Frege: The Basic Laws of Arithmetic, Russell's paradox, and geometry

Published: Feb 7, 2024
Vangelis Triantafyllou

This article is primarily concerned with Logic, Philosophy, Mathematics, and their history. Specifically, we delve into the emergence of Russell’s paradox within Frege’s Basic Laws of Arithmetic, Frege’s initial attempts to address the paradox, and his eventual rejection of extensions in favor of embracing Geometry as the foundational basis for mathematical objects. The article begins by outlining the basic characteri- stics of Frege’s Laws system, and then investigates the paradox’s emer- gence. We then examine Frege’s efforts to come to terms with this fact and deal with the paradox. His stance is particularly revealing, as specific aspects of his approach, that distinguish him from later thinkers, become evident here. These features will eventually lead him to adopt a distinctive, geometricocentric approach, which we briefly explore in the concluding sections of the article.

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