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The difficulty in understanding a proof I think is partly because whoever wrote the proof omitted many "trivial" steps in the proof. But they were trivial to
by galaxyLogic 3y ago
The difficulty in understanding a proof I think is partly because whoever wrote the proof omitted many "trivial" steps in the proof. But they were trivial to the person who wrote the proof, not to whoever is reading the proof.
To understand the proof you have to accumulate enough mathematical knowledge that the trivial, omitted, steps in it become trivial to you as well.
This has to do also something with the fact that
to understand the proof it cannot be longer than what fits into your memory. That is why proofs omit the trivial steps because that way mathematicians can understand the proof both in terms of remembering its outline, but also understanding how each step in it follows from the previous ones.
- chongli 3y agowhoever wrote the proof omitted many "trivial" steps in the proof This is not an issue specific to mathematics. Listen to a couple of surgeons talking shop in a hospital cafeteria. They're going to be using all kinds of arcane anatomical and medical terms the average person would have no hope of understanding. Same goes for a pair of programmers talking through a complex bug deep within a massive codebase. Economy of language is critical for communication among experts in a field. If everyone was forced to use the vocabulary of a 6th grader there'd be a whole hell of a lot of repeated explanations of concepts the other person already knows and it would just be frustrating.
- galaxyLogic 3y agoTrue. I wonder if there was a way to study what are the hierarchies of concepts needed to explain higher-level concepts in any given field of study. That is not the same as "most often used" concepts, but really "most often assumed to be understood concepts". Such an exposition might help students, if there was such an analysis of their field of study. What are the most useful concepts, including proofs and facts to know and understand to be able to learn more "higher level" concepts.
- chongli 3y agoI don’t know about other fields of study, but such an exposition exists for math [1]. I heard about this book on the YouTube channel The Math Sorcerer [2]. [1] https://www.amazon.ca/All-Math-You-Missed-Graduate/dp/1009009192 https://www.amazon.ca/All-Math-You-Missed-Graduate/dp/100900... [2] https://youtu.be/ur0UGCL6RWc https://youtu.be/ur0UGCL6RWc