Measure of Morality: A Mathematical Theory of Egalitarian Ethics

This paper develops a rigorous mathematical framework for egalitarian ethics by integrating formal tools from economics and mathematics. We motivate the formalism by investigating the limitations of conventional informal approaches by constructing examples such as probabilistic variant of the trolley dilemma and comparisons of unequal distributions. Our formal model, based on canonical welfare economics, simultaneously accounts for total utility and the distribution of outcomes. The analysis reveals deficiencies in traditional statistical measures and establishes impossibility theorems for rank-weighted approaches. We derive representation theorems that axiomatize key inequality measures including the Gini coefficient and a generalized Atkinson index, providing a coherent, axiomatic foundation for normative philosophy.