What has been the historical relationship between set theory and logic? On the one hand, Zermelo and other mathematicians developed set theory as a Hilbert-style axiomatic system. On the other hand, set theory influenced logic by suggesting to Schröder, Löwenheim and others the use of infinitely long expressions. The questions of which logic was appropriate for set theory - first-order logic, second-order logic, or an infinitary logic - culminated in a vigorous exchange between Zermelo and Gödel around 1930.
This paper explores how the Generalized Continuum Hypothesis (GCH) arose from Cantor's Continuum Hypothesis in the work of Peirce, Jourdain, Hausdorff, Tarski, and how GCH was used up to Gödel's relative consistency result.
Nietzsche, Biology and Metaphor explores the German philosopher's response to the intellectual debates sparked by the publication of Charles Darwin's Origin of Species. By examining the abundance of biological metaphors in Nietzsche's writings, Gregory Moore questions his recent reputation as an eminently subversive and (post) modern thinker, and shows how deeply Nietzsche was immersed in late nineteenth-century debates on evolution, degeneration and race. The first part of the book provides a detailed study and new interpretation of Nietzsche's much disputed relationship (...) to Darwinism. Uniquely, Moore also considers the importance of Nietzsche's evolutionary perspective for the development of his moral and aesthetic philosophy. The second part analyzes key themes of Nietzsche's cultural criticism - his attack on the Judaeo-Christian tradition, his diagnosis of the nihilistic crisis afflicting modernity and his anti-Wagnerian polemics - against the background of fin-de-siècle fears about the imminent biological collapse of Western civilization. (shrink)
Hilbert’s unpublished 1917 lectures on logic, analyzed here, are the beginning of modern metalogic. In them he proved the consistency and Post-completeness (maximal consistency) of propositional logic -results traditionally credited to Bernays (1918) and Post (1921). These lectures contain the first formal treatment of first-order logic and form the core of Hilbert’s famous 1928 book with Ackermann. What Bernays, influenced by those lectures, did in 1918 was to change the emphasis from the consistency and Post-completeness of a logic to its (...) soundness and completeness: a sentence is provable if and only if valid. By 1917, strongly influenced by PM, Hilbert accepted the theory of types and logicism -a surprising shift. But by 1922 he abandoned the axiom of reducibility and then drew back from logicism, returning to his 1905 approach of trying to prove the consistency of number theory syntactically. (shrink)
What gave rise to Ernst Zermelo's axiomatization of set theory in 1908? According to the usual interpretation, Zermelo was motivated by the set-theoretic paradoxes. This paper argues that Zermelo was primarily motivated, not by the paradoxes, but by the controversy surrounding his 1904 proof that every set can be wellordered, and especially by a desire to preserve his Axiom of Choice from its numerous critics. Here Zermelo's concern for the foundations of mathematics diverged from Bertrand Russell's on the one hand (...) and from Felix Hausdorff's on the other. (shrink)
This paper combines personal reminiscences of the philosopher John Corcoran with a discussion of certain conflicts between historians of logic and philosophers of logic. Some mistaken claims about the history of the Bolzano-Weierstrass Theorem are analyzed in detail and corrected.
Hilbert’s unpublished 1917 lectures on logic, analyzed here, are the beginning of modern metalogic. In them he proved the consistency and Post-completeness of propositional logic -results traditionally credited to Bernays and Post. These lectures contain the first formal treatment of first-order logic and form the core of Hilbert’s famous 1928 book with Ackermann. What Bernays, influenced by those lectures, did in 1918 was to change the emphasis from the consistency and Post-completeness of a logic to its soundness and completeness: a (...) sentence is provable if and only if valid. By 1917, strongly influenced by PM, Hilbert accepted the theory of types and logicism -a surprising shift. But by 1922 he abandoned the axiom of reducibility and then drew back from logicism, returning to his 1905 approach of trying to prove the consistency of number theory syntactically. (shrink)
This is the first translation of Fichte's addresses to the German nation for almost 100 years. The series of 14 speeches, delivered whilst Berlin was under French occupation after Prussia's disastrous defeat at the Battle of Jena in 1806, is widely regarded as a founding document of German nationalism, celebrated and reviled in equal measure. Fichte's account of the distinctiveness of the German people and his belief in the native superiority of its culture helped to shape German national identity throughout (...) the nineteenth century and beyond. With an extensive introduction that puts Fichte's argument in its intellectual and historical context, this edition brings an important and seminal work to a modern readership. All of the usual series features are provided, including notes for further reading, chronology, and brief biographies of key individuals. (shrink)
A seminal figure in the philosophy of history, culture, and language, Johann Gottfried Herder also produced some of the most important and original works in the history of aesthetic theory. A student of Kant, he spent much of his life striving to reconcile the opposing poles of Enlightenment thought represented by his early mentors. His ideas influenced Hegel, Schleiermacher, Nietzsche, Dilthey, J. S. Mill, and Goethe. This book presents most of Herder's important writings on aesthetics, including the main sections of (...) one of his major untranslated works, Kritische Wälder. These notes, essays, and treatises, the majority of which appear here in English for the first time, show this idiosyncratic thinker both deeply rooted in the controversies of his day and pointing the way to future developments in aesthetics. Chosen to reflect the extent and diversity of Herder's concerns, the texts cover such topics as the psychology and physiology of aesthetic perception, the classification of the arts, taste, Shakespeare, the classical tradition, and the relationship between art and morality. Few thinkers have reflected so sensitively and productively on the cultural, historical, anthropological, ethical, and theological dimensions of art and the creative process. With this book, the importance of aesthetics to the evolution and texture of Herder's own thought, as well as his profound contribution to that discipline, comes fully into view. (shrink)
This volume shows Russell in transition from a neo-Kantian and neo-Hegelian philosopher to an analytic philosopher of the first rank. During this period his research centred on writing The Principles of Mathematics where he drew together previously unpublished drafts. These shed light on Russell's paradox. This material will alter previous accounts of how he discovered his paradox and the related paradox of the largest cardinal. The volume also includes a previously unpublished draft of an early attempt to solve his paradox, (...) as well as the earliest known version of his generalised relation arithmetic. It contains three articles which have never previously been published in English. (shrink)
This volume of Bertrand Russell's _Collected Papers_ finds Russell focused on writing _Principia Mathematica_ during 1905–08. Eight previously unpublished papers shed light on his different versions of a substitutional theory of logic, with its elimination of classes and relations, during 1905-06. A recurring issue for him was whether a type hierarchy had to be part of a substitutional theory. In mid-1907 he began writing up the final version of _Principia_, now using a ramified theory of types, and eleven unpublished drafts (...) from 1907-08 deal with this. Numerous letters show his thoughts on the process. The volume's 80-page introduction covers the evolution of his logic from 1896 until 1909, when volume I of _Principia _went to the printer. (shrink)