Neo-Kantians and Scientific Revolutions: Ruptures and Continuities within the Neo-Kantian Tradition (1860-1942)

To what extent can we find similarities between Hermann von Helmholtz's and Ernst Cassirer's philosophy of sciences facing the scientific revolutions in logic, mathematics and physics that started in the late 19th century? In spite of his rather empirist epistemology, calling on physiology and psychology, Helmholtz was an inspiring source for the latter because of what could be called his protostructuralism. Indeed, both found in structuralism resources to solve the problems raised by their common rejection of the given of intuition and the copy theory ("Abbildtheorie").
Cassirer is indebted to Helmholtz, mainly for the sign theory and their convergences about geometry. In sciences of nature, Cassirer opposes the copy theory and presents in a favourable light the sign theory, which holds that we cannot know the things in themselves but only the causal relations between them. Concerning the new geometries of space that challenged the Euclidian geometry in the mid-19th century, Helmholtz eventually rallied to the transcendental aspect of space and introduced the distinction between pure geometry and geometry of the physical space in "Ueber den Ursprung und die Bedeutung der geometrischen Axiome". Finally, Helmholtz claims to be Kantian, but, contrary to Cassirer, granted a very limited room to a priori faculties and knowledge, which are basic constituents of a Kantian philosophy.
Cassirer distinguishes itself from other neo-Kantians by abstaining from a complete rejection of intuition, whereas the other ones integrate it into understanding. Following Helmholtz, he keeps space as a pure form of sensibility in « Zur Einsteinschen Relativitätstheorie ». Like the other neo-Kantians of the Marburg school, Cassirer postulates the existence of a priori categories, and, among them, gives the first place to the category of relation, which a mark of structuralism. His structuralism is more systematic than Helmholtz's. Contrary to Helmholtz's structuralism, sometimes bordering on analogy, Cassirer's is a mathematical structuralism, close to the structuralism that emerged later in human sciences. Finally, unlike Helmholtz, who had no interest in history of sciences, Cassirer shared with the other neo-Kantians from Marburg a deep interest for this history, which they judged inseparable from philosophy of sciences. In "Substanz und Funktion", he presents this history as an evolution from the conceptualisation by substance to conceptualisation by function. 
In summary, Cassirer's structuralism presents analogies with Helmholtz's protostructuralism, but brings also significant new points when compared to the latter. This can be explained by other important influences from the other neo-Kantians (Hermann Cohen and Paul Natorp), the founders of modern logic, in particular Bertrand Russell, as well as from the physicists-philosophers, Pierre Duhem and Max Planck.

In this talk, I provide elements of definition for "Naturalized Epistemology" and evaluate if they apply to Helmholtz and Poincaré's thoughts on geometry and space. Contemporary commentators have frequently credited Helmholtz for putting together a naturalized epistemology Avant l'heure (Hatfield (1990), De Kock (2012), Patton (2018)); the same has not been attempted for Poincaré. I demonstrate in a two-part argument that Poincaré's thoughts on geometry and space are compatible with arguments found in Helmholtz's "Naturalized Epistemology". 
Historically, both thinkers have addressed the same problem: defining the geometrical relations in space and providing useful applications to physical objects. To this end, Helmholtz relied on the mobility of the rigid natural body and the notion of congruence to establish the real conditions for the measurement of physical space. In contrast, Poincaré opts for a classification of group displacement in geometrical space that is neither true nor false conventions indirectly guided by experience (or sensation in motion).
Despite this debate, I suggest that both thinkers shared a common scientific unregenerated realism and epistemological holism, typically associated with "Naturalized Epistemology". Both do not look outside of the modes and methods of explanations of natural sciences to justify mathematical propositions; the methodological continuity between epistemology and natural sciences allows Helmholtz to construct spatial geometries out of a psychophysiological interpretation of spatial perception and Poincaré to support his reconstruction of spatial geometry with a psychophysiological genesis. Also, both fall back on holistic criteria (ex. commodity, simplicity) to overcome the empirical equivalence of different types of geometry. Helmholtz embeds the geometrical axioms in a system of actual physical measurement (as sufficient preconditions) while Poincaré's geometrical conventions take part in the generalization process along verifiable physical hypotheses which have observational (thus true or false) consequences. 
That said, I argue that Helmholtz and (more explicitly) Poincaré used methodological structuralism to account for the validity of geometrical relations. However, this interpretation could be at odd with the previous "Naturalized Epistemology" label. The structural relations (described by geometry) provide a norm of invariance that must be conventionally adapted and empirically exemplified (Heinzmann & Stump, 2021). Yet, it seems this structuralism allows the relations an epistemic apriority over the modes and methods of explanations of natural sciences which could be prohibited by "Naturalized Epistemology". In response, I conclude that the Helmholtz/Poincaré debate on geometry could be interpreted as a debate from within scientific practices (or more precisely from within mathematical practices). Hence, I suggest that Helmholtz and Poincaré's methodological structuralism is compatible with a moderate version of "Naturalized Epistemology".

The two great scientific revolutions of the early 20th century -- relativity theory and quantum mechanics -- inspired new outlooks on theories of physics elaborated in the 18th and 19th centuries. A leader of the quantum revolution, Werner Heisenberg drew from first-hand experience of theory change an epistemological lesson: classical mechanics, like quantum mechanics, is both unsurpassable and true. Both mechanics are what he termed "closed" theories, the creation of which he believed, furthermore, to be the objective of theoretical physics. Heisenberg's epistemology of closed theories, cast in a neo-Kantian framework by a number of writers, including Chevalley (1998), Bokulich (2004, 2006), and Schiemann (2009), is considered here in relation to Poincaré's conventionalism, on the one hand, and to Hilbert's axiomatization program for the exact sciences (Hilbert's Sixth Problem), on the other hand. Heisenberg's epistemology of closed theories will be seen to incorporate elements from the programs of both Poincaré and Hilbert.