Expert, Intuitive, Organizing Analogies

Question

In learning a new area it is very helpful to have high-level intuitive analogies that keep track of the various parts of an important argument or strategy. Experts have a store of such things, and often the only way to hear about them is to talk with the experts or hear the intuition during their talks.

I am talking about very intuitive analogies with the property that one side of the analogy can be understood almost completely by the layperson, but after the "mapping" to the mathematics on the other side, the analogy outlines the argument or strategy reasonably well, of course missing most key details… . The point is to keep in mind a rough outline for the purposes of navigating the technical literature!

For example, in the theory of finite von Neumann algebras, Sorin Popa and his collaborators have made excellent use of his "deformation/rigidity strategy", which Sorin has described in at least one of his talks using the following physical analogy:

Consider a bucket of dark liquid in which you know there is a hard stone. If you put your hands in the water and swish them around and never feel the stone, then you know the stone must be located where you have not swished your hands.

In this picture, the bucket of liquid including the stone is the finite von Neumann algebra $M$. The "liquid part" of the von Neumann algebra can be "stirred" by a pointwise 2-norm deformation of the identity by normal, unital, trace-preserving completely positive maps relative to some subalgebra $A$ which was not "stirred". The hard stone can be, e.g. a subalgebra $B$ with relative property (T), since trying to pointwise "stir" the unit ball of such an algebra by maps of the above sort is not possible without moving the ball uniformly. If we can deform $M$ around $A$ and we know $M$ has a property (T) subalgebra $B$, we can conclude that $B$ must have been contained in $A$ (up to something like unitary conjugacy).

Anyone in the field can see the lies I've told in the above paragraph, but nevertheless the intuitive picture does a reasonably good job of communicating the parts of the strategy. In fact, you can immediately see the main limitations of the technique by asking what happens if (a) there is no stone in the liquid (or the stone is not large), e.g. as in a free group factor and (b) the whole algebra is a stone, i.e. the factor itself has property (T).

Question: What are your favorite such expert intuitive analogies for important parts of your subject? Please include the analogy and explanation of the "details", as I did above.

As in the example I include above, an analogy presented as an answer should include a reasonably complete explanation of the details on the "technical side". The best answers are those which encode and organize surprisingly many technical details in the intuitive analogy, and are not just intuitive mnemonics for remembering the existence of some theorem or other.

EDIT: The following are some helpful modifications to the question suggested by Aaron Myerowitz.

The question starts out with the claim (which is certainly unsupported): "Experts have a store of high-level intuitive analogies that keep track of the various parts of an important argument or strategy."

Question: Is this claim valid? Is it common for experts to have a store of such analogies? Is this more common in some fields than in others?

To loosen the very strict requirement of metaphor suitable (on one side) to the layperson, we may ask instead:

Question: What are some metaphors used to convey the gist of a topic or technique to people not in one's field?

In the latter question we'd like to focus on topics that are not necessarily part of the tool kit of "most" experts, and want to steer clear of descriptions and analogies used to communicate with fellow experts.

Answer 1

The analogy between proofs and games. This analogy is so strong that it can be formalized in two mathematical ways.

In (finite) model theory, it gives Ehrenfeucht–Fraïssé games. Logical truth becomes the existence of a winning strategy. This technique underlies many proofs of undefinability and of logical equivalence of structures.

In the Curry-Howard correspondence, it gives game semantics. There are specific analogies that express some very specific concepts e.g. Wadler's devil bargain illustrating how classical logic can backtrack on its choices.

Answer 2

Larry Guth has a nice paper on Metaphors in systolic geometry which I think fits the bill quite well.

In the context of systolic geometry, one such "organizing analogy" that I found helpful is that between a pair of invariants of a manifold $X$. One is the famous Lusternik-Shnirelman category (LS) of $X$, and the other the systolic category (SY) of $X$. The definition of LS is topological whereas that of SY is geometric (roughly, the greatest length $k$ of a product $sys_1\, sys_2 \ldots sys_k$ of systolic invariants of $X$ that can serve as a universal lower bound for the total volume). Here the existence of certain systolic inequalities has led one to conjecture similar lower bounds for LS which were eventually proved by Dranishnikov--Rudyak, Strom, and others.

Answer 3

Might functors between categories (as typed 'n-categories') be properly deemed 'analogies'? When I first learned of categories, I was inspired (having first read Hesse's Glass Bead Game) to deem Category theory as a possible basis for such a 'game'. The question that now remains for me is how properly to define the notion of 'deformation' for arbitrary functors and how to properly define 'approximate analogies' in terms of category theory (if, in fact, such notions can properly be defined in Category theory at all....)

Answer 4

Here is an example: Sullivan's analogy (or dictionary) between complex dynamics and Kleinian groups.