Infinite numbers have long posed a fundamental challenge for utilitarian ethicists. If the universe is boundless, it contains both infinite positive utility and infinite negative utility — making it impossible to define our current state, let alone meaningfully affect it through action. This problem, known as "infinite ethics," has been a persistent sticking point in moral philosophy, discussed by thinkers including Oxford's Nick Bostrom and Peter Singer, because if standard arithmetic renders every action morally meaningless under infinite stakes, the entire utilitarian framework that guides much of effective altruism and policy analysis falters.

Oxford philosopher Toby Ord proposes a potential resolution: employ hyperreal numbers, a mathematical construct in which infinite sums behave more like finite ones. Hyperreals, originally developed by mathematician Abraham Robinson in the 1960s through nonstandard analysis, extend the real number system to include infinitely large and infinitely small quantities while preserving familiar arithmetic rules. Under this framework, for example, the sum of infinitely many twos is twice as large as the sum of infinitely many ones — a property that standard transfinite arithmetic does not share.

As with many philosophical innovations, this approach resolves some existing problems while introducing new ones. Pinpointing certain hyperreals requires selecting what mathematicians call an "ultrafilter," but this choice is underdetermined, meaning multiple valid selections exist without a clear criterion for preferring one over another.

Nonetheless, Elias Schmied describes Ord's contribution as a genuine philosophical advance — the kind that emerges only rarely and makes tangible progress on a problem that previously seemed intractable. The stakes extend beyond academic curiosity: Ord, best known for his work on existential risk and author of The Precipice, is part of a broader community of philosophers and researchers in effective altruism and AI safety for whom resolving infinite ethics could provide firmer foundations for prioritizing actions that affect the long-term future.

The author of the post notes surprise that a framework for infinities in which they behave logically already exists, raising the question of why this approach was not adopted from the start, rather than the conventional teaching that infinity plus infinity equals infinity and should simply be accepted as such. An AI assistant, Claude, was consulted to help address these questions.

The content is cited from Scott Alexander, whose original post contains additional links, many of them related to AI safety.

Source: Marginal Revolution