 To simplify the motivation, mathematics, like many another science, encounters anomalies, puzzles and paradoxes. Paradoxes typically appear in the form of contradictions where there are grounds for accepting incompatible sides of the contradiction. Mathematics progresses at least in part by eliminating anomalies in favour of consistent reconstructions. This has typically been the method of addressing anomalies, and it is facilitated by noticing that the contradictions emerge in the foundations of mathematics. But it was noticed in the later twentieth century that there is another way, namely accept the contradiction and develop mathematical theories containing both sides of the contradiction. This would be impossible if the contradictory theory was erected on a logical foundation containing the Boolean principle Ex Contradictione Quodlibet ECQ, from a contradiction everything follows. So ECQ has to be abandoned, but fortunetely that proves possible, indeed mathematically straightforward. What remains is a rich field, of novel mathematical applications interesting in their own right, which sidestep the vexing questions of which foundational principles to adopt, by developing contradictions in areas of mathematics such as number theory or analysis which are far from foundations. This is inconsistent mathematics.