A note on fields of definition

2019

© Andrée C. Ehresmann et les auteurs, 1988, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

2009

Abstract. One of the fundamental and seemingly simple aims of math-ematical knowledge management (MKM) is to develop and standardize formats that allow to “represent the meaning of the objects of mathemat-ics”. The open formats OpenMath and MathML address this, but differ subtly in syntax, rigor, and structural viewpoints (notably over calculus). To avoid fragmentation and smooth out interoperability obstacles, effort is under way to align them into a joint format OpenMath/MathML 3. We illustrate the issues that come up in such an alignment by looking at three main areas: bound variables and conditions, calculus (which relates to the previous) and “lifted ” n-ary operators. Whenever anyone says “you know what I mean”, you can be pretty sure that he does not know what he means, for if he did, he would tell you. — H. Davenport (1907–1969) 1

Proceedings of the 28th Conference of the …, 2004

The definition-construction process is central to mathematics. The aim of this paper is to propose a few Situations of Definition-Construction (called SDC) and to study them. Our main objectives are to describe the definition-construction process and to design SDC for classroom. A SDC on "discrete straight line" and its mathematical and didactical analysis (with students' productions) will be presented too.

Cahiers de Topologie et Géométrie Différentielle Catégoriques, 1986

© Andrée C. Ehresmann et les auteurs, 1986, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

2016

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

1998

This paper studies the cognitive demands made on students encountering the systematic development of a formal theory for the first time. We focus on the meaning and usage of definitions and whether they are "operable" for the individual in the sense that the student can focus on the properties required to make appropriate logical deductions in proofs. By interviewing students

Mathematics in Computer Science, 2008

The Philosophy of Mathematical Practice, 2008

These are some of the rules of classification and definition. But although nothing is more important in science than classifying and defining well, we need say no more about it here, because it depends much more on our knowledge of the subject matter being discussed than on the rules of logic. (Arnauld and Nicole (1683/1996) p.128) I Definition and Mathematical Practice The basic observation structuring this survey is that mathematicians often set finding the "right" / "proper" / "correct" / "natural" definition as a research objective, and success-finding "the proper" definition-can be counted as a significant advance in knowledge. Remarks like these from a retrospective on twentieth century algebraic geometry are common:

2007

European Journal for Philosophy of Science, vol. 8 (2018), pp. 605-629., 2018

In the past century the received view of definition in mathematics has been the stipulative conception, according to which a definition merely stipulates the meaning of a term in other terms which are supposed to be already well known. The stipulative conception has been so absolutely dominant and accepted as unproblematic that the nature of definition has not been much discussed, yet it is inadequate. This paper examines its shortcomings and proposes an alternative, the heuristic conception.

Lecture Notes in Computer Science, 2008

Formalizing mathematical argument is a fascinating activity in itself and (we hope!) also bears important practical applications. While traditional proof theory investigates deducibility of an individual statement from a collection of premises, a mathematical proof, with its structure and continuity, can hardly be presented as a single sequent or a set of logical formulas. What is called "mathematical text", as used in mathematical practice through the ages, seems to be more appropriate. However, no commonly adopted formal notion of mathematical text has emerged so far.

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1979

This article treats three aspects of Frege's discussions of definitions. First, I survey Frege's main criticisms of definitions in mathematics. Second, I consider Frege's apparent change of mind on the legitimacy of contextual definitions and its significance for recent neo-Fregean logicism. In the remainder of the article I discuss a critical question about the definitions on which Frege's proofs of the laws of arithmetic depend: do the logical structures of the definientia reflect the understanding of arithmetical terms prevailing prior to Frege's analyses? Unless they do, it is unclear how Frege's proofs demonstrate the analyticity of the arithmetic in use before logicism. Yet, especially in late writings, Frege characterizes definitions as arbitrary stipulations of the senses or references of expressions unrelated to predefinitional understanding. I conclude by examining some options for conceiving of the status of Frege's logicism in light of this apparent tension, and outline a suggestion for a philosophically fruitful way of resolving this tension.

Educational Studies in Mathematics, 2006

The definition of 'definition' cannot be taken for granted. The problem has been treated from various angles in different journals. Among other questions raised on the subject we find: the notions of concept definition and concept image, conceptions of mathematical definitions, redefinitions, and from a more axiomatic point of view, how to construct definitions. This paper will deal with 'definition construction processes' and aims more specifically at proposing a new approach to the study of the formation of mathematical concepts. I shall demonstrate that the study of the defining and concept formation processes demands the setting up of a general theoretical framework. I shall propose such a tool characterizing classical points of view of mathematical definitions as well as analyzing the dialectic involving definition construction and concept formation. In that perspective, a didactical exemplification will also be presented.

… International Conference, MKM 2003, Bertinoro, Italy, …, 2003