HPS 0410 Einstein for Everyone

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Philosophical Significance of the General Theory of Relativity
What does it all mean, again?

Theory and Evidence

John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh

General relativity appeared in science at a fertile moment philosophically. In the mid 1910s, new, modern traditions were emerging in philosophy of science. The philosophers building this new tradition were especially impressed by Einstein's achievement. They took careful note of the theory and of Einstein's general views on scientific methodology. The influence of Einstein's success with general relativity remains today a popular example and test case for views on the relationship of theory and evidence. At the center of these considerations is a reappraisal of how evidence bears on theory. As we shall see below, there are two sides to the reappraisal. One overturns an older view and the other promotes replacement views.


Philosophers and philosophically-minded scientists have advocated traditionally an inductive method for science. The idea was that scientists would gather  particular facts of experience and combine them to form the generalizations that are the laws of science. In the seventeenth century, Bacon urged scientists to assemble tables of absence and presence of some feature of the world and from them to discern the underlying form or law of the feature. In the nineteenth century, John Stuart Mill urged scientists to see which instances agreed and which differed in some property and from them to infer to the property's cause. The key element is that this was both an inductive logic and a method of discovery.

Best known of Mill's expansive  pronouncements in his 659 page System of Logic are what are now known as "Mill's Methods." They consist of a collection of recipes for identifying the causes of effects. The first two are the "method of agreement" and the "method of difference." Here are their primary statements, in Mill's words, of his "methods of experimental inquiry":


If two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which alone all the instances agree, is the cause (or effect) of the given phenomenon.

If an instance in which the phenomenon under investigation occurs, and an instance in which it does not occur, have every circumstance in common save one, that one occurring only in the former the circumstance in which alone the two instances differ, is the effect, or the cause, or an indispensable part of the cause, of the phenomenon.

 J. S. Mill. A System of Logic. New York: Harper Bros., 1882. Book III, Ch. VIII, pp. 280.

These simple methods can be very effective. They enable us to identify when a particular microbial pathogen is responsible for some specific disease. For it will be present whenever the disease manifests and the missing factor when it fails to manifest. It is, by the lights of Mill's methods, the cause of the disease.


The methods, however, are strained when the relationship of cause and effect is probabilistic. For then the presence of the cause need not guarantee the effect. Mill described his methods in his 1843 System. More serious problems were coming.

Through the nineteenth century physical theories became successively more complicated and abstract. However one could still maintain the idea that they were derived somewhat mechanically from experience. Einstein's two principles of special relativity, for example, could be seen as some sort of simple generalization of experience. The principle of relativity, for example, might still be a simple generalization of the failure of ether drift experiments. All ether drift experiments have failed so far. Hence we infer that all ether drift experiments will fail.

With general relativity, however, the gap between experience and the core elements of the theory had become too great. To arrive at general relativity required creative leaps that seemed quite remote from the sorts of mechanical generalizations envisaged by Bacon and Mill. General relativity seemed to provide the refutation of both notions: that physical theories could be arrived at by a process of generalization and that there was an identifiable method governing the process.

Or so Einstein asserts in his 1933 lecture "On the Methods of Theoretical Physics":

"The natural philosophers of those days were, on the contrary, most of them possessed with the idea that the fundamental concepts and postulates of physics were not in the logical sense free inventions of the human mind but could be deduced from experience by "abstraction"--that is to say, by logical means. A clear recognition of the erroneousness of this notion really only came with the general theory of relativity, which showed that one could take account of a wider range of empirical facts, and that, too, in a more satisfactory and complete manner, on a foundation quite different from the Newtonian."

Here Einstein attributes the general theory of relativity, specifically, as precipitating a change of viewpoint. Einstein concludes the lecture by describing it:

"Physics constitutes a logical system of thought which is in a state of evolution, whose basis cannot be distilled, as it were, from experience by an inductive method, but can only be arrived at by free invention. The justification (truth content) of the system rests in the verification of the derived propositions by sense experiences, whereby the relations of the latter to the former can only be comprehended intuitively."

The idea that physical theories must be arrived at by "free invention" had already been proposed in the 19th century. In endorsing it, Einstein put the authority of his reputation behind it.


Modern readers may be tempted to interpret Einstein as advocating something beyond what he intended. It is what we would now call the "underdetermination thesis." Both Newtonian theory and general relativity fit empirically very well with the ordinary gravitational phenomena such as were available to Newton. Yet they proceed with very different concepts. Newton represents gravity as a force. Einstein represents it as an aspect of spacetime structure. Thus, we might conclude, the evidence available to Newton could not pick between the two and thus could not determine the theory. The underdetermination thesis asserts that this situation is widespread to varying degrees according to the version of the thesis. The strongest version asserts that, for any theory, there will always be alternatives equally adequate to the evidence, so that the evidence is powerless to decide among them.

