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2007-Aug-24, 11:24 PM
Special Relativity/The interpretation of special relativity
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Special Relativity

[edit] Geometrical interpretation

[edit] Natural mathematics, Einstein's "practical geometry" and the Ryskamp objection of "natural" coincidence
Note that the metrical interpretation of the 1905 paper disguises, but does not avoid, the "natural" coincidence of points M' and M ("fallt zwar...zusammen") which Einstein, following the program of natural mathematics which he had adopted from Poincare's SCIENCE AND HYPOTHESIS (1902), put in the train experiment in his book RELATIVITY (1916). This work receives little attention from scholars because it is regarded as a popularization. However, Einstein valued it highly and recently it has been claimed as the clearest exposition of the relativity of simultaneity--of "practical geometry," which was Einstein's term for natural mathematics.

Previously, Einstein's devotion to "practical geometry" was regarded as problematic in relation to relativity, because it could not be seen precisely where it operated in the mechanics of special relativity. John Ryskamp's contribution was to explicate this operation, in the course of which he drew a fundamental objection. Although RELATIVITY had been around for upwards of ninety years, he was the first commentator to draw attention to "natural" coincidence as THE application par excellence of "practical geometry," in "Paradox, Natural Mathematics, Relativity and Twentieth-Century Ideas" (May 19, 2007). Available at SSRN: http://ssrn.com/abstract=897085. See also the Discussion section for this issue. This paper also lodged a "natural" coincidence objection against many fundamental theories, and claimed that it invalidates the Pythagorean theorem--although to date he has not indicated where "natural" coincidence is lodged in any given proof of the Pythagorean theorem.

Here is the crucial passage from the paper (pp. 13-14):

Consider this passage from Lawson’s accurate translation of Einstein’s Relativity:

Are two events (e.g. the two strokes of lightning A and B) which are simultaneous with reference to the railway embankment also simultaneous relatively to the train? We shall show directly that the answer must be in the negative. When we say that the lightning strokes A and B are simultaneous with respect to be embankment, we mean: the rays of light emitted at the places A and B, where the lightning occurs, meet each other at the mid-point M of the length AB of the embankment. But the events A and B also correspond to positions A and B on the train. Let M1 be the mid-point of the distance AB on the traveling train. Just when the flashes (as judged from the embankment) of lightning occur, this point M1 naturally coincides with the point M but it moves…with the velocity…of the train.

This passage is by now so familiar that we think there can be nothing new to be seen in it. But there is: it is the term, “naturally coincides.” This term (“fällt zwar…zusammen” in the German) leaps out at us because we are looking at it with twenty-first century eyes, not twentieth-century eyes; indeed, perhaps the most difficult cultural task now before us is simply to realize that we are not living in the twentieth century.

“Natural” coincidence is otherwise known as a spacetime point. Einstein has already spent twenty-odd pages of this very brief book laying out the assumptions which underlie the train experiment. He is very careful about being consistent with them, and he is a devoted and very strict Euclidean. But Einstein was not, it appears, quite careful enough. We know that he is assuming, along with Euclid, that the definition of the coincidence of two points is a point. However, we have never gotten (and never get, in any of Einstein’s writings) a definition of a “natural” coincidence of two points. This alone prevents us from going on and this argument, which defined the twentieth century, abruptly ends. We also have a problem if we try to resolve the issue ourselves. If we simply drop the term “naturally” we run into a situation in which Einstein has told us to assume two Cartesian coordinate systems, but now leaves us with one, since, following from the definition of the coincidence of two points, if two parallel coordinate systems coincide at one point, they coincide at all points and are one coordinate system, not two. We have been led to a contradiction.

