Zeno and Black

Below are notes on Zeno and Black from 03/01. The Dichotomy and Black's argument are presented in numbered premise-conclusion form, as is the argument that Black's conclusion has vast implications for mathematical physics. Comments about which premise or premises are false and why are warmly encouraged.

The Dichotomy

The first asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal. (Aristotle Physics, 239b11)

Call the starting point 'Z' and the endpoint 'Z*'. We may present the argument as follows:

1. If A travels from Z to Z* and space is continuous, then A can perform an infinite number of tasks.

2. A cannot perform an infinite number of tasks.

3. Therefore, it's not the case that A travels from Z to Z*.

4. If (3), then motion in continuous space is impossible.

5. Therefore, motion in continuous space is impossible.

Note that a parallel argument would apply to continuous times and a similar idea is employed by the paradox of Achilles and the Tortoise. We might think of Achilles and the Tortoise as arguing that if space is continuous, Achilles can never reach his destination. We might think of the Dichotomy as arguing that if space is continuous, Achilles can never get started. Either way, assuming nothing is special about Achilles, we may conclude that motion in continuous space is impossible.

Zeno's paradoxes pose mathematical problems. But do they pose problems that extend beyond mathematics? Descartes, Pierce, Whitehead, and high school calculus teachers all across the land say 'no'. Black says 'yes'. Why does Black think that?

Here's a quote from Black:

"The series of distances traveresed by Achilles is convergent. This means that if Achilles takes enough steps whose sizes are given by the series 100 yards, 10 yards, 1 yard, 1/10 yard, etc., the distance still to go to the meeting point eventually becomes, and stays, as small as we please . . . But the distance, however much reduced, still remains to be covered; and after each step there are infinitely many steps still to be taken. The logical difficulty is that Achilles seems to be called upon to perform an infinite series of tasks, and it does not help to be told that the tasks become easier and easier, or need progressively less time in the doing. Achilles may get nearer to the place and time of his rendezvous, but his task remains just as hard, for he still has to perform what seems to be logically impossible. It is just as hard to draw a very small square circle as it is to draw an enormous one: we might say both tasks are infinitely hard . . . I think Zeno had enough mathematical knowledge to understand that if Achilles could run 111 1/9 yards -- that is to say, keep going for 11 1/9 seconds -- he would indeed have caught the tortoise. The difficulty is to understand how Achilles could arrive anywhere at all without first having performed an infinite number of tasks." (Black 1951: 122-123.)

"I am going to argue that the expression, 'infinite series of acts', is self-contradictory, and that failure to see this arises from confusing a series of acts with a series of numbers generated by some mathematical law." (Black 1951: 123.)

Black supposes, for the sake of argument, that we can construct an infinity machine. Here is a simple one (due to Thompson 1954-5): Suppose we have a push-button lamp. Suppose someone pushes it an infinite number of times by taking 1/2 min for the first push, 1/4 for the second, 1/8 for the third, and so on. Now Black argues (roughly) as follows:

1. If it is logically possible to perform an infinite series of acts, then the lamp was switched off each time it was on and on each time it was off.

2. If the lamp was switched off each time it was on and on each time it was off, then in its final state the lamp is neither on nor off.

3. Therefore, if it is logically possible to perform an infinite series of acts, then in its final state the lamp is neither on nor off.

4. It's not the case that in its final state the lamp is neither on nor off.

5. Therefore, it is logically possible to perform an infinite series of acts.

This is, in short, Black's argument. It follows from this that, on the assumption that the race can be run, the physical situation does not correspond to the mathematical operation of summing an infinite series. Here is the argument:

6. If it is logically impossible to perform an infinite series of acts, then the running of a race cannot be correctly described as completing an infinite series of tasks.

7. If the running of a race cannot be correctly described as completing an infinite series of tasks, then mathematical physics needs a radically different mathematical foundation in order to adequately describe physical reality.

The consequent of (7) follows from HS on (6,7) and MP on the resulting premise and (5).

Further Reading:

Benaceraff, Paul. 1962. "Tasks, Supertasks, and the Modern Eleatics." Journal of Philosophy 59:24, 765-84.

SEP entry on Supertasks (Warning: Much of this is difficult reading. However, it contains a discussion of Black and Benaceraff's reply which are not too hard to follow.)

