function fields are not the only ones. the most important ones are the p-adic ones. and they have a immensely rich analysis.
So we know that not all fields obey the Archimedean property. For instance, the field of all rational functions with real coefficients does not. But is there any utility in doing analysis in these fields?
Also suppose we have a positive cut and a negative cut . Then we should (and do) get a negative cut. But why do we write it as follows: ?