Epistemology and Probability pp 77-113 | Cite as

# Heisenberg’s Revolutions: New Kinematics, New Mathematics, and New Philosophy

## Abstract

This and the next chapter explore the physical, mathematical, and philosophical significance of Heisenberg’s discovery of quantum mechanics. This discovery was among the most momentous discoveries in the history of physics, comparable to that of classical mechanics by Newton, Maxwell’s discovery of his equations for electromagnetism, and Einstein’s discoveries of special and then general relativity. The comparison with Newton’s discovery of classical mechanics is especially apt, since in both cases at stake was the introduction of a new calculus—differential calculus by Newton and matrix (de facto Hilbert-space operator) calculus by Heisenberg. The relationships between the mathematical calculus deployed and the physical phenomena considered are, however, entirely different in the case of Heisenberg’s mechanics. Unlike Newton’s mechanics, Maxwell’s electromagnetic theory, or Einstein’s relativity, Heisenberg’s calculus does not describe the behavior of *quantum objects*, but only relates, in terms of probabilistic predictions, to *quantum phenomena*, manifested in our measuring instruments impacted by quantum objects. Section 3.1 offers a general introduction to Heisenberg’s quantum mechanics, developed into the matrix quantum mechanics shortly thereafter by Born, Jordan, and Heisenberg himself. Sections 3.2 and 3.3 discuss the key physical and philosophical principles that shaped Heisenberg’s approach—the principle of dealing with “quantities, which in principle are observable” and the correspondence principle. Section 3.4 considers the key features of Heisenberg’s quantum mechanics. It contains some technical mathematical details and may be skipped by readers unfamiliar with mathematics at this level. Its key conceptual points are explained in nontechnical terms in Section 3.1.