What Does “Existence” Mean in Mathematics?
Understanding what existence means in mathematics is the key to understanding what it means for concepts like “infinity” or “imaginary numbers” to exist–something that puzzles a lot of people when they first encounter these weird ideas! Mathematical objects do not exist in the same sense that a physical object exists; nobody has ever bumped their elbow on a number, for instance. Instead, mathematical objects are abstract concepts (often abstracted from a real world situation, by isolating just the part of the situation that is relevant for a particular discussion). When we ask whether or not a mathematical object exists, we must have in mind an appropriate context: a particular, precisely defined collection of concepts. Then we ask, “among these concepts, is there one which matches the object we are looking for?” If so, we say that the object exists; if not, it doesn’t exist. For example, the natural numbers (that is, the numbers 1, 2, 3, 4, and so on) are the concepts obtained by abs
