Informal proof, formal proof, formalism

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 metaphysical doctrine. I argue that there are strong reasons to doubt that all proofs are fully formalisable, if formal proofs are required to be finitary, but that, on a proper view of the way in which formal proofs idealise actual practice, this restriction is unjustified and formalism is not threatened

From Euclidean Geometry to Knots and Nets

This document is the Accepted Manuscript of an article accepted for publication in Synthese. Under embargo until 19 September 2018. The final publication is available at Springer via https://doi.org/10.1007/s11229-017-1558-x.This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if (a) it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, (b) the information thus displayed is not metrical and (c) it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions.Peer reviewe

A Case Study in Dependent Type Theory: Extracting a Certified Program from the Formal Proof of its Specification

Proofs are an important part of mathematics, but they are not without their flaws. Most proofs are written by humans, and humans make mistakes. In this thesis, we explore the use of proof assistants to construct formal versions of informal proofs, and to extract certified and correct programs from these formal proofs. We study a specific case of two problems from lattice theory, solved by Bezem and Coquand in [2]. Firstly, we ask ourselves: are the results from [2] correct? The informal proofs posed in [2], used to justify the correctness of the results, are complex enough that mistakes are possible. Using the Coq proof assistant, we formalize parts of the proofs from [2], to gain more confidence in the correctness of these informal proofs. Secondly, is the process of formalization a feasible one? Transforming an informal proof into a formal proof is not necessarily an easy or straightforward process, and there are several approaches to formalization of a proof. Lastly, is the process of formalization worth the effort? A fully formalized proof gives us almost complete confidence in the correctness of the proof, and hence the result. Moreover, if the proof is constructive, we get the added bonus of extracting a certified program from the proof. In our case, this program has some practical use cases in a real-world setting, mostly as a prototype. Our formalization, done using Coq, can be found in the GitHub repository at [1]. For convenience, the formal proof of the central result has been reproduced in Listing A.1.Masteroppgave i informatikkINF399MAMN-INFMAMN-PRO

Informal and formal proofs, metalogic, and the groundedness problem

When modeling informal proofs like that of Euclid’s Elements using a sound logical system, we go from proofs seen as somewhat unrigorous – even having gaps to be filled – to rigorous proofs. However, metalogic grounds the soundness of our logical system, and proofs in metalogic are not like formal proofs and look suspiciously like the informal proofs. This brings about what I am calling here the groundedness problem: how can we decide with certainty that our metalogical proofs are rigorous and sustain our logical system? In this paper, I will expose this problem. I will not try to solve it here

Relational Rippling: a General Approach

We propose a new version of rippling, called relational rippling. Rippling is a heuristic for guiding proof search, especially in the step cases of inductive proofs. Relational rippling is designed for representations in which value passing is by shared existential variables, as opposed to function nesting. Thus relational rippling can be used to guide reasoning about logic programs or circuits represented as relations. We give an informal motivation and introduction to relational rippling. More details, including formal definitions and termination proofs can be found in the longer version of this paper, [Bundy and Lombart, 1995]

On the formal justification of informal proofs

When modeling the informal proofs of Euclid’s Elements using the sound logical system E, we go from proofs seen as somewhat nonrigorous – even having gaps to be filled – to rigorous proofs. According to the ‘standard view’, the correctness of an informal proof is underwritten by the existence of a corresponding formal derivation. However, metalogic grounds the soundness of the logical system E, and proofs in metalogic are not like formal proofs and look suspiciously like informal proofs. In our view, they are informal proofs. This brings about what we are calling here the groundedness problem: how can we show with certainty that our metalogical proofs are correct and sustain our logical system? According to the ‘standard view’, we cannot. In this way, we would have to doubt the soundness of the formal system E. This in turn might lead us to doubt the justification of Euclidean informal proofs in terms of the corresponding formal proofs in E

Learning-assisted Theorem Proving with Millions of Lemmas

Large formal mathematical libraries consist of millions of atomic inference steps that give rise to a corresponding number of proved statements (lemmas). Analogously to the informal mathematical practice, only a tiny fraction of such statements is named and re-used in later proofs by formal mathematicians. In this work, we suggest and implement criteria defining the estimated usefulness of the HOL Light lemmas for proving further theorems. We use these criteria to mine the large inference graph of the lemmas in the HOL Light and Flyspeck libraries, adding up to millions of the best lemmas to the pool of statements that can be re-used in later proofs. We show that in combination with learning-based relevance filtering, such methods significantly strengthen automated theorem proving of new conjectures over large formal mathematical libraries such as Flyspeck.Comment: journal version of arXiv:1310.2797 (which was submitted to LPAR conference

Automation of Diagrammatic Proofs in Mathematics

Theorems in automated theorem proving are usually proved by logical formal proofs. However, there is a subset of problems which can also be proved in a more informal way by the use of geometric operations on diagrams, so called diagrammatic proofs. Insight is more clearly perceived in these than in the corresponding logical proofs: they capture an intuitive notion of truthfulness that humans find easy to see and understand. The proposed research project is to identify and ultimately automate this diagrammatic reasoning on mathematical theorems. The system that we are in the process of implementing will be given a theorem and will (initially) interactively prove it by the use of geometric manipulations on the diagram that the user chooses to be the appropriate ones. These operations will be the inference steps of the proof. The constructive !-rule will be used as a tool to capture the generality of diagrammatic proofs. In this way, we hope to verify and to show that the diagra..

Reliability of mathematical inference

Of all the demands that mathematics imposes on its practitioners, one of the most fundamental is that proofs ought to be correct. It has been common since the turn of the twentieth century to take correctness to be underwritten by the existence of formal derivations in a suitable axiomatic foundation, but then it is hard to see how this normative standard can be met, given the differences between informal proofs and formal derivations, and given the inherent fragility and complexity of the latter. This essay describes some of the ways that mathematical practice makes it possible to reliably and robustly meet the formal standard, preserving the standard normative account while doing justice to epistemically important features of informal mathematical justification