Rob Bensinger

Communications lead at MIRI. Unless otherwise indicated, my posts and comments here reflect my own views, and not necessarily my employer's.

Multiplicative Operations on Cartesian Frames

And now I've made a LW post collecting most of the definitions in the sequence so far, so they're easier to find: https://www.lesswrong.com/posts/kLLu387fiwbis3otQ/cartesian-frames-definitions

Additive and Multiplicative Subagents

I'm collecting most of the definitions from this sequence on one page, for easier reference: https://www.lesswrong.com/posts/kLLu387fiwbis3otQ/cartesian-frames-definitions

Multiplicative Operations on Cartesian Frames

For my personal use when I was helping review Scott's drafts, I made some mnemonics (complete with silly emojis to keep track of the small Cartesian frames and operations) here: https://docs.google.com/drawings/d/1bveBk5Pta_tml_4ezJ0oWiq-qudzgnsRlfbGJgZ1qv4/.

(Also includes my crude visualizations of morphism composition and homotopy equivalence to help those concepts stick better in my brain.)

Biextensional Equivalence

Scott's post explaining the relationship between and exists as of now: Functors and Coarse Worlds.

Controllables and Observables, Revisited

To get an intuition for morphisms, I tried listing out every frame that has a morphism going to a simple 2x2 frame

.

Are any of the following wrong? And, am I missing any?

Frames I think have a morphism going to :

Every frame that looks like a frame on this list (other than ), but with extra columns added — regardless of what's in those columns. (As a special case, this includes the five frames corresponding to the other five ensurables that can be deduced from the matrix: , , , , . If has more than four elements, then there will be additional frames / additional ensurables beyond these seven.)

Every frame biextensionally equivalent to one of the frames on this list.

Biextensional Equivalence

An example of frames that are biextensionally equivalent to :

... or any frame that enlarges one of those four frames by adding extra copies of any of the rows and/or columns.

Biextensional Equivalence

They're not equivalent. If two frames are 'homotopy equivalent' / 'biextensionally equivalent' (two names for the same thing, in Cartesian frames), it means that you can change one frame into the other (ignoring the labels of possible agents and environments, i.e., just looking at the possible worlds) by doing some combination of 'make a copy of a row', 'make a copy of a column', 'delete a row that's a copy of another row', and/or 'delete a column that's a copy of another column'.

The entries of and are totally different (, while , before we even get into asking how those entries are organized in the matrices), so they can't be biextensionally equivalent.

There *is* an important relationship between and , which Scott will discuss later in the sequence. But the reason they're brought up in this post is to make a more high-level point "here's a reason we want to reify agents and environments less than worlds, which is part of why we're interested in biextensional equivalence," not to provide an example of biextensional equivalence.

Additive Operations on Cartesian Frames

Scott's Sunday talk, covering content from this post and the Intro post: https://www.youtube.com/watch?v=H1tJdaCvcck

I've added the section-2 definitions above to https://www.lesswrong.com/posts/kLLu387fiwbis3otQ/cartesian-frames-definitions.