Grothendieck Universes

The foundation of the Mizar Mathematical Library [2], is first-order Tarski-Grothendieck set theory. However, the foundation explicitly refers only to Tarski’s Axiom A, which states that for every set X there is a Tarski universe U such that X ∈ U. In this article, we prove, using the Mizar [3] formalism, that the Grothendieck name is justified. We show the relationship between Tarski and Grothendieck universe.This work has been supported by the Polish National Science Centre granted by decision no. DEC-2015/19/D/ST6/01473.Institute of Informatics, University of Białystok, PolandGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Chad E. Brown and Karol Pąk. A tale of two set theories. In Cezary Kaliszyk, Edwin Brady, Andrea Kohlhase, and Claudio Sacerdoti Coen, editors, Intelligent Computer Mathematics – 12th International Conference, CICM 2019, CIIRC, Prague, Czech Republic, July 8-12, 2019, Proceedings, volume 11617 of Lecture Notes in Computer Science, pages 44–60. Springer, 2019. doi:10.1007/978-3-030-23250-4_4.N. H. Williams. On Grothendieck universes. Compositio Mathematica, 21(1):1–3, 1969.28221121

MDML: The Mathdoc Digital Mathematics Library

International audienceFollowing the steps of previous projects such as EuDML, Mathdoc is launching its Digital Mathematics Library. Based on a reliable infrastructure made for Numdam, learning from previous projects, and relying on a network of institutions we trust, we aim to push the ball further for accessing mathematical content online. We focus for a start on the aggregation part, aiming to reach a critical mass of mathematical content by harvesting various sources: OJS instances, preprint repositories , and locals DMLs. We thus build a database of mathematical documents, linking back to the source's website for accessing content

Discovering Mathematical Objects of Interest -- A Study of Mathematical Notations

Mathematical notation, i.e., the writing system used to communicate concepts in mathematics, encodes valuable information for a variety of information search and retrieval systems. Yet, mathematical notations remain mostly unutilized by today's systems. In this paper, we present the first in-depth study on the distributions of mathematical notation in two large scientific corpora: the open access arXiv (2.5B mathematical objects) and the mathematical reviewing service for pure and applied mathematics zbMATH (61M mathematical objects). Our study lays a foundation for future research projects on mathematical information retrieval for large scientific corpora. Further, we demonstrate the relevance of our results to a variety of use-cases. For example, to assist semantic extraction systems, to improve scientific search engines, and to facilitate specialized math recommendation systems. The contributions of our presented research are as follows: (1) we present the first distributional analysis of mathematical formulae on arXiv and zbMATH; (2) we retrieve relevant mathematical objects for given textual search queries (e.g., linking $P_{n}^{(\alpha, \beta)}\!\left(x\right)$ with `Jacobi polynomial'); (3) we extend zbMATH's search engine by providing relevant mathematical formulae; and (4) we exemplify the applicability of the results by presenting auto-completion for math inputs as the first contribution to math recommendation systems. To expedite future research projects, we have made available our source code and data.Comment: Proceedings of The Web Conference 2020 (WWW'20), April 20--24, 2020, Taipei, Taiwa

OntoMathPRO{}^{\mathbf{PRO}} 2.0 Ontology: Updates of the Formal Model

This paper is devoted to the problems of ontology-based mathematical knowledge management and representation. The main attention is paid to the development of a formal model for the representation of mathematical statements in the Open Linked Data cloud. The proposed model is intended for applications that extract mathematical facts from natural language mathematical texts and represent these facts as Linked Open Data. The model is used in development of a new version of the OntoMath${}^{\mathrm{PRO}}$ ontology of professional mathematics is described. OntoMath${}^{\mathrm{PRO}}$ underlies a semantic publishing platform, that takes as an input a collection of mathematical papers in LaTeX format and builds their ontology-based Linked Open Data representation. The semantic publishing platform, in turn, is a central component of OntoMath digital ecosystem, an ecosystem of ontologies, text analytics tools, and applications for mathematical knowledge management, including semantic search for mathematical formulas and a recommender system for mathematical papers. According to the new model, the ontology is organized into three layers: a foundational ontology layer, a domain ontology layer and a linguistic layer. The domain ontology layer contains language-independent math concepts. The linguistic layer provides linguistic grounding for these concepts, and the foundation ontology layer provides them with meta-ontological annotations. The concepts are organized in two main hierarchies: the hierarchy of objects and the hierarchy of reified relationships

Proofgold: Blockchain for Formal Methods

Proofgold is a peer to peer cryptocurrency making use of formal logic. Users can publish theories and then develop a theory by publishing documents with definitions, conjectures and proofs. The blockchain records the theories and their state of development (e.g., which theorems have been proven and when). Two of the main theories are a form of classical set theory (for formalizing mathematics) and an intuitionistic theory of higher-order abstract syntax (for reasoning about syntax with binders). We have also significantly modified the open source Proofgold Core client software to create a faster, more stable and more efficient client, Proofgold Lava. Two important changes are the cryptography code and the database code, and we discuss these improvements. We also discuss how the Proofgold network can be used to support large formalization efforts

Intelligent Computer Mathematics 12th International Conference, CICM 2019, Prague, Czech Republic, July 8–12, 2019, Proceedings

This book constitutes the refereed proceedings of the 12th International Conference on Intelligent Computer Mathematics, CICM 2019, held in Prague, Czech Republic, in July 2019. The 19 full papers presented were carefully reviewed and selected from a total of 41 submissions. The papers focus on digital and computational solutions which are becoming the prevalent means for the generation, communication, processing, storage and curation of mathematical information. Separate communities have developed to investigate and build computer based systems for computer algebra, automated deduction, and mathematical publishing as well as novel user interfaces. While all of these systems excel in their own right, their integration can lead to synergies offering significant added value