Thanks! This seems to me like another piece of the puzzle =D

In this case, this is one that I already had (at least, well enough for the hindsight bias to kick in :-p), and it’s on my list of trailheads next time I try to ground out the 2 in the Born rule. FWIW, some lingering questions I have when I take this viewpoint include “ok, cool, why are there no corresponding situations where I want to compare 3 vectorish thingies?” / “I see why the argument works for 2, but I have a sneaking suspicion that this 2 is being slipped into the problem statement in a way that I’m not yet quite following”. Also, I have a sense that there’s some fairly important fact about anointing some linear isomorphism between a vector space and its dual as “canonical” that I have yet to grasp. Like, some part of the answer to “why do I never want to compare 3 vectorish thingies” is b/c the relationship between a vector space and its dual space is somehow pretty special, and there’s no correspondingly special… triality of vector spaces. (Hrm. I wonder whether I can ground the 2 in the Born rule out into the 2 in categorical duality. That would be nuts.)

FWIW, one of my litmus tests is question “assuming we are supposed to measure distance in R2 using an L2 norm, why are we not supposed to measure distance in R3 using an L3 norm?”. And, like, I have a bunch of explanations for why this is (including “L3 isn’t invariant under most any change of basis” (per Alex’s argument above) and “b/c the natural notion of distance between two points in R3 factors into two questions of distance in R2, so using L2 in R2 pins down using L2 in Rn”), but I still feel like there’s some… surprising fixation on 2 here, that I can’t yet explain to the satisfaction of young-Nate who guesses that cuberute(x^3 + y^3 + z^3)=r is the equation for a sphere. Like, I still feel kinda like math said “proof by induction: starting in the case where n=2, …” and I’m like “wat why aren’t we starting at 0” and it’s like “don’t worry, n < 2 will work out as special cases” and I’m like “ok, sure, this argument is valid, but also wtf are you doing”. My wtfs aren’t all ironed out yet.

And, maybe I’m chasing shadows (eg, seeking a logical explanation where there is none, which is the sort of thing that can happen to a poor sap who lacks an explicit understanding of logical causality), but my suspicion is that I’m still missing part of the explanation. And both this route (which I’d gloss as “understand why it’s so important/natural/??? to have/choose/determine a canonical isomorphism between your vector space and its dual, then appeal to bilinearity”, with a prereq of better understanding why we have/care-about duality but not triality in Vect) and Alex’s (which I’d gloss as “explain why 2 obviously-should-be the balance point in the Lp norms”) both feel like good trailheads to me.

(And, in case this wasn’t clear to all readers, none of my assertions of confusion here are allegations that everyone else is similarly confused—indeed, Alex and Adele have already given demonstrations to the contrary.)

<3 hooray.

Thanks! I expect I can stare at this and figure something out about why there is no reasonable notion of “triality” in Vect (ie, no 3-way analog of vector space duality—and, like, obviously that’s a little ridiculous, but also there’s definitely still something I haven’t understood about the special-ness of the dual space).

ETA: Also, I’m curious what you think the connection is between the “L2 is connected to bilinear forms” and “L2 is the only Lp metric invariant under nontrivial change of basis”, if it’s easy to state.

FWIW, I’m mostly reading these arguments as being variations on “if you put anything else than a 2 there, your life sucks”, and I believe that, but I still have a sense that the explanation I’m looking for is more about how putting a 2 there is

positivelynatural, not just the best of a bad lot. That said, I’m loving these arguments, and I expect I can mine them for some of the intuition-corrections I seek :-)