May 23, 1997

Continue discussion from problem in last lesson:

Construct an ellipse and, given amount of angle traversed in a month, calculate the length of a year using Kepler's second law.

Answer these questions:

1. What tools did you need to do this problem?

2. Which of these tools do you think Kepler had?

3. How do you think Kepler would have done the problem?

4. What do you suppose gave him the idea for his second law?

5. Why did Kepler's second law stick?

Brief discussion of questions 1 and 2, as well as explanation of how problem was done by each group. (Two presentations by groups.)

Class discussion of questions 3 and 4, exploring ways people might have gotten stuck.

1. Kepler was measuring angles only. The distance had to be inferred. Why did he use "area" as a basis for law rather than "square of the distance"?

2. Did Kepler have tables of sines and cosines? If so, where were they from?

3. Draw a picture of the Copernican schema on the board. How does Kepler's schema resolve problems?

4. Looking at things geometrically is a cultural preference. Kepler was also a platonist. What does that mean?

Harmonies of the world

1. What exactly has Kepler calculated here and why?

2. Kepler is looking for rational relationships between natural cycles. From what assumptions does this approach spring? p.177 about platonic solids.

... As regards the ratio of the planetary orbits, the ratio between two neighboring planetary orbit is always of such a magnitude that it is easily apparent that each and every one of them approaches the single ratio of the spheres of one of the five regular solids, namely, that of the sphere circumscribing to the sphere inscribed in the figure. Nevertheless it is not wy equal, as I once dared to promise concerning the final perfection of astronomy. Epitome of Copernican Astronomy & the Harmonies of the World, Kepler, p. 177

On geometry:

But since God has established nothing without geometrical beauty, which was not bound by some other prior law of necessity, we easily infer that the periodic times have got their due lengths, and thereby the mobile bodies too have got their bulks, from something which is prior in the archetype, in order to express which thing these bulks and periods have been fashioned to this measure, as they seem disproportionate ... And because all these things are variable in the planets, there can be no doubt but that, if these things were allotted any geometrical beauty, then, by the sure design of the highest Artisan, they would have been received that at their extremes, at the aphilial and perhelial intervals, not at the mean intervals lying in between. Epitome of Copernican Astronomy & the Harmonies of the World, Kepler, p. 185

3. He is looking for harmonies -- literally among celestial phenomena. He knows that any measurement of angular velocity will depend on the position of the viewer, so he tries the periodic times of the planets, and finds the ratios aren't nice enough to suit him.

Accordingly, in these periodic times there are no harmonic ratios, as is easily apparent... All the last numbers as you see are counter to harmonic ratios and seem as it were irrational. Epitome of Copernican Astronomy & the Harmonies of the World, Kepler, p. 184.

How does he explain this?

There for with every thing reduced to one view, I concluded rightly that the true journeys of the planets through the ether should be dismissed, and that we should turn our eyes to the apparent diurnal arcs...Epitome of Copernican Astronomy & the Harmonies of the World, Kepler, p. 190,

So he concludes it is reasonable to look for harmonies that would be visible to a viewer at a particular viewpoint: diurnal motions of planets as viewed from the sun. If he finds them, he has more ammunition for the Copernican viewpoint. why?

4. This table is the basis for his argument. Note the fudging of data.

... Accordingly, I could mentally presume, even from the ratios of the diurnal eccentric arcs given above that there were harmonies and concordant intervals between these extreme apparent movements of the single planets, since I saw that everywhere there the square roots of harmonic ratios were dominant, but knew that the ratio of the apparent movements was the square of the ratio of the eccentric movements. Epitome of Copernican Astronomy & the Harmonies of the World, Kepler, p.191.

5. What about these ratios? related to music. Should read ratios as string lengths. He represents the alteration in the diurnal motion of a single planet as a musical interval and then attempts to tune them against each other, up to octaves at least. A little bit about string length and how it affects tuning.

6. Check Mars' top note with earth's bottom note. It's a perfect fifth, exactly two thirds of the way up the string. Kepler discusses Mars and times when these harmonies are sounded.

