Significado y enfoques del individualismo liberal

In this chapter we introduced the common mathematical language term which stands for a mathematical language that we are taught from the early years of secondary school to the university level. We also presented existing tools for com- puterising and formalising mathematics. We discussed different approaches of ex- isting proof systems. Finally we gave an overview of the MathLang project and its aspect oriented design. Furthermore, we presented MathLang origins, goals and current overall situation of development. We also illustrated the MathLang framework in terms on encodings and covered that illustration with a number of examples annotated in MathLang.

In Section 2.5 we shortly described the Document Rhetorical aspect which is a part of MathLang aspect-oriented design. The DRa aspect is the main contribution of this PhD and has been developed during this PhD studies.

Chapter 3

The Mizar Project and Some of

Our Formalisations in Mizar

In this chapter we present a short description of the Mizar project. Section 3.1 presents a brief overview of the whole project, where we discuss the overall goals of the project, different tools and third-party development approaches. In Section 3.2 we discuss the Mizar Language and its similarities to the CML. The following Section 3.3 describes the Mizar Article which stores the authors input of the math- ematical knowledge in the Mizar language and which the Mizar system verifies. In Section 3.5 we present the Mizar Mathematical Library (MML), which is the biggest collection of computer verified mathematical facts. Primarily, the MML is a library of Mizar articles which form a graph of connected and related mathematical facts that form the integral library. Moreover, the library is centrally maintained and upgraded very often.

In the same section we provide a general overview of a number of complex theories and books formalisation in Mizar. Furthermore, we present the MML Query a semantic search engine, that was built to provide semantical search over the Mizar Mathematical Library. In the same section we shortly present the Mizar types structure.

The following Section 3.6 discusses formalisation attempts done by K. Retel. Those formalisations resulted in a number of articles submitted in the MML and later on automatically translated into English language and published in the For- malized Mathematics journal.

Section 3.7 presents a short description of history use of the Mizar system in a number of teaching experiments conducted at many universities. K. Retel has been involved in some of those experiments and he placed important role of tutor

and teacher of the Mizar system.

Finally in Section 3.8 we present the Formal Proof Sketch (FPS) term, invented by F. Wiedijk, which applies to any formal proof system and language, and partic- ularly to the Mizar.

It is important to note that this chapter is mainly a survey about the Mizar. This means that some parts of the content of this chapter might contain a direct or indirect relations, citations, examples and express similar or very close ideas, thoughts and explanations that were written in a number of publications regarding the Mizar. The chapter also contains number of short examples taken directly or indirectly from the MML.

During the course of the PhD of this student, we had used many versions of the Mizar system. This was due to a rapid enhancements and development of the software, as well as lots of submissions, revisions and major improvements of the Mizar library (Mizar Mathematical Library). Each time we present bigger examples of formalised texts in Mizar language it should come from the latest, at the date of writing the thesis, version 7.9.03 of the Mizar system, which is distributed with version 4.108.1028 of the Mizar Mathematical Library. If the example was verified and used different MML version, we mention the version of the Mizar system and the MML on which this example was composed. This is presented usually as a footnote to the main example. We will talk about the Mizar by providing number of toy examples from original articles taken randomly from the MML. We would also use some articles that have been written by this student, K. Retel, and have been published in the MML and electronic journal FM.

3.1

Overview of The Project

The main goal of the Mizar project1 _{is to create a system for computer-aided}

formalisation of mathematics [Try77, Try78, Try80, Try82, RT99, Rud92, MR05]. The project started in 1973 at the University of Warsaw, Poland, when A. Trybulec envisioned computer assistance in the process of writing mathematical papers. The idea was motivated in the latest stages of writing A. Trybulec’s PhD [MR05]. The name, Mizar, of the project appeared in late 1972, picked up by the wife of A. Trybulec. A. Trybulec asked her for a good name of a project, while she was looking through an astronomical atlas, and she suggested Mizar, the name of a star in the familiar Big Bear constellation [MR05].

1

Since 1973, A. Trybulec has been leading the group of mathematicians that forms the Mizar Group. The Mizar Group mainly refers to people from the Univer- sity of Bialystok, which is the heart of the Mizar project. The ongoing development of Mizar has resulted in several things:

1. the Mizar language, 2. the Mizar system, 3. the Mizar library,

4. Mizar software utilities for working with Mizar documents and the Mizar library,

5. and an electronic hyper-linked journal of mathematics that is also published in the paper edition.

The Mizar language is used for recording mathematics whereas the Mizar system is used for checking the correctness of texts written in this language. The Mizar language is a language suitable for practical formalisation of mathematics. It is based on first–order logic with free second-order variables. Proofs are written in the style of natural deduction as proposed by Ja´skowski [Ja´s34]. The language itself is also an attempt to approximate in a formal way the mathematical vernacular used in publications. On one hand, the Mizar language inherits the expressiveness, naturalness and freedom of reasoning of CML. On the other hand it is formal enough to allow mechanical verification and computer processing.

The development of the Mizar language resulted in the Mizar system. The system allows to check mathematical publications written in the Mizar language and to verify the logical structure and reasoning of such formalised documents.

The essential achievement of the project is a centrally maintained library of mathematical knowledge (the Mizar Mathematical Library – MML). This corpus of formalised mathematics is accompanied to the Mizar system and it is distributed with the system. Moreover, the MML library together with the Mizar system play the integral unit. For many years to nowadays, the Mizar Mathematical Library is the biggest collection of digitalized mathematical texts verified by com- puter [Wie03].

The MML consists of Mizar documents, which are called Articles within the Mizar community. This library is based on two axiomatic Articles: HIDDEN [Com89] which consists of built-in notions, and TARSKI [Try89] which presents axioms of

the Tarski-Grothendieck set theory [Tar39]. All other Articles of the MML are consequences of those axioms and are verified by the Mizar system. The user while writing a new Mizar Article reuses the notation, definitions and theorems and other constructs stored in the library. The Mizar system assists the author while formalising new terminology and results. It verifies the claims of the new Article and extracts facts and definitions for inclusion into the library. The task of building a rich mathematical library is currently the main effort of the Mizar community. At the time of writing this thesis, the library includes 1011 Articles contributed by 209 authors2_{, a number of whom have been active on a long term}

basis. There is a number of introductory papers and manuals on Mizar [Miz, RT99] as well as practical hints for writing Mizar Articles.