## Digitisation, Representation and Formalisation Digital Libraries of Mathematics

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@MISC{Adams_digitisation,representation,

author = {A. A. Adams},

title = {Digitisation, Representation and Formalisation Digital Libraries of Mathematics},

year = {}

}

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### Abstract

Abstract. One of the main tasks of the mathematical knowledge management community must surely be to enhance access to mathematics on digital systems. In this paper we present a spectrum of approaches to solving the various problems inherent in this task, arguing that a variety of approaches is both necessary and useful. The main ideas presented are about the differences between digitised mathematics, digitally represented mathematics and formalised mathematics. Each has its part to play in managing mathematical information in a connected world. Digitised material is that which is embodied in a computer file, accessible and displayable locally or globally. Represented material is digital material in which there is some structure (usually syntactic in nature) which maps to the mathematics contained in the digitised information. Formalised material is that in which both the syntax and semantics of the represented material, is automatically accessible. Given the range of mathematical information to which access is desired, and the limited resources available for managing that information, we must ensure that these resources are applied to digitise, form representations of or formalise, existing and new mathematical information in such a way as to extract the most benefit from the least expenditure of resources. We also analyse some of the various social and legal issues which surround the practical tasks. 1

### Citations

712 |
Handbook of mathematical functions with formulas, graphs, and mathematical tables
- Abramowitz, Stegun
- 1964
(Show Context)
Citation Context ... It is run by a consortium of (primarily) academics most of whom were present at the MKM ’01 workshop. – NIST Digital Library of Mathematical Functions Based on the Handbook of Mathematical Functions =-=[AS72]-=-, this NIST project aims to digitise and extend the utility of this seminal mathematical reference text. Two papers were presented about this project at MKM ’01: [Loz01, MY01]. – Digital Library of Ma... |

433 | Isabelle: A Generic Theorem Prover
- Paulson
- 1994
(Show Context)
Citation Context ...n” framework. In addition, a fair amount of work has been done on reconciling the differences between the underlying logics of these various systems (HOL, NuPrl, MetaPRL, PVS, Coq [BB + 96], Isabelle =-=[Pau88]-=-. . . ) where possible, and on identifying the difference where not. Thus, a theorem proved in one system may be identified as compatible or incompatible with the basic logic of another (such as the i... |

354 |
Intuitionistic Type Theory
- Martin-Löf
- 1984
(Show Context)
Citation Context ...ranslation of digitised mathematics texts into a representational format.) Seminal works may still go out of print, or be available from small publishers and almost unobtainable, try to buy a copy of =-=[ML84]-=- for instance. University libraries periodically cull their stock, and printed matter is always susceptible to accidental destruction. Material is still subject to copyright and, as we shall discuss i... |

98 |
Proofs from THE
- Aigner, Ziegler
- 1998
(Show Context)
Citation Context ...er to languish in the annals of the history of science, older mathematics is rarely completely replaced. Newer proofs may be shorter or more elegant, even appearing to approach some divine perfection =-=[AZ99]-=-. However, when a new area of study opens up it is often the case that older proofs, in their less refined and honed state, may give more insight into approaches, than the newer subject-specific ones.... |

97 | The Coq proof assistant reference manual : Version 6.1 - Barras, Boutin, et al. - 1997 |

47 | Automated Deduction Techniques for the Management of Personalized Documents
- Baumgartner, Furbach
(Show Context)
Citation Context ...e abstraction layer of \label and \ref in a L ATEX file, such internal links may be obscure and are almost certainly incomplete. Some very good initial work on these kinds of problems can be found in =-=[BB01]-=-, which looks at the problem of developing new documents from a variety of originals, through the use of text-slicing. Some automation is present, but most of the ‘”meta-information” (information abou... |

43 | OMDoc: Towards an internet standard for the administration, distribution, and teaching of mathematical knowledge
- Kohlhase
- 2000
(Show Context)
Citation Context ...heir scope and utility. A full exploration of these varied systems would require at least one, probably many, papers so by necessity this is only a brief overview. OpenMath [Dew00b, Dew00a] and OMDoc =-=[Koh01]-=- form an interesting starting point for consideration of formats for representing mathematics. As pointed out in [APSCS01, APSC + 01], however, the source of OpenMath as primarily a transport layer fo... |

43 | CoFI: The Common Framework Initiative for Algebraic Specification and Development
- Mosses
- 1997
(Show Context)
Citation Context ...ul as a format for generic representations. The same can be said of the various theorem provers and computer algebra systems. Their languages, and even more cross-system developments such as the CASL =-=[Mos97]-=- are still more focussed on a single kind of mathematical knowledge. In bringing MKM into sharper focus there must be a review of the various languages in terms of their capabilities, their current us... |

