Infinity Paradox and contineous variables in physical quantities

[black_hole]

The sum of the series is 1.

 

But even if it were only a "tendency"' date=' that wouldn't matter. The point is that it is [b']finite[/b]. You seem to be thinking that crossing the distance would take an infinite amount of time because it would take an infinite amount of time to add the numbers. But that is not right, because the series is the time it would take to cross the distance. It doesn't matter that it has an infinite number of terms, and it doesn't matter what real number it converges to. As long as it is finite (which it is: even if you don't believe the series sums to 1, it' s certainly less than, say, 1.1), then the time to cross the distance is finite.

 

Mr. Tom,

 

I think it is only a tendency. The series only tells IF 1 is divided by 2, then by 2, and so on up to infinity THEN the answer would be 1. This 1 answer is the meaning that answer CANNOT be more than 1 and it will be 1 ONLY AND ONLY IF process of dividing by 2 has been performed up to infinity.

 

Thus the formula assumes that level of infinity has reached.

 

 

Now the series is just like that;

 

 

{1/2+1/4+1/8.......+1/(infinity-1)+1/infinity} = 1

 

if 1/infinity is equal to zero (i.e. any number divided by infinity has to be zero), it means the answer of 1 we already have achived at the level of 1/(infinity-1).

 

But (infinity-1) is such a term which is possible to occur only if the infinity is a finite number.

 

So if infinity is in fact a finite number, then this formula shall be applicable to a real situation, other wise it is applicable only to an assumed situation that "infinity is a finite number."

 

This formula cannot be proved experimentally. We shall never be able to reach the final term i.e 1/infinity which would be equal to zero. On an unlimited capacity calculator, we always find smaller and smaller answer and we shall never find the answer coming to be exact zero i.e. indication of final term which is infinity. We however, shall find answers nearer and nearer to zero.

 

A formula which is incapable of being experimentally proved, is it all right...???

 

Note: I only have said that sum of series shall remain less than 1 and it shall never reach to the figure 1 due to involvement of infinite terms.

 

Your point still stands that less than 1 is also a finite number, so the finite time is required to cross the distance.

 

I think I have to deeply consider this situation (which is new for me and thanx for that), in order to make comprehension of just how this finite number i.e. less than 1, is the time required to cover the distance.

[black_hole]

Mr. Tom,

 

I am still waiting the answer of:

 

If linear motion of car is the result of rotation motion of its wheels, then why is that rotation motion is quantized but linear motion, which is directly linked to that quantized rotation motion, is not quantized...???

[AntiMagicMan]

We do not need to prove experimentally maths. Maths is not an experimental science. In fact you are right that we shall never reach the number 1, but mathematical limits are not expressed in that way, the true definition of a limit is that you can get as close as possible to the limit to within any degree of accuracy.

 

As for your next post, it is widely believed that eventually space time and therefore motion will be quantised, the reason we do not talk of quantised space time is because we have no working theory. As has been shown above, these "paradoxes" do not need quantised space time to be explained.

[Tom Mattson]

I think it is only a tendency. The series only tells IF 1 is divided by 2' date=' then by 2, and so on up to infinity [b']THEN[/b] the answer would be 1. This 1 answer is the meaning that answer CANNOT be more than 1 and it will be 1 ONLY AND ONLY IF process of dividing by 2 has been performed up to infinity.

 

The summation of the series does involve taking a limit as n-->∞, but that does not mean that we can not say what the sum is. The sum of the series literally equals 1, despite the fact that we can not manually perform all the steps in the addition.

 

So if infinity is in fact a finite number, then this formula shall be applicable to a real situation, other wise it is applicable only to an assumed situation that "infinity is a finite number."

 

No, the formula holds even with the correct definition of infinity.

 

This formula cannot be proved experimentally. We shall never be able to reach the final term i.e 1/infinity which would be equal to zero. On an unlimited capacity calculator, we always find smaller and smaller answer and we shall never find the answer coming to be exact zero i.e. indication of final term which is infinity. We however, shall find answers nearer and nearer to zero.

 

A formula which is incapable of being experimentally proved, is it all right...???

