FastSaying

A good example of the archetypal ideas which the archetypes produce are natural numbers or integers. With the aid of the integers the shaping and ordering of our experiences becomes exact. Another example is mathematical group theory. ...important applications of group theory are symmetries which can be found in most different connections both in nature and among the 'artifacts' produced by human beings. Group theory also has important applications in mathematics and mathematical physics. For example, the theory of elementary particles and their interactions can in essential respects be reduced to abstract symmetries.
[The Message of the Atoms: Essays on Wolfgang Pauli and the Unspeakable]

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The prime number 137 had continuously occupied Pauli's mind. It is an approximate value for a constant appearing in the fine structure theory of atomic spectra which in its theoretical expression ties together electromagnetism, relativity and quantum theory. Pauli saw the fine structure theory of spectra as a key in understanding the deepest contemporary problems of theoretical physics. For that reason the number 137 possessed a mysterious attraction for him.
— Kalervo V. Laurikainen
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... it should be remembered that the atomicity of electric charge has already found its expression in the specific numerical value of the fine structure constant, a theoretical understanding of which is still missing today.
— Wolfgang Ernst Pauli
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The strength of the familiar electromagnetic force between two electrons, for example, is expressed in physics in terms of a constant known as the fine structure constant. The value of this constant, almost exactly 1/137, has puzzled many generations of physicists. A joke made about the famous English physicist Paul Dirac (1902-1984), one of the founders of quantum mechanics, says that upon arrival to heaven he was allowed to ask God one question. His question was: "Why 1/137?
— Mario Livio
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The unsolved problems of the physical world now seem even more formidable than those solved in the twentieth century.

Though in application it works splendidly, we do not even understand the physical meaning of quantum mechanics, much less how it might be united with general relativity.

We don't know why the dimensionless constants (ratios of masses of elementary particles, ratios of strength of gravitational to electric forces, fine structure constant, etc.) have the values they do, unless we appeal to the implausible anthropic principle, which seems like a regression to Aristotelian teleology.
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Here the attention of the research workers is primarily directed to the problem of reconciling the claims of the special relativity theory with those of the quantum theory. The extraordinary advances made in this field by Dirac ... leave open the question whether it will be possible to satisfy the claims of the two theories without at the same time determining the Sommerfeld fine-structure constant.
— Werner Heisenberg
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