[In front of a blackboard, in an office at Real College]
Skeptic: And why should I care about holomorphic functions?
Holomorphic enthusiast:$\;$ Can you compute $\quad$ $\sum_{n={-\infty}}^{\infty} \frac{1}{(a+n)^2}$ ? Here $a$ is one of your cherished real numbers, but not an integer.
Skeptic: Well, hm...
Holomorphic enthusiast, nonchalantly: Oh, you just get
$$\sum_{n={-\infty}}^{\infty} \frac{1}{(a+n)^2}=\pi^2 cosec^2 \pi a $$
It's easy using residues.
Skeptic: Well, maybe I should have a look at these "residues".
Holomorphic enthusiast (generously): Let me lend you this introduction to Complex Analysis by Remmert, this one by Lang and this oldie by Titchmarsh. As Hadamard said: "Le plus court chemin entre deux vérités dans le domaine réel passe par le domaine complexe".You can look for a translation at Mathoverflow. They have a nice list of mathematical quotations, following a question there.
Skeptic: Mathoverflow ??
Holomorphic enthusiast (looking a bit depressed) : I think we should have a nice long walk together now.
[Exeunt]
You are comparing apples and organges. Model theory should be compared with categorical logic, not category theory. Conversely, category theory should be compared with algebra, not model theory.
Model theory is the study of set-theoretic models of theories expressed in first-order classical logic. As such it is a particular branch of categorical logic, which is the study of models of theories, without insistence on set theory, first order, or classical reasoning.
Best Answer
Let me give a very concrete analogy between analogies.
1:2 :: 2:4
is to
p:q :: kp:kq
as
x2+2x+1 : 0 :: x : -1
is to
ax2+bx+c : 0 :: x : (-b +- sqrt(b2-4ac))/2a.
And this analogy is called generalization.