I like your argument, Rafal. If I might summarise, I think this is what you
- There are universes N which occur naturally, from "pure mathematics",
which contain humanlike civilisations (enough like our universe for purposes
of this argument).
- There are universes S which occur as simulations inside other universes,
which are humanlike civilisations (as above).
- The space of all mathematics of which N is the same order is larger than
the space of simulations S, thus we are more likely to be in N than S, and
thus not in a simulation.
Is that about right?
If so, I would counter by saying that the space N cannot be larger than the
space S, and in fact should be of a lower order (aleph thingy?) than S.
Take this from the hierarchical conception of the full space of applicable
universes. Note that universes in N spring out of "mathematics per se".
Whereas elements of S can derive from elements of N, or from other elements
So, if each n in N has some non-zero probability of creating a sub-hierarchy
of elements of S, where the depth of that hierarchy is infinite, we get an
infinite number s's in S for each n. ie: the order of S is higher than N, so
that you are vanishingly unlikely to actually exist in an n, rather than an
Now, I am not sure if your argument shows below that the order of N is as
high as it can be (equating it with the space of all mathematics?). I'd
question that claim, if you are making it (not sure). If that was the case,
it would follow, I guess, that the space of simulations must be constrained
in some way to match it.
From: Smigrodzki, Rafal [mailto:SmigrodzkiR@MSX.UPMC.EDU]
Sent: Tuesday, 11 December 2001 5:36
Subject: RE: New website: The Simulation Argument
I just read Nick Bostrom's article:
Not being a physicist or mathematician I feel a bit apprehensive about
commenting on it, lest I say something embarrassing, but let me try to
explain how I could try to disagree with the article's conclusion:
1) Let's assume that the whole of reality is a mathematical construct - a
self-evolving formal system, not constrained by the laws of physics (such as
conservation laws), whose infinitesimally small subset is our observable
physical world, with it's laws ultimately derived from a set of axioms and
rules of production. No entity external to the totality of this system
exists, hence it is not a simulation, but rather an explosion of pure
mathematics, arising spontaneously and unavoidably from a true and absolute
nothingness, in all the unimaginable ways pure math could develop, with new
extensions of the axiom set adding themselves whenever undecidable theorems
2) We cannot place ourselves with absolute certainty within any level of the
system - there is an unknown and perhaps unknowable number of layers
separating our physical reality from the mathematical constructs which we
are capable of understanding (contain within our minds), such as the roots
of number theory, and the empty set.
3) Since the number of levels of the system (its axiomatic extensions) is as
large as mathematics itself, infinite in a way beyond the reach of our
minds, it follows that subsets of the system capable of supporting our
physics should arise in an infinity of levels, over and above any of the
branching patterns predicted by the multiuniverse interpretation of QM. It
is somewhat similar to the infinite formation of patterns in a fractal, such
as the Mandelbrot set, where the central black rosette comes up over and
over again, arbitrarily large or infinitesimally smaller than the first
instance you calculate, except the whole system is not fractal - new
extensions are not blowups of lower levels, although they do contain
repetitions of the smaller sets as their constituent parts.
4) Some of the instances of worlds indistinguishable from ours will occur
within computational devices used by sentient entities living on higher
levels, farther removed from the root of the system. Let's call them S
5) Some of the instances of worlds indistinguishable from ours will occur as
subsets of branches of mathematics per se, like fractals, without any
conscious entities producing them. Let's call them N.
6) Both N and S are infinities, and if most of the system consists of
branches which are not sentient, then N>>S.
7) If N>>S, then it is indeed possible that a vast majority of
instantiations of our minds would exist without being simulated, while
allowing a high likelihood of developing into posthumans, infinitely large
numbers of whom would run ancestor simulations (e.g. as part of applied
cosmology experiments, without any moral motivations).
So this is the fourth proposition. Metaphysics at it's worst, I'm afraid,
probably unprovable and unfalsifiable, but pleasant.
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