Large Language Models (LLMs) are often misleadingly recognized as having a personality or a set of values. We argue that an LLM can be seen as a superposition of perspectives with different values and personality traits. LLMs exhibit context-dependent values and personality traits that change based on the induced perspective (as opposed to humans, who tend to have more coherent values and personality traits across contexts). We introduce the concept of perspective controllability, which refers to a model's affordance to adopt various perspectives with differing values and personality traits. In our experiments, we use questionnaires from psychology (PVQ, VSM, IPIP) to study how exhibited values and personality traits change based on different perspectives. Through qualitative experiments, we show that LLMs express different values when those are (implicitly or explicitly) implied in the prompt, and that LLMs express different values even when those are not obviously implied (demonstrating their context-dependent nature). We then conduct quantitative experiments to study the controllability of different models (GPT-4, GPT-3.5, OpenAssistant, StableVicuna, StableLM), the effectiveness of various methods for inducing perspectives, and the smoothness of the models' drivability. We conclude by examining the broader implications of our work and outline a variety of associated scientific questions. The project website is available at https://sites.google.com/view/llm-superpositions .