Frogs and birds seem to straightforwardly correspond to bottom-up and top-down thinking
This seems like an intuitive summary to me. Let me elaborate on why.
One way the birds might build theories is by noticing the possibility of large patterns, and then asking themselves questions about such patterns. Perhaps they might be thought of as solving blurry conceptual problems like “How are algebraic geometry and algebraic number theory connected?”. To build theory they would have to ask progressively more specific questions, until their answers could be written as mathematical definitions.
The frogs could instead start by asking a very concrete question like “Do the prime numbers contain arbitrarily long arithmetic progressions?”. If the question couldn’t be solved with elementary means they would have to develop new concepts, and understand the relationships between these concepts. To do this they could ask progressively more abstract questions, until they had built enough theory to solve their original problem.
This seems like an intuitive summary to me. Let me elaborate on why.
One way the birds might build theories is by noticing the possibility of large patterns, and then asking themselves questions about such patterns. Perhaps they might be thought of as solving blurry conceptual problems like “How are algebraic geometry and algebraic number theory connected?”. To build theory they would have to ask progressively more specific questions, until their answers could be written as mathematical definitions.
The frogs could instead start by asking a very concrete question like “Do the prime numbers contain arbitrarily long arithmetic progressions?”. If the question couldn’t be solved with elementary means they would have to develop new concepts, and understand the relationships between these concepts. To do this they could ask progressively more abstract questions, until they had built enough theory to solve their original problem.