Sunday, December 03, 2006

Inductive Epochs

...These primary movements, when the Inductive process, by which science is formed, has been exercised in a more energetic and powerful manner, may be distinguished as the Inductive Epochs of scientific history; and they deserve our more expressed and pointed notice. They are, for the most part, marked by the great discoveries and the great philosophical names which all civilized nations have agreed in admiring. But, when we examine more clearly the history of such discoveries, we find that these epochs have not occurred suddenly and without preparation. They have been preceded by a period, which we may call their Prelude, during which the ideas and facts on which they turned were called into action;--were gradually evolved into clearness and connection, permanency and certainty; till at last the discovery which marks the epoch, siezed and fixed forever the truth which had till then been obscurely and doubtfully discerned. And again, when this step has been made by the principal discoverers, there may generally be observed another period, which we may call the Sequel of the Epoch, during which the discovery has acquired a more perfect certainty and a more complete development among the leaders of the advance; has been diffused to the wider throng of the secondary cultivators of such knowledge, and traced into its distant consequences. This is a work, always of time and labor, often of difficulty and conflict. To distribute the History of science into such Epochs, with their Preludes and Sequels, if successfully attempted, must needs make the series and connections of its occurrences more distinct and intelligible. Such periods form resting places, where we pause till the dust of confused march is laid, and the prospect of the path is clear.

[Whewell, History of the Inductive Sciences, Volume I. D. Appleton and Company (New York: 1858) p. 47.]

An example might suffice to clarify. Whewell identifies an early inductive epoch in astronomy, which he calls the Inductive Epoch of Hipparchus. The scientific problem with which it dealt was the problem of the wandering bodies, i.e., planets, which appeared to defy the otherwise rigid order of the heavens. Bit by bit people had begun to develop rules for describing their motions, tracing over long years the various cycles that the various planets undergo. But this on its own doesn't get you very far; it just gives you Bradshaw, not the train. By thinking about the planets on analogy with wheels and the like, they were able to come up with the notion of an epicycle. This handled otherwise puzzling problems like retrograde motion; but astronomers were forced to extend it by further anomalies uncovered by close examination of the data, such as the peculiarities involved in the paths taken by the moon and the sun across the sky. It doesn't take much to see that the notion of an epicycle can easily handle this sort of problem; so it was extended. Thus we have a progress in conceptions of the epicycle going with progress in acquaintance with facts. All these were prerequisites for the first great theory of astronomy, that of Hipparchus.

The scientific problem as developed in the prelude to the epoch, then, was to reconcile the celestial phenomena by means of equable circular motions. Whewell notes that, while we tend to dismiss this problem (since the circularity condition is inconsistent with nature), as a part of the prelude it not only makes sense, it is a reasonable thing to try, since, if it panned out, it would give you the phenomena by way of the simplest and most manageable conjecture. The bad name epicycles have received is due not to the work done in the prelude, nor Hipparchus's advance in constructing a fully successful theory of epicycles and eccentrics, but to the quirks of the sequel to the epoch, in which the circularity condition was held with great tenacity even in the face of mounting evidence against it, precisely because it was such a simple and elegant supposition. There's an ambivalence to it: Whewell puts it forward as an example of the love for simplicity that both drives scientific progress and creates impediments to the same.

In any case, Whewell identifies Hipparchus as the cardinal point in astronomical progress during this epoch, on the basis of the maxim that he who proves, discovers. The epicycle was nothing new when Hipparchus came along; it had already been in use for the purposes of explaining anomalies in the wandering bodies. Similarly with the eccentric. To have a genuine theory of epicycles and eccentrics, however, you need to be precise: you need to identify the magnitudes, distances, and positions of the of the circles you are positing, in such a way that the circles capture the irregular and anomalous motions for which you are trying to account. One of the signs of Hipparchus's genius was his ability to come up with this on the basis of surprisingly limited data; the tables he constructed stood up to the test of predicting eclipses, the most serious and important test of any astronomical model at the time. By doing so, they showed that they were an adequate representation of the path of the sun to the level of precision required for tracking eclipses. Hipparchus did the same with the moon, to the same level of precision, on the basis of only six recorded eclipses. This formed a clear and definite basis for people to extend the same idea to the bodies of the planets. Hipparchus did not complete this task, but famously gathered together much of the material for it; according to Ptolemy the whole mass of astronomocial observations left to posterity by Hipparchus' time was dwarfed by the mass left to posterity by Hipparchus himself.

