Online research in philosophy

Much work in MKM depends on the application of formal logic to mathematics. However, much mathematical knowledge is informal. Luckily, formal logic only represents one tradition in logic, specifically the modeling of inference in terms of logical form. Many inferences cannot be captured in this manner. The study of such inferences is still within the domain of logic, and is sometimes called informal logic. This paper explores some of the benefits informal logic may have for the management of informal mathematical knowledge.

‘Semantic technologies’ are touted as the next big wave in Educational Technology and as the solution to many problems in this arena. Interdisciplinary work between the fields of Knowledge Management (KM) and Educational Technology (ET) is booming. But the crop of actual systems and semantically enhanced learning objects is still meager, maybe KM and EL they are lacking a consensus on the underlying notions e.g. of ‘semantics’, yielding specific problems in their interplay. In this paper we take a look at semantic educational technologies and draw conclusions for their approach in KM. In particular, we (re)-evaluate the notions of ‘semantics’, ‘knowledge’, ‘learning’, their role for learning materials in ET, and how they interact with the contexts involved in the learning/teaching process. Based on this, we distill a list of conditions the underlying knowledge representation format must fulfil to support these. As these conditions are still rather abstract, we show how they can be realized in a concrete language design, taking in our OMDoc (Open Mathematical Documents) format as a point of departure.

In the last two decades, the World Wide Web has become the universal, and — for many users — main information source. Search engines can efficiently serve daily life information needs due to the enormous redundancy of relevant resources on the web. For educational — and even more so for scientific information needs, the web functions much less efficiently: Scientific publishing is built on a culture of unique reference publications, and moreover abounds with specialized structures, such as technical nomenclature, notational conventions, references, tables, or graphs. Moreover, many of these structures are peculiar to specialized communities determined by nationality, research group membership, or adherence to a special school of thought. To keep the much-lamented “digital divide” from becoming a “cultural divide”, we have to make online material more accessible and adaptable to individual users. In this paper we attack this goal for the field of mathematics where knowledge is abstract, highly structured, and extraordinarily interlinked. Modern, content-based representation formats like OpenMath or content MathML allow us to capture, model, relate, and represent mathematical knowledge objects and thus make them context-aware and machine-adaptable to the respective user contexts. Building on previous work which can make mathematical notations adaptable we employ user modeling techniques to make them adaptive to relieve the reader of configuration tasks. We present a comprehensive framework for adaptive notation management and evaluate it on an implementation integrated in the e-learning platform panta rhei.

The OMDoc (Open Mathematical Documents) format is a content markup scheme for (collections of) mathematical documents including articles, textbooks, interactive books, and courses. OMDoc also serves as the content language for agent communication of mathematical services on a mathematical software bus.

We explore the social context of mathematical knowledge: Even though, the community of mathematicians may look homogeneous from the outside, it is actually structured into various sub-communities that differ in preferred notations, the choice of basic assumptions, or e.g. in the choice of motivating examples. We contend that we cannot manage mathematical knowledge for human recipients if we do not take these factors into account. As a basis for a future extension of MKM systems, we analyze the social context of information in terms of Communities of Practice (CoP; a concept from learning theory) and present a concrete extensional model for CoPs in mathematics.

We present a search engine for mathematical formulae. The MathWebSearch system harvests the web for content representations of formulae and indexes them with substitution tree indexing. In version 0.4 we have parallelized and distributed the search server and augmented the web interface with a new JavaScript-based visual editor for content math formulae. Furthermore, we have extended the query language by generalization, variants, unification, and text search facilities, which can also be mixed. Our experiments show that this architecture results in a scalable application.

What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions. Written by some of the world's leading philosophers of mathematics, psychologists, and mathematicians, Mathematical Knowledge gives a lively sense of the current state of debate in this fascinating field. Contents 1. Mary Leng: Introduction 2. Michael Potter: What is the problem of mathematical knowledge? 3. Tim Gowers: Mathematics, memory, and mental arithmetic 4. Alan Baker: Is there a problem of induction for mathematics? 5. Marinella Cappelletti and Valeria Giardino: The cognitive basis of mathematical knowledge 6. Mary Leng: What's there to know? A fictionalist account of mathematical knowledge 7. Mark Colyvan: Mathematical recreation versus mathematical knowledge 8. Alexander Paseau: Scientific platonism 9. Crispin Wright: On quantifying into predicate position: Steps towards a (new)tralist position.

The interest of the field of Mathematical Knowledge Management is predicated on the assumption that by investing into markup or formalization of mathematical knowledge, we can reap benefits in managing (creating, classifying, reusing, verifying, and finding) mathematical theories, statements, and objects. This global value proposition has been used to motivate the pursuit of technologies that can add machine support to these knowledge management tasks. But this (rather naive) technology-centered motivation takes a view merely from the global (macro) perspective, and almost totally disregards the user’s point of view and motivations for using it, the local (micro) perspective. In this paper we go a first step into a more principled analysis of the MKM value proposition by focusing on motivations for mathematical search engines from the micro perspective. We will use a table-based method called the “Added-Value Analysis” (AVA) developed by one of the authors. Even though we apply the AVA only to mathematical search engines, the method quickly leads to value considerations that are relevant for the whole field of MKM.

Although knowledge is a central topic for MKM there is little explicit discussion on what ‘knowledge’ might actually be. There are specific intuitions about form and content of knowledge, about its structure, and epistemological nature that shape the MKM systems, but a conceptual model is missing. In this paper we try to rationalize this discussion to give MKM a firmer footing, to start a discussion among MKM researchers and help relate the MKM intuitions and discourses to other communities. Starting from the observation that many concrete realizations of mathematical knowledge objects are considered equivalent, we propose a conceptual model of the space of (mathematical) knowledge objects graded by levels of abstraction and presentational explicitness and draw conclusions for MKM markup formats.

Version Control systems like CVS and Subversion have transformed collaboration workflows in software engineering, and made possible the globally distributed project teams we know from the Open Source Phenomenon. On the other hand, XML is coming of age as a basis for document formats, and even though XML as a text-based format is amenable to version control in principle, the fact that version control systems work on files makes difficult the integration of fragment access techniques like XPath, XQuery that are currently revolutionizing XML workflows. In this paper we present the TNTBase system, an open-source versioned XML database obtained by integrating Berkeley DB XML into the Subversion Server. The..