead Britney Spears. Anything she might say would be about as relevant.


"Incidentally, I have always supported your claim that in our Universe space is not continuous - but on the basis of our understanding of Quantum Mechanics."


I understand that - always did. This line of discussion was always tangential.


"["I know it doesn't prove anything other than this: my intuitive sense that it is wrong to use the "limit" to measure infinite distances has support among better minds than are likely to be found around here. "] I don't think you should be so quick to dismiss real live posters here over people who have written in other forums. There is no reason to assume that some of the posters here (potentially _even_ including yourself) aren't the intellectual equal of those you call 'better minds'."


I was attempting to inflate the credibility of the "authorities" (my posse) referenced in my "appeal to authority" argument. And I wasn't referring to "people who have written in other forums."; I was referring to philosophers and mathematicians with funny, hard to pronounce names (that always helps) most of whom are long dead (that helps even more). This is vitally important since the argument alone proves nothing; the stature of the cited authorities makes the argument sound more plausible (and we know how important that is to me). Indirectly denigrating the intellectual capabilities of 2+2 posters was unavoidable but necessary. It was dirty and lowdown but you made me do it.

And, I would never make an unfavorable comparison of 2+2 posters to those on other forums. Especially not the ones in this topic section; they are as bright or brighter than any I’ve seen anywhere. Mostly civil, too.

“potentially _even_ including yourself” - Hey… wait just a minute. I came up with Zeno’s key paradox on my own without ever having heard of it. And I took it to the same logical conclusions… in my spare time and outside my field. That’s why he’s in my posse and I’m not in his. (I was 2500 years late, but let’s not quibble.) potentially? even? I’m insulted. I expect an immediate apology.


"In the case of the motion from A to B using half distances, the maths proves that the time taken can never be larger than some specific value - which, coincidentally enough, happens to be the distance divided by the velocity."


Yes, the time taken can never be larger than some specific value (1 in our case) but it also can never BE 1. So we (conveniently) leap over infinity (wasn’t that fun!) to the limit and (coincidentally) arrive at a proof.

In business models, we call that a simplifying assumption; "well, it can never be greater than 1, but we know it isn't 1, but it's really, really close to 1. Oh, what the hell, let's assume it's 1."

It is NOT 1 and we KNOW it is not 1 and we KNOW it can never be 1. But let's go ahead and use it anyway and call the result a "proof."

No. Never. Unacceptable.

I could never get away with that line of reasoning about a critical parameter in a business model. When we get that loosey-goosey with the inputs (and we frequently do); we have to qualify the outputs - attach probabilities to ranges of outcomes, that sort of stuff. But here we are talking about a theoretical proof; right or wrong… yes or no. I just don’t see how you get loose with the input in this context.

The proof fails using any value other than 1.

So how can we accept a proof that uses a number that we KNOW is not exactly right? [I just had an appalling thought: are business geeks more rigorous than math geeks?]


"Undoubtedly, there are some very useful mathematical systems which limit themselves to finite sets, but a 'feeling' or 'intuition' about something is irrelevant in any formal system."


Not fair. Out of context. My "intuitive sense" was simply what got me started. I did my homework after that and I think I've played the game fairly since. And surely you are not suggesting that 'feeling' or 'intuition' is irrelevant to discovery.


"...you can't use the paradox you started with to deduce that space in our Universe isn't continuous."


I can if I reject the "proof." And I just did, as others (my posse) before me have.


“Be content that there is other evidence to suggest that space in our Universe isn't continuous.”


I know there is. But that’s not the point here. We are on a tangent. This is about the “proof.”


“I accept your (unconditional) withdrawal (/ surrender)”


Say WHAT???

I withdrew to regroup. And I never surrendered. I conceded a draw on the original issue of continuous vs. non-continuous only because I became aware there are multiple, contradictory proofs out there and I don’t have the time or inclination to investigate and try to understand all of them. Basically, I am accepting Hawking’s assistant’s answer: "…when things get that small, we can no longer measure them so we don't know what the hell is going on."

Recap….

The discussion veered off to the issue of using the “sum to infinity” to prove an infinite series of discrete distances can be traversed in a finite amount of time.

