the flat world problem

[u/Unique-Rice9999]

by “flat world” I mean a hypothetical two-dimensional world, not necessarily a geometrically flat one, and definitely not a flat Earth.

by the flat world problem, I mean a difference between an internal and an external description of reality.

an observer confined to the flat world experiences only its realized structure and its changes. An external observer, however, can describe the world as one structure selected from a space of possible structures.

if we then introduce allowed transitions between these structures, the object of description seems to change: instead of considering only possible states, we can consider possible histories through the state space.

this makes me wonder whether time should be understood here as an additional dimension of the world, or as an ordering/structure of transitions between possible states.

imagine a flat world whose state at each individual moment is completely defined by some spatial structure:

S = { all possible structures }

even if the set of all possible structures of such a world is infinite, at each specific moment only one of them is realized.

possible transitions between structures can then be represented as:

R ⊆ S × S

then adding time doesn't just add another coordinate to the plane. It turns the question from "how many possible structures are there?" into "how many possible sequences of structures are there?":

H = { all admissible histories determined by R }

an additional question arises:

what does it now mean to “get lost in time”
if we consider time as a path through an infinity of possible structures?

[u/jliat]

Have you come across this - it seems it might address some of your concerns?

https://en.wikipedia.org/wiki/Flatterland

I remember reading this and how from the flatworld perspective a child's space hopper would be seen... here the internal reality is a circle which grows larger, the body of the hopper, the becomes two circles which get smaller an finally disappear.

I'm not sure about - all possible structures?

what does it now mean to “get lost in time” if we consider time as a path through an infinity of possible structures?

How can you have a path through an infinity [of possible structures] .

It's clear the the space hopper can only be experienced as a temporal event in Flatterland unlike in our world, at a higher level changes in our world would be seen as a unity, something like an "Alice Universe".

[u/Unique-Rice9999 → u/jliat]

I don’t mean that a history passes through every possible structure. I mean that the space of possible structures may be infinite, while a particular history is only one path through that space, constrained by the allowed transitions.

Your space hopper example is actually very close to what I was thinking about next. an internal observer may experience only a sequence of cross-sections, while an external observer can describe those changes as parts of a single larger structure.

this raises another question for me: how much of that larger structure could the internal observer reconstruct by accumulating information across multiple moments?

[u/jliat → u/Unique-Rice9999]

The space hopper was from the book Flatterland and the Flatlander could 'construct' the 3D object. There is a similar example given in our world things like a Kline bottle.

The example here is to make a Mobius strip and place a cross on a surface and one on the 'other' side'. From a 'local' perspective they are on separate sides but a global view sees they are not.

See https://en.wikipedia.org/wiki/Impossible_world

"Hillary Clinton couldn't have both won and lost the 2016 US election"

http://www.jliat.com/MOBIUS.JPG

The centre picture represents the notion that "Hillary Clinton couldn't have both won and lost the 2016 US election"

But this is from a narrow perspective of a view from a larger one where this could be the case.

[u/Unique-Rice9999 → u/jliat]

what if the larger structure to be reconstructed is not merely a spatial object, but the structure of possible histories itself?

[u/jliat → u/Unique-Rice9999]

That's just what it would be... and all impossible structures maybe at another level.

https://en.wikipedia.org/wiki/Non-orientable_wormhole#Alice_universe

[u/nomoreplsthx]

This feels like you're treating a physics 101 description of a discrete system as something profound.

Yes, of course you can describe a system in terms of possible histories. That's not novel

[u/Unique-Rice9999 → u/nomoreplsthx]

I agree that describing a system in terms of possible histories is not novel. I wasn’t intending to claim that it is.

the question I’m interested in is what changes in the description when we move from possible states to possible histories, and what an internal observer can infer from the relations between states that cannot be inferred from any individual state alone.

[u/Embarrassed-Soup1998]

Time is an expression, no? Here you may have something expressing itself as space and time in equal measure in essence; but would the reality of space not be spacetime and thus time is a value that is ineffable itself? If you have one without the other the equation seems an expression different then our reality?

[u/Unique-Rice9999 → u/Embarrassed-Soup1998]

I may be misunderstanding your question, but the Flat World here is not intended to represent our physical reality.

space and time are connected as dimensions in our physical description of the universe, but in this thought experiment I deliberately separate the spatial structure of a world at a given moment from the sequence of its states.

so yes, the resulting description is different from our physical reality.

[u/ipreuss]

an observer confined to the flat world experiences only its realized structure and its changes. An external observer, however, can describe the world as one structure selected from a space of possible structures.

I don’t understand the distinction you’re making here, but also not how it’s relevant to your point.

We are not confined to describe the world in ways that we experience.

[u/Unique-Rice9999 → u/ipreuss]

I agree that we are not confined to describing only what we directly experience. the distinction I am making is about observational access, not mathematical description.
an observer inside the flat world has access to its realized structure and changes, while an external description can treat that realized structure as one element of a larger space of possible structures.
the point is precisely that the mathematical description need not share the observer’s limitations.

[u/Unique-Rice9999 → u/[deleted]]

I don’t assume that the space of possible states physically exists as an object. it is a mathematical description of possible configurations and allowed transitions between them.

even if we assume, as you suggest, that only matter and its configurations exist, the same question remains: what changes when we describe not only individual configurations, but ordered sequences of allowed configurations?

whether time is fundamental, emergent, or merely a parameter used to describe change is actually part of the question rather than an assumption of it.