Defining “dangerous anthropogenic interference with the climate system” in the context of Article 2 of the UN Framework Convention on Climate Change (UNFCCC) presents a complex challenge for those developing long-term climate policy. Natural science has a key role to play in quantifying vulnerabilities of elements of the Earth system and estimating the risks from a changing climate. But attempts to interpret Article 2 will inevitably draw on understanding from social science, psychology, law, and ethics. Here I consider the limits of science in defining climate “danger” by focusing on the potential disintegration of the major ice sheets as an example of an extreme impact. I show that considerations of timescale, uncertainty, and learning preclude a definition of danger drawn purely from natural science. Decisionmakers will be particularly challenged by one characteristic of global problems: answers to some scientific questions become less accurate over decadal timescales, meandering toward the wrong answer, a feature I call negative learning. I argue for a precautionary approach to Article 2 that would be based initially on current, limited scientific understanding of the future of the ice sheets.