"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?
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.
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 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...
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."
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.
