reperiendi

Theocosmology

Posted in Theocosmology by Mike Stay on 2007 July 23

The View from the Center of the Universe says we’re the first generation that can tell stories about the creation that might actually be true. We should take the mythological imagery we have around us and reinterpret it so that it corresponds to something real instead of something imagined, thus giving people a sence of place in the universe.

Their stories, by most accounts, weren’t very good. But I believe they’re on the right track: stories are how humans understand the world. It’s part of why I like John Baez’ teaching style so much: he tells stories. A search on his site for the word “story” turns up hundreds of occurrences!

Gödel, Escher, Bach talks about string rewrite rules and theorem proving, trying to illustrate the difference between syntax (structure) and semantics (meaning). He states some rules like

xpyqz => sxpyqsz

and then suggests that “sx” is the “successor” of “x”, and thus one interpretation of “p” and “q” is “plus” and “equals,” respectively. But it could just as easily be “equals” and “taken from.” How do we know what a structure “really means?” Does that question even make sense?

Universal algebra is a language for describing simple structures like sets, monoids, groups, rings, and modules (but not fields). Lawvere showed that every algebraic theory corresponds to a cartesian category, and that a functor from the cartesian category to Set gives a model of the theory. The Yoneda lemma says that there’s a one-to-one correspondence between the set of models of a theory and the set of functors out of the theory. So any model has to arise as a functor out of the theory.

This lets us separate syntax from semantics, structure from meaning. The theory is a Platonic ideal that has models in the “real world” of sets and functions. Of course, now that the “real world” has been revealed to be quantum, we should consider functors into other categories, too: a functor into Hilb assigns quantum meaning to the structure. This is how Feynman diagrams are used for quantum field theory: every line represents a Hilbert space of states instead of a set, and each vertex represents a linear operator instead of a function.

Usually, what a structure “means” can only really be answered by considering all of the possible meanings together and looking at the relationships among them. But that’s a very hard task in most cases. It’s easier to have a single, typical instance in mind: what do you think of when I say “undirected graph?” It’s probably an image of a not-so-special graph, not the mathematical definition.

Choosing some example is essential to the way we learn. Adam and Christ, the “last Adam,” are the prototypes used by the Gospel. But there are a lot of details left out.

I’m trying, in these Theocosmology posts, to build up a backstory for my existence, which is the story I’m most interested in; I’m choosing details that may or may not be true, but are as true-to-life, as typical as I know how to make them.

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4 Responses

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  1. Alma said, on 2007 July 25 at 12:37 pm

    I remember Bro. Nibley dropping the phrase “symbolic as far as the man and the woman are concerned” in a Pearl of Great Price class. You are probably a few years too young to have met that phrase in its original setting, but I remember it, and miss it.

  2. mike said, on 2007 July 25 at 2:47 pm

    Welcome! The phrase in question, for anyone else who might be reading, is “They will organize man in their own likeness and image, male and female. This, however, is simply figurative, so far as the man and the woman are concerned.”

    How did you interpret it? Why do you miss it?

  3. ATW said, on 2007 July 26 at 9:35 pm

    Yeah, “figurative” is right. So what we have is not primarily a description, or even a particular story, but rather a kind of stem cell for a class of story. I miss the reminder that truths about agents are organic much more than crystalline.

  4. ATW said, on 2007 July 26 at 10:32 pm

    John Baez is a rich source of counterexamples to the unmotivated describing that often passes for professional demeanor in mathematical disciplines—Baez sure can turn a story. I envy you.


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