Remarks on the Foundations of Mathematics

Remarks on the Foundations of Mathematics is a posthumously published collection of notes by the philosopher Ludwig Wittgenstein, compiled from manuscripts written between 1937 and 1944. Edited by his literary executors G. H. von Wright, Rush Rhees, and G. E. M. Anscombe, the work was first published in 1956 by Basil Blackwell. It presents a radical critique of established views in the philosophy of mathematics, challenging the programs of logicism, formalism, and Platonism that dominated the field following the work of Gottlob Frege and Bertrand Russell.

Historical Context and Publication

The remarks were composed during a highly productive yet turbulent period in Ludwig Wittgenstein's life, following his return to Cambridge University and his succession to the chair once held by G. E. Moore. This era was marked by intense philosophical development, distancing himself from the ideas of his earlier Tractatus Logico-Philosophicus. The manuscripts were written against the backdrop of rising tensions in Europe, including the Anschluss and the onset of World War II, events that deeply affected Wittgenstein. The editorial work by G. H. von Wright, Rush Rhees, and G. E. M. Anscombe involved organizing a vast array of notes and typescripts, resulting in a publication that has since become a cornerstone of twentieth-century analytic philosophy.

Central Philosophical Themes

A central theme is the rejection of mathematics as a body of eternal, objective truths discoverable by a priori reasoning. Instead, Ludwig Wittgenstein argues that mathematical propositions are rules of grammar or norms of representation within a given language-game, akin to moves in a game like chess. He emphasizes the role of human practice, agreement, and training in establishing mathematical certainty, famously exploring scenarios like following the rule "+2" in the sequence 2, 4, 6, 8. This perspective places the activity of calculating and proving at the forefront, challenging the metaphysical foundations sought by figures like Gottlob Frege and David Hilbert.

Critique of Logicism and Formalism

Wittgenstein mounts a sustained attack on the logicism of Gottlob Frege and Bertrand Russell, which sought to reduce mathematics to logic, and the formalism of David Hilbert, which treated mathematics as the manipulation of uninterpreted symbols. He argues that logic cannot provide a secure foundation for mathematics because logical proofs themselves rely on mathematical practices and human agreement. His critique extends to Kurt Gödel's incompleteness theorems; Wittgenstein controversially questioned the standard interpretation of Gödel's results, suggesting they reveal limitations in certain formal systems rather than profound metaphysical truths about mathematics itself.

Rule-Following and Private Language

The discussions on rule-following are deeply intertwined with arguments later central to his Philosophical Investigations. Wittgenstein investigates what it means to follow a mathematical rule correctly, arguing that no mental state or internal interpretation can fix its application indefinitely. This leads to his famous rejection of the possibility of a private language; the correctness of a rule is determined by its public, customary use within a form of life. This analysis undermines the idea of a solitary mathematician discovering truths in isolation, instead highlighting the necessity of a community and shared practices, a view that influenced later thinkers like Saul Kripke.

Influence on Later Philosophy

The publication of Remarks on the Foundations of Mathematics had a profound impact on subsequent philosophy, particularly within the movements of ordinary language philosophy and post-analytic philosophy. It inspired radical reconsiderations of the nature of necessity and logical truth by philosophers such as G. E. M. Anscombe and Peter Winch. Its challenges to foundationalism contributed to the development of quasi-empiricism in mathematics and influenced the sociology of scientific knowledge, as seen in the work of David Bloor and the Edinburgh School. The text remains a vital, though contentious, reference point in debates within the philosophy of mathematics and Wittgensteinian scholarship.

Category:Philosophy of mathematics literature Category:Books by Ludwig Wittgenstein Category:1956 non-fiction books