III - ALTERNATIVE LINES
Summary
We turn to alternative real lines, number systems that contain the real numbers and have the same algebraic and geometric properties as the reals. If these systems are so similar, same algebra, same geometry, and even, to a certain extent, the same numbers, then the reader may well ask, why bother with them?
These systems are interesting not so much because they contain different numbers. They embody different ideas of number, radically different philosophies of mathematics. They have conflicting visions about what should be allowed to be a number, what properties are possible for numbers, and what tools should be permitted to prove those properties.
The three systems we present portray a kind of political spectrum of mathematical philosophies, from the radical right (the constructive reals), through moderately liberal (the hyperreals), to the radical left (the surreals). Each embodies a distinctive vision of what numbers are, how to calculate with them, and how to prove theorems about them.
- Type
- Chapter
- Information
- Which Numbers are Real? , pp. 95 - 96Publisher: Mathematical Association of AmericaPrint publication year: 2012