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Philosophical systems have developed for centuries, but only in the nineteenth century did the notion of scientific philosophy emerge. This notion presented two dimensions in the early twentieth century. One dimension arose from scientists' concern with conceptual foundations for their disciplines, while another arose from philosophers' appetite for more rigorous philosophy. In the current paper, I will focus on David Hilbert's construct of "critical mathematics" and Edmund Husserl' and Jules Vuillemin's systematic philosophy. All these three thinkers integrated Kant's legacy with the axiomatic method. However, they did so in different ways, with Hilbert's goal being the opposite of that of Husserl or Vuillemin. Specifically, I will show how the scientism of Hilbert's mathematical epistemology aimed at shattering the ambition of philosophy to submit mathematical practices and problems to philosophy's own principles and methods, be they transcendental or metaphysical. On the other hand, phenomenology promoted the idea of a non-exact philosophical rigour and highlighted the need of a point of view encompassing positive sciences, ontology, and ethical values in connection with the dominant category of sense/meaning, and Jules Vuillemin built on from the inseparability of thought—scientific or philosophical—from the metaphysics of free will and choice.

DOI: 10.1007/978-3-319-93733-5_2

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