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People have written about mathematical creativity and the joy and beauty of mathematics. What about the sleazy underside of mathematics? Has anyone written essays about Mathematical Confusion?
As I recall my teenage studies of mathematics, the most common experience was "getting stuck". I would read math textbooks on my own in a systematic manner, going through them in order, page by page, until I would hit some statement or "step" that I didn't understand. Then I would make no progress at all till I understood that passage - if I ever did.
I recall trying to read "An Introduction To Number Theory" by Uspensky and Heaslet. I got stuck when I hit the exposition of the Euclidean Algorithm. It has lines with many subscripted Greek letters. That type of exposition was completely new to me. There weren't any personal computers in those days so I had no experience in reading precise prescriptions for algorithms. I returned the book to the library. A few weeks later, I was in English class and I looked over at my friend Calvin, who was known as "a brain". He had "An Introduction To Number Theory" among his books. I asked him if he was reading it. He said "The first part made sense, but look, how can anyone understand this?". He opened the book and showed the page with the symbolic statement of the Euclidean Algorithm on it.
The "getting stuck" experience characterized my mathematical studies through my undergraduate years. Then the experience of mathematical confusion gradually changed. Instead of a step function ( -understand everything up to page 202 section 6 completely, understand nothing afterward) , it change into a more gradual decline. Maybe I accepted the fact that I wouldn't understand any subject completely. So I continued to "progress" through material while understanding less and less of it.
In very abstract courses there was another type of confusion. Various definitions were made that seemed unmotivated to me. Proofs of theorems were demonstrated. The steps all made sense the logic was undeniable. But why would anyone care about the result? The fact that everything seemed to be without a purpose contributed to the difficulty of remembering the exact definitions and theorems.
As life experiences go, I prefer the "getting stuck" feeling to "gradual decline" and I am nostalgic for it.
As I recall my teenage studies of mathematics, the most common experience was "getting stuck". I would read math textbooks on my own in a systematic manner, going through them in order, page by page, until I would hit some statement or "step" that I didn't understand. Then I would make no progress at all till I understood that passage - if I ever did.
I recall trying to read "An Introduction To Number Theory" by Uspensky and Heaslet. I got stuck when I hit the exposition of the Euclidean Algorithm. It has lines with many subscripted Greek letters. That type of exposition was completely new to me. There weren't any personal computers in those days so I had no experience in reading precise prescriptions for algorithms. I returned the book to the library. A few weeks later, I was in English class and I looked over at my friend Calvin, who was known as "a brain". He had "An Introduction To Number Theory" among his books. I asked him if he was reading it. He said "The first part made sense, but look, how can anyone understand this?". He opened the book and showed the page with the symbolic statement of the Euclidean Algorithm on it.
The "getting stuck" experience characterized my mathematical studies through my undergraduate years. Then the experience of mathematical confusion gradually changed. Instead of a step function ( -understand everything up to page 202 section 6 completely, understand nothing afterward) , it change into a more gradual decline. Maybe I accepted the fact that I wouldn't understand any subject completely. So I continued to "progress" through material while understanding less and less of it.
In very abstract courses there was another type of confusion. Various definitions were made that seemed unmotivated to me. Proofs of theorems were demonstrated. The steps all made sense the logic was undeniable. But why would anyone care about the result? The fact that everything seemed to be without a purpose contributed to the difficulty of remembering the exact definitions and theorems.
As life experiences go, I prefer the "getting stuck" feeling to "gradual decline" and I am nostalgic for it.
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