As with the physical entities described by the math in Baez's book, I feel that I have a leg up on understanding the math but not so much on the relationship to the described entities. It must be my aversion to the real world.
--- Frank C. Wimberly 140 Calle Ojo Feliz, Santa Fe, NM 87505 505 670-9918 Santa Fe, NM On Tue, May 26, 2020, 8:21 PM Jon Zingale <[email protected]> wrote: > 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' > -- --- .-. . .-.. --- -.-. -.- ... -..-. .- .-. . -..-. - .... . -..-. . > ... ... . -. - .. .- .-.. -..-. .-- --- .-. -.- . .-. ... > FRIAM Applied Complexity Group listserv > Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam > un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com > archives: http://friam.471366.n2.nabble.com/ > FRIAM-COMIC http://friam-comic.blogspot.com/ >
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