Zeno's Racetrack Paradox

[Hippasus]

The paradox: Suppose a runner is supposed to cover a distance of one unit. First she has to get halfway across the unit. Then she has to get halfway across the remaining expanse of the unit. Then she has to get halfway across the new remainder. Ad infinitum. So the runner does not traverse the unit after all and change is unreal. Being and logic are real though (Zeno of Elea was a Parmenidean student).

The modern response: The sigma notation of Riemann sums suggests a summation of discrete terms coinciding with the interval [0,1] through a limiting process. We suppose the discrete summation of terms to morph into the continuity and magnitude constituted by the interval [0,1]. We logically require the notion of infinity, that is, a hypothetical abstract limiting process where an infinite number of arithmetical operations is permissible. This is a convergent geometric series where the individual terms are summed as the summation coincides with the interval [0,1]. The first term of this series is (1/2) and the common ratio of the series is (1/2). In expanded form the series would look as follows: [1/2 + 1/4 + 1/8 + . . . 1/2^n].

This is the general form of how this series is defined:

Riemann sum:
(Upper-case sigma), index: n=1,2,3, . . . infinity
(1/2)^n

This is saying the series equals the number one:

= (1/2)/(1-(1/2)) = interval [0,1]

We are only able to say this because the common ration of 1/2 is less than 1. Having a common ratio less than 1 makes the series convergent. Often series diverge to infinity. This one does not, although it involves an infinite number of operations.

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[JMCx4]

Why was the runner a "she"?

 

[Hippasus]

Why was the runner a "she"?

I wanted to go with the other sex to balance things out a bit and to picture a girl or woman running rather than a guy.

 

[JMCx4]

I wanted to go with the other sex to balance things out a bit and to picture a girl or woman running rather than a guy.

There are a LOT of "other sex" choices these days.

 

[Hippasus]

There are a LOT of "other sex" choices these days.

Okay, so what do you think about the Parmenideans? They seem a bit out to lunch to me, to be honest, in the sense that they ascribe the highest designation of reality to being or the absence of number. Yet they are enormously influential for Western Civilization through ontology. They were perhaps the ancient equivalents of the idealists (Berkeley), as opposed to the empiricists (Hume) or the transcendental idealists (Kant).

 

[tarheelhockey]

The paradox: Suppose a runner is supposed to cover a distance of one unit. First she has to get halfway across the unit. Then she has to get halfway across the remaining expanse of the unit. Then she has to get halfway across the new remainder. Ad infinitum. So the runner does not traverse the unit after all and change is unreal. Being and logic are real though (Zeno of Elea was a Parmenidean student).

The modern response: The sigma notation of Riemann sums suggests a summation of discrete terms coinciding with the interval [0,1] through a limiting process. We suppose the discrete summation of terms to morph into the continuity and magnitude constituted by the interval [0,1]. We logically require the notion of infinity, that is, a hypothetical abstract limiting process where an infinite number of arithmetical operations is permissible. This is a convergent geometric series where the individual terms are summed as the summation coincides with the interval [0,1]. The first term of this series is (1/2) and the common ratio of the series is (1/2). In expanded form the series would look as follows: [1/2 + 1/4 + 1/8 + . . . 1/2^n].

This is the general form of how this series is defined:

Riemann sum:
(Upper-case sigma), index: n=1,2,3, . . . infinity
(1/2)^n

This is saying the series equals the number one:

= (1/2)/(1-(1/2)) = interval [0,1]

We are only able to say this because the common ration of 1/2 is less than 1. Having a common ratio less than 1 makes the series convergent. Often series diverge to infinity. This one does not, although it involves an infinite number of operations.

I'm not familiar enough with the math to express it in those terms.

However, operating from a common sense level there doesn't seem to be a paradox here at all. Yes, the runner must cover 1/2 of the Unit (and its remainders) before the whole can be completed. But that doesn't imply that the runner can only cover 1/2 of any given Unit (or remainder).

Quite simply, the Unit is only 1/2 of a larger distance (let's call it 2U). Even though the runner's goal might be to complete the race at 1U, she is still in effect running 1/2 of 2U. There is no distinction between those measures. It is just as rational for her to complete the segment [1/2 of 2U] as it is for her to complete the segment [1/2 of 1U]. When she completes [1/2 of 2U], she has reached her goal of running 1U. Nothing regresses ad infinutum.

