In this article we propose the following definition for advanced mathematical thinking: Thinking that requires deductive and rigorous reasoning about mathematical notions that are not entirely accessible to us through our five senses. We argue that this definition is not necessarily tied to a particular kind of educational experience; nor is it tied to a particular level of mathematics. We also give examples to illustrate the distinction we make between advanced mathematical thinking and elementary mathematical thinking. In particular, we discuss which kind of thinking may be required depending on the size of a mathematical problem, including problems involving infinity, and the types of models that are available.