Philosophical reflections on physics and its theories have been shaping the agenda of general philosophy of science, including issues such as the nature of scientific theories, the meaning of scientific terms, and scientific modeling. But physics is also well worth being considered a special science of its own, whose methods and tools raised specific questions for philosophers of science: this is the approach followed in this chapter, which has as its main thread the role of mathematics in physical theories. It tackles classical issues concerning measurement and determinism and long-standing controversies in the philosophy of statistical mechanics (how may the reversible laws of mechanics account for the irreversible principles of thermodynamics?) and in the philosophy of quantum physics (does quantum mechanics provide us a with a complete description of quantum phenomena?). It also addresses emerging issues in the field, such as computer simulations and their role between theory and observation.
The 'unreasonable effectiveness of mathematics in the natural sciences' may at times give the impression that natural science, especially modern theoretical physics, is all but a formal apparatus. However, a key challenge of physics remains to align everyday notions of experience -- be it directly or mediated through instruments of arbitrary sophistication -- with their mathematical formalisation. This applies in particular to quantum theory, arguably the most successful theory of physics ever invented. One of its core assumptions postulates:
Observables correspond with self−adjoint operators. Here, we argue that already this most basic premise about the quantum formalism falls apart under careful scrutiny, at least when understood for all observables at once. In other words, we challenge the role of the noncommutative algebra of bounded operators as an adequate formalisation of the 'algebra of observables', understood as (equivalence classes of) measurement procedures that admit a coherent empirical interpretation.
We know from physics education research that, for effective physics teaching, knowledge needs to be framed within a context where the nature of science plays an essential role. This work presents the framework we have developed for introducing pre-service and in-service teachers to quantum mechanics via an active learning approach. It starts with some selected elements of the history of quantum mechanics, develops a set of physical principles, and then motivates constructing the formal aspects of quantum mechanics, such as states, linearity, superposition, observables, and probability.