I am looking for any material dealing with the evolution of what now are standard mathematical definitions. One example what I am looking for:
Let $(a_n)_{n\in\mathbb N}$ be a sequence with $a_n\in\mathbb R$ for all $n\in\mathbb N$. We call $a\in\mathbb R$ the limit of the sequence $(a_n)_{n\in\mathbb N}$ if the following holds:
for every $\varepsilon>0$ there exists $N\in\mathbb N$ such that, for every natural number $n\geq N$, we have $|a_n-a|<\varepsilon$.
This definition of the limit of a sequence is one of the basic definitions and the concept of sequences one of the foundations of modern analysis, thus every student of mathematics gets to work with this quite early when studiyng at college/university. And of course this definition makes sense and is useful and describes excactly what we want to have.
But how did this definition develop? And is it neccesary to write it using all these $\varepsilon$s and $n$s or $N$s which can be quite confusing? Why not use the easier definition which defines the limit as the number our sequence gets closer and closer to as we increase $n$?
I often discuss these questions or similar questions like this with students which often leads to them not only learning the definition but really understanding what they are reading and doing there (at least that's what I hope).
As for the definition of the limit, I usually ask (using the "easy definition") what it means for a sequence to get closer to a number following up with the statement that by this definition we have $\lim\limits_{n\to\infty} \frac{1}{n}=-42$, as $\frac{1}{n}$ gets closer to $-42$ when we increase $n$. This usually motivates a precise definition of "getting closer" using the absolute value $|a_n-a|$ while the "getting closer and closer part" takes care of the introduction of $\varepsilon$ as we want to make $|a_n-a|$ really small.
I hope I made clear what I'm looking for. Any book, link, article, answers dealing with that are appreciated.