Again, we have a tendency to underrate the importance of this theory of eccentrics and epicycles; we now know that perhaps its key postulate is false, and we tend to think of the theory as inordinately complicated and tangled. In fact, this is in great degree an illusion of hindsight, and to the fact that we are separated from Hipparchus's actual epoch of discovery by a long and complicated sequel. As Whewell notes, the value of a true part of a theory may far outweight its error; and the usefulness of a rule does not always depend on its simplicity. As he notes, "The first steps of our progress do not lose their importance because they are not the last; and the outset of the journey may require no less vigor and vitality than the close" (p. 152). What is true about the theory of Hipparchus is its resolution of the phenomena into circles; it as true that they can be so resolved today as it was then. This resolution allows us to construct precise tables by which we can clearly determine the position of the planets at any time. The assumption that made this resolution possible, and thus the precise predictions, was false; but some assumption is needed for any theory of motions, and it was a simple and straightforward assumption to make. We tend to look at the result and say, "How complicated!" But the simplicity of the theory was not in its final conclusions but in the fact that from a very small set of data, from just a few features of planetary motion, you could derive the principles that would allow you to explain a an immense array of extremely disparate observations. Eccentrics and epicycles were perfectly capable of representing the quantity of the inequalities in planetary motion; and this is not a small thing. So well does it do so, in fact, that for the convenience of calculation of this inequality, the theory first put forward by Hipparchus is very difficult to beat; as Whewell notes, if we complain about the complexity of an unusually simple method of calculation for a given natural quantity, what we are really complaining about is nature, not astronomy. Moreover, we tend to assume that astronomers gave the same faith to epicycles and eccentrics that we do to Kepler's ellipses; when in fact they were much more ambivalent, and were usually quite explicit about its being simply the best hypothesis on hand. The precise representation of apparent motion provided by the theory, however, allows you to collect the data that is needed before you can identify the actual motion of the planets; without Hipparchus, there would be no Kepler.

After the epoch of induction, we entered the sequal, a period of development, verification, application, and extension. This period was started by Hipparchus himself, who developed a catalogue of stars for more accurate record-keeping, showed with greater precision the length of years and days. He also began discovering points that did not confirm the theory -- parallaxes, for instance -- but which astronomers at the time lacked the ability to be precise enough to handle properly. (One of the problems parallax shows in the theory is that while it can accurately chart apparent position, it has serious trouble getting the distances correct.) The most famous person in the sequel, however, is naturally Ptolemy, whose development, extension, and popularization of the Hipparchian theory is what became dominant, and is, in fact, our primary source for knowing anything about Hipparchus at all. One of Ptolemy's contributions was the discovery of yet another celestial inequality, the evection of the moon, and his accounting for it by the theory of epicycles. This discovery was quite crucial to astronomy, or would become so much later on, because it suggested that there might be numerous inequalities that were yet remaining to be discovered, and that the confirmation of the theory lay in its power to explain such residual phenomena. It was also one of the first in a long line of discoveries of inequalities in the moon's motion that are due to the pull of the sun; and the importance of those for the Newtonian discovery of universal gravitation can hardly be underestimated. From Ptolemy the sequel extends through very long years, until the rise of a new inductive epoch.

Thus we have a sort of stage in astronomical progress, involving a prelude of preparation, an epoch of induction, and a sequel of development.

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