I have paraphrased Nick Bostrum's three options in the chart below. A posthuman stage is when we are able to make sims. A sim is a human being simulated by a computer that is indistinguishable from ourselves.
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| A posthuman state is not possible. | 1. We CAN'T make sims. |
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| A posthuman state is possible. | 2. We DON'T make sims. |
| | 3. We DO make sims. |
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The OP notes the following about the probability of us being sims.
...if there exist trillions and trillions of simulated worlds that are
similar to ours and there is only one real world, we are probably
living in a simulated world.
Bostrum observes something similar (page 6):
Posthuman civilizations would have enough computing power to run
hugely many ancestor‐simulations even while using only a tiny fraction
of their resources for that purpose.
Bostrum's calculation for the fraction of sims is the following (page 7):
fsim = fp N H / ((fp N H) + H)
fp is the fraction of civilizations able to make sims. This would be 0 if option 1 holds. N is the average number of simulations per civilization. This would be 0 if option 1 holds. H is the average number of humans who lived in the civilization prior to reaching posthuman stage. And fsim is the fraction of sims out of all humans, simulated or real. Again, this would be 0 if option 1 holds.
Assume also the "blind indifference principle" that "you have no
information that bears on the question of which of the various minds are
simulated and which are implemented biologically" (page 8), then the chance of any of us being a sim is high.
Now consider the OP's questions:
If the claim that one of the three mentioned propositions is true
holds, the example I gave should hold as well, but why would it? Why
would pressing a button determine whether or not we're living inside a
simulation?
If we actually CAN press a button and therefore DO want to turn on a simulation capable of making a sim as defined above, then that is evidence that we are in option 3. If that should happen, by assumption we can make a lot of sims. If that should happen, by assumption we cannot tell sims from real people, that is, we would not be able tell whether we are sims or not.
Given the large number of sims compared to real humans and our inability to distinguish between the sim and the real human, we are more likely to be sims than real.
Some things need to be noted:
- No one has made any sims to date.
- John Searle's Chinese Room Argument suggests that we cannot make sims.
- Michael Rescorla raises further issues about the underlying computational theory of mind:
Advances in computing raise the prospect that the mind itself is a
computational system—a position known as the computational theory of
mind (CTM). Computationalists are researchers who endorse CTM, at
least as applied to certain important mental processes. CTM played a
central role within cognitive science during the 1960s and 1970s. For
many years, it enjoyed orthodox status. More recently, it has come
under pressure from various rival paradigms. A key task facing
computationalists is to explain what one means when one says that the
mind “computes”. A second task is to argue that the mind “computes” in
the relevant sense. A third task is to elucidate how computational
description relates to other common types of description, especially
neurophysiological description (which cites neurophysiological
properties of the organism’s brain or body) and intentional
description (which cites representational properties of mental
states).
Given these three points above, of the three options Bostrum presents only option 1 appears to be possible.
Reference
Bostrom, N. (2003). Are we living in a computer simulation?. The Philosophical Quarterly, 53(211), 243-255.
Rescorla, Michael, "The Computational Theory of Mind", The Stanford Encyclopedia of Philosophy (Spring 2017 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/spr2017/entries/computational-mind/.
John Searle, "Minds, Brains and Programs" reprinted in Haugeland, John, ed. Mind design II: philosophy, psychology, artificial intelligence. MIT press, 1997.