| Let me give an example from calculus - a continuous function. How do you define the concept? The definiton of the concept changed quite a lot in the past 250 years or so. (Please take the following explanation with a grain of salt, I am not writing a thesis on the topic, just pointing out stuff.) For Euler, continuous function was pretty much intuitive notion. He only composed functions with only occasional point discontinuities, so it wasn't a big deal for him to even not have a proper definition. Then people like Bolzano and Dirichlet came along and realized they need a better definition, because there can be some really weird cases. So they formalized the continuity with limits (which is typical way how to define it in basic calculus). Later yet, people understood better what it means to be a real number by looking at notions such as countability and measurability. While this doesn't affect continuity itself, it does affect understanding of what is a real fuction. Then came more abstraction, to metric spaces and eventually topological spaces, which redefined "continuous function" yet again as a morphism between topological spaces. Another shift in thinking about continuity happened when theory of distributions was invented. This actually completely reverses the intuition - instead of properly definining reals and then on top of real function define what it means to be continuous, you define the "function" itself in an entirely different way, in which the continuity becomes somewhat irrelevant. Finally, modern mathematics is quite obsessed with category theory and various ways to make everything into some algebra. In a way, we care less what reals really are, only what we can do with them (or their sets). So I think to understand the intuitive relation of all these different definitions, you need to understand a little bit of history. |