Numbers and Geometry

A. Gardiner
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A. Gardiner: University of Birmingham, Department of Mathematics

Chapter Chapter III.1 in Infinite Processes, 1982, pp 157-161 from Springer

Abstract: Abstract In Parts I and II we have gone out of our way to stress the enormous difference between the finite procedures of ordinary arithmetic, and those mathematical concepts whose very meaning depends on the introduction and interpretation of infinite processes. In contrast, you have in the past been encouraged to use real numbers (whether rational or irrational) in a naive, unquestioning way—especially in geometry: for example, you have been quietly encouraged to assume that, if we measure the length of a line segment AB in terms of some given unit segment CD, then its length AB/CD can obviously be expressed as a real number. While this is obvious when CD fits into AB a whole number of times leaving no remainder, or when CD and AB have some common measure MN which fits into CD precisely b times with no remainder and into AB precisely a times with no remainder (in which case AB/CD = a/b), it is not at all obvious in general. In Chapter 11.13 we saw one way of justifying the belief that AB can always be measured in terms of CD, but it was not exactly obvious!

Date: 1982
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DOI: 10.1007/978-1-4612-5654-0_16

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