In 1919, Sir Frank Watson Dyson and Sir Arthur Eddington made an expedition to the island of Príncipe off the west coast of Africa to observe a total solar eclipse. They were hoping to test a prediction of Einstein’s general theory of relativity, that light from a distant star will appear to bend by a specifiable amount as it passes by a massive stellar object, such as the sun. During a total solar eclipse, it is possible to see distant stars that are normally hidden from view and thus observe whether the light they emit does in fact bend — and by how much — as it passes by the sun.

Their observations matched Einstein’s prediction exactly. It was the first major test of general relativity, and it showed that the new theory was superior to the Newtonian physics that had held sway for centuries. The news appeared on the front page of major newspapers all across the world: Einstein became the world’s most famous physicist literally overnight. This event is often considered to be an experimentum crucis, or “crucial experiment” — an empirical test that definitively confirms a scientific theory over its rivals. Was it?

A closer examination suggests a more complicated story, but one that illuminates some key features of the scientific enterprise.

On what basis did Einstein make his prediction about the apparent bending of light? According to his general theory of relativity, gravity is not a force but rather the very curvature of space-time itself. A massive object, such as the sun, will create a kind of divot in space-time, just like a bowling ball on a rubber sheet. Thus, as light passes by the object, it will trace the curvature of that divot. Imagine rolling a ball bearing across the rubber sheet, right past the bowling ball. If the ball bearing is going quickly enough, it will roll down into the divot and come up on the other side; hence its trajectory will be curved. In this example, the rubber sheet is space-time, the bowling ball is the sun, and the ball bearing the light whose trajectory curves as it passes by.

To describe this kind of phenomenon, Einstein had to use a relatively new branch of mathematics: non-Euclidean geometry — a kind of geometry in which space is curved, not flat. This was an important break from classical physics, which used Euclidean geometry. Einstein did not invent non-Euclidean geometry, but until his general theory of relativity it was considered to be a mathematical curiosity, at best.

Euclidean geometry — the kind familiar to all from high school — is an axiomatic system that dates back to the ancient Greek mathematician Euclid. An axiomatic system begins with a set of “axioms,” self-evident postulates from which all other truths may be deduced, following a few general principles. One of Euclid’s axioms states that two parallel lines never meet. This “parallel postulate” irked mathematicians from the beginning, because it did not seem self-evident. (How can we be certain that infinitely long lines will never meet?) As a result, mathematicians over the course of millennia tried to reduce the parallel postulate to the remaining Euclidian axioms. These efforts proved futile. Ultimately, it was demonstrated that the parallel postulate was logically independent, that is, it could not be reduced to the other axioms.

What would happen then, if we simply denied this parallel postulate, some mathematicians began to wonder? Could we have a geometry, for instance, in which parallel lines do meet?

It turns out, the answer is yes. Nineteenth-century mathematicians produced several — non-Euclidean — geometries for which the parallel postulate does not hold. The two most famous of these are geometry with a space of negative curvature (think of a saddle), named for the Russian mathematician Nikolai Lobachevsky, and geometry with a space of positive curvature (think of a globe), named for the German mathematician Bernhard Riemann.

It was taken for granted at the time, thanks to classical physics and the evidence of everyday experience, that real, physical space is Euclidean. After all, no building could be constructed without assuming that parallel lines are, well, parallel in the conventional, Euclidean sense. But some scientists began to wonder if we could test empirically whether physical space is truly Euclidean. In the early twentieth century, the French scientist, mathematician, and philosopher Henri Poincaré conducted a thought experiment in order to determine if this was indeed possible.

Euclidean geometry has several well-known properties that distinguish it from non-Euclidean geometries. One of these, known as the Angle Sum Theorem, states that the interior angles of a triangle sum to 180 degrees. This is not true in non-Euclidean geometry. (In Lobachevskian geometry the interior angles of a triangle sum to less than 180 degrees; in Riemannian geometry, they sum to more than 180 degrees.) Thus, Poincaré surmised, we should be able to test whether physical space is Euclidean by measuring the angles of a triangle in physical space. If the interior angles of that triangle sum to 180 degrees, physical space is Euclidean; otherwise it is not. It would have to be a very large triangle, though, since the difference between the two kinds of geometry is unnoticeable on small scales.

So Poincaré devised an experimental test. One very large triangle would have as its three points the Earth at two locations in its orbit around the sun — comprising the base of the triangle — and the location of a distant star (the apex of the triangle). It is possible, using simple astronomical techniques, to measure the interior angles of such a triangle relying on what is known as stellar parallax — thanks to the light emitted by the distant star. Say we take such a measurement, Poincaré says, and we add up the angles of the cosmic triangle, and the result is greater than 180 degrees. It would appear that we have demonstrated empirically that physical space is non-Euclidean. But have we?

