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In this paper, we try to establish that some mathematical theories, like  K -theory, homology, cohomology, homotopy theories, spectral sequences, modern Galois theory (in its various applications), representation theory and character theory,  etc ., should be thought of as (abstract) machines in the same way that there are (concrete) machines in the natural sciences. If this is correct, then many epistemological and ontological issues in the philosophy of mathematics are seen in a different light. We concentrate on one problem which immediately follows the recognition of the particular status of these theories: the demarcation problem between ‘natural kinds’ and ‘artefacts’.

Abstract Mathematical Tools and Machines for Mathematics †