togglesbloggle replied to your post “It’s a bit unsettling to me how much of mathematics is grounded in the…”

Depending on how far you want to stretch ‘spatial’, you could include integers here as well- certainly we first got interested in numbers because they can be used for quantifying physical objects. So I’m not even sure how much number theory and algebra are in separate bins.

Huh, maybe.  When I think about why the definition of a field (like the naturals) includes multiplication but not exponentiation, the first argument that comes to my mind is spatial (grouping things in squares and cubes).

But do you get anything interesting out of the other “hyperoperations” (exponentiation, tetration, etc.), the way you get primes and stuff out of multiplication?  Everything I can find about discrete logs/roots (analogue of factorization for ^ rather than *) is in modular arithmetic.  I’ve never thought about this but I have the feeling there’s some trick that makes all these reduce to factorization, or something.  In which case, just having + and * makes sense.

The reason I am unsettled by these things is that sometimes I hear math characterized as “the study of abstract structures” or something like that, and I always wonder about that – if there are different types of “abstract structure,” do we know about all of them?  Are we grouping them in a natural way?

I try to sit down and think of “mathematical structures,” imagining that I am about to tell someone all about this exciting “study of abstract structures,” and I’m like, “well, there’s my old friend Squishy Space, and of course there’s Unsquishy Space, there’s Space That Holds Stuff, there’s Shapes You Had to Draw on Graph Paper in School, there’s Especially Spacey Space and its Special Hills, um,”

The author’s note is by Palmer, not Canner, so I don’t think it would have anything to do with Bridger per se. It’s just an expression of the author’s values- that the most important thing about the book is that people read it, and care enough about / discuss it to make sure it’s read for timescales longer than human lifespans.

“Conquer time’s diaspora” as in “communicate with people otherwise separated by non-overlapping lifespans.”

That was my understanding of her reference to a “diaspora” earlier in the note, it just startled me to see her write about “the next part” a sentence after mentioning the second part of Mycroft’s history.  Although now that I look at the exact wording, the phrase she uses in the previous sentence is “second half,” not “second part,” so there isn’t the verbal parallelism I thought there was