Einstein's endorsement of the notion of "free invention" is explicitly not also an endorsement of the underdetermination thesis. We see this in another version of the same general idea, this time expressed by Einstein in a 1918 lecture, "Principles of Research." It culminates in an explicit denial of the underdetermination thesis:

"The supreme task of the physicist is to arrive at those universal elementary laws from which the cosmos can be built up by pure deduction. There is no logical path to these laws; only intuition, resting on sympathetic understanding of experience, can reach them. In this methodological uncertainty, one might suppose that there were any number of possible systems of theoretical physics all equally well justified; and this opinion is no doubt correct, theoretically. But the development of physics has shown that at any given moment, out of all conceivable constructions, a single one has always proved itself decidedly superior to all the rest. Nobody who has really gone deeply into the matter will deny that in practice the world of phenomena uniquely determines the theoretical system, in spite of the fact that there is no logical bridge between phenomena and their theoretical principles; this is what Leibnitz described so happily as a"pre-established harmony."

In a 1936 article, "Physics and Reality," Einstein gives a more figurative denial of underdetermination:

First crossword by Arthur Wynne, December 21, 1913.  https://en.wikipedia.org/wiki/Crossword#/media/File:First_crossword.png

"The liberty of choice, however, is of a special kind; it is not in any way similar to the liberty of a writer of fiction. Rather, it is similar to that of a man engaged in solving a well-designed word puzzle. He may, it is true, propose any word as the solution; but, there is only one word which really solves the puzzle in all its parts."

Einstein's view is a curious hybrid of notions. The concepts of a theory are free in the sense that we freely invent them; but they are at the same time uniquely determined by the phenomena. We may well wonder whether this hybrid notion is coherent. Are our inventions truly free? Or is it that our methods are haphazard, involving much free play; and that most of it fails until we eventually hit upon results that are determined by the phenomena?

Severe Tests

If inductivism is to be discarded, we need a new account of how evidence bears on theory. Einstein's last remark already showed how he conceived the relation: "The justification (truth content) of the system rests in the verification of the derived propositions..." This is a version of the nineteenth century notion of hypothetico-deductive confirmation: a theory is confirmed when we find that it makes true predictions.

Einstein, however, added an extra twist in his practice that he did not mention above. He sought predictions that would be unlikely to be the case if the theory were false. That is, he chose predictions designed to expose his theory to the greatest chance of failure when the prediction was tested. This was a standard practice in Einstein's writing. A paper proposing some novel science would end with three empirical tests. This is how Einstein's 1905 special relativity paper ended; how his 1905 light quantum paper ended; and how his 1916 review article on general relativity ended.

Some of the tests became celebrated in their own right. His Nobel Prize winning relationship governing the photoelectric effect was one of the tests indicated in the 1905 light quantum paper. The eclipse test for the deflection of starlight and the red shift of light from massive bodies like the sun are two of the tests indicated in his 1916 review article on general relativity.

Einstein's strategy impressed a young Karl Popper immensely. Popper was uncomfortable with psychoanalysis and other fringe theories, since no matter what happened, the theorists seemed to have an explanation for it. The theories could not fail. It was quite otherwise with general relativity. Popper recalled his experience in 1919 reflecting on the eclipse test of general relativity. (In a 1953 lecture, subsequently published as Science: Conjectures and Refutations.)
"Now the impressive thing about this case is the risk involved in a prediction of this kind. If observation shows that the predicted effect is definitely absent, then the theory is simply refuted. The theory is incompatible with certain possible results of observation--in fact with results which everybody before Einstein would have expected.[1] This is quite different from the situation I have previously described, when it turned out that the theories in question were compatible with the most divergent human behaviour, so that it was practically impossible to describe any human behaviour that might not be claimed to be a verification of these theories.

These considerations led me in the winter of 1919-20 to conclusions which I may now reformulate as follows."

What followed was an account of Popper's notion that science proceeds through a cycle of bold conjectures and refutation. It is powered by the theorist seeking to expose the theory to severe tests.

Einstein's Caution

In Popper's telling, science is an adventurous romance of audacious researchers making bold and dangerous conjectures and then modestly and meekly submitting to the negative verdict of experience. It is an appealing and even flattering story for science and captures something of the dynamic of theory construction and testing. However it is  at best an oversimplified picture of something much more complicated.

