Eddington aim for the fifth chapter of his 1938 ‘Philosophy of the Physical Sciences‘ was to show that Einstein’s General Relativity presents an example of how epistemological methods can offer positive contributions to physics. He already offered some negative examples in the previous chapters. But given that Eddington made the claim that all of physics is derivable with these methods, the reader now requires some better examples.
To begin, he quotes the famous geometer, Poincaré, as a parade-case for, what he considers to be, a common misconception regarding the philosophical status of Einsteinian relativity:
…what we call a straight line in astronomy is simply the path of a ray of light. If, therefore, we were to discover negative parallaxes, or to prove that all parallaxes are higher than a certain limit, we should have a choice between two conclusions: we could give up Euclidean geometry, or modify the laws of optics, and suppose that light is not rigorously propagated in a straight line. It is needless to add that every one would look upon this solution as the more advantageous. Euclidean geometry, therefore, has nothing to fear from fresh experiments.
(p 67) from: Poincaré H. (1905) ‘Science and Hypothesis’ [Chapter V: Experiment and Geometry] p 73
Of course, we must note that Poincaré was speaking before Einstein published Special Relativity, and long before General Relativity’s non-Euclidean geometry. His comments were in relation to the mathematical discovery of non-Euclidean geometry, and how this would impact future physical discoveries.
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I must confess that I am of a similar view, thinking for many years that ‘curved space-time’ was a methodological (and unfalsifable) assumption that relativists adopted from an argument of simplicity.
Sure, we could adjust the laws of physics to accommodate new observation (as Poincaré suggests), but relativists found the Einsteinian view to offer a greater synthesis of our latest physics. And, while simplicity is not the primary goal of scientific inquiry, we are clearly interested in predicting and controlling our environment with the least possible effort. Thus, simplicity becomes a secondary goal.
Therefore, in order to preserve the possibility of a theory of gravitation which is universal (in space and time), Einstein abandoned Euclidean space in favor of a different geometry.
Eddington disagrees:
I do not see how anyone who accepts the theory of relativity can dispute that there has been some replacement of physical hypotheses by epistemological principles; nor do I think that those who accept the theory with understanding will be inclined to dispute it. (pp 53-54)
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Which epistemological principles?
It has come to be the accepted practice in introducing new physical quantities that they shall be regarded as defined by the series of measuring operations and calculations of which they are the result. Those who associate with the result a mental picture of some entity disporting itself in a metaphysical realm of existence do so at their own risk; physics can accept no responsibility for this embellishment.
The innovation made by Einstein in his relativity theory was that the physical quantities involved in the measurement of space and time were brought under this rule. (p 67)
This is quite a bold claim. Eddington asserts that Einstein’s main innovation for general relativity was not merely to recognize that the Euclidean three-dimensional space is a presumption. No – Einstein wasn’t claiming how we define space is an arbitrary axiom we must adopt in order to do physics. Rather, in considering what is possible to measure, he replaced an abstract impossibility (Euclidean space) with a well-defined measuring scheme.
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This is perhaps why Einstein was so sure of his theory. For example, consider the following anecdote, as told by one of Einstein’s research assistants.
Suddenly Einstein interrupted the reading and handed me a cable that he took from the window-sill with the words, ‘This may interest you.’ It was Eddington’s cable with the results of the famous eclipse expedition. Full of enthusiasm, I exclaimed, ‘How wonderful! This is almost the value you calculated!’ Quite unperturbed, he remarked, ‘I knew that the theory is correct. Did you doubt it?’ I answered, ‘No, of course not. But what would you have said if there had been no confirmation like this?’ He replied, ‘I would have had to pity our dear God. The theory is correct all the same.’
Rosenthal-Schneider, Ilse (1980) ‘Reality and Scientific Truth‘ p 74
Without Eddington’s interpretation, we might consider Einstein’s remarks to be another example the arrogance of theorists. But with Eddington’s perspective, Einstein’s boldness takes on a new meaning.
Did he consider his geometry to be a proven consequence of epistemological considerations, and, therefore, beyond any empirical doubt?
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Einstein’s innovation, claims Eddington, was in demanding that we specify a measuring procedure to define the unit length and the unit duration. Without this, we are left to the whims of the pure mathematician to imagine the geometry of the objective world.
If, instead of length being defined observationally, its definition were left to the pure mathematician, all the other physical quantities would be infected with the virus of pure mathematics. (p 68)
But, Eddington reminds us, we can only form patterns between our observations, and so our physics speaks of our knowledge of the objective world – not the world itself. How then can we define such units?
Let us consider from a general point of view the problem of specifying a reproducible standard of length. Obviously we must not employ lengths in the specification; for that would be a vicious circle. Nor can we use any of the other “dimensional” physical quantities, for their definitions presuppose that standards of length, time, and mass are already defined. The quantitative part of the specification must therefore consist of undimensional quantities, i.e. pure numbers. (p 70)
And where are we to find such pure-number specifications?
“A purely numerical description of material structure is elaborated in quantum theory.” (p 70)
“A spatial extension of the quantum-specified structure provides the standard of length; a time-periodicity of the same structure provides the standard of time-extension.” (p 71)
So, quantum theory, in using dimensionless (quantum) numbers, provides a means to specify a reproducible definition of length. Eddington seems to think that only quantum theory is so far capable of achieving this. And this implies, to Eddington, that a synthesis between quantum and relativity theory is required.
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Next, he makes a subtle, but crucial, point:
The failure to define long distances observationally, or in mathematical language the non-integrability of displacement, is the foundation of Einstein’s theory of gravitation. According to the usual outlook gravitation is the cause of the trouble; gravitation produces the strains which render long standards useless. But Einstein’s outlook is more nearly that the “trouble” – the non-integrability of displacement – is the cause of gravitation. I mean that in Einstein’s theory the ordinary manifestations of gravitation are deduced as mathematical consequences of the non-integrability of displacement.
(pp 77-78)
Here Eddington is referring to the fact that, given that we need to measure distances and times with quantum theory:
…it is necessary to develop a system of description of the location of events based wholly on infinitesimal distances and time intervals… (p 79)
Therefore, we should remember that the addition of small distances in a linearly intuitive way is an assumption. And one that:
…becomes a special hypothesis which requires defending. One does not accept hypotheses gratuitously. Proceeding from this rational basis of space-time measurement we find that the phenomenon of gravitation appears automatically – unless we deliberately introduce a hypothesis of integrability to exclude it – and in this way we are led immediately to Einstein’s theory of gravitation. (p 79)
Thus, when we measure large distances, there is almost always a discrepancy between the raw observation and the straightforward addition of incremental distance measurements. Whereas this measurement problem is often considered to be a consequence of ‘gravitation’, Eddington argues that this ‘non-integrability of displacement’ is gravitation!
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Eddington makes a compelling case. Perhaps I have been wrong in regarding geometry as a methodological assumption. And I think he is correct in asserting that we should not leave questions of the nature of space to the geometers – or our imaginations. But is he correct in supposing that General Relativity can be entirely derived from empirical considerations? I am still doubtful, but curious to read on.
Next up: Quantum Theory.
