h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

http://www.bbc.co.uk/h2g2/guide/A524242

This article not only has a really cool number, but it's informative to boot.

Lochinvar

A well written entry. Well illustrated and discussed. I could do without the opening comment, or just change it so it seems less subjective. Also I object to your spacing, the sections seem squeezed together. Are you using the tags to enclose your text? Maybe a slightly more coherent intro might do it some good as well.

I think the purely arithmetical version of Xeno's paradox is worth mentioning: The sum of the infinite series of fractions 1/2 + 1/4 + 1/8 + 1/16 + . . . is equal to 1.

Xeno argues that an infinite number of quantities cannot be passed through in a finite time, but he is ignoring the above fact of arithmetic. Your analysis that he is considering space and time differently, which is unfair, is right. If the arrow can get to its target in one second, then it gets halfway there in half a second, a quarter of the way there in a quarter second, etc.

(Aristotle provides a rather thorough answer to Xeno in The Physics, Book VI, chapter 2.)

GTB

Xeno was a philosopher in greek times - a LONG time ago, and a LONG time before we had any idea that an infinite sum of finite elements could be anything other than infinite in size. Calling him stupid is kinda like calling Newton stupid for not figuring out general relativity....

nitpicks:
There are two versions of Xeno's arrow: one where it can never reach its target, and one where it can never get started.

Where is the block-quote quoted from? You don't say, but it'd be interesting to know...

Was the Greek who thought up this paradox called Xeno or Zeno? I always thought it was Zeno.

I think the crux of Zeno's paradox was:

Motion is impossible because you have to cover an infinite number of different sections of your journey and you can't do an infinite number of things in a finite time.

The modern solution was produced Newton:

You can if you do they're small enough.

If each distance to be covered is vanishingly small, then it is possible to do an infinite number of them in a finite time. The paradox is only a paradox to someone who does not have the modern understanding of infinity.

Lucinda wrote: "Xeno was a philosopher in greek times - a LONG time ago, and a LONG time before we had any idea that an infinite sum of finite elements could be anything other than infinite in size. Calling him stupid is kinda like calling Newton stupid for not figuring out general relativity...."

Lucinda, regarding your assertion that the ancient Greeks didn't know that an infinite sum of finite elements could be finite, I could refer you to Euclid's Elements, Aristotle's Physics, and several proofs by Archimedes in which each of these men demonstrated that they understood that magnitudes could be finite in extension, yet divisible into an infinite number of disjoint parts. The clearest examples that spring immediately to mind are the passage in Aristotle cited in my last post, and Euclid, Book X, Proposition 1, in which he shows that any magnitude equals the limit of partial sums of a geometric series.

I would not call Xeno stupid, but I think it IS worth noting that the fallacy in his paradox was correctly identified by his near-contemporaries.

GTB

Gnomon wrote: "If each distance to be covered is vanishingly small, then it is possible to do an infinite number of them in a finite time. The paradox is only a paradox to someone who does not have the modern understanding of infinity."

- In defense of the ancients -

Aristotle had the "modern" understanding of infinity, if you want to call it that. He said, in response to Xeno, "There are two senses in which length and time and generally anything continuous are called 'infinite': they are called so either in respect of divisibility or in respect of their extremities. So while a thing in a finite time cannot come in contact with things quantitatively infinite, it can come in contact with things infinite in respect of divisibility: for in this sense the time itself is also infinite"

Nobody since, however modern, has come up with a more succinct, complete, and definitive reply to Xeno's paradox.

GTB

The citation of the quote from my last post is Aristotle's Physics, Book VI, Chapter 2; I was quoting the Hardie/Gaye translation, and the above post might be moderated for copyright reasons. If so, I'll post a paraphrase.

GTB [not wanting to tread on anyone's copyright!]

Can't copyright Aristotle, he lived too long ago. Except for the translation part, which can be copyrighted. But then again how would we know you didn't translate it yourself?

