The inconsistency of mathematics and the mathematics of inconsistency

No one will dispute, looking at the history of mathematics, that there are
plenty of moments where mathematics is "in trouble", when paradoxes and inconsistencies
crop up and anomalies multiply. This need not lead, however, to the view that
mathematics is intrinsically inconsistent, as it is compatible with the viewthat these are
just transient moments. Once the problems are resolved, consistency (in some sense
or other) is restored. Even when one accepts this view, what remains is the question
what mathematicians do during such a transient moment? This requires some method
or other to reason with inconsistencies. But there is more: what if one accepts the
view that mathematics is always in a phase of transience? In short, that mathematics
is basically inconsistent? Do we then not need a mathematics of inconsistency? This
paper wants to explore these issues, using classic examples such as infinitesimals,
complex numbers, and infinity.

Publication year:2014

Keywords:philosophy, mathematics, inconsistency