Results

**11 - 20**of**20**### mArachna – Ontology Engineering for Mathematical Natural Language Texts

"... The knowledge contained in the growing number of scientific digital publications, particularly over the internet creates new demands for intelligent retrieval mechanisms. One basic approach in support of such retrieval mechanisms is the generation of semantic annotation, based on ontologies describi ..."

Abstract
- Add to MetaCart

The knowledge contained in the growing number of scientific digital publications, particularly over the internet creates new demands for intelligent retrieval mechanisms. One basic approach in support of such retrieval mechanisms is the generation of semantic annotation, based on ontologies describing both the field and the structure of the texts themselves. Many current approaches use statistical methods similar to the ones employed by Google to find correlations within the texts. This approach neglects the additional information provided in the upper ontology used by the author. mArachna, however, is based on natural language processing techniques, taking advantage of characteristic linguistic structures defined by the language used in mathematical texts. It stores the extracted knowledge in a knowledge base, creating a low-level ontology of mathematics and mapping this ontology onto the structure of the knowledge base. The following article gives an overview over the concepts and technical implementation of the mArachna prototype. 1

### The Sources of Certainty in Computation and Formal Systems

, 1999

"... In his Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences, Rene Descartes sought \clear and certain knowledge of all that is useful in life." Almost three centuries later, in \The foundations of mathematics," David Hilbert tried to \recast mathematical deniti ..."

Abstract
- Add to MetaCart

In his Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences, Rene Descartes sought \clear and certain knowledge of all that is useful in life." Almost three centuries later, in \The foundations of mathematics," David Hilbert tried to \recast mathematical denitions and inferences in such a way that they are unshakable." Hilbert's program relied explicitly on formal systems (equivalently, computational systems) to provide certainty in mathematics. The concepts of computation and formal system were not dened in his time, but Descartes' method may be understood as seeking certainty in essentially the same way. In this article, I explain formal systems as concrete artifacts, and investigate the way in which they provide a high level of certainty| arguably the highest level achievable by rational discourse. The rich understanding of formal systems achieved by mathematical logic and computer science in this century illuminates the nature of programs,...

### Coordination of Mathematical Agents

, 2001

"... Mathematical Services . . . . . . . . . . . . . . . . . . . 10 2.3.2 Autonomy and Decentralization . . . . . . . . . . . . . . . . . . . 11 2.3.3 Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Distributed Articial Intelligence 12 3.1 Agent-Oriented Programming . . . . . ..."

Abstract
- Add to MetaCart

Mathematical Services . . . . . . . . . . . . . . . . . . . 10 2.3.2 Autonomy and Decentralization . . . . . . . . . . . . . . . . . . . 11 2.3.3 Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Distributed Articial Intelligence 12 3.1 Agent-Oriented Programming . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 The Knowledge Query and Manipulation Language . . . . . . . . . . . . 13 3.3 Coordination in Multi-Agent Systems . . . . . . . . . . . . . . . . . . . 13 4 Agent Technology for Distributed Mathematical Reasoning 15 4.1 MathWeb Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Communication between MathWeb agents . . . . . . . . . . . . . . . . . 18 4.2.1 Technical Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2.2 Characterization of Reasoning Capabilities . . . . . . . . . . . . 18 4.2.3 Context in Mathematical Communication . . . . . . . . . . . . . 19 4.3 Coordination of MathWeb Agents . . . . . . . . . . . . . . . . . . . . . 20 5 Summary and Work Plan 22 5.1 Work Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1

### The Sources of Certainty in Computation and Formal Systems

"... In his Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences, Rene Descartes sought \clear and certain knowledge of all that is useful in life." Almost three centuries later, in \The foundations of mathematics," David Hilbert tried to \recast mathematical deniti ..."

Abstract
- Add to MetaCart

In his Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences, Rene Descartes sought \clear and certain knowledge of all that is useful in life." Almost three centuries later, in \The foundations of mathematics," David Hilbert tried to \recast mathematical denitions and inferences in such a way that they are unshakable." Hilbert's program relied explicitly on formal systems (equivalently, computational systems) to provide certainty in mathematics. The concepts of computation and formal system were not dened in his time, but Descartes' method may be understood as seeking certainty in essentially the same way. In this article, I explain formal systems as concrete artifacts, and investigate the way in which they provide a high level of certainty| arguably the highest level achievable by rational discourse. The rich understanding of formal systems achieved by mathematical logic and computer science in this century illuminates the nature of programs,...

### The Mathematical Infinite as a Matter of Method

, 2010

"... Abstract. I address the historical emergence of the mathematical infinite, and how we are to take the infinite in and out of mathematics. The thesis is that the mathematical infinite in mathematics is a matter of method. The infinite, of course, is a large topic. At the outset, one can historically ..."