Notice, Poincaré says, that we made a crucial, if implicit, assumption when conducting our experiment. We inferred, based on our measurements, that the light from the distant star followed a curved trajectory on its way toward us, creating a triangle whose lines appear to bow outward — as if the triangle were traced on the surface of a globe. In other words, we assumed that this case was analogous to the ball bearing rolling across the curved rubber sheet. And so we concluded that the underlying space was itself curved, that is, non-Euclidean. But who is to say that the path of light must coincide with the lines of the underlying geometrical figure? It could just as well be that the underlying space is flat — Euclidean — while the light itself was bent on its way toward us from the distant star, perhaps by some undetected force. If we reject the assumption that the light’s path coincides with geometrically straight lines, then what our observations suggest is not that space itself is curved (non-Euclidean), but only that the trajectory of the light is curved. In that case, we have no reason to abandon Euclidean geometry.

In other words, our experiment does not settle the issue, but may be interpreted in multiple ways. On one interpretation, our empirical results show that physical space is non-Euclidean; on the other interpretation, we assume that space is Euclidean and so conclude that there must be some (other) empirical explanation for why the light is bending. To use a technical term from philosophy of science, our observations underdetermine which theory is correct, whether space is Euclidean or non-Euclidean. How are we to choose?

According to Poincaré, the choice came down to one of convenience. Is it more convenient to describe the experiment using Euclidean or non-Euclidean geometry? Thus, Poincaré believed, this thought experiment illustrates that we cannot determine empirically whether space is Euclidean or non-Euclidean. There is no empirical fact of the matter, since observations cannot resolve the issue for us. Instead, the choice between these two mathematical descriptions is conventional, not empirical, akin to picking between two languages. He believed (wrongly) that scientists would always prefer Euclidean geometry on account of its familiarity and simplicity and so make whatever other theoretical adjustments were needed to do so.

Note the similarity between Poincaré’s thought experiment and the real-world experiment undertaken by Dyson and Eddington. Though they were not directly testing whether space is non-Euclidean, but rather Einstein’s prediction about the bending of light, they nevertheless had to make the same assumptions about straight lines and light rays for the same reasons, because Einstein’s theory employs non-Euclidean geometry. Like Poincaré’s imagined experimenter, Dyson and Eddington had to make certain non-empirical choices in order to make sense of their observations. It follows, then, that a Euclidean (i.e., a non-Einsteinian) interpretation of their observations is logically possible. To be sure, their observations appear to contradict Newtonian physics. But why assume that Newton’s physics is therefore incorrect? Why not instead keep Newtonian physics and make the necessary theoretical adjustments to accommodate this recalcitrant piece of data? In other words, why not do what Poincaré thought scientist would, in fact, do when faced with such empirical evidence and find a way to hang on to the Euclidean description of physical space?

Einstein was well aware of — and even indebted to — Poincaré’s thinking about geometry and the ineradicable role of convention in science. Like Poincaré, Einstein emphasized the underdetermination of theory by observation and the attendant need for non-empirical assumptions in science. So why, then, when asked what he would have done if Dyson and Eddington had made observations that contradicted rather than confirmed his prediction, did Einstein quip, “Then I would feel sorry for the dear Lord. The theory is correct anyway”? If Poincaré is right, Dyson and Eddington’s observations underdetermine whether Einstein’s prediction was correct, just as Poincaré’s thought experiment underdetermined whether physical space was non-Euclidean. And Einstein agreed with Poincaré that such empirical tests could never be entirely definitive. At the very least, why didn’t Einstein say something like, “My theory is the simplest — most convenient — interpretation of the phenomena,” instead of calling it “correct”?

Because Einstein saw more clearly than Poincaré that the underdetermination of theory by observation does not imply that the choice between theories is merely one of convenience, that there is no fact of the matter. What it does imply is that particular claims of a theory cannot be tested in isolation; there is no single, definitive, empirical test of a scientific claim — no experimentum crucis. Instead, a scientific theory must be evaluated as a whole, on the basis of how well and perspicaciously it predicts and explains a wide range of empirical phenomena within the broader context of other battle-tested theories. On this score, Einstein’s theory was and remains unrivaled. Einstein’s theory is “correct” — not because of one experiment, but because of its overall explanatory success.

If for Poincaré the theoretician’s choice is akin to choosing between two languages, for Einstein, the labor of constructing a scientific theory is akin to the creation of a work of art — of a representation that truly captures reality. So Einstein did more than make a correct prediction; he furnished us with a new scientific image, one to which we remain indebted, at least until it becomes necessary to create another one that better pictures reality. Dyson and Eddington observed the solar eclipse with Newtonian eyes; some one hundred years later, we observe another solar eclipse with eyes habituated to a theoretical universe created by Einstein.

One Response

  1. Fascinating and interesting! Some things we may never know, but to ponder and question keeps the unknown alive and vital in this sometimes mundane and predictable world.

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