 

I am not making a scientific prediction with this series, I am telling you that the finite sum of the series is the reason that Zeno's paradox is not a paradox. In the argument leading up to the paradox, he assumes that the sum is infinite, when it is really finite.

 

Note: I only have said that sum of series shall remain less than 1 and it shall never reach to the figure 1 due to involvement of infinite terms.

 

Your point still stands that less than 1 is also a finite number, so the finite time is required to cross the distance.

 

OK, so we agree that Zeno was wrong.

[Tom Mattson]

If linear motion of car is the result of rotation motion of its wheels' date=' then why is that rotation motion is quantized but linear motion, which is directly linked to that quantized rotation motion, is not quantized...???[/quote']

 

Well in that case, the linear motion would be quantized, but not because spacetime is quantized. It would be quantized because of the constraint v=rw, where:

 

v=linear speed of the car

w=angular speed of the tire

r=radius of the tire

 

In the absence of such a constraint, we can have quantized angular momentum in continuous spacetime as in, for example, the electron in a hydrogen atom.

[black_hole]

We do not need to prove experimentally maths. Maths is not an experimental science. In fact you are right that we shall never reach the number 1' date=' but mathematical limits are not expressed in that way, the true definition of a limit is that you can get as close as possible to the limit to within any degree of accuracy.

 

As for your next post, it is widely believed that eventually space time and therefore motion will be quantised, the reason we do not talk of quantised space time is because we have no working theory. As has been shown above, these "paradoxes" do not need quantised space time to be explained.[/quote']

 

Yes maths do not need to be experimentally proved but its results shall be applicable to physical world when it is experimentally proved in a real i.e. actual situation.

 

Pure mathematics which is absolutele abstract in nature, if not required to be experimentally proved, so is also not applicable to real situations. Pure mathematics is a pure intellectual activity.

 

Zeno's paradoxed are concerned with a real situation and not with an abstract situation. So we should have a real solution for it. Abstract solution is sufficient for assumed condition that level of infinity has reached. But the real situation is that level of infinity cannot come in a real situation.

[black_hole]

The sum of the series literally equals 1, despite the fact that we can not manually perform all the steps in the addition.

 

No, the formula holds even with the correct definition of infinity.

 

Is it not the difference between pure abstract mathematics and a real situation. I mean pure mathematics can find the solution with the correct definition of infinity i.e. without any assumption, whereas, in a practical situation i.e. when we have to move a distance of 1, what maximum we can do is to reach more and more near to 1. This is also incorrect to say as I shall say in last paragraph.

 

I am not making a scientific prediction with this series, I am telling you that the finite sum of the series is the reason that Zeno's paradox is not a paradox. In the argument leading up to the paradox, he assumes that the sum is infinite, when it is really finite.

 

In your previous reply you said that the finite sum i.e either 1 or less than 1, whatever the case may be, this finite number is the time required to cover the distance 1.

 

I considered that point. I reached at the conclusion that this "finite" number is NOT the time required to cover the distance 1.

 

This "finite" number, which, in a real situation, shall always remain less than 1, (instead of time required) is the distance covered, which is "finite" i.e. less than 1. I have concluded this because series is the breakup of distance and not of time.

 

In this way, note that the finite distance of 1 has not been covered fully, i.e, the actual distance covered is still less than 1. It means that finite distance has not been covered fully. However, this is also incorrect to say as would become clear in my next paragraph.

 

OK, so we agree that Zeno was wrong.

 

As I already have said that in a real situation, the stage of smallest possible number shall not come. If we divide a real distance of 1 by 2 and continue the process, we shall get smaller and smaller distance. And the smallest possible distance shall never come.

 

Now according to Zeno's paradox, taking the sum of series is not important. Only thing required is to cover the smallest possible distance first, then we shall be able to cover the greater distances. Since smallest possible distance is not reachable in a real situation, we shall remain unable to cover the smallest possible distance. So to cover greater distances is out of question.

 

Also note that your formula also do not tell the smallest possible distance.

 

Sum of series is only the breakup of distance of 1. It has nothing to do with the time required to cover this distance because it has no relationship with time.