We can see that in Einstein's own deliberations. They were not purely driven by Popper's falsificationist dynamic. Einstein's greatest empirical coup was his recovery in November 1915 by general relativity of the missing 43 seconds of arc in Mercury's perihelion motion.

What might Einstein have done if his theory had proved to be unable to recover these missing seconds of arc?

Generally such questions are unanswerable. This one, however, has an answer. During his work on the "Entwurf..." theory, the near complete but flawed earlier version of the general theory of relativity, Einstein exchanged a manuscript of calculations with his friend Michele Besso. We learn from Michel Janssen's analysis of the document that Einstein and Besso computed the adjustment the theory required for the motion of mercury. Through a calculation error, they found a huge and implausible result, 1821 seconds of arc. That error was corrected and eventually Einstein settled on a more plausible 17 seconds of arc.

Einstein did not follow the Popperian model: he did not retract his theory. Rather he seemed to find the result unremarkable and passed over it. Prior to November 1915, there was no pressing need to capture precisely those elusive 43 seconds of arc. The older theory's smaller result was well within the sorts of bounds one might expect when the measured value suffered a tiny aberration of then unknown origin.

More broadly, Einstein did respect the verdict of experiment. But  that verdict was not accepted in haste. If the experiment did not conform with his theoretical expectations, he would not abandon his theory, but wait with the expectation that further experiments would vindicate him.

Examples include his reaction in the 1900s to experiments that gave a velocity dependence of mass that contradicted relativity theory; and his reaction to the Miller experiments of the early 1920s that purported to show that the earth's motion through the ether did have a measurable effect, after all.

Novel Evidence and Adjustable Parameters

This last example of the recovery by general relativity of the anomalous motion of Mercury has played an outsized role in studies of confirmation in philosophy of science.

It is an important example. Einstein's investigations into gravity were both conceptually and mathematically intriguing or challenging, according to your inclination. But was his the right theory? For someone who failed to follow Einstein's conceptual analysis, the recovery of the anomalous motion of Mercury was the first external indication that Einstein had found something truly important. One did not need to understand the mathematics of curved spacetimes or fall in with Einstein's program of relativization to see it.

For this reason, it is an example that should fit well with any philosopher's account of evidential support. The difficulty is that it did not sit well with two foundational ideas within different accounts of inductive inference.

Problems for Predictivism

What I shall call "predictivist" accounts emphasize the importance of the prediction of novel facts. That is, imagine that some new theory makes a prediction that we would otherwise find unlikely and the prediction turns out to be correct. This success provides strong confirmation of the theory.

The difficulty is that the anomalous 43 seconds of arc in Mercury's motion was already well-known at the time of  Einstein's computation. He knew of it. He had even wondered years before if his developing theory might eventually explain it. To predict something that is already well-known has all the flavor of cheating by predictivists' standards. The cheating is sometimes called making an "ad hoc" posit. That is, you tune your theory to the data. It is analogous to throwing a dart onto a board and then drawing in the bull's eye round the point where the dart struck.

The standard solution offered within predictivism is to refine the notion of novelty. A successful prediction can still provide evidential support for a theory if the novel fact predicted is not used in the deriving of the prediction. This notion is called "use-novelty." Einstein's recovery of the anomalous motion of Mercury within general relativity accords with use-novelty. For the known value of the anomaly was never used in the derivation.

The literature on this topic is enormous. A good entry to it is provided by Vincenzo Crupi, "Confirmation", The Stanford Encyclopedia of Philosophy https://plato.stanford.edu/entries/confirmation/ The example of Mercury's perihelion appears repeatedly in it.

Problems for Incremental Confirmation

A related form of this problem arises in accounts of confirmation theory that assess the bearing of evidence incrementally. A very few simple formulae make the issue easy to see. Let us write:

"[T|B]" means the degree of support theory T accrues on some body of evidence B.

The term "degree" here is used loosely. In most cases, however, the degree of support will be a real number. The most prominent example is Bayesian confirmation theory. There [T|B] is just the conditional probability P(T|B).

In the incremental notion of support, learning E supports a theory T evidentially if adding it to B increases the degree of support for T. That is,

Evidence E supports theory T
with respect to body of evidence B

just if

[T|E&B] > [T|B]

Let us apply this to the case of Mercury

T = the general theory of relativity
B = the totality of astronomical knowledge in 1915
E = the anomalous motion of Mercury

Since the evidence E is already contained within prior total evidence B, adding it to B formally by conjunction does not change it. That is

E&B = B

We conclude that

[T|E&B] = [T|B]

so that the evidence of Mercury's anomalous motion provides no support for the general theory of relativity.

