[FRIAM] privacy games

[Jon Zingale]

Glen,

I am really enjoying steelmaning and getting steel-mansplained,
so thank you for the discussion up to this point. For me the
images are that of the constructions for free modules
<https://en.wikipedia.org/wiki/Free_module> over a ring,
dependent types, and algebraic varieties
<https://en.wikipedia.org/wiki/Algebraic_variety>. In the first case, there
is a natural interplay between the category of sets and the category
of modules, equipped with a map describing *inclusions* of bases into
the collection of vectors they span, and a map describing how to
*evaluate* a vector in a context to return a single number. The thing
I find relevant here is that while the evaluation function does
much to *found* the module (a vector space say), the *evaluation* is
not what is interesting about the module. What is interesting is that
we have a playground to talk about *dimension*, to make metaphors about
phenomenological experiences of space, and most importantly to play and
entertain one another. The whole game comes to an end the minute we
finally concatenate the *evaluation* function onto the end of our
compositions.
The entire notion of space collapses and we are left with a single number.
To your comments on free/bound variables, I can interpret these bases as
bindings for the underlying ring and the coefficients as representing free
variables (do I have that right?).

I don't have much to write that is specific to dependent types
that would be all that different from the algebraic variety image,
so let me jump next to there. Varieties are often described in terms
of comprehension or inverse images. For instance in pseudo-Haskell
I can write:

conicVariety = [ (x,y,z) | (x,y,z) <- R3, x² + y² + z² == 1]

Varieties as you can image get pretty nasty (singularities, cusps, etc..)
This no doubt made the development of algebraic geometry much more
treacherous than the study of manifolds, its tamer sibling. What is
novel about the varieties like conicVariety above is that it can be
understood in terms of sections. We can interpret the function above
as asking for the collection of all triples which all map to 1, and
when we do we have a fiber in hand. What is relevant here is the image
that as we collapse states-of-affairs onto objects of designation,
we get variety-like objects in the space of affairs. As conversants
collaboratively build fibers over designations, they are constructing
eidetic variations of concepts. Somehow in the sense of EricS, the
collection of these variety-like concepts are personal and irreducible
complexes of meaning.

Jon

ps. I will look up 'Dies the Fire'
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