[Diogenesian]

  This scale sometimes goes to 11, for example the size of the solar system is ↑11 meters, but even the size of the entire universe is only ↑27 meters. Which should indicate both how “universal” this scale is, and also how big the universe is!

This feels like the problem of understanding scale gets pushed around syntactically without actually aiding understanding.

"Got it, so the universe is 3x bigger than the solar system."

"No no, remember these are base 10 logarithms."

"Ah... so it's 16x!"

"No, 10^16x."

"Ah... so it's totally incomprehensible."

[saulpw]

It feels like that at first glance because you haven't done the work to internalize the notation. This is like someone who uses Roman Numerals being introduced to decimal notation, and getting confused that "22" isn't just twice "2". Yet no one these days would say that decimal notation is merely syntactic shuffling.

The notation helps, but you still have to do the work.

[Diogenesian]

This is uncharitably condescending because I myself am not confused by the notation. Obviously I "did the work."

It absolutely is syntactic shuffling of scientific notation and does nothing to solve the problem of intuitive scale that scientific notation doesn't already handle. The problem is that, unlike 2 and 22, 10^16 is simply not an intuitive quantity, and pretending it is because of Knuth notation / whatever is only displacing the misunderstanding.

[saulpw]

Sorry it felt condescending. I'm sure you think it's obvious that you "did the work" because you know about scientific notation and understand the notation. I thought I had done it too, but more and more I see how my internal number line is/was still fragmented, and gets better when I think in terms of exponents first/foremost.