No doubt, Niels Bohr would have been horrified to see his fluid,
natural-language-based thinking translated into clunky mathematical
formalism.^{1} Nonetheless, Bohr himself claimed that having a certain
kind of mathematical formalism shows that quantum mechanics provides a
consistent scheme that can be applied to all the different kinds of
experiments that can be performed. (We would prefer it if he had said
“can be applied in all contexts” to clarify that Bohr does *not* think
that quantum theory applies only to the results of measurements.)

A mathematical reconstruction of Bohr’s way of talking was first supplied by Don Howard in his PhD thesis of 1979 (Howard 1979). That approach lay mostly fallow until the 1990s, when Jeffrey Bub and collaborators started thinking about the implications of the Kochen-Specker theorem for the interpretation of QM. While the KS theorem shows that not all “observables” are “beables”, Bub’s question was: just how big can the set of beables be? A complete answer to that question is given by the Bub-Clifton classification of no-collapse interpretations of quantum mechanics (Bub and Clifton 1996).

Treating Bohr’s interpretation as “no collapse” doesn’t make sense of what he says about the state of an object being an idealization that only applies in certain contexts.

This reconstruction doesn’t make so much sense of how central a role entanglement plays in Bohr’s considerations about quantum mechanics.

Bub, Jeffrey, and Rob Clifton. 1996. “A Uniqueness Theorem for ‘No Collapse’ Interpretations of Quantum Mechanics.” *Studies in History and Philosophy of Modern Physics* 27 (2): 181–219. https://doi.org/10.1016/1355-2198(95)00019-4.

Howard, Don Armin. 1979. “Complementarity and Ontology: Niels Bohr and the Problem of Scientific Realism in Quantum Physics.” PhD thesis, Boston University Graduate School.

Operator algebras is hardly “clunky”, as far as mathematics go. But it’s still math, and so misses some of the richness of expression in natural language.↩︎