Hm; offhand, I would think level 2 should be split up. There’s the level where you can see the analogies to other areas, but can’t formalize them—you can use them to reason analogically, but can’t be quite sure that what you’re doing make sense. Then there’s the level where you actually, well, understand the connections to other areas. Does this distinction still make sense outside of mathematics?
Well, like I clarified to pjeby, the levels aren’t intended to be discrete: you can be partway toward completion of one. But what you’ve described still fits right in with Level 1: until you know why the analogy holds, and therefore can correctly predict if any given analogical inference will hold, your’e still at Level 1.
That’s not to say that apparent analogies can’t be usefully suggestive, it’s just that they’re not a higher level of understanding.
Hm; offhand, I would think level 2 should be split up. There’s the level where you can see the analogies to other areas, but can’t formalize them—you can use them to reason analogically, but can’t be quite sure that what you’re doing make sense. Then there’s the level where you actually, well, understand the connections to other areas. Does this distinction still make sense outside of mathematics?
Well, like I clarified to pjeby, the levels aren’t intended to be discrete: you can be partway toward completion of one. But what you’ve described still fits right in with Level 1: until you know why the analogy holds, and therefore can correctly predict if any given analogical inference will hold, your’e still at Level 1.
That’s not to say that apparent analogies can’t be usefully suggestive, it’s just that they’re not a higher level of understanding.