Reading such stuff the other week-end I noticed (a) a heap of geometrical metaphors and (b) limited geometrical vocabulary.
— isomorphismes (@isomorphisms)
In my opinion functional analysis (as in, precision about mathematical functions—not practical deconstruction) points toward more appropriate geometries than just the
[0,1] of fuzzy logic. If your goal is to escape “either/or” then I don’t think you’ve escaped very much if you just make room for an “in between”.
ℝ→ℝ functions (even continuous ones; even smooth ones!) can wiggle out of definitions you might naïvely try to impose on them. The space of functions naturally lends itself to different metrics that are appropriate for different purposes, rather than “one right answer”. And even trying to define a rational means of categorising things requires a lot—like, Terence Tao level—of hard thinking.
I’ll illustrate my point with the arbitrary function ƒ pictured at the top of this post. Suppose that ƒ∈𝒞². So it does make sense to talk about whether ƒ′′≷0.
So “fits the binary” is rarer than “doesn’t fit the binary”. The “borderlands” are bigger than the staked-out lands. And it would be very strange to even think about trying to shoehorn generic 𝒞² functions into
- one type,
- the other,
- or “something in between”.
Beyond “false dichotomy”, ≶ in this space doesn’t even pass the scoff test. I wouldn’t want to call the ƒ I drew a “queer function”, but I wonder if a geometry like this isn’t more what queer theorists want than something as evanescent as “liminal”, something as thin as "boundary".