This seems to be another case of “reverse advice” for me. I seem to be too formal instead of too lax with these spatial metaphors. I immediately read the birds example as talking about the relative positions and distances along branches of the Phylogenetic tree, your orthogonality description as referring to actual logical independence / verifiable orthogonality, and it’s my job to notice hidden interaction and stuff like weird machines and so I’m usually also very aware of that, just by habits kicking in.
Your post made me realize that instead of people’s models being hard to understand, there simply may not be a model that would admit talking in distances or directions, so I shouldn’t infer too much from what they say. Same for picking out one or more vectors, for me that doesn’t imply that you can move along them (they’re just convenient for describing the space), but others might automatically assume that’s possible.
As others already brought up, once you’ve gotten rid of the “false” metaphors, try deliberately using the words precisely. If you practice, it becomes pretty easy and automatic over time. Only talk about distances if you actually have a metric space (doesn’t have to be euclidean, sphere surfaces are fine). Only talk about directions that actually make sense (a tree has “up” and “down”, but there’s no inherent order to the branches that would get you something like “left” or “right” until you impose extra structure). And so on… (Also: Spatial thinking is incredibly efficient. If you don’t need time, you can use it as a separate dimension that changes the “landscape” as you move forward/backward, and you might even manage 2-3 separate “time dimensions” that do different things, giving you fairly intuitive navigation of a 5- or 6-dimensional space. Don’t lightly give up on that.)
Nitpick: “It makes sense to use ‘continuum’ language”—bad word choice. You’re not talking about the continuum (as in real numbers) but about something like linearity or the ability to repeatedly take small steps and get predictable results. With quantized lengths and energy levels, color isn’t actually a continuous thing, so that’s not the important property. (The continuum is a really really really strange thing that I think a lot of people don’t really understand and casually bring up. Almost all “real numbers” are entirely inaccessible! Because all descriptions of numbers that we can use are finite, you can only ever refer to a countable subset of them, the others are “dark” and for almost all purposes might as well not exist. So usually rational numbers (plus a handful of named constants) are sufficient, especially for practical / real world purposes.)
This seems to be another case of “reverse advice” for me. I seem to be too formal instead of too lax with these spatial metaphors. I immediately read the birds example as talking about the relative positions and distances along branches of the Phylogenetic tree, your orthogonality description as referring to actual logical independence / verifiable orthogonality, and it’s my job to notice hidden interaction and stuff like weird machines and so I’m usually also very aware of that, just by habits kicking in.
Your post made me realize that instead of people’s models being hard to understand, there simply may not be a model that would admit talking in distances or directions, so I shouldn’t infer too much from what they say. Same for picking out one or more vectors, for me that doesn’t imply that you can move along them (they’re just convenient for describing the space), but others might automatically assume that’s possible.
As others already brought up, once you’ve gotten rid of the “false” metaphors, try deliberately using the words precisely. If you practice, it becomes pretty easy and automatic over time. Only talk about distances if you actually have a metric space (doesn’t have to be euclidean, sphere surfaces are fine). Only talk about directions that actually make sense (a tree has “up” and “down”, but there’s no inherent order to the branches that would get you something like “left” or “right” until you impose extra structure). And so on… (Also: Spatial thinking is incredibly efficient. If you don’t need time, you can use it as a separate dimension that changes the “landscape” as you move forward/backward, and you might even manage 2-3 separate “time dimensions” that do different things, giving you fairly intuitive navigation of a 5- or 6-dimensional space. Don’t lightly give up on that.)
Nitpick: “It makes sense to use ‘continuum’ language”—bad word choice. You’re not talking about the continuum (as in real numbers) but about something like linearity or the ability to repeatedly take small steps and get predictable results. With quantized lengths and energy levels, color isn’t actually a continuous thing, so that’s not the important property. (The continuum is a really really really strange thing that I think a lot of people don’t really understand and casually bring up. Almost all “real numbers” are entirely inaccessible! Because all descriptions of numbers that we can use are finite, you can only ever refer to a countable subset of them, the others are “dark” and for almost all purposes might as well not exist. So usually rational numbers (plus a handful of named constants) are sufficient, especially for practical / real world purposes.)