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The issues in dispute are: (1) about the nature of such a simulation, if reality is indeed a simulation; (2) about the final disposition of such a simulation; and (3) about crucial other factors not considered in defining the terms used in the mathematics. The conclusions to be shown here will be, correspondingly, (1) that simulations are probabilistically likely to be "What You See Is What You Get" (WYSIWYG) or “real” universes; (2) that reaching the level of being able to conduct such simulations ourselves may be a good, and not a fatal step, for humanity as Bostrom argues; and (3) that there may be many other kinds of universes with humans in them besides those created specifically to simulate humans, i.e. what Bostrom calls "ancestor-simulation" universes.

Bostrom's concept is fascinating to me as a computer scientist and science fiction writer, even if couched in gloomy language. There may be dispute as to the mathematics of the equation itself, but let’s take that for granted. It’s the reduction of the equation's terms into English meaning, both in definitions and in conclusions drawn, that are faulty – allowing for even more interesting interpretations.

On the Nature of a Simulated Universe

My first source of unease with Bostrom's argument is his assumption about the nature of the simulation, namely that the simulation is in some way inferior to the original. He mentions e.g. that one could omit the microscopic structure of the earth's core, or that the stars in our galaxy or beyond could be "highly compressed representations" just enough to fool our telescopes and space probes. He makes the fundamental statement, "Simulating the entire universe down to the quantum level is obviously infeasible, unless radically new physics is discovered."

The first question is why run a simulation at all? If one were a Suitably Advanced transhuman, and wanted to run a simulation, you'd pop open a "Petri Dish Universe^TM", twiddle the physical constants to match the ol' home universe, and let it run. No need for "simulation": The simulated space would be space.

Given that it’s conceivable that computational power of some flavor will keep increasing roughly per Moore's Law[7] of doubling every 18 months, it also seems likely we'll use that to advance knowledge in physics as well, and thus it seems not unlikely we’ll eventually whip up a workable theory of everything, and quite possibly learn to create universes. (Of course, all bets may be off if we first hit a Vingean technological singularity![9]) The computational power Bostrom suggests for running a simulation is 10³⁶ computations, which at current trends would be doable in roughly 100 years. We're already talking about several ways we might create universes[2,3,4,5,6], so it doesn't seem far off, in geological time anyway, that we could be creating real universes, not just simulating them. As shown below, acquiring the computational power needed to simulate ourselves would only be a brief stop on the road that quickly leads to creating universes "exactly" like our own. (Give or take a few mere centuries/millennia until we could do it.)

Thus if one is fixated on Bostrom's equation assuming an imperfect simulation, then you'd need to introduce a term to account for what percent of simulations are "perfect", i.e., truly biological WYSIWYG ("what you see is what you get") flesh&blood people in a universe of atoms, quarks, and black holes, vs. pretending to have all that (with or without having a Matrix-movie-like ability to "wake up" and see backstage into the simulation). The simulation might be achieved – but the “real,” the WYSIWYG, would likely be achieved soon thereafter and used ever more commonly, leaving the crummy "old style" simulations as historical curiosities. Thus, rendering the percent of imperfect simulations extraordinarily small compared to a large percent of “perfect” simulations, AKA real universes. While it would remain true that most humans live in someone else's creation and "simulation" per the "for a purpose" component of the definition of simulation, those simulations would not really differ from the reality they model.

For a transhuman sufficiently capable of simulating billions of minds, it should just be trivial to provide the microscopic level to the center of the earth just in case the simulants might look. Otherwise, those transhumans aren't very "trans" yet. (And will thus be a stepping stone to ones that are, shortly thereafter, who will do it right, and for much longer.)

Alternatively, applying Bostrom's argument that the odds are high that any given entity is in another entity's creation, then any such just-barely-trans-human would be a (better) simulation in a more-trans-human's Petri dish, and they in an even-more-trans-human's, etc.

Thus there is presumably a hierarchy of how close to reality each layer of simulation is. Indeed, following Bostrom’s logic that most universes are simulated, then one can go further when considering simulations within simulations (ad infinitum): At each layer the probability is that it, too, is a simulation, thus presumably simulated by a more powerful simulator. It’s simulated turtles all the way down – but the turtles get more and more real.

The question becomes: How powerful (i.e. “real”, “WYSIWYG”) is the simulator I'm in.