But Mars, which got a greater interval as its own, received some amplitude of tuning. Mercury received an interval great enough for it to set up almost all the consonances with all the planets within one of its periods. Earth and Venus much more so, on account of the smallness of their intervals, limit the consonances, which they form not merely with the others but with one another in especial, to visible fewness. But if three planets are to concord in one harmony, many periodic returns are to be awaited; nevertheless there are many consonances, so that the may so much the more easily take place, while each nearest consonance follows after its neighbor, and very often threefold consonances of four planets are seen to exist between Mars, the Earth, and Mercury. But the consonances of four planets now begin to be scattered throughout centuries, and those of five planets through thousands of years ...201

7. Kepler declares that human harmonies imitate God--this is why we think they are beautiful.

Hence it is no longer a surprise that man, the ape of his Creator, should finally have discovered the art of singing polyphonically [per concentum], which was unknown to the ancients, namely in order that he might play the everlastingness of all created time in some short part of an hour by means of an artistic concord of many voices and that he might to some extent taste the satisfaction of God the Workman with His own works, in that very sweet sense of delight elicited from this music which imitates God. p. 208

8. Is this science? science fiction? Can you categorize it exactly?


Axiom. It is reasonable, that wherever in general it could have been done, all possible harmonies were due to have been set up between the extreme movements of the plainest taken singly and by twos, in order that variety should adorn the world. p211

231, bottom Below he describes his process of discovery.

... I recognized nothing more beautiful than the ratio of equality. But this ratio is without head or tail: for this material equality furnished no definite number of mobile bodies, no definite magnitude for the intervals. Accordingly, I meditated upon the similarity of the intervals to the spheres, i.e., upon the proportionality. But the same complaint followed. For although to be sure, intervals which were altogether unequal were produced between the spheres, yet they were not unequally equal, as Copernicus wishes, and neither the magnitude of the ratios nor the number of the spheres was given. I passed on to the regular plane figures: intervals were formed for them by the ascription of circles. I came to the five regular solids: here both the number of the bodies and approximately the true magnitude of the intervals was disclosed, in such fashion that I summoned to the perfection of astronomy the discrepancies remaining over and above. P 238

To sum up: One the one hand Kepler was a platonist and believed that his mathematics was a "discovery." On the other hand, his choice of the ellipse, the geometric version of the second law, the putting of the sun at the focus, and all the tools at his disposal were all culturally skewed decisions. So:

Is mathematics something that exists independently until we discover it, or is mathematics something that is constructed by human beings?

If it is out there, then by what mechanism do we interact with it? Mysticism? "If it is in here" then why does it model nature as well as it seems to do? Again, mysticism and the "unreasonable effectiveness of mathematics".

In conclusion:

Planets move in ellipses around the sun.

It is convenient to assume that planets move in ellipses around the sun.

These are both religious, extremely pious statements.

Quotes from Pi in the Sky.

A reality completely independent of the spirit that conceives it, sees it or feels it, is an impossibility. A world so external as that, even if it existed, would be for ever inaccessible to us." Henri Poincare

It by no means follows, however, that (intuitions), because they cannot be associated with actions of certain things upon our sense organs, are something purely subjective... Rather, they too may represent an aspect of objective reality, but, as opposed to the sensations, their presence in us maybe due to another kind of relationship between ourselves and reality. Kurt Godel

"What empirical scientists would be impressed by an explanation this flabby? " ...It is like appealing to experiences vaguely described as `mystical experiences' to justify belief in the existence of God." Charles Chihara

"I imagine that whenever the mind perceives a mathematical idea, it makes contact with Plato's world of mathematical concepts... When mathematicians communicate, this is made possible by each one having a direct route to truth, the consciousness of each being in a position to perceive mathematical truths directly, through this process of 'seeing'." Roger Penrose

In the Reality Game religion has always been science's toughest opponent, perhaps because there are so many surface similarities between the actual practice of science and the practice of most major religions. Let's take mathematics as an example. Here we have a field that emphasizes detachment from worldly objects, a secret language comprehensible only to the initiates, a lengthy period of preparation for the "priesthood, holy missions (famous unsolved problems) to which members of the faith devote their entire lives, a rigid and somewhat arbitrary code to which all practitioners swear their allegiance, and so on." John Casti.

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