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Proofs as Objects

In: Axiomatic Thinking I

Author

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  • Wilfried Sieg

    (Carnegie Mellon University, Department of Philosophy)

Abstract

The rigor of mathematics lies in its systematic organization that supports conclusive proofs of assertions on the basis of assumed principles. Proofs are constructed through thinking, but they can also be taken as objects of mathematical thought. That was the insight prompting Hilbert’s call for a "theory of the specifically mathematical proof" in 1917. This pivotal idea was rooted in revolutionary developments in mathematics and logic during the second half of the 19-th century; it also shaped the new field of mathematical logic and grounded, in particular, Hilbert’s proof theory. The derivations in logical calculi were taken as "formal images" of proofs and thus, through the formalization of mathematics, as tools for developing a theory of mathematical proofs. These initial ideas for proof theory have been reawakened by a confluence of investigations in the tradition of Gentzen’s work on natural reasoning, interactive verifications of theorems, and implementations of mechanisms that search for proofs. At this intersection of proof theory, interactive theorem proving, and automated proof search one finds a promising avenue for exploring the structure of mathematical thought. I will detail steps down this avenue: the formal representation of proofs in appropriate mathematical frames is akin to the representation of physical phenomena in mathematical theories; an important dynamic aspect is captured through the articulation of bi-directional and strategically guided procedures for constructing proofs.

Suggested Citation

  • Wilfried Sieg, 2022. "Proofs as Objects," Springer Books, in: Fernando Ferreira & Reinhard Kahle & Giovanni Sommaruga (ed.), Axiomatic Thinking I, chapter 0, pages 165-183, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-77657-2_9
    DOI: 10.1007/978-3-030-77657-2_9
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