I took issue with that. Then, every poster contributing to the thread at that time – half a dozen or so - disagreed with me. Every one of them. None more emphatically than you.

So I withdrew to do some homework – something I clearly should have done earlier.

What I found is that competent and well-regarded philosophers and mathematicians (my posse) have taken – and take – exactly the position I take. They reject the proof based on the handling of the sum to infinity.

All of which proves nothing except this: my position is reasonable, has merit and it is defensible. It cannot be dismissed out-of-hand, as it was.

Which gets us to what this is really all about….

I took your response to me on the issue as more than a little disrespectful. I won’t re-quote it here; I think you know what I’m talking about. Let’s just call it your little OTTR [Over-The-Top-Rant] for short. It was as though you were screaming in my face: “PAY ATTENTION, YOU MORON. CAN’T YOU GET THIS?”

I know you were (understandably) exasperated with my mangling of the terminology and misunderstanding of certain concepts, but I don’t believe I deserved that.

I am way outside my field here and that should have been obvious and indicated a little patience was in order. I would also hope it was obvious I am not a moron in need of having some concept hammered into me. I may have been a bit slow out of the blocks but I think I “get it” well enough now to simply say I disagree with you on the issue and I am confident I have good reason to do so.

Very simply, it turns out that the essential message of your OTTR (“This is universal truth. It is beyond debate. Accept it or die.”) may… just may, be a little off the mark.

Hell, it’s plain wrong.

This is a case where “appeal to authority” works exquisitely… you say the matter is beyond debate… meet my posse…. (nonexistent) debate to follow. It demonstrates that “There is NO ROOM for question” is patently false. The simple existence of my posse is all I need to prove there IS ROOM for question.

So, no, I don’t surrender. I disagree.

Please feel free to retract your OTTR and acknowledge that my position – while not necessarily provably right - has merit and is worthy of respect. (Groveling apologies not required.)

If you choose not to… that’s OK. I know what I know and – in the immortal words of Tom Petty – I won’t back down.

Soooo….

Thank you very much for taking the time to help me along this little journey of discovery.

Sorry I made the ride bumpier than it needed to be. I should have done more homework but somehow this whole thing went further and faster than I anticipated.

And there are no hard feelings. The OTTR, taken in context (I was frustrating to deal with), was a bit annoying but understandable.

And, thanks to you challenging me, I actually now know a lot of stuff I didn’t know a few days ago. Hell, gotta be happy with that.

It’s been infinitely interesting. (I know… that was lame… and to make matters worse, I think I already used it once.)

Later…

[FNHinVA]

Yeah... if I understand what you're saying - and I'm not sure I do - I am back where I started.

Hawking's assistant's original answer: "…when things get that small, we can no longer measure them so we don't know what the hell is going on."

Guess I'll have to live with that.

Thanks...

[usmhot]

I think you're still missing the core point.
All the eminent mathematicians who reject the concept of infinity would still completely accept that if you assume infinity in the first place then the infinite series proof is unassailable.
What they say is they don't accept infinity at all - so they don't even start with an assumption such as 'space is continuous' because the very definition of 'continuous' involves infinity.

What you did was assume space is continuous and then use an argument against which there is a complete proof in that context that travel in continuous space in finite time is possible. You set up the context in the first place in which that proof is beyond reproach. I.e. if you start with the concept of infinity then you automatically have sums of infinite series.

This is exactly what I've been saying all along - in Universe in which space is continuous travel between any two points in finite time is possible.

Incidentally, some further rebuttal
1. when I said

"I don't think you should be so quick to dismiss real live posters here over people who have written in other forums. There is no reason to assume that some of the posters here (potentially _even_ including yourself) aren't the intellectual equal of those you call 'better minds'."

by "other forums" I meant publication in any manner whatsoever, and was referring directly to the eminent minds you spoke of. Just because someone is dead, has a difficult name and is published doesn't make him/her any more intelligent or insightful than those of us who are still alive.
And, I assumed you would realise I was being jocular, and even subtly complimentary with "(potentially _even_ including yourself)"

2. "No. Never. Unacceptable."

