Monday, July 7, 2008

New Book: The Philosophy of Mathematical Practice

The Philosophy of Mathematical Practice, edited by Mancosu, is now out, and I am hopeful that it will push the much-discussed turn to practice in new and exciting directions. In his introduction Mancosu helpfully situates the volume by saying "What is distinctive of this volume is that we integrate local studies with general philosophy of mathematics, contra Corfield, and we also keep traditional ontological and epistemological topics in play, contra Maddy". Here we have an approach to practice that finds philosophical issues arising from within mathematical practice, as was ably demonstrated in Mancosu's earlier book on the 17th century.

For me, the highlight of the volume is the excellent essay by Urquhart on how developments in physics have been "assimilated" into mathematics. This assimilation is not limited to putting the mathematics on a more rigorous foundation (or foundations, as several rigorizations are often possible), but also has led to new mathematics of intrinsic mathematical importance. As Urquhart puts it,
The common feature of the examples of the Dirac delta function, infinitesimals, and the umbral calculus is that the explications given for the anomalous objects and reasoning patterns involving them is what may be described as pushing down higher order objects. In other words, we take higher order objects, existing higher up in the type hierarchy, and promote them to new objects on the bottom level. This general pattern describes an enormous number of constructions.
The essay finishes with an intriguing case of "spin glasses" where the so-far unrigorized "replica method" has proven extremely successful.

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