Husserl and Frege on Imaginary Numbers and the Deep Nature of Things

Abstract

This paper examines the fundamentally different responses of Edmund Husserl and Gottlob Frege to the epistemological problem posed by imaginary entities in mathematics, including negative, irrational, complex, and transfinite numbers. It argues that their divergent approaches to the justification of symbolic procedures involving apparently non-denoting or impossible objects played a decisive role in shaping the subsequent trajectories of Continental and Analytic philosophy. The study reconstructs Husserl’s intellectual struggle with the foundations of arithmetic and shows how questions concerning imaginary numbers led him to develop a broader theory of symbolic thinking, formal systems, and manifolds. Husserl’s solution is presented as an attempt to explain how deductive reasoning can legitimately employ “imaginary” elements while still yielding valid results within complete axiomatic domains. In contrast, Frege’s insistence that mathematical expressions must denote genuine objects is examined in relation to his logicist program and his reliance on extensions of concepts. The paper contends that Frege’s treatment of logical objects and extensions contributed to the difficulties that culminated in Russell’s paradox and the collapse of his foundational project. By comparing these two responses to the problem of imaginaries, the article highlights their significance for the philosophy of mathematics, the theory of meaning, and the search for secure foundations of arithmetic. It concludes that Husserl’s analysis of symbolic procedures and formal structures offers a more promising framework for understanding the epistemological status of mathematical entities and the deeper logical structure of reality.