Isn't this a little extreme? - or closed mined? You're saying that no matter how convincing an argument is made in this case, you will not accept it because of your 'feeling'

3. "?I accept your (unconditional) withdrawal (/ surrender)?

Say WHAT???"

That was a joke.

4. "Hey? wait just a minute. I came up with Zeno?s key paradox on my own without ever having heard of it. And I took it to the same logical conclusions? in my spare time and outside my field. That?s why he?s in my posse and I?m not in his. (I was 2500 years late, but let?s not quibble.) potentially? even? I?m insulted. I expect an immediate apology."

I'm not sure from your tone here if you genuinely feel insulted - as I pointed out above there was no need to, but you'll react how you react. If you thought that was insulting then wait till you hear this lol ...
you are by no means the first to have independently come up with similar thoughts to Zeno's paradox, nor will you be the last. For example, when I was about 10 I came up with it and was then pointed to various (simpler) writings about it by my father. I'm sure there are many people with the same story.

You want another interesting one?

Imagine a train traveling at 100kph in one direction and a fly traveling at 10kph in exactly the opposite direction. They crash head-on. In going from 10kph in one direction to 100kph in the exact opposite direction the fly must, at some point, be at a standstill - 0kph - but at this point the train must also be at a standstill. So how is it that a fly can stop a train?

[FNHinVA]

I guess I should start using those damn little “smiley” things. Many of my comments were facetious or jocular.


"I think you're still missing the core point."


And I think you are avoiding my key question about your core point. I'll get to that.


“ if you start with the concept of infinity then you automatically have sums of infinite series.

This is exactly what I've been saying all along - in Universe in which space is continuous travel between any two points in finite time is possible.”


It has been my understanding all along that this assertion depends on a proof that depends on:

1/2 + 1/4 + 1/8 + [infinity] = 1

This is the gut issue for me. But you did not directly respond. I will quote myself to save time (insert smileys in appropriate places):

“Yes, the time taken can never be larger than some specific value (1 in our case) but it also can never BE 1. So we (conveniently) leap over infinity (wasn’t that fun!) to the limit and (coincidentally) arrive at a proof.

In business models, we call that a simplifying assumption; "well, it can never be greater than 1, but we know it isn't 1, but it's really, really close to 1. Oh, what the hell, let's assume it's 1."

It is NOT 1 and we KNOW it is not 1 and we KNOW it can never be 1. But let's go ahead and use it anyway and call the result a "proof."

The proof fails using any value other than 1.

So how can we accept a proof that uses a number that we KNOW is not exactly right?”


Please explain to me where you think I am going wrong here.



“Just because someone is dead, has a difficult name and is published doesn't make him/her any more intelligent or insightful than those of us who are still alive.”


Smiley missing. I was being a bit facetious.



“And, I assumed you would realise I was being jocular, and even subtly complimentary with "(potentially _even_ including yourself)"”



I did. I responded in kind (mock outrage at the "insult")



“2. "No. Never. Unacceptable."

Isn't this a little extreme? - or closed mined? You're saying that no matter how convincing an argument is made in this case, you will not accept it because of your 'feeling'”


The operative term being: “…that no matter how convincing an argument…”

As for the “…, you will not accept it because of your 'feeling'.” I have offered reasoned argument for my position. You may not like it or accept it but to dismiss it as ‘feeling’ is gratuitous, out of context and grossly unfair. (And you should be shot for it.)


“3. "?I accept your (unconditional) withdrawal (/ surrender)?

Say WHAT???"

That was a joke.

4. "Hey? wait just a minute. I came up with Zeno?s key paradox on my own without ever having heard of it. And I took it to the same logical conclusions? in my spare time and outside my field. That?s why he?s in my posse and I?m not in his. (I was 2500 years late, but let?s not quibble.) potentially? even? I?m insulted. I expect an immediate apology."

I'm not sure from your tone here if you genuinely feel insulted…”



That was mock outrage. And a shameless attempt to affirm what a smart guy I am. (AKA: “Sklanskyism”)



“…you are by no means the first to have independently come up with similar thoughts to Zeno's paradox, nor will you be the last. For example, when I was about 10 I came up with it and was then pointed to various (simpler) writings about it by my father. I'm sure there are many people with the same story.”



