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Foreword

Published online by Cambridge University Press:  06 July 2010

F. William Lawvere
Affiliation:
State University of New York, Buffalo
Robert Rosebrugh
Affiliation:
Mount Allison University, Canada
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Summary

Why Sets for Mathematics?

This book is for students who are beginning the study of advanced mathematical subjects such as algebra, geometry, analysis, or combinatorics. A useful foundation for these subjects will be achieved by openly bringing out and studying what they have in common.

A significant part of what is common to all these subjects was made explicit 100 years ago by Richard Dedekind and Georg Cantor, and another significant part 50 years ago by Samuel Eilenberg and Saunders Mac Lane. The resulting idea of categories of sets is the main content of this book. It is worth the effort to study this idea because it provides a unified guide to approaching constructions and problems in the science of space and quantity.

More specifically, it has become standard practice to represent an object of mathematical interest (for example a surface in three-dimensional space) as a “structure.” This representation is possible by means of the following two steps:

  1. (1) First we deplete the object of nearly all content. We could think of an idealized computer memory bank that has been erased, leaving only the pure locations (that could be filled with any new data that are relevant). The bag of pure points resulting from this process was called by Cantor a Kardinalzahl, but we will usually refer to it as an abstract set.

  2. (2) Then, just as computers can be wired up in specific ways, suitable specific mappings between these structureless sets will constitute a structure that reflects the complicated content of a mathematical object. For example, the midpoint operation in Euclidean geometry is represented as a mapping whose “value” at any pair of points is a special third point.

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Sets for Mathematics , pp. ix - xii
Publisher: Cambridge University Press
Print publication year: 2003

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  • Foreword
  • F. William Lawvere, State University of New York, Buffalo, Robert Rosebrugh, Mount Allison University, Canada
  • Book: Sets for Mathematics
  • Online publication: 06 July 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511755460.001
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  • Foreword
  • F. William Lawvere, State University of New York, Buffalo, Robert Rosebrugh, Mount Allison University, Canada
  • Book: Sets for Mathematics
  • Online publication: 06 July 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511755460.001
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Foreword
  • F. William Lawvere, State University of New York, Buffalo, Robert Rosebrugh, Mount Allison University, Canada
  • Book: Sets for Mathematics
  • Online publication: 06 July 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511755460.001
Available formats
×