Quantum mechanics is perhaps the most successful physical theory we have ever had in terms of predictive power—and it is hardly an overstatement to say that the Born rule is almost single-handedly responsible for its predictions. The task of understanding how the mathematical formalism is related to the physical world, and why probabilities arise in quantum theory, is therefore tied to the task of understanding the Born rule, its origin and its nature—but the connection does not by itself determine the arrow of explanation. The traditional assumption that the Born rule was originally postulated rather than derived, as if it were somehow handed down from heaven, has to some extent mystified the physics behind it and has thereby fixed a very definite direction of explanation in foundational discussions: it apparently follows from it that one must resolve the interpretative questions first, so that an understanding of the nature of the Born rule will follow. But this assumption about the Born rule amounts to a historical claim—it may be mistaken. In which case, discussion is long overdue. In this talk, I will share some of the findings of a new and very rigorous mathematical investigation of the conceptual development of quantum mechanics that has led to the identification of important facts constitutive of the Born rule which up to now have remained hidden from sight. Such facts are not merely clues as to how the Born rule was discovered, however: they supply the very physical premises from which the rule itself can be mathematically deduced. They reveal the foundation of the proposition that, in the quantum theory, probabilities are completely determined by the modulus squared of the amplitude associated to the state of the system.

For if Max Born introduced the so-called Born rule nonchalantly, in a footnote; if his interpretation of the wave function was received without any surprise by many of his contemporaries—“we never imagined it could be anything else,” Bohr said in this regard—this is for a definite reason. There were no wave functions before 1926, of course, but the rule itself had long been fixed—it is only the mathematical object representing it that changed in 1926. The rule is there, sitting in Max Planck’s 1913 The Theory of Heat Radiation, for anyone who cares to look.