Then I guess there is no point in mentioning figuring out the Doppler effect when I was 8? (Silly me… I just did.)



“You want another interesting one?”


No.


Always fun… Later…

[BTW: Am I to assume you refuse to retract your OTTR? It was offensive, demeaning, insulting and you should be shot for it.]

[K C]

The simplest way to explain this seeming paradox is this. This example is construed such that it is set up to measure an infinite series short of completion. It's certainly much more of a "riddle" than any scientific problem. The reason of course that the series does not lead to completion is that the possibility of such is not built in to the example.

Given A and B as points in space of course there are an infinite amount of points in between, just as there are infinite points in time. I can for instance postulate the same thing from the time it takes from this very moment to a minute later : 10:26 to 10:27 for instance. Yet oddly enough 10:27 did occur as I'm writing this. And it did not involve an infinite amount of time for this to occur.


KC

[FNHinVA]

Welcome to our little ininite thread on issues of infinity.

All of the points you make have been thrashed about pretty well. The issue now is use of the "sum to infinity" in a "proof." Here is the key question I am trying to get answered:

usmhot:

"...in Universe in which space is continuous travel between any two points in finite time is possible.”

FHNinVA:

"It has been my understanding all along that this assertion depends on a proof that depends on:

1/2 + 1/4 + 1/8 + [infinity] = 1

This is the gut issue for me. But you did not directly respond. I will quote myself to save time (insert smileys in appropriate places):

“Yes, the time taken can never be larger than some specific value (1 in our case) but it also can never BE 1. So we (conveniently) leap over infinity (wasn’t that fun!) to the limit and (coincidentally) arrive at a proof.

In business models, we call that a simplifying assumption; "well, it can never be greater than 1, but we know it isn't 1, but it's really, really close to 1. Oh, what the hell, let's assume it's 1."

It is NOT 1 and we KNOW it is not 1 and we KNOW it can never be 1. But let's go ahead and use it anyway and call the result a "proof."

The proof fails using any value other than 1.

So how can we accept a proof that uses a number that we KNOW is not exactly right?”


Please explain to me where you think I am going wrong here."

K C:

??


Feel free to join in...

Thanks...

[usmhot]

Right, now that I understand your tone a little better and can see that you are being jocular in various places I feel a little more comfortable.

"It has been my understanding all along that this assertion depends on a proof that depends on:

1/2 + 1/4 + 1/8 + [infinity] = 1

This is the gut issue for me. But you did not directly respond."

I've been avoiding using mathematics so far, as you said you didn't want to see any, but now you've introduced it so here goes - this is really quite straight forward, so don't be put off.
You've accepted that proving that the (upper) limit is 1 is valid (incidentally, in itself that's all that's needed and I'll explain why shortly). But, you're saying that the proof says its not quite 1 and is some 'imaginary' number other than 1 so you can't accept it. Well, actually, it's easy to prove that it can't be less than 1. So, in fact its easy to prove that the sum is exactly 1.

First thing is, you wrote

1/2 + 1/4 + 1/8 + [infinity] = 1

thats not quite right - it should be

1/2 +1/4 + 1/8 + ... + [an infinitesimally small value] = 1

And the difference in formulation is vital. We show that the sum is not less than 1 by simply pointing out that whatever value less than 1 you get to, no matter how close to 1 it is does not complete the series - there is always another tiny value that can be added to it to bring you even closer to 1. In other words, we assume that the sum is a value less than 1 and show that there is no such value that it can be.

To put it succinctly, we can prove that the sum of the given series is exactly 1 - not some imaginary value, and not some value thats achieved by skipping out infinite terms. And the crux of the whole thing is this is all possible because you start with the assumption that space is continuous.
Starting with an assumption that space is continuous does not lead to a paradox in the form proposed by Zeno.

Anyway, back to the point that showing that the upper limit is 1 is sufficient - remember, your original problem was that you can't get from A to B in finite time. You got around to accepting that the upper limit of the series is 1 - i.e. that the result is at very most 1 - but even if its some silly little value less than 1 that means its clearly finite, so you travel the distance in a clearly finite time.

