New Book: Mancosu, The Adventure of Reason

While it came out in 2010, some of the readers of this blog may still not be aware of this important collection of Mancosu's articles on the interactions between logic and philosophy in the 1900-1940 period. I recently completed a summary of the book's contents for Zentralblatt Math that might be useful.

Revised SEP Entry: Mathematical Explanation

The Stanford Encyclopedia Entry on "Mathematical Explanation" has just been updated and revised. Thanks to Paolo Mancosu for this important resource!

Mancosu on Mathematical Style

Paolo Mancosu continues his innovative work in the philosophy of mathematics with a thought-provoking survey article on Mathematical Style for the Stanford Encyclopedia of Philosophy. From the introductory paragraph:

The essay begins with a taxonomy of the major contexts in which the notion of ‘style’ in mathematics has been appealed to since the early twentieth century. These include the use of the notion of style in comparative cultural histories of mathematics, in characterizing national styles, and in describing mathematical practice. These developments are then related to the more familiar treatment of style in history and philosophy of the natural sciences where one distinguishes ‘local’ and ‘methodological’ styles. It is argued that the natural locus of ‘style’ in mathematics falls between the ‘local’ and the ‘methodological’ styles described by historians and philosophers of science. Finally, the last part of the essay reviews some of the major accounts of style in mathematics, due to Hacking and Granger, and probes their epistemological and ontological implications.

As Mancosu says later in the article "this entry is the first attempt to encompass in a single paper the multifarious contributions to this topic". So it is wide-open for further philosophical investigation!

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.