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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.

Jon Bannon
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    Regarding the vote to close, there is probably something I am not communicating clearly. Wouldn't it be useful to have organizing analogies available for more rapidly learning new areas? This would benefit research by allowing us to more rapidly learn important ideas in other areas we are not familiar with. This is very much about research-level mathematics, as it is about facilitating cross-pollination of ideas to accelerate progress. Please let me know what about this is not appropriate or clear! – Jon Bannon Mar 20 '16 at 15:31
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    So something like the analogy between number fields and function fields does not qualify (since both sides are mathematical)? – Sam Hopkins Mar 20 '16 at 16:09
  • @Sam Hopkins: Probably not. The motivation for the question is roughly that human intuition and experience can be used to internalize and enliven mathematics. "Mapping" mathematics to everyday experience as an intuition pump may be an effective way to remember or understand a bit of mathematics. The image of Popa I cite is telling, as it is salient and makes it rather easy to remember the thrust of the technique. Perhaps this position supports the votes to close... – Jon Bannon Mar 20 '16 at 21:49
  • OK. If this question is closed, I can bring it to math.se, perhaps. I fear that the high-level organizing viewpoints will not be available (I'm interested in research-level topics, not undergraduate homework topics). I can move it to matheducators stackexchange, but the same problem remains. We are not going to get good intuition pumps for, say, algebraic geometry, at either of these places. – Jon Bannon Mar 21 '16 at 12:34
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    With the level of analogy that you have asked for, this question can be mapped into Thurston's question, "Thinking and Explaining", of course, missing some key details :) – Amir Asghari Mar 21 '16 at 14:59
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    @Amir: This made me smile, and I see your point! This helps me understand the votes to close a bit more, thank you! – Jon Bannon Mar 21 '16 at 15:14
  • My guess is that this question may undergo a round of closing/reopening or two and would eventually survive (there are deep wellsprings in the community that welcome this type of question) -- unless it is convincingly shown this is effectively a duplicate. This type of question does seem somehow familiar to me, but I haven't been able to pin down any duplicates. Other than that, I think the question is fine and interesting and potentially useful. – Todd Trimble Mar 21 '16 at 17:51
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    @Amir: It may be that this question embeds in Bill Thurston's question, or perhaps that Bill Thurston could have provided a nice answer to this question. I find myself rather missing Bill Thurston right now... – Jon Bannon Mar 21 '16 at 19:17
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    Does "you can't comb a hairy ball flat without creating a cowlick" or "you can't comb the hair on a coconut" count as an answer? Hairy Ball Theorem – Amir Asghari Mar 22 '16 at 23:55
  • @Amir: It might, if you include the parts of the analogy in more or less precise detail. It certainly makes the result salient. Can you use it to remember the mathematics and reproduce an argument or strategy? – Jon Bannon Mar 23 '16 at 00:12
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    To be honest, I heard that about fifteen years ago when I was a math student and I've never had any professional reason to get back to it. I've just mentioned it here to help the question to be better understood. However, the mere fact that I remembered it after fifteen years perhaps says something about the importance of your question. – Amir Asghari Mar 23 '16 at 00:21
  • I'd say that the hairy ball theorem, aside from the obvious amusement, seems plausible and is more easily remembered than "there is no nonvanishing continuous tangent vector field on the 2-sphere." It is a convenient answer to "what does an algebraic topoligist do?" or "what is fixed point theory" but it doesn't really, in my opinion, do anything deeper. – Aaron Meyerowitz Mar 23 '16 at 19:28
  • @LarryGuth, can you comment on metaphors in systolic geometry? – Mikhail Katz Mar 25 '16 at 09:41
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    The idea that this is not about research-level mathematics is absurd. If this is closed it should be reopened immediately. – Mikhail Katz Mar 25 '16 at 09:44
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    I would be happier if the question were much tighter, exactly because there are a million "hairy ball theorem" or "hear the shape of a drum" type answers possible, whereas I think the question is more interesting if it excludes these by demanding that the examples be "organizing" in a strict sense. – Daniel Moskovich Mar 27 '16 at 13:10
  • @DanielMoskovich: This is the reason I included the example, in hope that similar answers would be spurred by it. If you can think of a good edit to "tighten" the question, I'd welcome it! – Jon Bannon Mar 28 '16 at 20:36
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    @DanielMoskovich: I put an italicized paragraph after the original question, in order to try to tighten the question and screen the hairy ball answers... – Jon Bannon Mar 29 '16 at 14:16

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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.

logicute
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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.

Mikhail Katz
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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....)