Finally, and perhaps most importantly, I will stand against a wall and allow you to personally shoot me for my grievous slights against your character, if and only if you can prove irrefutably, either that space is not continuous or Santa Claus doesn't exist.

[ChromePony]

FNHinVA

Ive been following this thread for a while now and feel like its about ime to chime in. I think the fundamental disagreement here is whether or not this infinite sum is actually equal to 1. Well from a mathematical standpoint it quite clearly is, exactly 1, not something close or approximate. I tutor freshman calculus and this is always a hard concept for them to grasp, the idea that an infinite number of terms can add up to such a simple number, but thats the beauty of math.

Now I feel that you may be willing to accept this mathematical side of this but assert that in real life you could never actually add up all of these terms, and therefore practically speaking you will never actually get to 1. But you also talk about dividing the distance between two points into an infinite number of intervals, can you see the contradiction here. You allow yourself to divide something into an infinte number of pieces but dont allow yourself to use the available tools to add them all together. Just as in real life you could never physically add an infinite number of terms one at a time, you could also never physically divide space an infinite number of times. Therefore both are abstract, theoretical concpets, and the situation must be dealt with accordingly. If you allow yourself to visualize space as infinitely divideable you must also allow yourself to use the proven mathematical methods that have been developed to deal with such situations.

I hope you dont feel that you are being attacked by any of the posters here, I think this is a great discussion and you are handling the math nerds very well considering your lack of background in the area.

[BluffTHIS!]

umhot, you are wasting your time with him. He just doesn't seem to want to understand either limit theory, or the fact that in a travel anaology, the simplest case being a person walking, that the person's average stride will eventually be greater than the distance left.

[hobbsmann]

I think it would be helpful for you to take a look at some of the definitions that go into infinity and infite series:

Infinity (http://mathworld.wolfram.com/Infinity.html)
Geometric Series (http://mathworld.wolfram.com/GeometricSeries.html)
Continuum (http://mathworld.wolfram.com/Continuum.html)

Also, further browse mathworld for some more interesting stuff.

[drudman]

This thread is kind of like James Brown at this point. It keeps collapsing of exhaustion, and galantly being led off the stage, the cape draped over its shoulders, only for the music to swell and the cape to be cast off over and over again.

I feel good. Hey!

[PairTheBoard]

_________________________________ = 1

________________|________________ = 1
________________|________|_______ = 1
________________|________|___|___ = 1
________________|________|___|_|_ = 1
.
.
.
= 1


Or.

1=1
1/2 + 1/2 = 1
1/2 + 1/4 + 1/4 = 1
1/2 + 1/4 + 1/8 + 1/8 = 1
The sum = 1 as far out as you want to go. No matter how small you make the pieces the sum still = 1 and you can make the pieces AS SMALL AS YOU LIKE.

If you imagine doing this "ad infinitum"
you still have what you started with. 1.

PairTheBoard

[Darryl_P]

Very good post IMO...gets to the nuts and bolts of it.

FNH,

As I said before, infinity is not something in the observable world, so to ask a question involving it you must define it. Turns out the same definition that lets you divide an interval into infinitely many parts also lets you add up infinitely many numbers to EQUAL (not approximately) but EXACTLY 1 in this case, by definition.

When you say it adds up to "almost" one, you are talking about finite partial sums, not the infinite sum itself. Finite partial sums can never reach one, but the infinite sum can and does.

[FNHinVA]

I was doing some more searching on the Web and ran across some stuff that gave me a new way to think about this issue.

I believe I would be correct to say that there exist in abstract math, higher physics, etc., contradictory proofs that are – for all appearances - each valid but seem mutually exclusive… sort of a “can’t be… but is” deal. In fact, I believe you alluded to this in an earlier post.

It was inferred in some of the very materials I cited:

“Nevertheless, Zeno's paradoxes are still hotly debated by philosophers in academic circles. Infinite processes have remained theoretically troublesome.”

“It would be incorrect to say that a rigorous formulation of the calculus … has resolved forever all problems involving infinities, including Zeno's.”

I started from (using a convenient reformulation I ran across):

0.9r .NE. 1

(where .NE. can be either “NOT EQUAL” or, (my fave) “NEVER EQUALS.”)

