the concept of mathematical proof has been controversial for the past few decades. different philosophers have offered different theories about the nature of mathematical proof, among which theories presented by lakatos and hersh have had significant similarities and differences with each other. it seems that a comparison and critical review of these two theories will lead to a better understan...

Increases in the use of automated theorem-provers have renewed focus on the relationship between the informal proofs normally found in mathematical research and fully formalised derivations. Whereas some claim that any correct proof will be underwritten by a fully formal proof, sceptics demur. In this paper I look at the relevance of these issues for formalism, construed as an anti-platonistic ...

Paul Erdős famously remarked that ‘a mathematician is a machine for turning coffee into theorems’ [9, p. 7]. The proof of mathematical theorems is central to mathematical practice and to much recent debate about the nature of mathematics. This paper is an attempt to introduce a new perspective on the argumentation characteristic of mathematical proof. I shall argue that this account, an applica...

The main aim of this paper is to promote a certain informal but rigorous style of doing coinductive proofs, similar to inductive proofs as commonly done by mathematicians. For this purpose we develop a theory which may serve as a reasonably direct justification for coinductive proofs written in this informal style, i.e., converting an informal coinductive proof into a non-coinductive argument m...

We present a generic mediator, called PlatΩ, between text-editors and proof assistants. PlatΩ aims at integrated support for the development, publication, formalization, and verification of mathematical documents in a natural way as possible: The user authors his mathematical documents with a scientific WYSIWYG text-editor in the informal language he is used to, that is a mixture of natural lan...

Powerful proof techniques, such as logical relation arguments, have been developed for establishing the strong normalisation property of term-rewriting systems. The first author used such a logical relation argument to establish strong normalising for a cut-elimination procedure in classical logic. He presented a rather complicated, but informal, proof establishing this property. The difficulti...

The Agora system is a prototype “Wiki for Formal Mathematics”, with an aim to support developing and documenting large formalizations of mathematics in a proof assistant. The functions implemented in Agora include in-browser editing, strong AI/ATP proof advice, verification, and HTML rendering. The HTML rendering contains hyperlinks and provides on-demand explanation of the proof state for each...

The topic of this thesis is the relationship between formal and informal proofs. Chapter One opens the discussion by examining what a proof is, when two proofs are identical, what the purpose of proving is and how to distinguish the two categories of proof. Chapters Two and Three focus in on informal and formal proof respectively, with the latter also including descriptions of various computati...

We define the notion of formal proof sketch for the mathematical language Mizar. We show by examples that formal proof sketches are very close to informal mathematical proofs. We discuss some ways in which formal proof sketches might be used to improve mathematical proof assistants.