Evolution of Definitions

Question

I try to understand how the definitions of mathematics have evolved (or formulated)... I'll use the epsilon-delta continuity definition as an example to ask my question... It may seem trivial, but it's important for me to understand the matter of definitions in math. Ok. It's clear that, if we look at an continuous curve, we "intuitively" say, we can draw it in one continuous stroke... And a little thought gives us the epsilon-delta definition. I mean, we see that our curve apparently satisfies the epsilon-delta definition. Now my core question: Once we fixed the definition of continuity, how can we sure that, in the future we won't come across a weird function, which satisfies our def. of continuity but doesn't fit into our intuitive continuity expectation? Is it a matter of natural selection in math history? I mean, let's say at first we formulate a rigorous definition for a geometric phenomenon and then, if we see some weird and unintuitive example, we change our definition to exclude that example? Is that the way in which the definitions evolve throughout math history? If so, can you give some examples of definitions that had been formulated for some geometric (or maybe abstract concept) but didn't work later on?

Answer 1

Your assumption seems to be that mathematicians start with a definition and then investigate it to determine whether it fits their intuitive understanding. It is arguable whether, historically speaking, this assumption is justified. Rather, throughout the history of analysis and until about 1870 mathematicians often (though not always) thought that they drew their concepts from reality and proceeded to manipulate them according to their understanding of how these concepts should behave. The idea of formalisation in terms of definitions was a fairly late idea.

More specifically, in the case of the notion of continuity that you are interested in, Cauchy introduced it in 1821 unusually by giving a definition, by saying that a function is continuous if every infinitesimal increment assigned to the variable always produces an infinitesimal change in the function. In Cauchy's notation and terminology, if $\alpha$ is infinitely small then $f(x+\alpha)-f(x)$ must also be infinitely small.

Your write that "a little thought gives us the epsilon-delta definition" but this in fact took about 50 years, with Weierstrass developing such definitions around 1870.