This… to me… is a “self-evident truth.” It requires no proof. I can keep adding 9s until the end of time (and we don’t want to go there) and it will always be true that 0.9r .NE. 1.

But, I now understand that you (and your fellow-travelers – I’d like to shoot some of them too) can use “theoretically troublesome” abstract concepts and “prove” that 0.9r = 1. (And BTW, I don’t blame you; who wants to wait until the end of time to find out what 0.9r comes out to? You guys have stuff to do and need answers NOW.)

This abstract math stuff is your world; you get to make up the rules… even if you can’t always agree on them among yourselves.

I accept that in your world, 0.9r = 1 and that you can prove it. I admit I don’t fully understand some of the concepts used but I’m much closer to “getting it” than I was before I (foolishly) started this thread.

BTW: One “proof” I ran across used set theory to eliminate all numbers smaller than 1 (the limit I acknowledge can’t be exceeded) and leave no possibility other than 1. To the extent I understood it, I liked it better than yours. You are free to feel insulted.

The problem I’ve had is that I was obstinate about rejecting the abstract math proof because it was threatening to my understanding of real stuff. In other words, both can’t be right; I have to reject yours.

Now, as I see it, we have “contradictory proofs” and I can accept it because that kind of stuff happens. It “can’t be… but is.” Both can be right on their own terms and in their own worlds. I just have to accept that you get to do things in your world that I can’t do in my world.

I see the merit in yours; but I’ll keep mine. The calculus works wonderfully in the practical world so that’s cool too. Long live the “limit.”

So it seems futile for me to point out where – in your last post – it still seems to me that you are “jumping the infinity” or “terminating the infinity” or making “simplifying assumptions.” (Although, I have to mention I did get a little chuckle from: “…the upper limit is 1 is sufficient.”) I would just be flogging the “theoretically troublesome” horse and nobody needs that.

In closing…

“Finally, and perhaps most importantly, I will stand against a wall and allow you to personally shoot me for my grievous slights against your character, if and only if you can prove irrefutably, either that space is not continuous or Santa Claus doesn't exist.”

I don’t have to prove anything. If I ever get to Ireland (and I hope to) I’m going to hunt you down like the math dog you are and shoot you anyway. I don’t need no stinkin’ proofs.

And, I KNOW Santa Claus exists; I see him every year at the Mall.

I don’t have to prove space is not continuous; I BELIEVE in the non-continuity. It’s my new religion. I call it Planckism.

Every Full Moon, we will gather around a bonfire of math textbooks and dance nude until the Sun come up. (Although we have no understanding of how that Sun thing works.)

Always fun…

Seeyadowntheroad…

[FNHinVA]

I posted another response to usmhot before I read yours.

If I understand what you are saying, I believe I acknowledge your essential points. To some degree... possibly... maybe... hell, I don't know.

Thanks for your input and thanks esp. for the (rare) kind words.

[FNHinVA]

My last post to usmhot went up before I read this one so - if you are still interested in how my understanding is ever-evolving - check it out. Thx...

[FNHinVA]

Love it... very funny.

About how I feel... I have tried a few times to "... get off stage."

[FNHinVA]

Hey Pair...

I thought you were pissed off at me.

Anyway, your offering is too trivial for me now; I'm on to set theory and other stuff I REALLY can't understand.

Oh, and Planckism... It's the next BIG THING.

Thx...

[PairTheBoard]

[ QUOTE ]
Hey Pair...

I thought you were pissed off at me.

Anyway, your offering is too trivial for me now; I'm on to set theory and other stuff I REALLY can't understand.

Oh, and Planckism... It's the next BIG THING.

Thx...

[/ QUOTE ]

Set Theory eh? Better be careful with that Axiom of Infinity. That's the rabbit hole my friend.

PairTheBoard

[FNHinVA]

I am intellectually fearless. (Translation: too dumb to know when to stop.)

[usmhot]

"Now, as I see it, we have ?contradictory proofs? and I can accept it because that kind of stuff happens. It ?can?t be? but is.? Both can be right on their own terms and in their own worlds. I just have to accept that you get to do things in your world that I can?t do in my world."

