Section3.6Session 6: Two general aspects or uses of maps
This week’s reading didn’t have any official exercises in it, but it also seemed important so I wanted to make sure I gave it the same level of attention as the others. I find that I need to continually reminding myself that math is not a race and that I should take advantage of current privilege to truly take things slow here. My purpose isn’t to get through the excercises in this book; it’s to understand the deeper ideas behind it.
This session is filled with beautiful mathematical writing.
I think math courses need more content like this. At the same time, both the student and teacher in me recognize that this is one of the first things that would be cut from the syllabus if the class ran short on time. It makes me sad that our system’s focus on measurable learning objectives means that philosophical discussions are treated as dispensible.
In much the same way as the authors describe science as maps being used to understand reality, the education system uses assessments to understand what’s going on inside a student’s head. Since we can’t observe someone’s state of mind, we alter their environment stimuli and see how they respond. This presents problems when we’re using map composition to turn one of many complex mental models down to a multiple choice question for the exam. As long as a student could distinguish between "sort" and "partition" by the end of this unit, we don’t really care how it impacted their broader understanding of world around them.
I think there’s a hidden exercise in this session. I’ll mark it as an example here so you can avoid spoilers if desired.
Example3.6.1.Session 6 Hidden Exercise.
What do you notice? What do you wonder?
Solution.
After reading through this session, I noticed something very interesting about the last diagram. I’ll recreate it here for reference.
Figure3.6.2.The last diagram in L&S Session 6
What struck me as interesting about this map was how closely it described a book. There’s only a finite number of "words" in the "language" that the book is written using, and we can index them with positive integers describing their sequence in the book.
This observation struck me as interesting because my reading of this book works the same as the sampling process. You could generate a map from time to the index of the word that I’m reading at the time, and compose this map with the book map to find out what word I’m reading. This would reveal my non-linear path through the book, rereading sections as needed to answer the problems I’m working on.
It makes me wonder if this session is actually an exercise in meta-mathematics. Maybe the reason the authors intended to sow seeds of curiousity about their ommission and for me look at the book as a system that can be studied using category theory itself. The chapters of the book act as a sort for the exercises, but the fact that this chapter doesn’t exercises have any means it’s not a partition.
I think it’s kind of funny that this section of the book about sections would be a counter-example to the book’s exercises having a section.
