Kant's case for the objectivity of at least some of our experience is more threatened by the indeterminate than the indeterministic character of modern physics. Indeterminancy is a complex notion. It can be understood, ultimately, in terms of the failure of a “separability” principle, that objects can be individuated only with respect to non-vanishing spatial-temporal intervals. Its failure seems to follow from the fact that it is indispensable to the derivation of Bell's Theorem and that the conclusion of the Theorem is incompatible with well-established empirical results. But Kant's case for objectivity depends on it. The result is unsettling.

This essay discusses the character of the concepts used in modern physical theories and the symbolic structure they define. Are these symbols interchangeable free constructs and the theories artificial devices with in and out slots to accomodate observations? Are the symbols “read” up - that is, interpreted - from nature and are the physical theories directly referring to a final reality? Did our reason by itself fix immovable rules to build up these concepts? None of the “fundamentalist” stances hereto appears supported by the scientific process of developing physical knowledge. A more differentiate approach is suggested, which could offer for these questions a sustainable, less strong but may be more fruitful point of view.

The paper deals with an attempt to present an “abductive-transcendental” argument in favour of a particular version of structural realism (SR), dubbed Intermediate SR. In the first part of the paper the general structure of transcendental arguments is scrutinized with a close view on Kant's original version and the prospect of their abductive variation. Then the role of symmetries in modern physics, especially symmetries without real instantiations and in particular gauge symmetries is discussed. This is combined with a presentation of SR as a promising current version of scientific realism. The discussion is supported by various arguments from gauge theories in modern physics. Intermediate SR, a realist position about all and only structurally derivable entities located between the extremes of Epistemic and Ontic SR turns out as the best fit to our current fundamental gauge physics and this finally leads to an abductive-transcendental reasoning concerning this position.

Moderne prolongement de la classique théorie des équations différentielles, la théorie des équations intégrales rejoint, à travers les espaces de Hilbert, l’analyse matricielle qui s’est révélée un outil important dans la mécanique ondulatoire de Louis de Broglie. M.-M. V.

In this Chapter, the authors present two sets of lecture notes by Hilbert in which he reflects on epistemological consequences of modern physics. The opening passages of both lectures set the theme in almost identical words as a reflection on the role of mathematics in the exact sciences. Indeed, both lectures cover similar ground and deal with epistemological implications of general covariance such as time-reversal invariance and the need to formulate what Hilbert calls a “principle of objectivity,” the independence from coordinates for all meaningful statements in physics. The second set of lectures takes the theme further and discusses the implications of what Hilbert called the “world equations,” i.e., the field equations of his 1915 communication on the “Foundations of Physics.” Hilbert considered the question of completeness of physical theory and introduced the concept of “accessorial” laws, arguing that in a proper sense such laws no longer exist in modern physics based on general relativity and quantum theory. This contention is discussed in its philosophical implications in an attempt to position Hilbert’s axiomatic method in distinction to Kantian apriorism and against Poincaré’s conventionalism.