Zeno of Elea

[paulhanke]

Originally Posted by VideCorSpoon

Also, I do not agree with your supposition that though we are familiar with a broad notion of an “instant,” it is a mathematical impossibility because “a point in time has zero temporal extent.” This statement is extremely problematic. It is extremely axiomatic and tautological. It can very well be argued that zero is an “extended” numerical value.

... but if we argue that zero is an "extended" numerical value, doesn't that render the description of the arrow paradox nonsensical?

1. When the arrow is in a place just its own size, it’s at rest.

2. At every moment of its flight, the arrow is in a place just its own size.

3. Therefore, at every moment of its flight, the arrow is at rest.

That is, if we define zero as an "extended" numerical value then it is impossible that "2. At every moment of its flight, the arrow is in a place just its own size." ... a "moment" endures for zero time - but because we have defined zero as an extended numerical value, some amount of time passes within a given moment, which implies that the arrow moves within a given moment, which in-turn implies that at every moment of its flight, the arrow is in a place that is larger than its own size, thus rendering the description of the arrow paradox nonsensical.

[VideCorSpoon]

Arguing that “zero can be an extended numerical value” was a juxtaposition to the supposition you posed which said, “a point in time has zero temporal extent.” First, I am unaware of any paradox that is sensical to any degree. Paradoxes illuminate “holes” in normative frameworks, but mostly never patch those frameworks up.

If we did argue that zero is an extended numerical value, we only uphold the abstract values of Xeno’s paradox. But more to the point, that very fact helps us provide some level of solution to the problem. The short summary of Xeno’s paradox is that the arrow never gets half way because it has to go through half of that half, etc. The arrow is moving but not moving. The paradox lies in the fact that the arrow (theoretically) does not move because it is essentially traveling half of a half of a half, etc. It is in a sense going backwards while going forwards. It is the quintessential reductio ad absurdum example.

The quote that you mention is an loose account of Xeno’s paradox from Aristotle’s Physics. But look very carefully at the quote you posted. If we were to define zero as an extended numerical value, it would not negate the second premise because they are both (#1 and #2) bi-conditionally concurrent (i.e. A=B & B=A). So any axiomatic argument that necessarily follows is illogical because the two premises cancel each other out.

So I don’t follow your assumptions which are based off of the #2 (assumed premise) and then disregard the #1 premise and the overall context of the summary. Zero is the underline feature….the extended feature… in the paradox because all three points underline a “zero” concept. Also of note is the fact that the ancient Greeks had no conception of zero the way we did. The Ancient Vedanta Indians conceived it and the Romans brought it into the western mind, but the Greek were not privy to our understandings. It was more like the void or something else of that nature.

[paulhanke]

Originally Posted by VideCorSpoon

First, I am unaware of any paradox that is sensical to any degree.

... agreed - that's why I said it renders the description of the arrow paradox nonsensical ...

Originally Posted by VideCorSpoon

The short summary of Xeno’s paradox is that the arrow never gets half way because it has to go through half of that half, etc.

... is it? ... here's (Aristotle on) Zeno on the arrow: "The arrow in flight is at rest. For, if everything is at rest when it occupies a space equal to itself, and what is in flight at any given moment always occupies a space equal to itself, it cannot move." ... maybe you have the arrow paradox confused with another of Zeno's paradoxes?: "You cannot cross a race course. You cannot traverse an infinite number of points in a finite time. You must traverse the half of any given distance before you traverse the whole, and the half of that again before you can traverse it. ..." (translations by J. Burnett)

Originally Posted by VideCorSpoon

But look very carefully at the quote you posted. If we were to define zero as an extended numerical value, it would not negate the second premise because they are both (#1 and #2) bi-conditionally concurrent (i.e. A=B & B=A). So any axiomatic argument that necessarily follows is illogical because the two premises cancel each other out.

I'm not sure I follow - how is a nonsensical antecedent #2 (from the standpoint of zero being an extended value) necessarily made sensical by virtue of antecedent #1? ... as far as I can see, the only relationship between the two is the (paradoxical) consequent (#3).

Originally Posted by VideCorSpoon

Also of note is the fact that the ancient Greeks had no conception of zero the way we did.

... which could be one big reason why this was such a paradox for them - Zeno was messing around with infinities, but was not mathematically equipped to deal with the implications of those inifinities (e.g., an infinite sum of zeros is still zero)

[VideCorSpoon]

To posit anything nonsensical implies that it may have been sensical to begin with. I’m glad we both agree on a fundamental notion of paradoxes and semantic composition.

On the short summary of Xeno’s paradox. It is first and foremost a paradox. Perhaps that is why the notion of a half’s half becomes relevant because it is contradictory (hence the biconditional). Your translation of a basic Aristotelian text is very much different than my own, from the deeper contexts of Aristotle’s terminology to the way his name is spelled. I think this is the greatest point of our discontention which is the translation aspect. But even in the Burnett example you provide, it underlines the same abstract notion I (and I would suspect we) are trying to convey, albeit through different standards.

The example you provided had 3 lines. These lines display a logical deductive proof formation. I am quite sure of this because the website where this was quoted from displays Xeno’s proof in predicate proof form. The “nonsensical” antecedent #2 is made “sensical” in virtue of antecedent #1 because both #1 and #2 are inextricably connected because they form a bi-conditional. In a biconditional, an apple is red and red is the color of the apple. The consequent #3 extrapolates on the biconditional premises (#1 and #2) illuminating that it is in fact a biconditional (#3). But as a side note, the word “sensical” and “nonsensical” are problematic because they are relative terms and insufficient for the terms of the discussion.

But was Xeno messing around with infinites??? That seems a bit premature to say at this point. Also, the concept of zero for the Greeks is rudimentarily established in cosmogony, not mathematics.

(I would have done the winky emoticon too, but I thought this was far more irrelevant. LOL!)

[paulhanke]

Originally Posted by VideCorSpoon

Your translation of a basic Aristotelian text is very much different than my own, from the deeper contexts of Aristotle’s terminology to the way his name is spelled. I think this is the greatest point of our discontention which is the translation aspect.

... which is one reason I pointed to the source of the translation - in going from one translation to another, I'm finding large discrepancies ... in fact, if I remember correctly Burnett states in his preface to "Early Greek Philosophy" that he did his own English translations because he couldn't agree with any of the existing English translations!

Originally Posted by VideCorSpoon

... an apple is red and red is the color of the apple.

... given this example, wouldn't the bi-conditional of antecedent #1 be "When an arrow is at rest, it's in a place just its own size." ... and for what it's worth, it doesn't sound like bi-conditionals make very good antecedents in a syllogism:

1. Apples are colored red.
2. Red is the color of apples.
3. Therefore, ???