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Warrigal comments on Generalizing From One Example - Less Wrong
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Comments (386)
Surely eidetic imagery isn't absolute. Who can imagine a sine curve and then zoom in on the least positive root in order to calculate pi? Less than five percent of people, I would think.
How do you draw a sine curve (on paper, say), without knowing the value of pi, in order to take the measurement of pi from it when you've finished? This example is broken. Unrolling half a circle should work though.
Check out the Feynman lecture #22 - the one in which he starts with the laws of algebra and ends up with de Moivre's theorem. With a calculation of pi/2 = 1.5709 along the way. Prettiest thing I've ever seen.
Incidentally, Feynman did it the hard way, since he didn't have computers. You can compute pi on a spreadsheet simply by simulating a harmonic oscillator.
Before anyone else complains: yes, there were computers in 1961, and had been for over twelve years, but Feynman doesn't use any in the lecture. And certainly Henry Briggs, who calculated the first fourteen-place common log tables and whom Feynman cites in the relevant section, didn't use any in 1620, and the results Feynman presents are far less precise.
And Lecture #22 - "Algebra" - is a thing of beauty. Anyone who likes mathematics will like it.
Disagreed -- if you know the general shape and you know the derivative at 0 is 1, then while you can't calculate pi very accurately, you can find out that it's closer to 3 than to 5.
If you know the derivative at 0 is 1, then you know the value of pi... just sayin'.
That's not strictly true, seeing as...
...but I agree that general-shape + derivative-at-zero is not really enough to form estimate of pi.
Yeah, I thought about that, but this information doesn't exactly define the curve, and so it becomes unclear which portion of the work is done by visual imagination, and which just fits the known result, taking a few obvious bounds into account. Unrolling half a circle, on the other hand...
It took me a little while to think of a definition of the sine function that does mention pi, though it turned out to be the first one taught in (my) school: "the y coordinate after going t/2pi times counterclockwise around the unit circle starting at (1,0)". If I were to draw the curve, I'd use Euler's method or roll a circle, both of which use the derivative going between -1 and 1 instead of pi for the frame of reference.
Since the derivative is also a sine curve, it helps only very approximately.
well, you'd have to be decently well-trained in math to picture a sine curve that isn't, say, a series of parabolas glued together.
I may not be using the same imagery you are. My mental eye appears to work from something my real eye has seen. I can look at a drawing of a sine curve and later imagine it in my head. This is not the same as recalling the original curve I saw. I can toy with the curve in my head but I do not sit there and draw the line from point A to B. It just poofs into my mind's eye.