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Sewing-Machine comments on One possible issue with radically increased lifespan - Less Wrong Discussion
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If your civilization expands at a cubic rate through the universe, you can have one factor of linear growth for population (each couple of 2 has exactly 2 children when they're 20, then stop reproducing) and one factor of quadratic growth for minds (your mind can go as size N squared with time N). This can continue until the accelerating expansion of the universe places any other galaxies beyond our reach, at which point some unimaginably huge superintelligent minds will, billions of years later, have to face some unpleasant problems, assuming physics-as-we-know-it cannot be dodged, worked around, or exploited.
Meanwhile, PARTY ON THE OTHER SIDE OF THE MILKY WAY! WOO!
You're picturing the far-future civilization as a ball, whose boundary is expanding at a constant rate. But I think a more plausible picture is a spherical shell. The resources at the center of the ball will be used up, and it will only be cost-effective to transport resources from the boundary inwards to a certain distance. If the dead inner boundary expands at the same rate as the live outer boundary, we'll be experiencing quadratic, not cubic growth.
You know, you're right. I will change my reply accordingly henceforth - linear population growth, linear increase in energy usage / computing power, and quadratic increase in (nonenergetically stored) memories.
Don't you get some pretty nasty latency on accessing those memories?
You get a linear increase in low-latency memory and a quadratic increase in high-latency memory.
And a linear increase in the latency of the high latency memory.