Ethics subsumes many difficult problems that are discussed in formal epistemology, philosophy of probability, decision theory, and game theory. No one denies that mathematics is useful in those areas, so it would display a curious inability to connect the dots to dispute its usefulness in ethics.
There are many ways in which mathematics is invaluable. They include
- Thinking about the complex dynamical systems that social structures form.
- Modelling uncertainty in a way that helps us make better decisions.
- Formulating natural ethical ideas precisely enough to admit them as axioms.
- Working out the implications of sets of plausible axioms.
First, we have been contributing to the development of expected utility theory in ways that enable it to accommodate the popular view in philosophy that there are significant limitations to welfare comparisons.
Second, we have been studying how axioms for aggregation that are weak enough to allow for such limitations can nevertheless fully determine social welfare.
One of many examples of how subtle and counterintuitive the interaction of several apparently plausible axioms can be is discussed further here. Future work will incorporate wider and more realistic ways of representing uncertainty.