32 | Formal verification of floating point trigonometric functions
- Harrison
- 2000
(Show Context)
Citation Context ... is built primarily on the value of certain classes of intellectual property (such as Intel Corporation) recognise that other intellectual developments (such as the formalised mathematics of Harrison =-=[Har00]-=-) are more valuable in free dissemination than in closed-source exploitation. Formalised mathematics still mostly follows this ideal and MKM tool developers need to be aware of the desires of the math... |

29 | and N.Mitchell, Optical Character Recognition and Parsing of Typeset
- Fateman, Tokuyasu, et al.
- 1996
(Show Context)
Citation Context ...esting work has been done on recovering some of the author’s mathematical intent from the printed or digital graphic format of published mathematics. A very interesting paper is that of Fateman et al =-=[FTBM96]-=-. The main issues in this topic are the recognition ofssymbols in the rich language of mathematics, and the parsing of the two dimensional presentation used in typesetting. Take the simple instance of... |

15 |
The Mizar-QC/6000 Logic Information Language
- Trybulec
- 1978
(Show Context)
Citation Context ...generic formal representation system for mathematical information. There are a number of individual systems with a good underlying format for the representation of parts of mathematics, such as Mizar =-=[Try80]-=-. However, these systems tend to be tied very strongly to a particular system and a particular viewpoint of mathematics. So, while these systems are worthwhile and form a highly useful and usable sour... |

10 | NIST Digital Library of Mathematical Functions - Lozier - 2003 |

5 | OpenMath: An Overview - Dewar |

5 | LaTeX: A Document Preparation System, 2/e - Lamport - 1994 |

4 | Special Issue on OPENMATH - Dewar - 2000 |

4 |
Project Gutenberg
- Hart
- 2000
(Show Context)
Citation Context ...ely to be true for a wide range of material, languishing in the hands of uninterested publishers refusing to relinquish rights gained from highly biased contracts. A project such as Project Gutenberg =-=[Har71]-=- should be created as part of a large-scale investigation into MKM, not to transcribe works of mathematics, but simply to capture their graphic essence, to guard against loss and degradation of the ph... |

4 |
TeX and Metafont: New Directions
- Knuth
- 1979
(Show Context)
Citation Context ...itisation effort. Given this idea, that older mathematics may be as useful as the latest up-to-date material, it is imperative that efforts at MKM do not begin from 1962 (LISP 1.5 [McC62]); 1979 (TEX =-=[Knu79]-=-); 1988 (Mathematica [Wol]). . . and work forward with existing represented and formalised mathematics. (In Sect. 3, we will discuss the work on automatic translation of digitised mathematics texts in... |

1 | Exchange of mathematical information on the web: Present and Future
- Athale, Athale
(Show Context)
Citation Context ...ole idea by many in the mathematics community. This raises a side question as to who is part of this “mathematical community”. Athale and Athale presented a modest categorisation of this community in =-=[AA01]-=-. In the sense of developing tools and techniques for mathematical knowledge management(MKM), we must be as inclusive as possible in defining this community. The narrower our definition, the less use ... |

1 |
On the Theoretical Principles of the Machinery for Calculating Tables
- Babbage
(Show Context)
Citation Context ...paper we present an overview of the use of information technology in mathematics. Some of the earliest uses of computers was the automation of mathematics. Babbage’s Difference and Analytical Engines =-=[Bab23]-=- were designed to engineer tedious, error-prone, mathematical calculations. Some of the first digital computers were designed for the purpose of breaking encryption codes. In the early days of digital... |

1 |
Mathematical Knowledge Representation (Extended Abstract
- Davenport
(Show Context)
Citation Context ... between formalised mathematics and digitally displayed mathematics, which is the simple representation of the syntax without necessarily linking this to an underlying semantics. Davenport’s overview =-=[Dav01]-=- provides useful background to both this and the following section. Typeset mathematics comprises two primary types of text: the mathematical text and the explanatory text. The richness of the combina... |

1 |
More is less
- Kealey
- 2000
(Show Context)
Citation Context ...o be doubling every ten years). This compares with the science in general where it is estimated that only the number of new works published every ten years doubles (as opposed to the total published) =-=[Kea00]-=-. Mathematics is a very hierarchical subject. It is rare that new work completely supersedes older work. Unlike physics, for example, where new theories expand upon and quite often replace older work,... |

1 |
How to Build a Prototype for a Distributed Mathematics Archive Library
- Michler
(Show Context)
Citation Context ...s, in their less refined and honed state, may give more insight into approaches, than the newer subject-specific ones. So, mathematics, and mathematical knowledge, is rarely, if ever, superseded. See =-=[Mic01]-=- for an interesting look at a specific digitisation effort. Given this idea, that older mathematics may be as useful as the latest up-to-date material, it is imperative that efforts at MKM do not begi... |