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An unbridled and passionate interest in foundations has often been singled out as a characteristic trait of both philosophy and science in this century. Nowhere has this trend been more rampant than in mathematics. Yet, foundational studies, in spite of an auspicious beginning at the turn of the century, followed by unrelenting efforts, far from achieving their purported goal, found themselves attracted into the whirl of mathematical activity, and are now enjoying full voting rights in the mathematical senate. As mathematical logic becomes ever more central within mathematics, its contributions to the philosophical understanding of foundations wane to the point of irrelevance. Worse yet, the feverish technical advances in logic in the last ten years have dashed all hope of founding mathematics upon the notion of set, which had become the primary mathematical concept since Cantor. Equally substantial progress in the fields of algebra and algebraic geometry¹ has further contributed to cast a shadow on this notion. At the other end of the mathematical spectrum, the inadequacy of naive set theory had been realized by von Neumann² since the beginning of quantum theory, and to this day the physicist's most important method of research remains devoid of adequate foundation, be it mathematical, logical, or philosophical.

DOI: 10.1007/978-94-010-1999-6_14

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