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Conceptual Mathematics, Second Edition by Stephen H. Schanuel, F. William Lawvere

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SESSION 7

Isomorphisms and coordinates

1.  One use of isomorphisms: Coordinate systems

The idea of ‘subjective contained in the objective,’ or ‘familiar contained in the general,’ discussed in the last session, is especially simple if the ‘naming’ map is an isomorphism. That is, to have an isomorphism from a ‘known’ object A to an object X allows us to know X as well. To fit with the applications, let’s give the isomorphism and its inverse these names:

image

Here is an example. Imagine a geometrical line L, extending forever in both directions. It is often useful to choose an isomorphism from the set of real numbers to the line L. The usual way to ...

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