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Problems in the foundations of mathematics have caused major upheavals in the subject from time to time. In this programme one of the special problems associated with the nature of 'number' is inve...stigated in detail. The problem is posed in a present day form by Graham Flegg, Reader in Mathematics at the Open University. Its historical setting and its solution in Greek times and then in the 19th century is discussed by David Fowler, Manager of the Research Centre, the Mathematics Institute, University of Warwick. Greek Mathematics was extensively based on the counting numbers and fractions, all of which can be represented by the length of a line, or the ratio between two lengths. The discovery that the length of the diagonal of a square with unit sides (that is Root Two) does not correspond to any such number, led to philosophical problems in Greek Mathematics. These problems were eventually circumvented by Eudoxus whose work is recorded in Euclid's Elements. But his definition is very subtle. In the 17th and 18th centuries The problem of irrational numbers undermined the rigorous development of the Calculus, but with the work of Richard Dedekind in the 19th century the problem was again solved.
Metadata describing this Open University video programme
Module code and title: AM289, History of mathematics
Item code: AM289; 02
First transmission date: 1975
Published: 1975
Rights Statement:
Restrictions on use:
Duration: 00:24:30
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Producer: Jean Nunn
Contributors: Graham Flegg; David Fowler
Publisher: BBC Open University
Keyword(s): Euclidean geometry; Incommensurables; Real numbers; Richard Dedekind
Footage description: Graham Flegg introduces the programme with a discussion on the problem of measuring a line. Several ruler animations illustrate the possible difficulties. Graham Flegg demonstrates that no ruler, no matter how finely calibrated, can measure the diagonal of a unit square. Graham Flegg and then David Fowler describe the crisis in Greek mathematics brought on by the discovery of incommensurables. Several animations are used. David Fowler and Graham Flegg look at the evidence in Euclid's Book 5 which shows how Eudoxus circumvented the problem of incommensurability. Shots of pages from Euclid's book and animations used as visual aids. Graham Flegg discusses the crisis in mathematics in the 18th century brought on by the demand to put the differential calculus on a firm logical foundation. Flegg and David Fowler discuss another fault of this crisis, the attempt to find a rigorous foundation for the principles of infinitesimal analysis. The method of Dedekind to define real numbers, which include irrational numbers is explained. Several animations are used to illustrate points in the discussion. Fowler sums up.
Master spool number: 6HT/71507
Production number: 00525_3164
Videofinder number: 936
Available to public: no