You are important. Every small action counts.
Exponential impact, Chaotic systems, Discreteness and Turing Completeness
“You are important. Every small action counts.”
That’s what they say. And I believe them. I just didn’t know why.
Until I realized that this world is the ultimate system that exhibits chaotic properties.
Systems that repeatedly use their own output as input often become chaotic because small differences in the initial state grow exponentially through recursive feedback.
That means small actions that we take now, have an exponential effect on the future. In our world at least, exponential functions grow fast, so really, once sufficient time passes, all actions have an huge impact, and the relative differences in magnitude in the initial state is less important.
My only question is how does this help if the system is really chaotic? The impact is large but unpredictable. If the system is truly chaotic then any initial intention would be subsumed into a vortex and spit out in an unrecognizable form.
The answer would be that the system is not truly chaotic in the sense that we understand it. Maybe intentions are atomic in a way that while they can sometimes be reduced to smaller “pieces”, there is a minimal “size” of intention that do not get warped or decomposed while within the vortex, and they end up more or less the same when they come out on the other side.
This aligns with the stories that are told about karma, where intentions purportedly originating from thousands of years ago come to surface now, and somehow affect us.
This would be a very interesting kind of system to experiment with actually. I wonder whether LLMs can help design artificial systems that exhibit this behavior?
Spoiler warning
(To the reader: I think this is a very interesting puzzle, and I have a very interesting mathematical answer. To avoid spoiling yourself you can stop here and think a bit first)
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Anyway I thought about this myself as I was pasting the above to a LLM. And I thought the chaotic behavior where initial features get smeared into unrecognizable forms is caused by the system being built from real numbers (as opposed to integers). Maybe discrete systems would preserve atoms. What would be a system that feeds on itself and is discrete?
And then I received this thought -- Connor’s Game of Life is discrete!
I quickly searched online to see whether anyone commented on whether it is a chaotic system or not.
Then, I came across one of the best reddit comments/answers I’ve seen in a long time:
https://www.reddit.com/r/askscience/comments/7bt954/comment/dpm21tm/
tl;dr: GoL is Turing Complete, and we can simulate chaotic systems with that.
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One question still remain though:
- We now know that GoL can exhibit chaotic behavior if we set it up to be, and that initial states can potentially have exponential impact, also features (like gliders) can be preserved indefinitely... But are there systems that ALSO exhibit the behavior that *all* the initial states are exponentially impactful?
I don’t know enough about chaotic systems to fully understand how that would work.