Archimedes too far ahead of his time

by rrusczyk, Jan 26, 2009, 6:59 PM

Here's an article Sandor sent me that argues that there's new evidence that Archimedes was even closer to developing calculus (or basically had) than historians realized. I had a couple thoughts upon reading this, mainly wondering why it took a couple millennia to get from Archimedes to Newton:

1) Was it simply lack of good notation that caused the huge delay? By the time Newton came along, or, more to the point, at almost exactly the time Newton came along, humanity had developed a lot of notation that was good for expressing the concepts of calculus. This notation did not exist in the time of Archimedes.

2) Was the problem simply that Archimedes was way, way too far ahead of his time -- that is, can big ideas really only take root at certain times in history. (Of course, most often when someone claims to have an idea that fails because it's simply way ahead of its time, in truth, it's just a bad idea. But Archimedes' evaluation of his ideas very close to calculus as more significant than the results he derived with them certainly isn't an example of someone overestimating an idea whose time has not yet come.)

Archimedes

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I don't think it was a lack of notation. Anything could have sufficed as a notation. For example, Leibniz's double-d notation wasn't used immediately. People got along just fine using $f(x)$ instead of $\frac{d}{dx}$ .

by n0vad3m0n, Jan 26, 2009, 7:59 PM

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Which made less progress from 200BC to 1400AD: math or art?

by sandor, Jan 26, 2009, 8:09 PM

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They both made plenty of progress. It's just that this progress wasn't happening in Europe, and a great deal of it was applied to the needs of the merchant classes.
Notation is a huge reason, and it's why Leibniz deserves more credit than Newton for calculus. They both came up with notation for ideas that were already circulating, but Leibniz's notation was better, to the point that we still use much of it. Farther back, the wall between arithmetic and geometry really hurts; Archimedes simply couldn't ask very many interesting questions that his geometric calculus could answer.

by jmerry, Jan 26, 2009, 8:30 PM

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Curious. I was always under the impression the Greeks had in fact pretty much worked out integral calculus, and the reason they weren't really given any credit for developing calculus was that the true essence of calculus lay in uniting the disparate parts of the differential and the integral calculus through the FTC's. This article seems to specify the focus of what exactly constitutes calculus to this whole business of infinities ? I guess that's pretty valid, though.

by MysticTerminator, Jan 27, 2009, 2:15 PM

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Let's see you try to do calculus without polynomial functions, trigonometric functions and their inverses, or logarithms and exponentials. The questions we can ask are just as important as the methods for answering them in the development of mathematics.

by jmerry, Jan 28, 2009, 3:54 AM