POINCARÉ'S LAST WORD ON INFINITY.

POINCARÉ, HENRI. 1854-1912.

Autograph Manuscript Signed ("Henri Poincaré"), "La Logique de l'Infini," Poincaré's final word on infinity,

POINCARÉ, HENRI. 1854-1912. Autograph Manuscript Signed ("Henri Poincaré"), "La Logique de l'Infini," Poincaré's final word on infinity, 4to (311 x 205 mm), 9 pp, ink on paper, [Paris?], 1912, published in Scientia 12, pp 1 – 11 (and collected as "Les Mathematiques et la Logique" in Derniers Pensees, 1913),
Provenance: Printer's notes and markings (likely from Scientia publication, 1912).

AUTOGRAPH MANUSCRIPT, ORIGINAL PRINTER'S COPY, OF POINCARÉ'S FINAL WORD ON THE PARADOX OF MATHEMATICAL INFINITY. A SEMINAL STATEMENT IN THE IDEOLOGICAL DEBATE THAT SHAPED THE LOGIC OF MODERN MATHEMATICS.
At the beginning of the 20th century, mathematics experienced an "infinity crisis." Abstract new ideas, in particular, Georg Cantor's theory of sets and transfinite number, together with the emergence of logicism, introduced a radically different way of thinking about infinity and led to divergent views on mathematical proof. Amid the contemporary swirl of argument and debate surrounding these new ideas, Henri Poincaré, through a series of articles published from 1905 to 1912, brought an objective scientific eye to the intellectual turmoil. Acting with neutrality in the interests of truth, Poincaré analyzed the philosophic merit of each of these challenging new doctrines and illuminated the ground of the logical paradoxes plaguing the new notions of infinity.

The present manuscript contains Poincaré's final statement on the infinity crisis. Seeking to restore unity to the fractured philosophy of mathematics, Poincaré begins by elucidating the nature of the "irreducible divergence" (quotes supplied from the English translation) between the old and new schools of mathematical thought, referencing his previous arguments with "various eminent mathematicians," although not mentioned by name, primarily Bertrand Russell, David Hilbert, and Ernst Zermelo. He then poses the critical question: "Does a theorem which does not result in any verifiable conclusion have a meaning?" ("Or, more generally, does any theory have a meaning apart from the proof which it involves?") After answering this critical question in the negative – in contrast to the pro-infinity (Cantorian) faction -- Poincaré examines the unorthodox conception of mathematical definition and proof adopted by the Cantorians. He concludes in establishing that the "vicious circle" of logical paradox – so famously epitomized by Bertrand Russell's "set of all sets that are not members of themselves"– is a consequence of the Cantorians permitting unrestricted comprehension in their predications ("postulates").

Poincaré has been called "the deepest, most prolific, and most versatile mathematician of his time." Considered the father of algebraic topology, including the "Poincaré conjecture," the central unsolved problem of the field for nearly a century, Poincaré's substantial work in both pure and applied mathematics has fed the development of diverse fields from functional analysis to quantum theory. As much a physicist as a mathematician, he was an early contributor to relativity theory in advance of Einstein, and his work on the 3-body problem and the stability of the solar system laid the foundation for modern Chaos Theory and earned him a nomination for the Nobel Prize in Physics (in 1909). Poincaré is regarded as one of the greatest philosophers of science, and his several books on scientific creativity and method – especially his Science and Hypothesis, which was read eagerly and approvingly by a young Einstein, are considered classics in the field. Poincaré's specific interest in the logic of infinity was triggered by the emergence of logicism, a doctrine which ran counter to his neo-Kantian view that there is a qualitative epistemic difference between mathematics and logic (characterized by synthetic vs analytic activity of the mind). For his many significant and wide-ranging contributions to the several fields of mathematics, physics, and philosophy, Poincaré is often called "the last universalist."

Poincaré's thought played a central role in resolving the infinity crisis provoked by Cantor and logicism. His balanced and sober perspective essentially carried the day, his philosophic viewpoint entering the mainstream and becoming formalized in the decades following his death (1912). Poincaré's analyses and critiques were a direct and major factor in deciding Bertrand Russell's ramification of the logical theory of types, and they further motivated some of the revisions Russell made to the second edition of Principia Mathematica. Poincaré's work with infinity also prompted Zermelo's reframing of Axiomatic Set Theory; and it later influenced Wittgenstein's philosophy of mathematics and provided a foundation for Godel's celebrated proof of the incompleteness of arithmetic.

Scientific and mathematical manuscripts of this quality are exceedingly rare, and the present manuscript, one of Poincaré's most important statements on logic, infinity and proof, discusses key ideas central to the foundation of 20th-century mathematics.