Artificial intelligence (AI) is being applied in almost every field. At the same time, the currently dominant deep learning methods are fundamentally black-box systems that lack explanations for their inferences, significantly limiting their trustworthiness and adoption. Explainable AI (XAI) methods aim to overcome this challenge by providing explanations of the models' decision process. Such methods are often proposed and developed by engineers and scientists with a predominantly technical background and incorporate their assumptions about the existence, validity, and explanatory utility of different conceivable explanatory mechanisms. However, the basic concept of an explanation -- what it is, whether we can know it, whether it is absolute or relative -- is far from trivial and has been the subject of deep philosophical debate for millennia. As we point out here, the assumptions incorporated into different XAI methods are not harmless and have important consequences for the validity and interpretation of AI explanations in different domains. We investigate ontological and epistemological assumptions in explainability methods when they are applied to AI systems, meaning the assumptions we make about the existence of explanations and our ability to gain knowledge about those explanations. Our analysis shows how seemingly small technical changes to an XAI method may correspond to important differences in the underlying assumptions about explanations. We furthermore highlight the risks of ignoring the underlying onto-epistemological paradigm when choosing an XAI method for a given application, and we discuss how to select and adapt appropriate XAI methods for different domains of application.