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.
McCarthy, D., Mikkola, K., Continuity and completeness of strongly independent preorders. Mathematical Social Sciences, 93 (2018): 141–145.PDF | Abstract
McCarthy, D., Mikkola, K. and Thomas, T., Representation of strongly independent preorders by sets of scalar-valued functions. MPRA Paper No. 79284 (2017).Online Article | PDF | Abstract
McCarthy, D., Mikkola, K., Thomas, T., Representation of strongly independent preorders by vector-valued functions. MPRA Paper No. 80806 (2017).Online Article | PDF | Abstract
McCarthy, D., Mikkola, K., Thomas, T., Aggregation for general populations without continuity or completeness. MPRA Paper No. 80820 (2017).Online Article | PDF | Abstract
We generalize Harsanyi’s social aggregation theorem. We allow the population to be infinite, and merely assume that individual and social preferences are given by strongly independent preorders on a convex set of arbitrary dimension. Thus we assume neither completeness nor any form of continuity. Under Pareto indifference, the conclusion of Harsanyi’s theorem nevertheless holds almost entirely unchanged when utility values are taken to be vectors in a product of lexicographic function spaces. The addition of weak or strong Pareto has essentially the same implications in the general case as it does in Harsanyi’s original setting.
McCarthy, D., Mikkola, K., Thomas, T., Utilitarianism with and without expected utility. MPRA Paper No. 90125 (2018).Online Article | PDF | Abstract