ET Math: How Different Could It Be?

We like to think that intelligent aliens would have the same basic ideas about numbers and geometry as us, but, even if they do, they might express those ideas very differently. To illustrate what different forms a concept can take, I will show how differently the law ab=ba has been interpreted at different times in human mathematical culture. This seemingly basic law has several different origins -- in geometry, number theory, and set theory -- some of which seem alien even to experienced mathematicians.