So there's only a paradox involved if the Unit is infinite, but that's intuitive from the fact that there is no such thing as 1/2 of infinity.

 

[Hippasus]

I'm not familiar enough with the math to express it in those terms.

However, operating from a common sense level there doesn't seem to be a paradox here at all. Yes, the runner must cover 1/2 of the Unit (and its remainders) before the whole can be completed. But that doesn't imply that the runner can only cover 1/2 of any given Unit (or remainder).

Quite simply, the Unit is only 1/2 of a larger distance (let's call it 2U). Even though the runner's goal might be to complete the race at 1U, she is still in effect running 1/2 of 2U. There is no distinction between those measures. It is just as rational for her to complete the segment [1/2 of 2U] as it is for her to complete the segment [1/2 of 1U]. When she completes [1/2 of 2U], she has reached her goal of running 1U. Nothing regresses ad infinutum.

So there's only a paradox involved if the Unit is infinite, but that's intuitive from the fact that there is no such thing as 1/2 of infinity.

There is indeed no paradox if one ignores or denies the possibility of infinitely many arithmetical operations like many of the ancient Greeks did. Common sense or intuition suggests that continuity and real time is no problem but rather given. Calculus, it seems to me from my student perspective, bridges the gap between the discrete (the counting numbers) and the continuous (the real number line). I think it is somehow especially significant because of its emphasis on arithmetic, algebra, and trigonometry as opposed to geometry by itself. So numbers and formulae rather than images are deemed to be the proper medium of mathematics in modern times. The infinite is implicit in geometry through the real number line, but it has to be imported on the side of arithmetic through the rigorization of calculus. The latter occurred in the late nineteenth century through the work of Dedekind and Weierstrass. Physics, engineering, and even computer science are informed or influenced by calculus.

 

[Eisen]

The paradox: Suppose a runner is supposed to cover a distance of one unit. First she has to get halfway across the unit. Then she has to get halfway across the remaining expanse of the unit. Then she has to get halfway across the new remainder. Ad infinitum. So the runner does not traverse the unit after all and change is unreal. Being and logic are real though (Zeno of Elea was a Parmenidean student).

The modern response: The sigma notation of Riemann sums suggests a summation of discrete terms coinciding with the interval [0,1] through a limiting process. We suppose the discrete summation of terms to morph into the continuity and magnitude constituted by the interval [0,1]. We logically require the notion of infinity, that is, a hypothetical abstract limiting process where an infinite number of arithmetical operations is permissible. This is a convergent geometric series where the individual terms are summed as the summation coincides with the interval [0,1]. The first term of this series is (1/2) and the common ratio of the series is (1/2). In expanded form the series would look as follows: [1/2 + 1/4 + 1/8 + . . . 1/2^n].

This is the general form of how this series is defined:

Riemann sum:
(Upper-case sigma), index: n=1,2,3, . . . infinity
(1/2)^n

This is saying the series equals the number one:

= (1/2)/(1-(1/2)) = interval [0,1]

We are only able to say this because the common ration of 1/2 is less than 1. Having a common ratio less than 1 makes the series convergent. Often series diverge to infinity. This one does not, although it involves an infinite number of operations.

The paradoxon can be solved by adding the Planck distance. You can never go under a Planck distance as it is the smallest definable distance. So the additions are not infinite but have a maximum number, namely the unit divided by the Planck distance.
Use physics to crush logics.

 

[Eisen]

I'm not familiar enough with the math to express it in those terms.

However, operating from a common sense level there doesn't seem to be a paradox here at all. Yes, the runner must cover 1/2 of the Unit (and its remainders) before the whole can be completed. But that doesn't imply that the runner can only cover 1/2 of any given Unit (or remainder).

Quite simply, the Unit is only 1/2 of a larger distance (let's call it 2U). Even though the runner's goal might be to complete the race at 1U, she is still in effect running 1/2 of 2U. There is no distinction between those measures. It is just as rational for her to complete the segment [1/2 of 2U] as it is for her to complete the segment [1/2 of 1U]. When she completes [1/2 of 2U], she has reached her goal of running 1U. Nothing regresses ad infinutum.

