Einstein's Unification: General Relativity and the Quest for Mathematical Naturalness

The aim of the thesis has been to understand Einstein's development and see the historical coherence in his later attitude in physics. The lesson we learned has been straightforward: the key that unlocks the later Einstein lies in the road by which he arrived at the field equations of general relativity, ``the most joyous moment of my life.''

Einstein's later position was strongly embedded in his methodological
beliefs (chapter 2). These ideas took shape according to his experiences as a physicist. They also document his development and inspired and justified the directions he took in his work.

At the beginning of his career, the empiricism of Ernst Mach was instrumental in the construction of special relativity. When Einstein later formulated general relativity, he used a dual method, in which mathematical deduction played an important creative role, whereas his inductive arguments in hindsight appeared to have been a hindrance (chapter 1).
He subsequently gradually changed his epistemology accordingly. The
reshaped methodolo\-gi\-cal beliefs were easily invoked to justify the
further mathematisation of his work, and its further alienation from
the realm of experience --- they contributed to Einstein's dissociation of the quantum program and prompted his decision that an alternative route should be followed.
His guiding principles were the maxims of logical simplicity and mathematical naturalness; these inspired and justified his strive for unification (chapter 3).

In the elaborations of the semivector theory we have seen the direct
influence of Einstein's methodological ideas on the direction of his actual
research (chapter 4). The subject sprang from the feeling that the spinor was
incomprehensible and `unnatural'. Subsequently, the attempt to give the Dirac theory its simplest and most general formulation led Einstein to believe that he had uncovered a unified description of the electron and proton and thus a deep clue about the fundamental nature of these two particles. However, it soon became clear that it had in fact been a Pyrrhic victory. Nevertheless, Einstein did not retract his ideas on the method of theoretical physics.

From the work on unified field theory, a view emanates on what Einstein's ideal theoretical description of nature should look like and how
such a description should be formed. Experience should tickle the
mathematical intuition into forming a natural set of axioms; one may
think of `natural' in the sense of symmetry principles and the
like. Deduction from these fundamental axioms next gives the laws that
in a more direct way relate back to experience. Inductive methods on
the other hand usually fail in producing true understanding and at
best give mere phenomenological descriptions. In effect, the only
theories that would live up to Einstein's requirements on for instance
causality and his realist philosophy were classical field theories. These theories were to be put in their logically simplest form, which meant that the forces ought to be unified and that particles were to follow as solutions from the field equations. In this way one was to arrive at a complete description of reality.

From the perspective of today's physics, of all of the unified field
theories that Einstein elaborated, the theory that came closest to these ideals was the Kaluza-Klein theory (chapter 5). Einstein's treatment of the
work of Klein further exemplified how he assessed the epistemic status of
typical quantum relations. In a way one could say that Klein had
achieved a unified structure that related the discreteness of charge
to the inobservability of Kaluza's fifth dimension. A pivotal element
in his reasoning was the de Broglie relation. Despite the fact that
Einstein did consider a compactification of the fifth dimension and
wanted to explain the quantized nature of the electric charge, he did
not repeat Klein's argument. We believe that the reason was that he did not
want to accept the de Broglie relation as an independent axiom in
his theoretical structure. Rather, the de Broglie relation and the
quantum of charge were to follow from the field equations.

This was exemplary for Einstein's attitude to the whole quantum theory.
He would not deny its validity, but firmly believed that it was at
best a phenomenological theory and that an underlying structure can
and ought to be uncovered (chapter 6).
The inductive and eclectic conception of the theory had already estranged him from Heisenberg's matrix mechanics and its probabilities and incompleteness further convinced him that the theory would always fail to describe ``the real in all its depth.''

Einstein had quite entrenched himself in these positions; he acknowledged that in the eyes of his contemporaries he must have looked like an ostrich. Of the correspondence that we have presented, this metaphor probably best reflected Pauli's sentiments. He kept pointing out to Einstein the
shortcomings of his models, but to no apparent avail. Einstein did
not particularly solicit Pauli's opinion, nor anyone else's, and he comes
across as a highly independent thinker. It appears that in our story
only Paul Ehrenfest had had some direct influence on him; Ehrenfest
was of course much more than any other correspondent someone of the
same generation. Hermann Weyl may have had some more indirect
influence, but certainly not to the degree that one can say that he brought Einstein to alter his own line of thought.

The same image of Einstein comes forward when we look at his
collaborators: Walther Mayer, Valentin Bargmann and Peter Bergmann.
Bergmann was of course too much a junior to expect him to have
exerted any great influence on Einstein. Mayer and Bargmann on the
other hand appear to be handpicked exclusively because of their
mathematical qualities. Their correspondence with Einstein shows that
physical issues were hardly discussed. Every now and then Einstein
would patiently sit through Mayer or Bargmann's physical reasonings,
politely break off the discussion and then return to the mathematical
problems that he intended to address and for which he felt that he
could use their expertise.

Although we believe to have arrived at a historically coherent
picture of Einstein, we have also seen that some of his ideas turned
out to be at odds with each other, and we certainly do not wish to purport that under further close scrutiny, no more inconsistencies can appear. One
should not forget that Einstein was --- or disallow him to have been
--- a practicing physicist; an opportunist, and not a philosopher in
the process of constructing an all-embracing system of the world. He foremost followed his immediate intuitions, in both his physics and philosophy.

Einstein did of course have fair and deep reasons to substantiate both
his attitude to the quantum theory and his pursuit of unified field
theory. Yet, one cannot but marvel at his steadfastness. All his
attempts in field theory have fallen in oblivion and all were --- characterized by a lack of contact with the world of experience --- all were attempts at unification, to some extent almost exclusively for the sole sake of unification. These attempts were ultimately foremost moulded by the belief that ``nature is the realization of the simplest conceivable mathematical ideas''.

Such observations tempt one to make overarching statements. Should we
conclude that a fundamentally empiricist attitude eventually carries
the day in theoretical physics? The example of Einstein is sometimes
given to underline that a one-sided strive for mathematical
naturalness and logical simplicity in the face of a lack of empirical
data --- one does recognize today's attempts at unification --- leads
one easily astray. After all, the greatest mind of the twentieth
century had his biggest successes when he was under the spell of
Mach's philosophy, yet when he traded this in for the maxim of
mathematical naturalness and logical simplicity, he gradually lost his
grip. But then again, maybe it is really only Einstein's refusal of
the quantum theory that inhibited new successes. Furthermore, and
most importantly: one swallow does not make a summer. These
conclusions may appear tempting, given the story of this thesis, but
they require very much more evidence than this very thesis could
possibly have presented.

Yet we never intended to address such wide panoramas. We only wanted
to understand the historical development of Einstein and see how he
came to his dissenting position. The emotionally defining moment was
the discovery of general relativity; it reverberated in the methodological beliefs that he advocated and had a decisive influence on his later physics.