(Below I attempt to reproduce a chart I can't [?] embed, from Models and Analogies in Science by Mary Hesse; I don't know how it will display on all screens, but the source is this image at the top of the cross-post version here.)

At an elementary level we can set up the following correspondences: 

WATER WAVES         |       SOUND        |          LIGHT
Produced by motion  | Produced by motion | *Produced by moving
 of water particles |  of gongs, strings |   flame, etc.
Reflection          | Echoes, etc.       | Reflection in mirrors
Properties of dif-  | Hearing round      | Diffraction through
 fraction           |  corners           |  small slits, etc.
Amplitude           | Loudness           | Brightness
Frequency           | Pitch              | Color
Medium: Water       | Medium: Air        | *Medium: "Ether"

In the table, we see a lot of apparent correspondences between water waves, sound, and light. The “horizontal” notion of similarity lets us notice that sound echoes and light reflects, or that these things all have some sort of “intensity” and “flavor”.

But it’s important to introduce the “vertical” analogy—the items in each column are related by some causal, organizing principle, and there’s a correspondence between those principles. We expect all sorts of things to have similar traits entirely by accident. You won’t get very far filling in a table’s gaps by arguing “this is like that”—better to say “this model is like that model”. You’re taking advantage of a ready-made language (with entailed internal relationships) for a metaphoric redescription of a new subject. In this way analogies can be a useful guide to teaching and learning new models in new domains. (You’ll realize, for example, that “produced by moving flame” isn’t really the appropriate correspondence there, because the motion of the flame doesn’t have to do with color in the way that the motion of a gong has to do with pitch, and eventually you’ll learn something about the production of light.) But what’s this about the medium of light—“ether”?

Well, if we observe that the vertical analogy works for the first two models, and it works for the third up to that point, then light having a medium starts seeming plausible enough for us to start looking into it. And it additionally suggests to us how to go about looking. But while the vertical analogy gives us a stronger inductive inference than does the horizontal analogy alone, it’s still quite weak. Light, it turns out, doesn’t seem to propagate through an ether. (But even the “negative analogy”—the apparent hole in the vertical analogy where light’s medium would go—suggests that “why not?” is an interesting question.)

There are two parts to what I just said, so I’ll work them out a little further:

The use of analogy in science is partly pedagogical—it’s about explaining things [as in providing exposition, not reasons or causes] in terms of better-understood things, through their models’ shared structure, or their horizontal points of similarity or difference (positive and negative analogies). If the structure of the relation you’re trying to draw is somewhat confusing on its own terms, or not readily distinguished from a similar model, it could be easier to communicate with reference to another domain. It’s easier to understand what we could possibly mean by “light is a wave” if you already know about water waves. And if we’re being careful, we say “light is a wave like how water waves are waves,” not “light is like water”—we care about communicating the vertical relation, not arguing for horizontal similarity.

And the use of analogy is also about guiding discovery—“neutral” analogies becoming definitely positive or negative as they’re used to pinpoint places to investigate. It can be useful to make tentative inferences based on similarity of causal relations to those of a better-understood model, but these inferences really are provisional. You don’t know if light has a medium, but the analogy has worked so far, so you design experiments (guided by the analogy) that would detect such a medium. You get a handy working hypothesis and a set of questions to study, and not much else. Your analogy is usually not a very good piece of evidence about its subjects—not good enough to use for engineering—but often still good enough to help decide what’s worth investigating. (And, as often when people talk about history and philosophy of science, the big, obvious examples are recapitulated in the everyday work of science on much smaller scales. It’s not always about major physical models like electromagnetism and quantum mechanics, but rather implicit in the kind of reasoning that guides investigation from week to week.)

(Philosophers of science also [used to] like to argue about whether analogy is necessary for explanation and/or discovery—that’s the dialogue Hesse was participating in above. This is out of scope for us, unless I’m being sneaky.)

Can we take this understanding of analogy outside of science? When is it worthwhile to inject an argument by analogy into your internet? And when is it worthwhile to dispute an analogy?

First, when you need better exposition. If you’re making an argument that’s hard to spell out in its own language, or easily confused for a more common argument you decidedly don’t want to make, then an analogy might be clarifying. (And usually more compelling to read, with that stimulating sensation of insight we get from novel connections, which should make all analogies suspect.) This is where it helps to say “this argument is like that argument” rather than “this is like that”. But be careful not to mistake this for substantiating your argument.

Second, to point to questions to investigate. If you’re not sure how an argument should come out, you can find other arguments in other domains that look like they flow in the same way. Then the points of analogy are good places to look for the evidence your argument hinges on. And disputing the analogy—saying a point of analogy is neutral or negative—is how your interlocutor points to where they think the contrary evidence lies.

Maybe this is a tedious distinction to keep making, but the usefulness of analogy is not, primarily, in making an inductive inference based on the fact that one model looks mostly like another, where the correctness of that inference depends on the success of the analogy between models. Usually, rather than argument by analogy, you want exposition or guidance by analogy.

Along these lines, it could be generally useful to distinguish between different levels of putting forward and substantiating a claim. You can talk about a position, an argument for that position, or evidence that the argument hinges on. In doing so you can be anywhere between just pointing to, or describing, or actually demonstrating the bit you’re talking about. If someone thinks you’re further down the list than you are, then you’re liable to get mired in a bad discussion. (Most debates don’t get past pointing to where evidence can be found, and most blogging (including this post) doesn’t get past pointing to positions or arguments either. Maybe that’s fine. Pointing is cheap, both to write and to read. Going deeper can be superfluous, if you’re pointing to the obvious. Starting out by pointing could get you to the crucial evidence for resolving a disagreement faster. And so on. [And if you are really just pointing, please consider whether you need so many words.])

In this sense, analogies are for pointing.