Paradoxes of the Infinite

by Bernard Bolzano

Paradoxes of the Infinite presents one of the most insightful, yet strangely unacknowledged, mathematical treatises of the 19th century: Dr Bernard Bolzano's Paradoxien. This volume contains an adept translation of the work itself by Donald A. Steele S.J., and in addition an historical introduction, which includes a brief biography as well as an evaluation of Bolzano the mathematician, logician and physicist.

Genres: Mathematics, Philosophy, Physics, Logic

202 pages, Paperback

First published January 1, 1851

Reviews

[Douglas · Rating 3 out of 5]

This book set the historical foundations of set theory. In it Bolzano presents a very early(and very naive) treatment of infinity. The book is standard for its time, which may turn modern readers off. The style involves very long sentences(sometimes only two per page) and early 19th century metaphysical talk.

[William Bies · Rating 4 out of 5]

Bernard Bolzano is a multifaceted figure from the early nineteenth century. He wins everlasting fame among mathematicians for having discovered the Bolzano-Weierstrass property, a necessary and sufficient condition for countable compactness, but was a founding father of the analytic philosophy of the twentieth century and a priest and theologian, as well. His Paradoxien des Unendlichen constitutes a report on the state of knowledge about the infinite in his day. Bolzano is notable among mathematicians of his generation for having been the one who, more than any other, recognized the centrality of the concept of the infinite to the program of shoring up the foundations of analysis. Dedekind took up Bolzano’s cue and made his so-called cuts (Durchschnitte), which are nothing but certain ordered and gapless infinite sets of rational numbers, the key to the definition of the real number field, which accomplished for the first time the arithmetization of the continuum and enabled Weierstrass to establish his foundational results in real analysis and calculus, such as the rigorous proof—at last!—of the intermediate value theorem. In the present work under review, we find Bolzano’s own reflections on the infinite and the paradoxes one encounters with it. He was among the first to stress what is now regarded as a defining property of infinity, viz., the existence of a 1-1 mapping to a proper subset. Bolzano discusses other mathematical problems with the infinite—most of which originate in sloppy handling of mathematical concepts such as limits and infinitesimals we know better about today—before going on to paradoxes of time and space, as seen from the pre-relativistic point of view of the mid-nineteenth century. If one makes due allowance to update it to modern notation, Bolzano’s own preferred aether theory is by no means as obsolete as one might suppose. To one who is accustomed to Cantor’s later set theory and transfinite numbers and to modern measure theory along the lines of Carathéodory, Borel, Baire, Lebesgue et al., Bolzano’s views impress one with a certain antique charm, but one should keep in mind that the systematization we employ these days need not be the only logically consistent possibility and perhaps, if one were resourceful enough, one could recover something more like what Bolzano thought about comparative infinities. Food for thought, at least! The introduction and extensive annotation by Christian Tapp in the Felix Meiner edition are quite helpful in explicating Bolzano’s occasional obscurities as well as in making the connection to more recent mathematical ideas.