Thomas Benjamin
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  • This doesn't seem to at all be the sense in which Jon is using the word "analogy" – Yemon Choi Mar 29 '16 at 22:56
  • @YemonChoi: The question for me is, whether Jon's example of the "deformation/rigidity strategy" can be correctly expressed in terms of Category theory (I think it might be able to correctly be so expressed). If so, then I might have (in some sense) captured the sense in which Jon is using the word "analogy". In fact, each of the three previous examples can probably also expressed in Category-theoretic terms as well, so Category theory can be used as a language in which analogies can properly be expressed. In what sense do you believe Jon is using the word "analogy"? – Thomas Benjamin Mar 30 '16 at 00:03
  • So you are asking a different question. Jon is asking for examples of analogies, and you are speculating on a way to use category-theoretic ideas as a model for what he is looking for. Isn't this is like me asking for examples of cheese, and getting a learned treatise on casein? – Yemon Choi Mar 30 '16 at 00:56
  • @YemonChoi: I guess for me such homely examples like "Consider a bucket of dark liquid...." just don't seem helpful to me. Jon, however, makes a nice translation between Sorin's "Consider a bucket..." example and a finite von Neumann algebra $M$. It is, at least to me (the example of the finite von Neumann algebra $M$, that is...) the more fruitful example, because the exact same structure (or nearly so) may exist elsewhere (i.e. in some other category) and so may be able to be used elsewhere (to some degree). "Cheese" examples are a 'dime-a-dozen' (so to speak), but "learned – Thomas Benjamin Mar 30 '16 at 01:34
  • (cont.) treatises on casein" (like the three previous examples) are of great use to cheese-makers (mathematicians). However, Sorin's "Consider a bucket..." example has applicability also, but only in translation to finite von Neumann algebras, which translation must necessarily take precedence over "Consider a bucket..." examples for mathematicians.... – Thomas Benjamin Mar 30 '16 at 01:55
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    Although the discussion of categories is interesting, and the observation is a good one (I do believe that mathematical structure as expressed in categories precises the concept of mathematical analogy), what I am looking for are "structure preserving maps" from mathematics to the concrete world of human sensory experience, as an exploration of certain ways of stimulating native human thinking for organizing abstract techniques. The "cheese" examples I'm looking for are not a dime-a-dozen, as they should not be contrived and yet should encode and organize a lot of the technical structure – Jon Bannon Mar 30 '16 at 11:40
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    of a mathematical theory. Roughly, what I'm looking for are maps to common sense experience that capture the basic structure of a strategy or technique with a large kernel, namely the definition-baggage of the abstract theory. Admittedly, this is probably mostly for mnemotechnic purposes. – Jon Bannon Mar 30 '16 at 11:41
  • @JonBannon: Thanks for the clarification. – Thomas Benjamin Mar 30 '16 at 15:05
  • Jon, perhaps you could comment on the other answers, some of which do not seem to me to meet your original question. (That said, I seem to be in a minority of MO people who care about answering the question that was asked) – Yemon Choi Mar 30 '16 at 15:58
  • Thomas, it seems to me that Jon's clarification is mostly a repetition of what was written in his original question -- hence my frustration with people choosing to ignore it – Yemon Choi Mar 30 '16 at 15:59
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    @Yemon: I will consider commenting on the other answers, but I'm at a bit of a loss regarding how to do this in a way that won't transmogrify this thread into a discussion. For brevity, I'll keep quiet for a while and hope people read the comments associated with this answer. – Jon Bannon Mar 30 '16 at 19:30
  • @YemonChoi: Have you an analogy that meets Jon's criteria? If so, I'd like to see it posted as an answer – Thomas Benjamin Mar 30 '16 at 21:39
  • @JonBannon: Would treating Category theory as a discipline analogous to Hesse's Glass Bead Game as described in his novel an analogy more along the lines of what you were looking for (though Hesse's 'Game' is more of a metaphor, but then you did ask for metaphors that conveyed the gist of a topic to people not in the field--in this case nonmathematicians....)? – Thomas Benjamin Mar 30 '16 at 22:03
  • It might be, depending on whether the game lays out the "gist" in a way that would be useful for actually constructing the theory, so that each piece of the game corresponds to a particular aspect of category theory, and so that the pieces fit together in an appropriate and illuminating way. (I'm not familiar with the game, so I'd be interested to see...) – Jon Bannon Mar 30 '16 at 23:13
  • Andrée Ehresmann actually has thought about (& modified) the glass bead game in light of cat theory: http://www.glass-bead.org/article/the-glass-bead-game-revisited-weaving-emergent-dynamics-with-the-mes-methodology/?lang=enview – Trent Apr 11 '16 at 07:12
  • Trent: Would you consider adding Ehresmann's emergent system as an answer (I thought the article was great)? If not , would you allow me to use your comment to revise my answer to better fit Bannon's criteria? – Thomas Benjamin Apr 21 '16 at 10:33
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Here is an example: Sullivan's analogy (or dictionary) between complex dynamics and Kleinian groups.

Gil Kalai
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