Ryskamp's idea that the presence of "natural" coincidence disproves the theory, is new among the many lodged against special relativity over the decades. Remarkably, and unlike other objections, it disputes NONE of the hypotheses, assumptions or principles of special relativity--the way critics of relativity previously proceeded. This is because, in Ryskamp's view, natural mathematics imports ideas into arguments which have no logical relation to any part of the argument: according to the point of view of natural mathematics, this must be done. At the same time, if it is recognized that that is what is going on in any given natural mathematics argument, it leads us to wonder where this importation has occurred.

Ryskamp's accomplishment seems to be, to have definitively located such an importation into relativity. According to natural mathematicians, this is no great accomplishment, because such importations must occur. However, it is mortifying for advocates of relativity who feel relativity has been proved, because Ryskamp's objection is, if an accurate description of what is going on in the relativistic argument, manifestly and incontestably a disproof of relativity. According to natural mathematics itself, there are no grounds on which to contest it, which makes it the most alarming of all comments on relativity.

The physics community has found it extremely difficult to deal with the Ryskamp objection, apparently because physicists themselves have a natural mathematics point of view. They are now trying to disentangle themselves from this point of view, but it is proving to be a monumental task due to the mountainous physics terminology built on the foundation of "natural" coincidence. It is not surprising, then, that the objection has not yet been overcome by any commentator, no matter how high up in the physics hierarchy. At the very least, the objection has had two consequences: first, it anchors Einstein firmly within the natural mathematics polemic. Second, it clarifies other objections which, while not on point, are now evaluated as more or less unfocussed warnings that there may be something wrong with the geometry of special relativity. As the new mathematics historiography proceeds, more embarrassing material on the terms of natural mathematics is being revealed: it can now hardly be said to be a point of view; it is more akin to a religion or a cult. No scientist wants to be seen as partaking of any such doctrine.

Among the other contentions of Ryskamp's paper, he claims that Einstein, as a devotee of natural mathematics, could NOT have sought, in formulating relativity, to make relativity internally consistent, because it is the position of natural mathematics that arguments with a program of internal consistency lead invariably to paradox. While commentators had long noted problematic aspects of Einstein's argumentation, the universal approach to relativity was that Einstein had at least intended to make it internally consistent. Ryskamp contends that this was never the case, which if true would mean that an entirely different approach has to be taken to relativity. Indeed, it may be that Einstein's 1921 lecture, "Geometry and Experience," in which he used the phrase "practical geometry," constitutes Einstein's summary of the degree of refinement to which he was able, in RELATIVITY (1916), to bring his expression of natural mathematics.

Approaching relativity as an argument which was never intended to be internally consistent--an eye-opening approach, to say the least--and identifying "natural" coincidence by applying that approach, emerges from a recent trend in the historiography of physics: to examine a facet of Einstein's work which has not traditionally been examined in isolation--his rhetorical approach. Historians of physics have recently become much more sensitive to Einstein as a textual architect--an "artful" writer, as one commentator puts it--and the role this may play in evaluating the logic of special relativity. This would seem to be in order: if relativity was never intended to be internally consistent, then unification is a misdirected project based on a faulty understanding of relativity.

This approach is at the heart of two recent works: Don Howard and John Stachel, eds., EINSTEIN: THE FORMATIVE YEARS (2000) and Thomas Ryckman, THE REIGN OF RELATIVITY (2005). The approach is an attempt to break out of an apparent dead end in quantum electrodynamics by looking to see if new insights of a synthetic nature could be gleaned from the historical record. As Ryckman points out, terms coined decades ago are still used even by advanced physicists in order to frame and present physics research--without any consideration that such terms may be themselves problematic or incorporate problematic notions in relativity.

A recent phenomenon, historical perspective is finally being brought to bear on the early career of relativity. Ryskamp's own approach is novel and entirely unexpected: through the history of set theory. The result is that readers may now approach Einstein's work with the idea that it may need to be regarded as dated, as an historical anachronism addressed to a past audience which shared Einstein's general intellectual approach--closer, argument for argument, to Newton than to contemporary understanding. At the very least, the Ryskamp objection is waving a big red flag of caution.

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