The Foundations Of Mathematics

The Foundations of Mathematics is one of Paul Carus’s many philosophical works, written for an educated readership in the late nineteenth and early twentieth centuries. Carus was a German born American philosopher and prolific author, best remembered as the long time editor of the Open Court Publishing Company and of the journals The Open Court and The Monist. His philosophical work emphasized the integration of natural science, mathematics, and religious thought, and the foundations of mathematics as a subject was a particularly active area of late nineteenth century philosophical and mathematical work that Carus contributed to.

The foundations of mathematics as a philosophical subject had become particularly urgent in the late nineteenth century as the various developments in modern mathematics, including non Euclidean geometry, set theory, and the formal logical work that figures like Frege, Cantor, and others were producing, raised foundational questions about what mathematics was, how mathematical knowledge was possible, and what the relationship between mathematical truth and the physical world actually involved. Carus engaged with these foundational questions across multiple books and articles, with The Foundations of Mathematics representing one of his more sustained treatments of the subject.

Carus’s particular philosophical position involved a kind of monistic philosophy that tried to integrate the formal mathematical and logical work with a wider framework that included natural science and religious experience. The Foundations of Mathematics works through the various competing accounts of what mathematics actually is, with Carus drawing on the contemporary mathematical developments while also engaging with the historical philosophical tradition that ran from the Greek geometers through the early modern philosophical engagements of figures like Descartes, Leibniz, and Kant.

The book is in the formal nineteenth century philosophical style, dense by modern standards but carefully constructed for the educated general readership that Open Court’s publications served. Carus draws on the most current mathematical work alongside the wider philosophical tradition, with the comparative engagement that was his signature across his many books on philosophical and religious topics.

For students of late nineteenth and early twentieth century philosophy of mathematics, of the wider Open Court intellectual tradition, or of the history of how philosophical and scientific thought engaged with foundational mathematical questions during a particularly active period in mathematical history, The Foundations of Mathematics is worth knowing. Carus’s books are now historical artifacts more than living philosophical interventions, but they remain valuable as primary sources from a particular moment in American intellectual history when the foundational questions of mathematics were being actively worked through by mathematicians, philosophers, and the wider intellectual community.