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Hellman and Shapiro on Mathematical Structuralism

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CUP are publishing a series of short books (about 100 pages) under the title Cambridge Elements in the Philosophy of Mathematics. The blurb says that the series “provides an extensive overview of the philosophy of mathematics in its many and varied forms. Distinguished authors will provide an up-to-date summary of the results of current research in their fields and give their own take on what they believe are the most significant debates influencing research, drawing original conclusions.” Which sounds ambitious. So far, though, just two Elements have been published, Mathematical Stucturalism (2018) by Geoffrey Hellman and Stewart Shapiro, and A Concise History of Mathematics for Philosophers (2019) by John Stillwell. No further books are yet announced on the web-page for the series. The hyper-active Stillwell has already written a well-known and accessible Mathematics and Its History (3rd edn, 2010) as well as a number of other non-specialist books (alongside his hard-core maths texts). It seems a bit of a failure of imagination for the series editors to ask him to write another history; and to me, the result looks pretty unexciting. Again, getting Hellman and Shapiro to write on structuralism is hardly adventurous! But I have now read their book. It’s not clear, really, who the intended readership is. The series blurb — “up-to-date summary”, “current research”, “original conclusions” — might suggest a book aimed at e.g. graduate students. But little of the book reaches that sort of level. (One odd feature: we aren’t told who wrote what, even though some of the passages are in the first person and are characteristic of just one of the authors.) There is a short Introduction giving initial characterizations of some forms of mathematical structuralism, and setting out some questions that we’d want any structuralism to address. Chapter 2 gives Historical Background. This is over twice the length of the next longest chapter; it is nicely done, with a good. . .

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News source: Logic Matters

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