Celebratio Mathematica — Mac Lane

Saunders Mac Lane (1909–2005): His mathematical life and philosophical works

Works connected to Stewart Shapiro

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S. Mac Lane: “Structure in mathematics,” pp. 174–183 in Mathematical structuralism, published as Philos. Math. (3) 4 : 2. Issue edited by S. Shapiro. 1996. MR 1397752 Zbl 0905.18001 incollection

The article considers structuralism as a philosophy of mathematics, as based on the commonly accepted explicit mathematical concept of a structure. Such a structure consists of a set with specified functions and relations satisfying specified axioms, which describe the type of the structure. Examples of such structures such as groups and spaces, are described. The viewpoint is now dominant in organizing much of mathematics, but does not cover all mathematics, in particular most applications. It does not explain why certain structures are dominant, not why the same mathematical structure can have so many different and protean realizations. ‘Structure’ is just one part of the full situation, which must somehow connect the ideal structures with their varied examples.