44,437 research outputs found
Did Lobachevsky Have A Model Of His "imaginary Geometry"?
The invention of non-Euclidean geometries is often seen through the optics of
Hilbertian formal axiomatic method developed later in the 19th century. However
such an anachronistic approach fails to provide a sound reading of
Lobachevsky's geometrical works. Although the modern notion of model of a given
theory has a counterpart in Lobachevsky's writings its role in Lobachevsky's
geometrical theory turns to be very unusual. Lobachevsky doesn't consider
various models of Hyperbolic geometry, as the modern reader would expect, but
uses a non-standard model of Euclidean plane (as a particular surface in the
Hyperbolic 3-space). In this paper I consider this Lobachevsky's construction,
and show how it can be better analyzed within an alternative non-Hilbertian
foundational framework, which relates the history of geometry of the 19th
century to some recent developments in the field.Comment: 31 pages, 8 figure
From Euclidean Geometry to Knots and Nets
This document is the Accepted Manuscript of an article accepted for publication in Synthese. Under embargo until 19 September 2018. The final publication is available at Springer via https://doi.org/10.1007/s11229-017-1558-x.This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if (a) it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, (b) the information thus displayed is not metrical and (c) it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions.Peer reviewe
Doing and Showing
The persisting gap between the formal and the informal mathematics is due to
an inadequate notion of mathematical theory behind the current formalization
techniques. I mean the (informal) notion of axiomatic theory according to which
a mathematical theory consists of a set of axioms and further theorems deduced
from these axioms according to certain rules of logical inference. Thus the
usual notion of axiomatic method is inadequate and needs a replacement.Comment: 54 pages, 2 figure
Geometry in the Transition from Primary to Post-Primary
This article is intended as a kind of precursor to the document Geometry for
Post-primary School Mathematics, part of the Mathematics Syllabus for Junior
Certicate issued by the Irish National Council for Curriculum and Assessment in
the context of Project Maths.
Our purpose is to place that document in the context of an overview of plane
geometry, touching on several important pedagogical and historical aspects, in
the hope that this will prove useful for teachers.Comment: 19 page
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