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

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 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.”

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 confronted 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 book can be read along various tracks suggested by the author, from reading through the text in its worthwhile entirety to a summary appendix to fit each reader’s interest.

The first scene’s historical content largely focuses on the development of the concept of function. The mathematically indefinite — let alone infinite — is not deeply explored in the scores of pages in this portion. There is very little content that is technical; what is there, is in the few sections recommended to be passed over by the non-mathematician. This makes for nearly eighty pages of exposition prior to moment when the author addresses the parafinite for the first time. Bassler, “as a philosopher interested in mathematics”, continues a tradition, beginning with Galileo, Descartes and Leibniz, of confronting the mathematically indefinitely large, his parafinite. He surveys how this remote region has been understood as neither finite nor infinite (Galileo), a shadow of the infinite (Descartes), or as infinite (Leibniz).

Quoting these thinkers’ work highlights the long history of discussing the incomprehensibly large without admitting it to the infinite. As Galileo is quoted here, “if asked whether the quantified parts in the continuum are finite or infinitely many, the most suitable reply is to say neither finite nor infinitely many.” The sources quoted from Descartes and Leibniz find these great thinkers concerned on how admitting to infinite extents (geometrically and physically, generally) can possibly align with the Divine. As Descartes says here,

…not even being able to conceive, that the world has bounds, I call it indefinite. But I cannot deny on that account that there may be some reasons which are known by God, although they are incomprehensible to me: that is why I do not say absolutely that the world is infinite.

The author highlights modern exploration of this area, including Leibnizian infinitesimals considered by Abraham Robinson and Imre Lakatos through non-standard analysis and Jan Mycielski's "analysis without actual infinity."

Examining the intellectual insight and psychic wrestling of Leibniz is a significant and interesting thread of this work. Even more engaging than that is delving into controversial and foundations-questioning assault on arithmetic by Wittgenstein. This covers a significant portion of scene three — all of Chapter 6 and appendix materials — making this an excellent introduction to Wittgenstein’s views on the validity of even the most common mathematical concepts. Wittgenstein the “strict finitist” offers a “vigorous criticism of Cantorian set theory” that the author uses as philosophical motivation for Vopěnka’s Alternative Set Theory and from there for Abraham Robinson’s non-standard analysis. Culminating in “McCauley’s campaign against measure theory”, Chapter 6 is an essential and cogent kernel of this work.

A quote from van Dantzig frames the essential topic addressed by this book: “[t]he difference between finite and infinite numbers is not an essential, but a gradual one.” This self-contained consideration of the history, development, and implications of the parafinite is enlightening, educational, and thought-provoking.

Tom Schulte teaches mathematics at Oakland Community College in Michigan where he enjoys asking students to divide one by nine on the calculator as a prelude to discussing the infinitesimal and the infinite.