[FRIAM] The danger of a single story

[Jon Zingale]

I don't suppose that the existence of p-adic completion, via equivalence
classes of Cauchy-sequences, helps matters. Such a step moves the graph-
theoretic talk back in the direction of space-theoretic talk, along with
the arrival of derivatives and a meaningful way of doing calculus. What
such a space "looks like" in that completion is beyond me.

It appears to me that the role of "visualization" in the learning of
p-adic-normed spaces is (at least here in these humble beginnings) less
for *establishing* new hypotheses via visual intuitions, but rather for
*fortifying* knowledge that has been hard-won through more "operational"
reasoning. Something akin to *accounting* or *library science*.

That p-adic norms are rooted in logarithms makes me wonder about how
easily they might be deployed in early childhood education, working with
a child's *natural* sense of scale [log]. I am also left wondering if
there is a meaningful connection to proprioception here, a connection
between the logarithmic and the operational, analogous to moving from a
local group structure to an underlying Lie algebra [exp].

Continuing on with these musings just a bit longer than I should, I am
imagining factorization in the form of spectra. Choosing a prime p is
like choosing a hue of light with which to illuminate an image, the
closeness of two values being a closeness in saturation. Illumination
orders the image, rendering some aspects indistinguishable and rendering
others distinct. There is something "objective" about such a norm that
orders by "affinity", that structures space by abundances of a given kind,
like tuning an oscilloscope to 440hz and watching your favorite song.

[log] https://www.scientificamerican.com/article/a-natural-log/
[exp] https://en.wikipedia.org/wiki/Exponential_map_(Lie_theory)
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