Aye, but there's the rub. If you assume that 'infinity' is meaningless and doesn't exist, then you automatically reject the limit proofs - your world.
BUT, you started by assuming _infinity does exist_ so you came into my world - in which the proofs are irrefutable.

"I don?t have to prove anything. If I ever get to Ireland (and I hope to) I?m going to hunt you down like the math dog you are and shoot you anyway. I don?t need no stinkin? proofs."

Unfortunately in Ireland, that will make you just one more gun toting fanatic using a form of religion to justify his cause. I suggest you come in peace /images/graemlins/smile.gif.

"And, I KNOW Santa Claus exists; I see him every year at the Mall."

Hey!!! He's in Shopping Centres (read Malls) here every year too - do you think maybe there are infinite Santa Clauses?

"Every Full Moon, we will gather around a bonfire of math textbooks and dance nude until the Sun come up. (Although we have no understanding of how that Sun thing works.)"

You bring the chicks - I'll bring the math textbooks /images/graemlins/wink.gif


On a more serious note ...
"I don?t have to prove space is not continuous; I BELIEVE in the non-continuity. It?s my new religion. I call it Planckism."

Leaving all the maths stuff aside, you are subscribing to the Quantum Theory by believing that space is not continuous. (I myself am fully convinced of the results of Quantum Theory, though there remains a lot to be discovered.) However, you realise that the Quantum Universe is even more strange and counter-intuitive than the continuous Universe. If you're going to explore it then you'll have to be prepared to accept things that run contrary to your macro-level experiences and logic - so discounting infinity based maths will have to be the last time you get to do that on an 'intuition'.

[K C]

There's been countless posts after this and you'll have to excuse me as I just jump in here again.

What we're doing here is making infinite divisions of a fixed value. This doesn't speak at all to the fixed value though of course. We really don't need to go any further with this, although it's fun to do sometimes /images/graemlins/smile.gif

The fallacy of this paradox is of course seeking to objectify it such that we're measuring both distance and time divisively. Naturally by doing this we'll never reach completion, but that's because it's been excluded as a possibility.

KC

[FNHinVA]

“… you came into my world - in which the proofs are irrefutable.”


Understand. Don’t like Math World. Math World strange place.


“… just one more gun toting fanatic …”


I resent that. I am a “gun-stored-in-a-safe-and-secure-readily-accessible-location fanatic.” But I will leave it at home. I no longer have an overwhelming urge to shoot you; you are less annoying when I tolerate you doing strange things in your strange Math World.


Re: Quantum Theory

I think I’ll skip it and go straight to String Theory.

I hear it has 20+ dimensions. That would make the BB&C (baffle befuddle & confuse) factor awesome.

One last question…

What do math mystics do when the limit is 0 (zero) and turns up in some bewildering complex of equations as a divisor? (n/0 = “mathematical absurdity”)

That must be inconvenient. Even in Math World, that must be frowned upon.

I’m guessing you reach into your quiver of “proofs” and draw out one that works in a more convenient manner. Or maybe a “creative adjustment” of the parameters. Hell, even Einstein used that one.

Please answer with minimal BB&C, if possible.

Thanks…

Later…

Afterthought: Google -> Results 1 - 10 of about 103,000 for mathematical absurdity.

I was surprised it was so low. But then even Google doesn’t do infinite searches. It would take too long. We are not safe though... Google is full of PhD Math Mystics so I'm sure searching infinity is a trivial challenge to them. If it can be imagined, a Math Mystic has already done it... in theory.

[usmhot]

"I think I?ll skip it and go straight to String Theory.

I hear it has 20+ dimensions. That would make the BB&C (baffle befuddle & confuse) factor awesome."

My understanding was it was pretty much settled on 11 dimensions, though 27 (I think) was the other possibility.

Not at all confusing - you've seen one dimension, you've seen them all!!

"What do math mystics do when the limit is 0 (zero) and turns up in some bewildering complex of equations as a divisor? (n/0 = ?mathematical absurdity?)"

Its a very useful occurrence, in that it signifies the breakdown of your equation / theorem. What you do is go back to the beginning and find a proper solution. Mathematicians never sweep something under the carpet, and never invent something without rigorous justification.

