Analogies between analogies "A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies."
--Stefan Banach
See also here: Famous mathematical quotes
So, can someone give an example of an analogy between analogies?
 A: This question was posed here followed by some attempted answers.
A: 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.
A: Quoting Final dialgebras : from categories to allegories, the authors write that, according to Freyd and Scedrov:

Allegories are to binary relations between sets as categories are to
  functions between sets.

Assuming functions between sets can be regarded as analogies between sets, and assuming that relations between sets can also be regarded as analogies between sets, then the above quote is expressing, in a pretty direct way, an analogy between analogies.
A: I think the following is nice example. Classical Galois Theory gives an analogy between finite étale $k$-algebras and finite sets with an action of the absolute Galois group $\mathrm{Gal}(k)$ of $k$. Classical Theory of Covering Spaces gives an analogy between coverings of a nice topological space $X$ and sets with an action of the fundamental groups $\pi_1(X,x)$. Of course this two analogies can be made pretty precise with the a little use of category theory.
Now the analogy between this analogies is to call both analogies Galois correspondence and to compare $\pi_1(X)$ with $\mathrm{Gal}(k)$, to call certain coverings Galois covering, to compare the universal cover of $X$ with the seperable closure of $k$ and so on.
Tamás Szamuely's  beautiful book "Galois Groups and Fundamental Groups" gives a lot of details and much more about the connection of these things.
A: A functor is to an analogy as a natural transformation is to an analogy between analogies.
Does this count?  This is making my head spin a little bit...
A: It is unfortunate Jan Weidner asked his question -- "Can someone give an example of an analogy between analogies?" -- in such a way that he is unlikely to receive very many insightful responses of the kind offered by Joel David Hamkins and philip314. I don't know how insightful my own response will be, but I can provide a context for the topic of analogy, as I was the contributor who added the quotations by Stefan Banach and Stanislaw M. Ulam to the list of famous mathematical quotes.
It is true that Banach is the original source of the quotation, but I first encountered the topic in Ulam's memoir Adventures of a Mathematician (1976; 1991). From there I learned that the University of California Press published a collection of papers by Ulam titled Analogies between Analogies: The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators (1991). The collection includes the paper "On the Notion of Analogy and Complexity in Some Constructive Mathematical Schemata" (1981). Rather than attempting to summarize Ulam's results, I'll simply provide an excerpt from the paper's introduction so that interested readers can investigate the topic on their own:

Throughout the development of
  mathematics and with the growth of new
  concepts and more complicated notions,
  a cohesive tendency and organic
  structure have been guided by a
  feeling of analogy between the old and
  new ideas.
Historically, problems posed by the
  development of a new mathematical
  discipline, which originally was only
  metamathematical, coalesced into new
  parts of mathematics itself. One could
  cite, as obvious examples, the study
  of transformations of a space as
  points of a new space of such
  transformations, and the study of
  algorithms for solving equations as
  entities per se (group theory, for
  instance).
The increasing proliferation of
  notions in pure mathematics may
  suggest that the idea of analogy
  itself is amenable to mathematical
  discussion. One finds that old and
  elementary formulations of this idea
  are, in special cases, present in the
  definitions of the similarity of
  geometrical figures, more generally in
  the equivalence of figures-sets,
  through the elements of a group of
  transformations, or, more generally
  yet, through the identity of proximity
  of such sets in spaces which encompass
  them.
Two abstract sets of elements may be
  felt to be "analogous" if the
  difference between their cardinalities
  (in the finite case) is small compared
  to the cardinalities themselves. Two
  classes of such sets may be deemed to
  be analogous if the numbers of sets in
  the two classes differ by "little" and
  if the cardinalities of the
  corresponding sets also do not differ
  by much. Obviously, one needs to
  attempt to formulate a quantitative
  criterion, and it is clear a priori
  that the notion of analogy will not
  be, in general, transitive.
In this report we merely want to
  discuss some of the salient features
  of analogy and exhibit them on a class
  of examples where we shall attempt to
  define it, at first, as proximity in
  the sense of a metric distance in
  suitably defined spaces. (Ulam 1991; 514)

Hopefully, Ulam's paper will provide some tools to make the topic of analogy less "far out."
