Synonymy and Identity of Proofs - A Philosophical Essay

Abstract

The main objective of the dissertation is to investigate from a strictly philosophical perspective different approaches and results related to the problem of identity of proofs, which is a problem of general proof theory at the intersection of mathematics and philosophy. The author characterizes,compares and evaluates a range of formal criteria of proof-identity that have been proposed in the proof-theoretic literature. While these proposals come from mathematical logicians, the author’s background in both mathematical logic and philosophy allows him to present and discuss these proposals in a manner that is accessible to and fruitful for philosophers, especially those working in logic and philosophy of mathematics, as well as mathematical logicians.

The dissertation is structured into a prologue and five sections. In the prologue, the author traces the development of the concept of a proof in ancient philosophy, culminating in the work of Aristotle. In Section I, the author turns to the roots of proof theory in modern philosophy, offering a detailed interpretation of Kant’s “Die falsche Spitzfindigkeit der vier syllogistischen Figuren”, which uncovers interesting links between Kant’s inferences of understanding and of reason and modern proof-theoretic semantics. In Section II, the author turns from historical to systematic considerations concerning different kinds of identity-criteria of proofs, ranging from overly liberal criteria that trivialize proof identity to overly strict, syntactical criteria. In Section III, the heart of the dissertation, the author offers a thorough philosophical discussion of the normalisation thesis. In Section IV, the author considers the difficulties encountered in his discussion of identity of proofs --- particularly of the normalisation thesis --- through the lens of a discussion of the notion of synonymy, and compares this thesis with other possible formal accounts of identity of proofs. In particular, by recourse to Carnap’s notion of synonymy, developed in “Meaning and Necessity”, the author proposes a notion of synonymy of proofs. In Section V, the final substantial section, the author compares the normalisation thesis to the Church-Turing thesis, thereby adducing another dimension of evaluation of the former.