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Nov. 8th, 2005
Rereading Old Textbooks
Nov. 8th, 2005 03:51 pmAs I've already mentioned, I'm currently rereading - or re-studying - one of my old college textbooks, Abstract Harmonic Analysis, v.1 by Edwin Hewitt and Kenneth A. Ross. I'm a little over halfway through. It's been an... interesting experience, so far.
Parts have been dull. Of the three major branches of mathematics - algebra, geometry, and analysis - I find the first two very interesting; the third makes a nice spice when it's mixed with the others, but by itself doesn't interest me much. All three are needed in this subject area. Unfortunately, there is one long section - thankfully, finished now - which is almost pure analysis, and that was a real slog. (I've been insisting on not going on from any proof until I understand it, at least at some level.)
The cross-referencing system H&R use has been driving me bats. Every item - definition, theorem, comment - gets a threefold designator such as "II.14.29" (indicating item 29 of section 14 of chapter II). (Some items have subparts, and their designators are correspondingly longer.) This style I generally associate with the hive-mind known as Bourbaki and, though it makes locating cross-references relatively easy and thus improves the quality of the work as a reference, I think it's deeply flawed pedagogically. Instead of saying things like "Because the function is non-negative and countably additive...", it says things like "Because of II.13.5...", which forces the reader to flip back to the section in question instead of remembering or reasoning from the given information. It breaks the reader's rhythm and makes it harder to set up a coherent mental frame for the material.
Nonetheless, it's been rewarding. For example, there is one proof, absolutely central to the subject, which I remembered as being long, tedious, and incomprehensible. (I say "incomprehensible", meaning that though I could follow each step of the proof, I could not understand it as a whole. Trees, not forest.) I finally reached that proof a week or two ago, and found myself reading and rereading the same four or five pages. It was every bit as long and tedious as I remembered, but suddenly... "Hey. Wait. Umm... Yes. YES! Of course that's how you'd do it! YES!!!"
Don't you just love those lightbulb moments?
Parts have been dull. Of the three major branches of mathematics - algebra, geometry, and analysis - I find the first two very interesting; the third makes a nice spice when it's mixed with the others, but by itself doesn't interest me much. All three are needed in this subject area. Unfortunately, there is one long section - thankfully, finished now - which is almost pure analysis, and that was a real slog. (I've been insisting on not going on from any proof until I understand it, at least at some level.)
The cross-referencing system H&R use has been driving me bats. Every item - definition, theorem, comment - gets a threefold designator such as "II.14.29" (indicating item 29 of section 14 of chapter II). (Some items have subparts, and their designators are correspondingly longer.) This style I generally associate with the hive-mind known as Bourbaki and, though it makes locating cross-references relatively easy and thus improves the quality of the work as a reference, I think it's deeply flawed pedagogically. Instead of saying things like "Because the function is non-negative and countably additive...", it says things like "Because of II.13.5...", which forces the reader to flip back to the section in question instead of remembering or reasoning from the given information. It breaks the reader's rhythm and makes it harder to set up a coherent mental frame for the material.
Nonetheless, it's been rewarding. For example, there is one proof, absolutely central to the subject, which I remembered as being long, tedious, and incomprehensible. (I say "incomprehensible", meaning that though I could follow each step of the proof, I could not understand it as a whole. Trees, not forest.) I finally reached that proof a week or two ago, and found myself reading and rereading the same four or five pages. It was every bit as long and tedious as I remembered, but suddenly... "Hey. Wait. Umm... Yes. YES! Of course that's how you'd do it! YES!!!"
Don't you just love those lightbulb moments?