Einstein was the first to admit that he was not a mathematician. And, indeed, he also admitted that adding a term to his equations to satisfy an intuitive contradiction was the biggest mistake of his life. See - there you go - one of the most brilliant men ever pointed out that giving into his intuitions at the expense of mathematical rigor was a mistake. See? /images/graemlins/grin.gif

Actually, throughout history, the most brilliant men and women have learned, time and time again, that when it comes to a choice between 'intuition' and pure, hard, rigorous mathematics, its the latter that turns out to be correct. The whole reason that Relativity and Quantum Theory came about is because they followed where the maths and logic lead them.

[FNHinVA]

OK, I lied. Next last question.

"...you started by assuming _infinity does exist..."

I'm not sure what you mean by this.

I thought what I was saying was consistent with a rather "conventional" dictionary-like definition:

"Unbounded space, time, or quantity; an indefinitely large number or amount."

As opposed the a mathematical definition:

"The limit that a function f is said to approach at x = a
when f(x) is larger than any preassigned number for all x sufficiently near a."

... whatever that means.

Irrespective of whatever it means, the key difference ... at least to me ... is that the former cannot be quantified while the latter can, thus defining the key difference between "Real World" and "Math World."

Am I somehow being inconsistent?

[PairTheBoard]

FNMinVA --
"Am I somehow being inconsistent? "

Consistently so.

PairTheBoard

[FNHinVA]

usmhot is safe. But now I want to shoot you, PairTheBoard

[usmhot]

Its very difficult to answer that without appealing to mathematical rigor but I'll try ...

First the conventional dictionary-like definition that you quoted is not quite appropriate - or needs to be reworded to fit the problem you're addressing.

You start with two points - A and B - a finite distance apart. You assume that, as space is continuous, you can divide the distance by 2 repeatedly into an infinite number of points each of which must be passed. Now, that is a bounded infinity, as opposed to unbounded. It is clearly bounded as you started with the bounds - A and B. (Technically, its a 'closed bounded' infinity as you're including the two points - if you didn't include them it would be an 'open bounded' infinity.)

Anyway, the mathematical definition of a bounded infinity is that given the bound (B) and any arbitrary point (x where x is not B) there is at least one more point (y) such that the distance from x to B is greater than the distance from y to B.
Now, you need to convince yourself that this makes sense to you and that it says what you are getting at.
You are taking A and B and saying you'll start with the halfway point as that x point, but there's a point which is halfway from x to B. So, now you'll take that point as your point x and there's another point halfway , etc.
This is really the same as the mathematical definition that I gave you and its within this definition that "sum of series" maths is totally consistent.

In short, you're starting with the assumption that there are an infinite number of points within a finite distance / space. This puts you firmly within the system of mathematics that includes sum of series proofs.

[usmhot]

Incidentally, my deepest gratitude to you for this discussion. It has been many years since I've had to think this rigorously about mathematics, and its reawkening my appreciation for the beauty and awe of it.

[FNHinVA]

You are very welcome. You have been very generous with your time and I appreciate that. As I said before, I know stuff I didn't know a few days ago. Gotta be happy with that.

With respect to your last post/answer: I believe I essentially understand it but - as always - it leads to more questions.

But this has to end sometime. So I'll do some more reading & Googling on my own and let you get back to whatever it is you do when not tutoring math dolts.

Thanks again...

Later...

[spoohunter]

You guys are all really dumb, in a smart kind of way. I apologize if the answer has already been said.

You must move a minimum distance at one point. You are incapable of moving less than this distance. Problem solved.

[FNHinVA]

Apology accepted.

This thread is over.

Go to "Some More Infinite Series Jive" and tell them:

"You must move a minimum distance at one point. You are incapable of moving less than this distance. Problem solved."

Fun will ensue...

[K C]

Hehe. That's the paradox of this other problem though. Intuitively we would assume that.

KC

i think a good theory is that space a "quanta" just like other things and that there aren't infinite halves

[kpux]

This whole thread makes me very angry, as I answered the OP in the second post.

Well it would be possible is space had a certain quanta like other things in quantum mechanics i don't know if this is true and it kind of sounds like a dumbed down version of what the grad studendt said about planck but maybe thats what you were going for