Paradoxes of the Infinite (German: Paradoxien des Unendlichen) is a posthumously published treatise by the Bohemian philosopher, theologian and mathematician Bernard Bolzano (1781âÂÂ1848). Edited by his former student Frantià ¡ek Pà Âihonský and published in Leipzig in 1851, the book surveys âÂÂparadoxesâ connected with the infinite in mathematics, geometry, physics and metaphysics, with the aim of showing that such paradoxes are not genuine contradictions once the relevant concepts are made precise. The work is often discussed in the prehistory of set theory for its detailed defence of actual (completed) infinity and its analysis of one-to-one pairings between infinite âÂÂmultitudesâ and their parts, themes later central to DedekindâÂÂs and CantorâÂÂs work on infinite collections.
According to Pà ÂihonskýâÂÂs editorâÂÂs preface, Bolzano began writing the book in 1847 while staying with the editor in Liboch (near MÃÂlnÃÂk) and completed it in the summer of 1848, the final year of his life. After receiving the manuscript from BolzanoâÂÂs heirs, Pà Âihonský prepared it for publication by correcting passages, improving legibility, and supplying a detailed synopsis of contents; he dated his preface from Bautzen (Budissin) on 10 July 1850 and chose Leipzig partly for wider distribution. The first edition appeared in 1851 with the Leipzig publisher C. H. Reclam.
The work is organised into 70 numbered sections. A synopsis supplied by the editor indicates the bookâÂÂs main arc:
BolzanoâÂÂs stated project is to treat paradoxes surrounding infinity as apparent rather than real contradictions. In later historical discussion, this is often described as a systematic defence of actual infinity within mathematics, combined with an attempt to clarify which principles of finite reasoning fail (or require modification) in infinite contexts.
Bolzano distinguishes several kinds of âÂÂcollectionsâ in the opening portions of the treatise, separating the general idea of a collection (Inbegriff) from the special case of a multitude (Menge) in which the arrangement of parts is treated as irrelevant. Modern commentators often translate Menge as âÂÂmultitudeâ to avoid importing later Cantorian assumptions into BolzanoâÂÂs framework.
A central portion of the book analyses the possibility of pairing elements of two infinite multitudes so that each element of either multitude occurs in exactly one pair (what later mathematics calls a bijection). Bolzano notes that such a pairing can hold even when one multitude is (in an intuitive partâÂÂwhole sense) contained in the other, and he argues thatâÂÂunlike the finite caseâÂÂpairing alone is not always sufficient to conclude âÂÂequality in respect of plurality.â He therefore appeals to additional conditions (often glossed as âÂÂdetermining groundsâ or a shared âÂÂway of being formedâÂÂ) to justify claims of equality or ratio for infinite multitudes in specific cases.
This stance is frequently contrasted with later nineteenth-century set theory, in which Dedekind and Cantor elevated one-to-one correspondence to a primary criterion for comparing sizes of infinite collections.
In the sections dealing with âÂÂcalculation with the infinite,â Bolzano rejects a number of informal uses of infinitesimals and infinite magnitudes (including treating division by zero as meaningful), and attempts to state constraints under which operations involving infinitely small or large quantities can be made consistent. Recent scholarship has re-examined these parts of the book, arguing that BolzanoâÂÂs mature views are best understood not as a precursor of Cantorian cardinal arithmetic but as a theory of infinite sums with its own algebraic structure.
BolzanoâÂÂs book was published posthumously and (by some accounts) had limited immediate impact on the subsequent development of mathematics. Nevertheless, later writers in the history of logic and set theory have highlighted its role as an early, sustained attempt to analyse infinite collections and the principles governing their comparison. In particular, Encyclopædia Britannica notes that Dedekind and Cantor made explicit use of one-to-one mappings as a tool for âÂÂmeasuringâ sets, and that Dedekind formulated a definition of an infinite set in terms of correspondence with a proper partâÂÂan idea closely related to the correspondences treated in Paradoxien des Unendlichen.
The book has also attracted attention in mathematics education and philosophy of mathematics as a historically rich source of arguments and examples about infinity, especially where intuitive partâÂÂwhole reasoning conflicts with correspondence-based comparison.