I skimmed most of this because I can’t handle quote-dumps. I wanted to comment similarly on some of your earlier maths quote-dumps but since you said there that you were just trying to organise your thoughts, I assumed that you would convert more of the quotes into prose.
On the post itself: The ‘beaver’ part kind of appears out of nowhere, I suggest putting more foreshadowing/summarising at the beginning. I’m also not sure I understand what a beaver does that’s different to the other groups. Frogs and birds seem to straightforwardly correspond to bottom-up and top-down thinking, or Sensing versus Intuitive in Myers-Briggs jargon. Beavering seems quite top-down to me.
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.
I skimmed most of this because I can’t handle quote-dumps. I wanted to comment similarly on some of your earlier maths quote-dumps but since you said there that you were just trying to organise your thoughts, I assumed that you would convert more of the quotes into prose.
I’ve wondered what to do about this. I like the idea of quoting really good people verbatim at length because (a) they have more credibility than I do and (b) I feel a little squeamish about paraphrasing them for fear of skewing the truth.
Would appreciate more detailed suggestions here if you have any to offer.
On the post itself: The ‘beaver’ part kind of appears out of nowhere, I suggest putting more foreshadowing/summarising at the beginning.
Okay, right.
I’m also not sure I understand what a beaver does that’s different to the other groups. Frogs and birds seem to straightforwardly correspond to bottom-up and top-down thinking, or Sensing versus Intuitive in Myers-Briggs jargon. Beavering seems quite top-down to me.
As I said to ThomasR, my subjective impression is that there are examples both of bird/beaver hybrids and frog/beaver hybrids. Maybe there’s a bird vs. frog axis and an independent axis measuring beaver-likeness. There is something real that I’m trying to get at here, but I’ll have to think more about what it is.
Felix Klein may be seen as a “bird/beaver” hybrid, in view of the calculational view on complex multiplication and class fields in the 19th century, leading to modular equations etc. Klein’s “icosahedron” and Weber’s 3 vol. “Algebra” (the best until v.d. Waerden’s book and E. Noether’s school). link A more modern example may be this, but I know only a part of the story.
I skimmed most of this because I can’t handle quote-dumps. I wanted to comment similarly on some of your earlier maths quote-dumps but since you said there that you were just trying to organise your thoughts, I assumed that you would convert more of the quotes into prose.
On the post itself: The ‘beaver’ part kind of appears out of nowhere, I suggest putting more foreshadowing/summarising at the beginning. I’m also not sure I understand what a beaver does that’s different to the other groups. Frogs and birds seem to straightforwardly correspond to bottom-up and top-down thinking, or Sensing versus Intuitive in Myers-Briggs jargon. Beavering seems quite top-down to me.
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.
I’ve wondered what to do about this. I like the idea of quoting really good people verbatim at length because (a) they have more credibility than I do and (b) I feel a little squeamish about paraphrasing them for fear of skewing the truth.
Would appreciate more detailed suggestions here if you have any to offer.
Okay, right.
As I said to ThomasR, my subjective impression is that there are examples both of bird/beaver hybrids and frog/beaver hybrids. Maybe there’s a bird vs. frog axis and an independent axis measuring beaver-likeness. There is something real that I’m trying to get at here, but I’ll have to think more about what it is.
Felix Klein may be seen as a “bird/beaver” hybrid, in view of the calculational view on complex multiplication and class fields in the 19th century, leading to modular equations etc. Klein’s “icosahedron” and Weber’s 3 vol. “Algebra” (the best until v.d. Waerden’s book and E. Noether’s school). link A more modern example may be this, but I know only a part of the story.
Right, makes sense.
I just remember a very nice online docu on the Chudnovsky brothers, an old NY’er article , the “one mathematician in two brains”.