Understating how remarkable a result is

• I was watching a 3b1b video on gradient descent, and at some point he mentions that because the gradient tells us the direction of greatest increase, the negative of the gradient tells us the direction of greatest decrease. I was thinking about why this was true for a second, and I was struck by how remarkable this is. It really says something rigid about the geometry of differentiable functions, in particular their local linearity.

The luxury of time

Sometimes I don’t have the luxury of time. My life doesn’t always revolve around math, and hasn’t always revolved around math.

• Several times when I’ve wanted to try working on a research problem with a professor they would tell me that I don’t have the necessary background, or that it would be useful to take these classes or read these books before doing what I want to do.
• In classes we’re expected to pursue myriad exercises (beyond the HW) and to cross reference i with several books.

Ending a sentence on a diagram

Many people really dislike ending a sentence on a diagram. For instance:

... and we want to show that the diagram 
--- (SOME DIAGRAM) ---
commutes.

I personally hate this, because I’m not sure what I should be getting out of the diagram until I read the words after it, at which point I go back and look at the diagram. My style is more like

... and we want to show that this diagram commutes:
--- (SOME DIAGRAM) ---