This problem was introduced by Clark Glymour as the "problem of old evidence" in his 1980 book, Theory and Evidence. He presented there only as a problem for Bayesian confirmation theory. His argument prompted many responses in a literature that still grows today.

For an entry to this literature, once again see Vincenzo Crupi, "Confirmation", The Stanford Encyclopedia of Philosophy https://plato.stanford.edu/entries/confirmation/

It is futile here to attempt a survey of the many responses. My view, for what it is worth, is that the existing body of evidence B used in the incremental analysis must be a counterfactual one. That is, it must be a mutilated body B from which the relevant item of evidence has been excised. If there is no plausible way to excise it, then the incremental conception of confirmation fails to apply to these cases. I do not find such a failure troubling. I do not believe that there is any single, formal account of confirmation that applies everywhere.

No Adjustable Parameters

In my view, what made Einstein's 1915 treatment of Mercury so successful was that there were no parameters Einstein could adjust in his theory to fit it to the anomalous motion. Either the theory fitted or it did not. When it did, Einstein gave philosophy of science one of the most powerful examples of how a single datum can lend strong evidential support to a theory.

What does that slogan "no adjustable parameters" mean? It is best understood by looking at the other ways that were tried to account for the anomalous motion.

The simplest supposition mathematically was a proposal by the astronomer Asaph Hall. Might it be that Newton's inverse square law needs just the tiniest of adjustments? The force of gravity, he supposed, might not dilute with distance r exactly as the inverse square law asserted:


Rather the force dilutes according to:


The adjustable parameter is δ. An easy calculation, drawing on a theorem first proved by Newton in his Principia, showed that the requisite value is

δ = 0.0000001574

The proposal however failed since the adjusted parameter gave incorrect results for the other planets.

The simplest supposition physically is that there is some mass in the solar system disturbing Mercury's orbit slightly, but that this mass has not yet been entered into the astronomers' calculations. This supposition employs a venerable strategy that had led to the discovery of new planets. Its application to the anomalous motion of Mercury, however, failed to produce usable results.

The first version of this supposition was that there was another planet "Vulcan," lying closer to the sun than Mercury. However the supposition failed when no new planet was found in the position Vulcan would have to occupy. The idea of novel matter persisted. If it did not reside in a whole planet, might it reside in a distributed cloud of matter orbiting the sun? Of might it be in equatorial bulges of the sun itself. None of these suppositions succeeded.

Each of these different proposals for novel matter involves setting freely adjustable parameters. Posit the matter and ask where it might be. The parameters are the amount of the matter and its spatial distribution.

The presence of these adjustable parameter--and their active adjustment to fit the datum--weaken the evidential strength of support provided, if they finally accommodate the anomalous motion of Mercury. For the success seems to provide no real test of alternative account. There is, we should expect, some combination of these parameters that would eventually fit the one datum. It seems to be more a test of the ingenuity of the theorists to find them, as opposed to a real test of the theory itself.

Einstein's treatment of Mercury is free from these concerns and provides  correspondingly strong support for his theory.

For more details of the history of the alternative accounts see my "7. Einstein and the Anomalous Perihelion of Mercury." in "Inference to the Best Explanation: Examples," Ch. 9 in The Material Theory of Induction.

Mathematical Platonism as a Method of Discovery

Einstein himself drew an even stronger lesson from general relativity. Having renounced all hope of an inductive method for discovering scientific theories, he announced in his 1933 Herbert Spenser lecture that there is another method. True laws are expressed in mathematically simple terms, so we can find them merely by sifting through the simple mathematical expressions. He wrote what is surely one of the most astonishing manifestos of a great scientist:

"Our experience hitherto justifies us in believing that nature is the realization of the simplest conceivable mathematical ideas. I am convinced that we can discover by means of purely mathematical constructions the concepts and the laws connecting them with each other, which furnish the key to the understanding of natural phenomena. Experience may suggest the appropriate mathematical concepts, but they most certainly cannot be deduced from it. Experience remains, of course, the sole criterion of the physical utility of a mathematical construction. But the creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed."

For my account of how Einstein derived his Platonism from his experience with general relativity, see "'Nature is the Realization of the Simplest Conceivable Mathematical Ideas¹: Einstein and the Canon of Mathematical Simplicity," Studies in the History and Philosophy of Modern Physics, 31 (2000), pp.135-170.

Copyright John D. Norton. February 23, 2013. November 16, 2019.