Reasoning with Conceptual Metaphor is an informal version of something done more formally in mathematics.
In mathematics, it’s not at all unusual to solve a problem in a complex or new Target Domain by translating it into the equivalent problem in a simpler or better-understood Source Domain, solve it there, then translate the solution back into the original domain.
In mathematics, the translation between domains is rigorous or formal. On Mastodon, Bob Apthorpe described a less rigorous version of the same idea:
> One of the really cool things I learned about at my last job was the Chilton & Colburn Analogy 
. Due to the similarity in the mathematics behind mass, momentum, and heat transfer, it turns out if you solve the problem of one of them, you can wave your hands a bit and get the solution to the other two. We'd often use this in cases where we understood heat transfer in a system fairly well but needed to know mass transfer behavior which was more complicated to calculate from scratch. It's not quite the same as a mathematical transform but it's in the spirit of indirectly solving a difficult problem by converting it to an easier problem then doing some hand-waving to convince people the solution is valid when you convert it back to the original domain.
--- Metaphorical reasoning has the same structure. We all have a lot of experience codified and organized in our brains. We’re good at getting around in the physical and social worlds, good at solving problems in those domains. It’s certainly worth a try translating novel hard problems into those domains, seeing if we can solve that equivalent problem there, and then checking how that solution might work for the original problem (by trying it or thinking about it).
The translations between the two domains are far from the rigor of mathematical transforms, but hey – the trick seems to work well enough that our brains have kept doing it. In fact, our brains seem to have specialized in it: this kind of problem-solving is fast, so fast that it’s usually unconscious when we’re just trying to understand – well enough – what other people mean.
When it comes to problem solving, the metaphor can lead to new ideas, as with Schön’s Paintbrush as Pump metaphor. It encourages attempts to match up things you find in pumps with things you find in brushes, leading to the team’s analogizing the space between bristles to the channel through which pump water flows.
But it can also encourage matches that are not, in Schön’s words, “generative.” Reddy argues the Conduit Metaphor directs attention *away* from the importance of the listener’s context and so makes it harder to understand some conversational problems.