Abstract
- Add to MetaCart

Abstract. I address the historical emergence of the mathematical infinite, and how we are to take the infinite in and out of mathematics. The thesis is that the mathematical infinite in mathematics is a matter of method. The infinite, of course, is a large topic. At the outset, one can historically discern two overlapping clusters of concepts: (1) wholeness, completeness, universality, absoluteness. (2) endlessness, boundlessness, indivisibility, continuousness. The first, the metaphysical infinite, I shall set aside. It is the second, the mathematical infinite, that I will address. Furthermore, I will address mathematical infinite by considering its historical emergence in set theory and how we are to take it in and out of mathematics. Insofar as physics and, more broadly, science deals with the mathematical infinite through mathematical language and techniques, my remarks should be subsuming and consequent. The main underlying point is that how the mathematical infinite is approached, assimilated, and applied in mathematics is not a matter of “ontological commitment”, of coming to terms with whatever that might mean, but rather of epistemological articulation, of coming to terms through knowledge. The mathematical infinite in mathematics is a matter of method. How we deal with the specific individual issues involving the infinite turns on the narrative we present about how it fits into methodological mathematical frameworks established and being established. The first section discusses the mathematical infinite in historical context, and the second, set theory and the emergence of the mathematical infinite. The third section discusses the infinite in and out of mathematics, and how it is to be taken. §1. The Infinite in Mathematics What role does the infinite play in modern mathematics? In modern mathematics, infinite sets abound both in the workings of proofs and as subject matter in statements, and so do universal statements, often of ∀ ∃ “for all there exists” form, which are indicative of direct engagement with the infinite. In many ways the role of the infinite is importantly “second-order ” in the sense that Frege regarded number generally, in that the concepts of modern mathematics are understood as having infinite instances over a broad range. 1 But

### unknown title

"... Searching for proofs (and uncovering capacities of the mathematical mind) 1 ..."

Abstract
- Add to MetaCart

Searching for proofs (and uncovering capacities of the mathematical mind) 1

### Natural Language Processing of Mathematical Texts in mArachna

"... Abstract-mArachna is a technical framework designed for the extraction of mathematical knowledge from natural language texts. mArachna avoids the problems typically encountered in automated-reasoning based approaches through the use of natural language processing techniques taking advantage of the s ..."

Abstract
- Add to MetaCart

Abstract-mArachna is a technical framework designed for the extraction of mathematical knowledge from natural language texts. mArachna avoids the problems typically encountered in automated-reasoning based approaches through the use of natural language processing techniques taking advantage of the strict formalized language characterizing mathematical texts. Mathematical texts possess a strict internal structuring and can be separated into text elements (entities) such as definitions, theorems etc. These entities are the principal carriers of mathematical information. In addition, Entities show a characteristic coupling between the presented information and their internal linguistic structure, well suited for natural language processing techniques. Taking advantage of this structure, mArachna extracts mathematical relations from texts and integrates them into a knowledge base. Identifying sub

### Why Husserl should have been a strong revisionist in mathematics ∗

, 2000

"... Husserl repeatedly has claimed that (1) mathematics without a philosophical foundation is not a science but a mere technique; (2) philosophical considerations may lead to the rejection of parts of mathematical practice; but (3) they cannot lead to mathematical innovations. My thesis is that Husserl’ ..."

Abstract
- Add to MetaCart

Husserl repeatedly has claimed that (1) mathematics without a philosophical foundation is not a science but a mere technique; (2) philosophical considerations may lead to the rejection of parts of mathematical practice; but (3) they cannot lead to mathematical innovations. My thesis is that Husserl’s third claim is wrong, by his own standards. To explain this thesis, let me first introduce the term ‘revisionism’. It is understood here, following Crispin Wright, as the term that applies to ‘any philosophical standpoint which reserves the potential right to sanction or modify pure mathematical practice ’ [Wright 1980, p.117]. I want to make a distinction between weak and strong revisionism. The point of reference is the actual practice of mathematics. Weak revisionism then potentially sanctions a subset of this practice, while strong revisionism potentially not only limits but extends it, in different directions. In strong revisionism, certain combinations of limitation and extension may lead to a mathematics that is no longer compatible with the unrevised one. ‘May lead’, not ‘necessarily leads’: it is all a matter of reserving rights; whether there is occasion to exercise them is a further question. To illustrate these categories, let me give examples of non-revisionism, weak revisionism, and strong revisionism. Non-revisionism can be found in Wittgenstein’s Philosophische Untersuchungen, where philosophy can neither change nor ground mathematics: Die Philosophie darf den tatsächlichen Gebrauch der Sprache in keiner Weise antasten, sie kann ihn am Ende also nur beschreiben. Denn sie kann ihn auch nicht begründen. Sie läßt alles wie es ist. Sie läßt auch die Mathematik wie sie ist, und keine mathematische

### University Press — Draft (November 2012); please do not quote! Frege’s Relation to Dedekind: Basic Laws and Beyond *

"... Among all of Frege's contemporaries, the thinker who is probably closest to him in terms of their basic projects is Richard Dedekind. Like Frege, Dedekind attempted to reconstruct the theories of the natural and real numbers in purely logical terms. As in Frege's case, this involved using a general ..."

Abstract
- Add to MetaCart

Among all of Frege's contemporaries, the thinker who is probably closest to him in terms of their basic projects is Richard Dedekind. Like Frege, Dedekind attempted to reconstruct the theories of the natural and real numbers in purely logical terms. As in Frege's case, this involved using a general theory of classes, relations, and