"circles"? what's that? oh u mean triangles.
"nonlinear"? what's that? oh u mean u is tryna projecting it to lower dimensions u adorable earthling.
"non-differentiable"? what's... huh? *why did you put a zero there?!*
@rime i... I don't get it
I'm the earthling
fundamental theorem of calculus: "every function is the derivative of its integral"
(aka "a function is at every point equal to the rate at which its total sum up to that point grows", duh)
take f=x^2.
can visualize it as a square, x*x.
... darnit. bedtime. hmm.
so here: u can visualize f as a 2d curve, OR as the red dotted line through the cube below.
@niplav and the fact that ∫ x² dx = ⅓x³ can be seen if u turn the cube so the function grows directly out of the page: now, the integral is the blue region and the rest of the cube is the red region.
i think i figured thr was smth wrong abt this visualization but i can't recall rn...
@rime i might be unhelpful here, but this sounds a bit like you want the upper half of the surreal numbers?
@rime tho the construction can be hard if you can't start from the empty set
(if you try to abolish 0 then you're also at odds with the empty set, which is quite tricky)
@niplav interesting! I no grok q "surreal numbers" bon quick lookup, but i'll now keep eye open for it lurn.
my orig motivation for it is feeling lk normie-graph fail capture symmetries btn interval (0,1) & (1,∞), and mk it seem lk (0,1) is the exception. normie-graph force u to lurn diff heuristics for sim shapes in ea region. ey squish (0,1) into tiny vortex* u can't even visualize anyth in. it's big-number bias.
*(1,∞) is j as much vortex as (0,1), tho it sure feels diff on normie-graph.
@niplav ok fine, nvm. (0,1) is more vortexy in most ways, at least if u hv a universe w additive interaction-laws... i is silly for saying otherwise, and repent my lack of forethink.
...hm, sometimes failing to see obv counter-arguments while excitedly follow crazy train-of-thought can be usefwl as long as no attachment and easily rewind. can recycle nuggets got.
@rime i get cirlces are triangles now! Half circles are overweight triangles
@rime also it sounds like you're developing a really deep geometric intuition for derivative and integral, which I absolutely don't have
So learning late could be an advantage
@niplav and the idea "why did you put zero there!?!!?" is from thinking "hmm, there are no absolute quantities; all quantities are relative, defined by smth else", and my hitherto-unsuccessfwl desire to replace the standard number-line with "deictic graphs" which go from 1/∞ to ∞. (no negative numbers or zero!)