Steve, you write: *I might instead say that the source domain of the metaphorical description of "bent" or "curved" space IS the formal mathematical construction of "a manifold"? *
What about Eddington's measurement or gravitational lensing? These both appear situated in a phenomenological domain, and so we seem to have another candidate domain for talking meaningfully about *bent* or *curved* space. My preference is to grant that the metaphor maker has a domain in mind, but one that I will only come to understand through investigation†. It is perfectly acceptable to me that the metaphor maker intends a domain which includes, but is not limited to, both phenomenological and mathematically formal domains (say). Still, I suspect I am missing something important in your emphasis on *apprehension.* Can you say a bit more about what you mean? you write: *Which I think is analogous or at least similar to Guerin's "least action paths"?* Yeah, I suspect so too. MacLane's book intentionally focuses on developing Lagrangian and Hamiltonian mechanics. Mechanics, as far as I am concerned, is the prototypical home for these ideas. You write: *feels a bit more to me like an "algebra of cliche's"?* Thanks for that. Upon further reflection, I completely agree with you. Jon †) At the unintended risk of moving the conversation into the *meta*††, I am including a link here to a page motivating the development of sheaves <https://ncatlab.org/nlab/show/motivation+for+sheaves%2C+cohomology+and+higher+stacks#TheBasicIdeaOfSheaves> . In section 2 the author invents a game where he thinks up a space and the player can query the author about how other spaces map into it. †† *Meta* in that sheaves themselves offer a more flexible paradigm for reasoning about generalized spaces than we get from manifold theory.
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