The value of brain-inspired neuromorphic computers critically depends on our ability to program them for relevant tasks. Currently, neuromorphic hardware often relies on machine learning methods adapted from deep learning. However, neuromorphic computers have potential far beyond deep learning if we can only harness their energy efficiency and full computational power. Neuromorphic programming will necessarily be different from conventional programming, requiring a paradigm shift in how we think about programming. This paper presents a conceptual analysis of programming within the context of neuromorphic computing, challenging conventional paradigms and proposing a framework that aligns more closely with the physical intricacies of these systems. Our analysis revolves around five characteristics that are fundamental to neuromorphic programming and provides a basis for comparison to contemporary programming methods and languages. By studying past approaches, we contribute a framework that advocates for underutilized techniques and calls for richer abstractions to effectively instrument the new hardware class.