The problem of human trust in artificial intelligence is one of the most fundamental problems in applied machine learning. Our processes for evaluating AI trustworthiness have substantial ramifications for ML's impact on science, health, and humanity, yet confusion surrounds foundational concepts. What does it mean to trust an AI, and how do humans assess AI trustworthiness? What are the mechanisms for building trustworthy AI? And what is the role of interpretable ML in trust? Here, we draw from statistical learning theory and sociological lenses on human-automation trust to motivate an AI-as-tool framework, which distinguishes human-AI trust from human-AI-human trust. Evaluating an AI's contractual trustworthiness involves predicting future model behavior using behavior certificates (BCs) that aggregate behavioral evidence from diverse sources including empirical out-of-distribution and out-of-task evaluation and theoretical proofs linking model architecture to behavior. We clarify the role of interpretability in trust with a ladder of model access. Interpretability (level 3) is not necessary or even sufficient for trust, while the ability to run a black-box model at-will (level 2) is necessary and sufficient. While interpretability can offer benefits for trust, it can also incur costs. We clarify ways interpretability can contribute to trust, while questioning the perceived centrality of interpretability to trust in popular discourse. How can we empower people with tools to evaluate trust? Instead of trying to understand how a model works, we argue for understanding how a model behaves. Instead of opening up black boxes, we should create more behavior certificates that are more correct, relevant, and understandable. We discuss how to build trusted and trustworthy AI responsibly.