Sorry, that was off-topic.

Lochinvar, good news!

This entry has been recommended to the Editors and is now going forward to become a part of the edited Guide.

Congratulations!

The editing process may take a little while. You'll receive an e-mail when the finished piece is about to hit the front page.

In the meantime, well done, and keep up the good work!

MDS

Gnomon, you asked me to enter the debate on the spelling of Zeno. To be honest, I had no idea, but it appealed! If GTB had only quoted the sentence before, he would have answered your question about the spelling:

'Hence Zeno's argument makes a false assumption in asserting that it is impossible for a thing to pass over or severally to come in contact with infinite things in a finite time.'

This is also from the Hardie/Gaye translation. So, Aristotle spelled it 'Zeno' (or that's the way that Hardie and Gaye transliterated it). I can't find a copy of the original Greek on the internet (not present at perseus), but I can't see why it would have been transliterated that way unless the first letter were zeta.

I can also find:
- references to a philosopher Zeno, but not Xeno.
- far more references to "Zeno's Paradox" on sites in the .edu domain than to "Xeno's", but both exist.

I also found references to both in the research. I thought the X looked better, though so I used that throughout. The Z, I suppose, should get mention.

Lochinvar

I think what I'm implying is that the "Zeno" spelling has a bit more credibility.

I zecond that

Just to be sure, I did a quick lookup over at the Perseus project, and sure enough, "Zeno" starts with zeta, the same letter which "Zeus" starts with. The Greek word "xenos" (strange), whence we derive "xenophobe" starts with a chi, not a zeta. Like "chronos". So the real question is, why do we pronounce "xenophobe" that way?

*whispers* xenos starts with a xi, rather than a chi, which is pronounced like an 'x'.

How did you find Zeno on perseus? I couldn't find it there, except as a reference in the encyclopaedia (in English rather than Greek).

I've now done a wider search on Perseus and have found a copy of Aristotle's Metaphysics, where he appears to be refuting something similar, but not quite the same as the Zeno's Paradox recorded here. I really don't know much avbout Aristotle, but is it possible that he didn't like Zeno?

"Further, if absolute Unity is indivisible, by Zeno's axiom it will be nothing. For that which neither when added makes a thing greater nor when subtracted makes it smaller is not an existent thing, he says; clearly assuming that what exists is spatial magnitude. And if it is a spatial magnitude it is corporeal, since the corporeal exists in all dimensions, whereas the other magnitudes, the plane or line, when added to a thing in one way will increase it, but when added in another will not; and the point or unit will not increase a thing in any way whatever. But since Zeno's view is unsound, and it is possible for a thing to be indivisible in such a way that it can be defended even against his argument (for such a thing when added will increase a thing in number though not in size)--still how can a magnitude be composed of one or more such indivisible things?"

This Zeno definitely starts with 'Z', zeta.

yes, of course... I completely forgot there was a letter called xi... that was silly wasn't it... wow.

I found Zeno's name at the Perseus Project in Plato's Parmenides. (ah, sweet alliteration!) Zeno was Parmenides' student and is a character in the dialogue of that name. When I noticed this, while doing a little research during my last round of postings here, I told myself I'd read the Parmenides real soon and see what Zeno is portrayed like by Plato. I'll still read it real soon. Aristotle seems to find Zeno's thinking specious in every way, but still worth responding to. Why?

I gather that our Mr. Zeta wasn't really a mathematician, but he spent some time trying to use mathematics to prove his teacher's somewhat mystical, non-technical assertions. He was wrong a lot, at least according to Aristotle...?

I'll read that dialogue real soon, and try to get an idea about what Plato thought of Zeno...

On the subject of what constitutes a mathematician, I think our distinctions would probably seem absurd to the Ancient Greeks. Zeno, Aristotle and Pythagoras all 'dabbled' in philosophy, mathematics and other sciences, but I suspect all would have described themselves first and foremost as philosophers ('lovers of wisdom').