Parsons and Prosser papers (especially Parsons)

Below are notes on Zeno and Black from 03/01. The Dichotomy and Black's argument are presented in numbered premise-conclusion form, as is the argument that Black's conclusion has vast implications for mathematical physics. Comments about which premise or premises are false and why are warmly encouraged.

The Dichotomy

The first asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal. (Aristotle Physics, 239b11)

Call the starting point 'Z' and the endpoint 'Z*'. We may present the argument as follows:

1. If A travels from Z to Z* and space is continuous, then A can perform an infinite number of tasks.

2. A cannot perform an infinite number of tasks.

3. Therefore, it's not the case that A travels from Z to Z*.

4. If (3), then motion in continuous space is impossible.

5. Therefore, motion in continuous space is impossible.

Note that a parallel argument would apply to continuous times and a similar idea is employed by the paradox of Achilles and the Tortoise. We might think of Achilles and the Tortoise as arguing that if space is continuous, Achilles can never reach his destination. We might think of the Dichotomy as arguing that if space is continuous, Achilles can never get started. Either way, assuming nothing is special about Achilles, we may conclude that motion in continuous space is impossible.

Zeno's paradoxes pose mathematical problems. But do they pose problems that extend beyond mathematics? Descartes, Pierce, Whitehead, and high school calculus teachers all across the land say 'no'. Black says 'yes'. Why does Black think that?

Here's a quote from Black:

"The series of distances traveresed by Achilles is convergent. This means that if Achilles takes enough steps whose sizes are given by the series 100 yards, 10 yards, 1 yard, 1/10 yard, etc., the distance still to go to the meeting point eventually becomes, and stays, as small as we please . . . But the distance, however much reduced, still remains to be covered; and after each step there are infinitely many steps still to be taken. The logical difficulty is that Achilles seems to be called upon to perform an infinite series of tasks, and it does not help to be told that the tasks become easier and easier, or need progressively less time in the doing. Achilles may get nearer to the place and time of his rendezvous, but his task remains just as hard, for he still has to perform what seems to be logically impossible. It is just as hard to draw a very small square circle as it is to draw an enormous one: we might say both tasks are infinitely hard . . . I think Zeno had enough mathematical knowledge to understand that if Achilles could run 111 1/9 yards -- that is to say, keep going for 11 1/9 seconds -- he would indeed have caught the tortoise. The difficulty is to understand how Achilles could arrive anywhere at all without first having performed an infinite number of tasks." (Black 1951: 122-123.)

"I am going to argue that the expression, 'infinite series of acts', is self-contradictory, and that failure to see this arises from confusing a series of acts with a series of numbers generated by some mathematical law." (Black 1951: 123.)

Black supposes, for the sake of argument, that we can construct an infinity machine. Here is a simple one (due to Thompson 1954-5): Suppose we have a push-button lamp. Suppose someone pushes it an infinite number of times by taking 1/2 min for the first push, 1/4 for the second, 1/8 for the third, and so on. Now Black argues (roughly) as follows:

1. If it is logically possible to perform an infinite series of acts, then the lamp was switched off each time it was on and on each time it was off.

2. If the lamp was switched off each time it was on and on each time it was off, then in its final state the lamp is neither on nor off.

3. Therefore, if it is logically possible to perform an infinite series of acts, then in its final state the lamp is neither on nor off.

4. It's not the case that in its final state the lamp is neither on nor off.

5. Therefore, it is logically possible to perform an infinite series of acts.

This is, in short, Black's argument. It follows from this that, on the assumption that the race can be run, the physical situation does not correspond to the mathematical operation of summing an infinite series. Here is the argument:

6. If it is logically impossible to perform an infinite series of acts, then the running of a race cannot be correctly described as completing an infinite series of tasks.

7. If the running of a race cannot be correctly described as completing an infinite series of tasks, then mathematical physics needs a radically different mathematical foundation in order to adequately describe physical reality.

The consequent of (7) follows from HS on (6,7) and MP on the resulting premise and (5).

Further Reading:

Benaceraff, Paul. 1962. "Tasks, Supertasks, and the Modern Eleatics." Journal of Philosophy 59:24, 765-84.

SEP entry on Supertasks (Warning: Much of this is difficult reading. However, it contains a discussion of Black and Benaceraff's reply which are not too hard to follow.)

Parsons and Prosser papers (especially Parsons)

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