Kuhn, Lakatos, and the Historical Turn in the Philosophy of Mathematics

Abstract

The paper deals with Kuhn’s and Lakatos’s ideas related to the so-called “historical turn” and its application to the philosophy of mathematics. In the first part the meaning of the term “postpositivism” is specified. If we lack such a specification we can hardly discuss the philosophy of science that comes “after postpositivism”. With this end in view, the metaphor of “generations” in the philosophy of science is used. It is proposed that we restrict the use of the term “post-positivism” to two and only two philosophical “generations”: the one to which Kuhn, Lakatos and Feyerabend belong, and the previous “generation” to which Wittgenstein, Polanyi, Popper and Quine (as well as the major part of logical positivists) belong. From this point of view, Bloor, Latour, Pickering, Daston and Galison belong to the “third generation” which represents the philosophy of science “after post-positivism”. The characteristic feature of post-positivism is the combination of decisive impact of logical positivism and its severe criticism. This combination inevitably makes post-positivism a transitional form in the philosophy of science. In the second part the contribution of the “big four” of post-positivist philosophers (Popper, Kuhn, Lakatos, and Feyerabend) to the radical change in the philosophy of mathematics in the second half of the 20th century is analyzed. Primarily, they shifted philosophical interest from the logical analysis of formal systems to the historical dynamics of informal mathematics. They also reconsidered the sharp opposition between mathematics and the physical sciences. However, the transitional character of their philosophy manifests itself both in their treatment of mathematics and their way of understanding history. On the one hand, their “heritage” is ambiguous, on the other hand, it opens new perspectives. Neither Kuhn, nor Lakatos, have eliminated completely the methodological barrier positing the fundamental heterogeneity of mathematics and natural science. Neither Lakatos, nor Kuhn, adhered to the viewpoint of relentless historicism. Nevertheless, it is their work that has made these options open for today’s historians and philosophers of science, even for philosophers of mathematics.