## Position paper: A real Semantic Web for mathematics deserves a real semantics

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@MISC{Corbineau_positionpaper:,

author = {P. Corbineau and H. Geuvers and C. Kaliszyk and J. Mckinna and F. Wiedijk},

title = {Position paper: A real Semantic Web for mathematics deserves a real semantics},

year = {}

}

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### Abstract

Abstract. Mathematical documents, and their instrumentation by computers, have rich structure at the layers of presentation, metadata and semantics, as objects in a system for formal mathematical logic. Semantic Web tools [2] support the first two of these, with little, if any, contribution to the third, while Proof Assistants [17] instrument the third layer, typically with bespoke approaches to the first two. Our position is that a web of mathematical documents, definitions and proofs should be given a fully-fledged semantics in terms of the third layer. We propose a “Math-Wiki ” to harness Web 2.0 tools and techniques to the rich semantics furnished by contemporary Proof Assistants. 1 Background and state of the art We can identify four worlds of mathematical discourse available on the Web: – Traditional mathematical practice: a systematic body of knowledge, organised around documents written by experts, most often in L ATEX, to varying degrees of sophistication. The intended audience is an expert readership, and

### Citations

752 |
Isabelle/HOL — A Proof Assistant for Higher-Order Logic
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Citation Context ... non-trivial formalisations: – The HOL Light system [8] is an LCF-style checker for higher-order logic; Harrison recently announced a proof of the analytic Prime Number Theorem; – The Isabelle system =-=[15]-=- is also LCF-like, but adds a generic twist in terms of an Automath-like theory of representation: it is a logical framework, that is, it is generic over the underlying choice of logic and axiomatisat... |

130 | OMDoc – An open markup format for mathematical documents [Version 1.2
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Citation Context ...but relatively shallow and of variable reliability, with little systematic development of larger theories, and little or no critical gloss on the material; – The Semantic Web, with the OMDoc standard =-=[9]-=- and tools like SWiM [11], for organising structured documents around a basic notion of “falling under a concept” (such concepts then further organised into content dictionaries); – The language and (... |

69 | HOL Light: A tutorial introduction
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(Show Context)
Citation Context ...dependently rechecked, or otherwise processed. Modern systems have elaborated these ideas with great sophistication, extensive libraries, and highly non-trivial formalisations: – The HOL Light system =-=[8]-=- is an LCF-style checker for higher-order logic; Harrison recently announced a proof of the analytic Prime Number Theorem; – The Isabelle system [15] is also LCF-like, but adds a generic twist in term... |

64 |
A computer-checked proof of the four colour theorem
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Citation Context ...[3] is type-theoretic, within which objects and proofs are λ-terms in a calculus of inductive and coinductive definitions; a notable development is Gonthier’s formalisation of the Four Colour Theorem =-=[6]-=-; – The Mizar system [13], a proof checker for a strong version of set theory, emphasises developing a formalised library of standard, classical mathematics. The decisive semantic advantage of all the... |

28 | SWiM – a semantic wiki for mathematical knowledge management
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Citation Context ...nd of variable reliability, with little systematic development of larger theories, and little or no critical gloss on the material; – The Semantic Web, with the OMDoc standard [9] and tools like SWiM =-=[11]-=-, for organising structured documents around a basic notion of “falling under a concept” (such concepts then further organised into content dictionaries); – The language and (checked) libraries of pro... |

27 |
The Open Math standard, version 2.0
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Citation Context ... documents, and their instrumentation by computers, have rich structure at the layers of presentation, metadata and semantics, as objects in a system for formal mathematical logic. Semantic Web tools =-=[2]-=- support the first two of these, with little, if any, contribution to the third, while Proof Assistants [17] instrument the third layer, typically with bespoke approaches to the first two. Our positio... |

20 | F.: C-CoRN, the constructive coq repository at Nijmegen
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(Show Context)
Citation Context ...gh not very focused. More focused examples arethe CompCert project in which a whole team participated in the verification of a C compiler and the Nijmegen repository of formalised mathematics (CoRN) =-=[5]-=-. The ongoing Flyspeck project [7] is another instance. Proof assistants proposed to be part of the MathWiki project in the initial phase are Coq, Isabelle and Mizar. They cover three different founda... |

17 | The Flyspeck project
- Hales, McLaughlin
- 2007
(Show Context)
Citation Context ...examples arethe CompCert project in which a whole team participated in the verification of a C compiler and the Nijmegen repository of formalised mathematics (CoRN) [5]. The ongoing Flyspeck project =-=[7]-=- is another instance. Proof assistants proposed to be part of the MathWiki project in the initial phase are Coq, Isabelle and Mizar. They cover three different foundational theories (Type Theory, High... |

17 |
An Outline of PC Mizar, Fondation Philippe le Hodey
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Citation Context ...ithin which objects and proofs are λ-terms in a calculus of inductive and coinductive definitions; a notable development is Gonthier’s formalisation of the Four Colour Theorem [6]; – The Mizar system =-=[13]-=-, a proof checker for a strong version of set theory, emphasises developing a formalised library of standard, classical mathematics. The decisive semantic advantage of all these systems over existing ... |

14 | An architecture for collaborative math and science digital libraries. Unpublished master’s thesis, Virginia Tech - Krowne - 2003 |

6 |
et al. The qed manifesto
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Citation Context ...rfaces. These technologies are important for our project since interactive proof development is by far the most popular way of using proof assistants. Already in 1993 the authors of the QED Manifesto =-=[1]-=- had this vision: to let the whole world participate in creating a shared repository of formalised mathematics. We can speculate as to why this was an idea before its time: inevitably, user communitie... |

6 |
A Description of Automath and Some Aspects of its Language Theory
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Citation Context ...Assistants The basic idea of using computer programs to check mathematical proofs goes back to the archaeology of AI research. The 1960s saw the emergence of two basic paradigms: de Bruijn’s Automath =-=[16]-=-, and Milner’s LCF. Both provide highly generic foundational approaches to representing mathematics: as a series of checked objects (definitions etc.) extending a body of knowledge from an initial axi... |

5 |
C.: Cooperative repositories for formal proofs
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- 2007
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Citation Context ...y exist for the included proof assistants, like the Coq user contributions (contribs) and the Archive of Formal Proofs for Isabelle, in order to facilitate access to them. We have created a prototype =-=[4]-=- that only supports Coq (without any semantic aspects yet), which suggests the project is technically feasible. In Figure 1 we sketch how the eventual system might look (including quoted material from... |

3 | SWiM - A Semantic Wiki for - Lange - 2008 |

1 |
The Coq Proof Assistant Reference
- Team
- 2006
(Show Context)
Citation Context ...or both higher-order logic, and for ZF set theory. It has been used to formalise Gödel’s completeness theorem, the consistency of the axiom of choice, the Prime Number Theorem, etc.; – The Coq system =-=[3]-=- is type-theoretic, within which objects and proofs are λ-terms in a calculus of inductive and coinductive definitions; a notable development is Gonthier’s formalisation of the Four Colour Theorem [6]... |