First: what’s the load-bearing function of visualizations in math?
I think it’s the same function as prototypical examples more broadly. They serve as a consistency check—i.e. if there’s any example at all which matches the math then at least the math isn’t inconsistent. They also offer direct intuition for which of the assumptions are typically “slack” vs “taut”—i.e. in the context of the example, would the claim just totally fall apart if we relax a particular assumption, or would it gracefully degrade? And they give some intuition for what kinds-of-things to bind the mathematical symbols to, in order to apply the math.
Based on that, I’d expect that non-visual prototypical examples can often serve a similar role.
Also, some people use type-tracking to get some of the same benefits, though insofar as that’s a substitute for prototypical example tracking I think it’s usually inferior.
First: what’s the load-bearing function of visualizations in math?
I think it’s the same function as prototypical examples more broadly. They serve as a consistency check—i.e. if there’s any example at all which matches the math then at least the math isn’t inconsistent. They also offer direct intuition for which of the assumptions are typically “slack” vs “taut”—i.e. in the context of the example, would the claim just totally fall apart if we relax a particular assumption, or would it gracefully degrade? And they give some intuition for what kinds-of-things to bind the mathematical symbols to, in order to apply the math.
Based on that, I’d expect that non-visual prototypical examples can often serve a similar role.
Also, some people use type-tracking to get some of the same benefits, though insofar as that’s a substitute for prototypical example tracking I think it’s usually inferior.