Review: The Long Shadow of the Parafinite: Three Scenes from the Prehistory of a Concept

The Long Shadow of the Parafinite: Three Scenes from the Prehistory of a Concept by O Bradley Bassler
My rating: 4 of 5 stars

The Long Shadow assays questions in the history and philosophy of mathematics in an area more often termed the transfinite. It is part of the author’s long-term project rethinking metaphysics in the modern European-American tradition, including examining the work of German philosopher and intellectual historian Hans Blumenberg. The three “scenes” in the work are: (I) relevant history of mathematics from antiquity, including Greek and Arabic advances, especially as framed by Jacob Klein and Reviel Netz; (II) early modern struggles by Galileo, Leibniz, and others as they confront the implications of infinity as an abstract idea; and (III) more recent post-Cantorian alternatives to considering infinity, especially by Bertrand Russell, Edmund Husserl, and Wittgenstein. More philosophy of mathematics than philosophically aware mathematics, the author suggests various tracks through the text from its worthwhile entirety to a summary appendix to fit each reader’s interest. The author’s use of the term parafinite allows separation of these investigations from traditional transfinite topics: “Cantorian set theory has no place for the parafinite because it is an infinite (or, more appropriately; transfinite) theory.”...

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