 

And the breakup of distance 1 has to be a total of 1 (i.e 1 = all breakups of 1), So the formula has no relationship with the time required. In simple terms it is only the breakup of 1. It is just like that breakups of 2 shall also be sumed at 2 and breakups of 3 shall be totalled at 3 and so on.

 

The point is that first of all we have to cover the smallest possible breakup of 1, 2 or 3 etc. We have no concern with the sum of breakups. we have interest only in finding out the value of smallest possible breakup.

 

Zeno's paradox also has nothing to do with the sum of series. Zeno's paradox is concerned only with covering the smallest possible distance first, so that we may be able to cover the greater distances.

[AntiMagicMan]

Yeah, but zeno's paradox is quite wrong because I walked to the shops today.

[black_hole]

Yeah, but zeno's paradox is quite wrong because I walked to the shops today.

 

And that is why it is called a "Paradox".

 

Walking to shops was recognized by Zeno. He was the disiple of Parmenides whose philosophy was that reality cannot be conceieved through senses.

 

Zeno's paradoxed intended to prove the theory of Parmenides. By giving the proof, through "reason", Zeno "proved" that information received through senses is mere "illussory" and in reality, any kind of motion do not exist.

 

The main essence of Greek Philosophy is the recognition of the superiority of "reason" over the "sense experience".

 

You walked to shops is your "sense experience" and the "reason" is that you cannot cover the minimum possible distance, so you are also unable to cover greater distances.

 

Now according to Greek Philosophy, "reason" is superior to "sense experience", so Zeno, denied the existence of any motion and held that walking to shops etc. is the illusion created by the senses.

 

The same position, i.e. superiority of "reason" over "sense experience" later on, also reflected in the philosophy of Plato in his theory of forms or ideas.

 

But if we consider your case i.e. walking to shops, it, in itself is not the proof against Zeno's arguments. Zeno's arguments are not consistent with our common sense experience. This important fact (As highlighted by you), can, however, serve as a guideline for us so that we should consider the "wrongness" of Zeno's arguments.

 

If we prove, through reason, that Zeno's arguments were invalid, then we shall have a solid base to believe in our sense experiences.

 

For this, as I have further considered that formula series showing the breakups of 1 totalling at 1. I now have reached a conclusion that this is an important argument in this respect.

 

The mathematical formula clearly indicates that all breakups, including the smallest possible, have a sum total of 1. This 1 is not the time required to cover the distance 1. But however, this exact 1, indicates that smallest possible distance is accountable. Because without accounting for the smallest possible distance, the answer could not become exact 1.

 

If smallest possible distance is accountable, as is clear in the exact 1 answer, it means it is also coverable.

 

This "reason" has two dimensions, i.e. Positive and Negative.

 

On positive side, it proves the possibility of covering shortest possible distance. Other positive aspact is that it is consistent with our sense experience.

 

On Negative side, it proves invalidness of Zeno's arguments by showing that Zeno's arguments are based on incomplete information, because the fact that breakups of 1 exactly equal to 1, was not considered by Zeno.

 

Therefore this, now, is not the case of "reason" versus "sense experience". Now it is the case of "reason based on incomplete information" versus "reason based on complete information". And reason based on complete information is also consistent with our sense experience.

 

In this way, Zeno's paradoxes were wrong, not just because I often go to shops. It is so because we have a better argument vailable which confirms our sense experience.

 

So thankyou very much Mr. AntiMagicMan and Mr. Tom for helping me in reaching that conclusion.

[AntiMagicMan]

Thankfully ancient greek philosophy has been discarded these days. A valuable lesson that sitting around and thinking is not the way to do science.

[black_hole]

Mr. AntiMagicMan,

 

Sitting around and thinking IS still very important activity. Planned experiments take their input from this activity. Einstien's both special theory of relativity and general theory of relativity are the product of just sitting and thinking/ imagining. He did not performed any experiment by himself. Only, his theories, later on, were confirmed by the experimental scientists.

 

Only thing added is that the conclusions of such thinkings, now, are required to be experimentally tested in order to get the status of "scientific theory".

 

You yourself said in one of your previous post that "Abstact Mathematics" need not be verified through experiments. Abstact Mathematics, as we know it today, also is the gift of those ancient Greeks.