# Mathematics

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.

My own work, especially in collaboration with Kalle Mikkola and Teru Thomas, has mainly focussed on the last two.

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.

#### Related papers

McCarthy, D., Mikkola, K., Continuity and completeness of strongly independent preorders. *Mathematical Social Sciences*, 93 (2018): 141–145.

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 | AbstractMcCarthy, D., Mikkola, K., Thomas, T., Representation of strongly independent preorders by vector-valued functions. MPRA Paper No. 80806 (2017).

Online Article | PDF | AbstractMcCarthy, D., Mikkola, K., Thomas, T., Aggregation for general populations without continuity or completeness. MPRA Paper No. 80820 (2017).

Online Article | PDF | AbstractMcCarthy, D., Mikkola, K., Thomas, T., Utilitarianism with and without expected utility. MPRA Paper No. 90125 (2018).

Online Article | PDF | Abstract | Discussion
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