The Logic Off Topic list

At 03:30 PM 11/11/01, you wrote:
> >>Then there was all that mumbo jumbo he had about cosmic solids. He
> >>really thought it beautiful. As a 'work of art'. maybe. But as a
> >>mathematical demonstration? Yet ... there were people who thought, at the
> >>time, that it was a demonstration of mathematics at its best.
>
>What mumbo-jumbo? Actually, the n-sphere is a good example of the
>abstractness of mathematics (the unit sphere in n-dimensions given by the
>equation (x_1)^2 + (x_2)^2 + ..... + (x_n)^2 = 1) - something that would
>take n dimensions to visualize, but nonetheless exists as though it were
>a household object for mathematicians.

I don't doubt any of that. However, that's not what so delighted Kepler. I 
meant to refer to his Aristotelian mumbo-jumbo ... all that stuff about the 
planets and their 'meaning'.

"The orbit of the Earth is a circle: round the sphere to which this circle 
belongs, describe a dodecahedron ... <snip> ... Now inscribe in the Earth's 
orbit an icosahedron ... <snip> Inscribe an octahedron in the orbit of 
Venus; the circle inscribed in it will be Mercury's orbit. This is the 
reason for the number of the planets". But ... he didn't stop there. He 
then went on to philosophise about the numerological significance of all 
those numbers, and therefore the role in life that each planet played 
because of its number and position. To him, that was a part of the beauty 
... not just the abstract mathematical patterning of it all.

This would not be so bad if it were not for the context in which it were 
written. Kepler was not in the least interested in mathematical beauty. He 
was after a Divine, harmonious order for the solar system, and if he could 
find it in prayer, good enough; if he could find it in witchcraft; good 
enough; and if he could find it in Platonic solids, also good enough.

In his defence, however, Kepler did also say the following:
"We do not ask for what useful purpose the birds do sing, for song is their 
pleasure since they were created for singing. Similarly,we ought not ask 
why the human mind troubles to fathom the secrets of the heavens. The 
diversity of the phenomena of Nature is so great, and the treasures hidden 
in the heavens so rich, precisely in order that the human mind shall never 
be lacking in fresh nourishment."

That bespeaks, I guess, an appreciation of abstract patterns. However ... 
Kepler had great difficulty in accepting that the model is completely false 
and that the interplanetary distances it predicts are inaccurate. The 
mathematician in him qua scientist had great difficulty accepting this 
because he wanted to cling to both the truth and the beauty of the pattern. 
This is probably a good example of the difficulties of equating truth and 
beauty because the two are not always equivalent -- but then again the 
whole point of Kepler's presentation was not just that it was a pretty 
pattern but that it was true.

Thomas Huxley referred to this kind of thing as: 'the great tragedy of 
science, the slaying of a beautiful hypothesis by an ugly fact'. I have 
always loved that quote!!

When Kepler's idea was no longer true, it also was not pretty any more. 
However, he was forced to accept their speciousness eventually because he 
himself found different laws that worked much better. When he found his new 
laws and published them, his whole attitude made them very difficult to 
find or appreciate. His New Astronomy was a complete mess and contained not 
only the valuable information about his three laws, but a lot of guff, 
philosophising -- and also intricate information about all the wrong 
guesses he had made, every bogus hypothesis he had had, every blind alley 
he had explored. He was not really that interested in the beauty of any of 
it. If one can conclude anything it was that his three laws of planetary 
motion were not particularly beautiful. They were just true.

All in all, the case of Kepler is confusing in this regard, and he is 
actually a much better candidate for discussing this whole issue of 
mathematical beauty because he really did stand at an important crossroads. 
Things moved from simple prettiness to their utility in terms of 
computational excellence and conciseness and a new kind of elegance -- 
which Kepler just didn't seem to appreciate. Even though he was a prime 
architect in introducing it.

Kool Musick
Keep Musick Kool


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