So there's only a paradox involved if the Unit is infinite, but that's intuitive from the fact that there is no such thing as 1/2 of infinity.

For instance, take the natural numbers. They are infinite. You can still make a set of even numbers and one of uneven numbers. Both are infinite but, logically, they are half as big as the natural numbers.

 

[Eisen]

There is indeed no paradox if one ignores or denies the possibility of infinitely many arithmetical operations like many of the ancient Greeks did. Common sense or intuition suggests that continuity and real time is no problem but rather given. Calculus, it seems to me from my student perspective, bridges the gap between the discrete (the counting numbers) and the continuous (the real number line). I think it is somehow especially significant because of its emphasis on arithmetic, algebra, and trigonometry as opposed to geometry by itself. So numbers and formulae rather than images are deemed to be the proper medium of mathematics in modern times. The infinite is implicit in geometry through the real number line, but it has to be imported on the side of arithmetic through the rigorization of calculus. The latter occurred in the late nineteenth century through the work of Dedekind and Weierstrass. Physics, engineering, and even computer science are informed or influenced by calculus.

If you want to deal with it mathematically, you could simply use the lowest quality (or level, I don't know the correct terms, I studied this all in German, so forgive me) of proof. That is in the end, the geometrical proof. That's what the ancient Greeks used to proof mathematical principles (or the reductio ad absurdum). Calculus is in a way more powerful (since you can solve more problems), but needs more rules that have to be observed than geometry. The KISS priciple applies (and I don't mean Knights in Satan's service).

 

[Hippasus]

The paradoxon can be solved by adding the Planck distance. You can never go under a Planck distance as it is the smallest definable distance. So the additions are not infinite but have a maximum number, namely the unit divided by the Planck distance.
Use physics to crush logics.

On a different level of reality than physics, one can go infinitely smaller. This is an abstract fact that is supported by experience. For instance, we can compute numbers like "e" to ever greater accuracy through numerical methods in calculus.

If you want to deal with it mathematically, you could simply use the lowest quality (or level, I don't know the correct terms, I studied this all in German, so forgive me) of proof. That is in the end, the geometrical proof. That's what the ancient Greeks used to proof mathematical principles (or the reductio ad absurdum). Calculus is in a way more powerful (since you can solve more problems), but needs more rules that have to be observed than geometry. The KISS priciple applies (and I don't mean Knights in Satan's service).

My mathematical knowledge is somewhat limited, but my sense is that geometry does not capture certain prominent examples of the infinite and infinitesimal. Geometry cannot properly define the infinitesimal in a way that can readily be used by other subject areas. Infinite sets like the natural numbers are not geometrical concepts, yet they are fundamental to mathematics. Calculus, however, does these things through rigorous mathematical analysis, and it is able to be used by the sciences.

Logic is what proves. This, at least in part, requires natural language as opposed to merely mathematical formalism. It should be natural language plus mathematical formalism.

Abstract form is interesting subject-matter and arguably a different form of reality from material existence.

 

[Eisen]

On a different level of reality than physics, one can go infinitely smaller. This is an abstract fact that is supported by experience. For instance, we can compute numbers like "e" to ever greater accuracy through numerical methods in calculus.

My mathematical knowledge is somewhat limited, but my sense is that geometry does not capture certain prominent examples of the infinite and infinitesimal. Geometry cannot properly define the infinitesimal in a way that can readily be used by other subject areas. Infinite sets like the natural numbers are not geometrical concepts, yet they are fundamental to mathematics. Calculus, however, does these things through rigorous mathematical analysis, and it is able to be used by the sciences.

Logic is what proves. This, at least in part, requires natural language as opposed to merely mathematical formalism. It should be natural language plus mathematical formalism.

Abstract form is interesting subject-matter and arguably a different form of reality from material existence.

I don't disagree. All I'm saying is that you don't need to use calculus if you can use a geometric solution. They come with the bonus, that they are often more intuitive. But sure, not everything can be solved that way, as a matter of fact, most things can't.