The Presence of Time

Response to Derek and Lisa.

I do agree that an object can hold a rigid definition. However, this definition takes place only once at certain time. For example on Monday we understand the Prof was discussion the trash can. Except that trash can on that day does not exist anymore. It only existed on that one day containing those particular molecules. Therefore, at one point in time the trash can does posses a rigid definition. Furthermore, there is no possible way to go back in time (to Monday) and claim that the trash can might posses some different definition than the one given on that day. This even includes if the trash can has exactly the same particles. This is due to the fact that the particles on Monday are not the same any time since or before because they existed in a different time.



Chad F

Entropy is the Answer:
A Response to Lisa


The way nature works kind of goes like this. Take a coke bottle and open it. Once opened all the trapped air, let’s call this set A, is released into the outside air. Let’s call this outside air set B. After a few moments most of the set A in now mixed into set B. Nature follows entropy which means it flows from high order to high disorder. This is crucial to understanding that you just might be Moses. Now if we watch the coke bottle for let’s say a billion years there is a finite chance that all of set A original in the coke bottle will exist there once again. But this chance may be so small that it would take longer than a billon years. It may take longer then the time universe has been into existence. So from this you can see there is a finite chance that every exact atom that made up Moses some 3400 years ago can exist in you. But this is a very small probability. But the most important part, it is possible. Furthermore, you may have every exact atom that made up Moses but arranged in a slightly different order that makes up Lisa.

Chad F

The Fourth Dimension


Before the twentieth century many people believed that every person experienced only three dimensions. This included most physicists who described the world within mathematics as mainly three dimensional and that time was absolute. In the beginning of the twentieth century came the reformulation of the basic principle of physics. This was the theory of special relativity.
The great mind of Albert Einstein has introduced to the world that time is no different than space, they are interchangeable. For example, what if you had an appointment in the heart of Manhattan? You are given instructions to meet someone as follows. Go to the corner of Fourth Street and Ninetieth Street. Then proceed in the building up to the tenth floor. Here you were given the three spatial directions to meet someone. However, you might be standing there for eternity because you never had the indication of when you were suppose to meet someone. Therefore, time is just as important as all three spatial dimensions.
In class discussion we talk about the arrow not moving at an instant. Now that we concluded that space and time are one in the same we have concluded that just as there no possible way to move an infinite amount of positions in a finite amount. Similarly, with time there is no way to capture an instant of a moment in a time equal to zero. This would violate the laws of special relativity. We can perceive this idea in our mind of a slice of the reality loaf and assume that it is an instant of time, but this is impossible. To capture a picture at an instant you would have to open and close you eyelids in an instant which would be physically impossible.
With the argument of Zeno I have concluded that it impossible to view an instant of a motion. It is easy to presume that at an instant something is not moving when flash the image in our mind, but with our human senses this impossible. If you looked at object moving and took a snap shot it would appear to be not moving in the photograph. The truth is that the object was moving during the time when the shutter opened and then closed. Furthermore, it is impossible to slice the reality loaf in an infinitely thin slice.

(Note: This is Chad's post.)

Essay Outline 2 Assignment

. . . is here.

Sets, Infinity, Limits, and Functions

I hope I have substantiated my claim that one does not need facility with modern mathematics in order to grasp the philosophical problems posed by Zeno's paradoxes. Regardless, some of you have asked about the mathematics employed in addressing the mathematical problems posed by Zeno's paradozes. I have intentionally not focused on the mathematics in class. In case you are curious, I have posted two discussions by Wesley Salmon that deal with the mathematical concepts involved. The first discusses the concept of a limit, the sum of an infinite series, and mathematical functions. The second discusses sets and infinity. Both expositions assume no background in modern mathematics. If you read either and have questions about them, please comment to this post.

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)

Chad on Zeno and Black

I agree with Black’s position that mathematics alone cannot break the argument put forth by Zeno. Moreover, his argument examines the point that trying to count an infinite number of steps in a finite amount of time as impossible. In nature, it is true that the faster runner will overtake the slower runner. In my opinion the answer does not lie in trying to prove that it is impossible to count an infinite number of something in finite amount of time. The answer lies in the whole definition of infinity.
To my knowledge no one has ever proved infinity can exists. It is just simply an approximation used to put at an end of what the human mind is unable to conceive of. Human beings have been approximating nature since the beginning of time. As the centuries have passed since the time of Zeno our approximations have become better and better.
A good example is from the time of Isaac Newton. He invented calculus and applied it to celestial and earth bound mechanics. This new formulation explained everything from the distance a baseball could be thrown in a finite amount of time to the motion of planetary objects. Except, one thing was unable to be explained by Newton’s mechanics. The orbit of Mercury did not fit the calculations or any other explainable reason for its peculiar orbit. However, Newton’s approximations were close on nearly everything else in nature, but they were still approximations none the less.
Three centuries later Albert Einstein formulated his own approximations and clearly explained the orbit of Mercury. More in particular he defined that nature does have finite values for what was previously thought to be infinite. For example, Newton believed that the speed of light and gravity are instantaneous or at a speed of infinity. Einstein discovered that the speed of light in not instantaneous. Therefore, light travels a finite distance in a finite amount of time.
Two of the greatest minds in all of history understood that mathematics is invented as just approximations. It is the best tool we have and the approximations are used to explain how we believe nature works. Therefore, their arguments, including Zeno’s, still hold until a better approximation can explain them otherwise.


note: This post is Chad's despite the fact that it is not posted by Chad.

Zeno Made Practical

The Eleatic Hangover Cure by Josh Parsons

The Eleatic Non-Stick Frying Pan by Simon Prosser

Assignment for 2/29

Please read Max Black's article, "Achilles and the Tortoise". It is available from the course website. In it he claims (and subsequently argues for) the following:

"(Though a) mathematical solution (to Zeno's paradox of Achilles and the Tortoise) has behind it the authority of Descartes and Pierce and Whitehead--to mention no lesser names--. . . I cannot see that it goes to the heart of the matter. It tells us, correctly, when and where Achilles and the tortoise will meet, if they meet; but it fails to show that Zeno was wrong in claiming that they could not meet."

(Black 1951: 122; editing by Chris Tillman, emphasis in the original.)

Is Black's suggestion that a purely mathematical solution is inadequate correct? Why or why not? For next class, please briefly state and explain/defend your answer.

Assignment for 2/29

Please read Max Black's article, "Achilles and the Tortoise". It is available from the course website. In it he claims (and subsequently argues for) the following:

"(Though a) mathematical solution (to Zeno's paradox of Achilles and the Tortoise) has behind it the authority of Descartes and Pierce and Whitehead--to mention no lesser names--. . . I cannot see that it goes to the heart of the matter. It tells us, correctly, when and where Achilles and the tortoise will meet, if they meet; but it fails to show that Zeno was wrong in claiming that they could not meet."

(Black 1951: 122; editing by Chris Tillman, emphasis in the original.)

Is Black's suggestion that a purely mathematical solution is inadequate correct? Why or why not? For next class, please briefly state and explain/defend your answer.

Assignment for 2/27

Please extract an argument from Zeno and present it in numbered premise-conclusion form. State which premise is least plausible and why. Extra credit will be given to extractions of more difficult arguments.