Pub Date : 2020-04-30DOI: 10.1093/oso/9780190641221.003.0001
E. Reck, G. Schiemer
The core idea of mathematical structuralism is that mathematical theories, always or at least in many central cases, are meant to characterize abstract structures (as opposed to more concrete, individual objects). As such, structuralism is a general position about the subject matter of mathematics, namely abstract structures; but it also includes, or is intimately connected with, views about its methodology, since studying such structures involves distinctive tools and procedures. The goal of the present collection of essays is to discuss mathematical structuralism with respect to both aspects. This is done by examining contributions by a number of mathematicians and philosophers of mathematics from the second half of the 19th and the early 20th centuries.
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Pub Date : 2020-04-30DOI: 10.1093/oso/9780190641221.003.0016
S. Morris
This chapter examines the development of and motives for Quine’s particular form of mathematical structuralism. It will argue that Quine, unlike many contemporary mathematical structuralists, does not appeal to structuralism as a way of accounting for what the numbers really are in any robust metaphysical sense. Instead, his structuralism is deeply rooted in an earlier structuralist tradition found in scientific philosophers such as Russell and Carnap, which emphasized structuralism as a critique of more metaphysical approaches to philosophy. On this view, a philosophy of mathematics answers, in a sense, only to mathematics itself. An account of mathematical objects requires only that the entities—whatever they are—serving as the mathematical objects satisfy the relevant postulates and theorems. Here we also see how Quine’s early work in the foundations of mathematics leads in a natural way to the more general naturalism of his later philosophy.
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Pub Date : 2020-04-30DOI: 10.1093/oso/9780190641221.003.0013
E. Reck
Ernst Cassirer was a keen observer of development in the mathematical sciences, especially in the 19th and early 20th centuries. In this essay, the focus is on his reception of Dedekind’s contributions to the foundations of mathematics, and with it, on Dedekind’s mathematical structuralism. Cassirer adopts that structuralism early on, defends it against a number of criticisms, and embeds it into a rich historical account of the structuralist transformation of modern mathematical science. He also adds some original elements to our understanding of structuralism, e.g., by relating it to the Kantian notion of the “construction of concepts” in mathematics, by introducing a basic distinction between “substance concepts” and “function concepts”, and by tracing the “unfolding” of structuralist aspects far back in the history of thought. Overall, Cassirer’s approach is guided by the conviction that the metaphysics of modern mathematics should be approached by way of its distinctive methodology.
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Pub Date : 2020-04-30DOI: 10.1093/oso/9780190641221.003.0012
J. Heis
Bertrand Russell was one of the first philosophers to recognize clearly the philosophically innovative nature of Richard Dedekind’s philosophy of arithmetic: a position we now describe as non-eliminative structuralism. But Russell’s response was deeply negative: “If [numbers] are to be anything at all, they must be intrinsically something” (Principles of Mathematics, §242). Nevertheless, Russell also played a significant positive role in making possible the emergence of structuralist philosophy of mathematics. This chapter explains Russell’s double role, identifying three positive contributions to structuralism, while laying out Russell’s objections to Dedekind’s non-eliminative structuralism.
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