The learned and luminously-penned Invisible Adjunct (who appears to leave a remarkably large and easily visible intellectual footprint, in the circles in which I move, at least) directs us to a website that tells us that the School of Zeno and Chrysippus has risen in the rankings from number 50 to number 22.

This is well-deserved. In fact, the Thirteen-Year-Old has already been thoroughly exposed to Zenoxian paradoxes and their solutions in his learning that one has two repeated-decimal representations: 1.00000..... and 0.99999....

Posted by DeLong at May 4, 2003 11:38 AM | TrackBack

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Wrong Zeno.

The School of Zeno and Chrysippus is the Stoa, and the Zeno is Zeno of Citium (333-261? BCE).

The Zeno of various logical conundrums, etc., is Zeno of Elea (c.490-c.425 BCE).

And, as Diogenes Laertius tells us (7.35), there were six other philosophers called Zeno, but none of them were terribly important.

Posted by: Chris Brooke on May 4, 2003 11:51 AMSo after every Zeno there would be another Zeno to read, who would yet again leave you short of the threshold of knowledge.

Posted by: Ben Vollmayr-Lee on May 4, 2003 01:51 PMOK, so if you really want to "get serious" about the 0.9999...=1.0 thing, you should try to come to peace with the wild-eyed constructivists who are not only serious mathematicians, but can sometimes deny the preceding equation. So try out Fred Richman:

http://www.math.fau.edu/Richman/html/999.htm

I think there are good reasons why you might not want to go with his ideas here, but it surprised me to find out that there *were* ideas about this kind of thing...

Posted by: Jonathan King on May 4, 2003 08:10 PMThere is, generally speaking, a difference in outlook between physicists and mathematicians, and while I'm something of a mathematical physicist, I find that I have pretty much the physics outlook. I can understand the mathematicians who only accept constructive proofs: they can even demonstrate they have a point. But I don't see what this guy (JK's link above) is talking about.

What does 0.999999... mean (or better, 0.9-bar)? Anything I can think of is clearly 1. Consider the infinite sum

0.9-bar = 9/10 + 9/100 + 9/1000 + ...

He uses a little sophistry here, I think, claiming that the ... in a sum of a series doesn't imply the infinite sum. The series came from the left-hand side which DOES imply the infinite sum. Any loss of that information on the right-hand side is a weakness of the notation, not something with mathematical significance. And since it is the infinite sum, it is a convergent sum of geometric series equal to 1.

He also talks about a limit, and I had no idea what he meant at first because he mentioned it out of context. We could construct a sequence

0.9, 0.99, 0.999, 0.9999, ....

and 0.9-bar is not any single member of this sequence, but it is the limiting value. Fine, but then no one would ever think of relating 0.9-bar to anything but the limit, so again I think the claims that the limit is not implied are unfair.

I suppose there is a valid point to this that is just not well described in the article. This reminds me of when I tried to figure what the commmotion was about fuzzy logic and never could. Everything I found was either trivial (algebra with continuous variables constrained to be between 0 and 1) or wrong (statements like "the negation of the statement 'P might be true' is 'P might not be true'"). It's highly likely there is some point to it, but it was sad that people writing books on it couldn't explain it correctly.

I suppose it's an example how sometimes we think better collectively than we do individually, so the individual's explanations of a topic aren't always so good.

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