Given Sir Arthur C. Clarke's oft-quoted maxim about sufficiently advanced technology appearing to be magic, it seems the limit of these increasingly more powerful simulators=universes is for them to be the real thing; thus the average simulator, assuming the seriously large and perhaps infinite number of layers this suggests, is itself so close to a "real" universe as to be inconsequential. As a likely simulation ourselves, being close enough to even be imagining how to create real universes, even the "entry level" simulation would be quite realistic, and could even be equivalently real to what we're in.

In other words, if the technology needed to create a "real" universe is T_r, and a simulation requires T_s, and a civilization's rate of technological improvement over time is i(t), and most new simulations are within an order of magnitude or two of i(now), the question of what is the ratio of "imperfect" simulations to "real" universes is expressed as:

f_sim ' =

(width of interval from t₁ such that. i(t₁) = T_s to t₂ such that i(t₂) = T_r)

(width of interval from t₂ to t₃ such that i(t₃) = T_{end_of_civilization}).

No, that’s not a well formed formula, but it’s shorter that way. In words, this says that the probability of being in a “poor” simulation is the ratio between (1) the time it takes from when you can start doing simulated universes at all until you can do them so realistically they are what they seem; and (2) the time from when your simulated universes are what they seem until the end of your civilization (more precisely, the point after which your civilization never creates another simulated universe). (Note that this is formula is simplified in a generous way, in that it ignores even an increase in the number of simulations per time due to increased population.)

Although our technology currently follows Moore's Law, which is to say i(t) = 2^(t/1.5) for t in years, this may not hold true over longer periods of time. Nonetheless, it seems reasonable to assume that i(t) is generally increasing, such that i(t_i) > i(t_j) for t_i >> t_j. (In other words, assuming that while some knowledge may be lost over time, generally some knowledge is retained and built upon.)

The calculations hinge on t₃, the end of such a civilization. Clearly if a civilization terminates quickly after t₂ (or before), then f_sim ' is a higher ratio, since more simulations of an imperfect nature will have taken place than of a "perfect" nature.

If i(t) exponentially follows Moore's law, for example, then t₃ need not be long after t₂ for most simulations to be “real”: Assuming, conservatively, the number of new simulations per time is constant, N(t) = kt for some constant k, then the break even point at which 50% of simulations are real and 50% are imperfect occurs t = (t₂-t₁) time after t₂. (That is, there were S imperfect simulations between time t₁ and t₂; after t₂, at the same rate, there will have been S perfect simulations after time t. This is conservative in that it seems more likely the pace of creating simulated universes will increase over time before the end of civilization, not remain the same.)

The question is then how much more technology is needed to achieve t₂ after t₁. (How much more technology will it take after simulating a first universe poorly until one can simulate them so perfectly that the simulation is a WYSIWYG reality, a real universe.) Assume perfect, "real" simulations require Q times more technology than minimal simulations: T_r = Q T_s. To find t, assuming technological, computational capability increases per something like Moore's Law, i(t) = 2^(t/1.5), we have

i(t) = Q

2⁽^^t/1.5) = Q

t/1.5 = ln₂ Q years

t = 1.5 * ln₂ Q years

This leads to asking how much more technological power is required to create a "real" universe vs. an imperfectly simulated one (but one that nonetheless simulates all of human existence as Bostrom suggests). Suppose it required ten million times more technological power. Then Q = 10⁸, and t = 1.5ln₂(10⁸) = 39.8 years. Thus, 40 years after being able to "imperfectly" simulate all of human existence (to a degree the simulants can hardly tell it's a simulation), then a mere 40 years later we would be able to actually create a real universe just for grins. Should civilization fail several thousand years later, the number of real universes created would likely dwarf the number of imperfect (but extremely good) "Bostrom" simulations.

Suppose even that Q were a trillion, that it required 10¹² more know-how and computational ability to create an actual universe than simulate one (which seems unlikely, given that we're already conceiving how to create universes before we've even conceived how to simulate one brain). But even at Q=10¹², t = a mere 60 years.

History suggests that the rate of technological improvement is an exponential function of some degree (even if just because population increases exponentially; which it certainly could continue to do if we admit for artificial intelligences being born, colonizing space, etc.). The number of years do not significantly change even if the degree of the exponent is lower; that's the nature of exponential growth.

Under even stringent requirements of how much more know-how is required to construct a universe than simulate one, the time between accomplishing the former and the latter is likely only on the order of decades or low number of centuries. Given the likelihood of self-destruction by a society that's attained the incredible powers of t1 doesn't seem to be exponentially more likely to vanish between then and t2, or t years after t2, and perhaps for a "civilization's life expectancy" afterward of more centuries or millennia, then the ratio of Bostrom simulations to real universes created would be small.

This argues, then, that there is only a vanishingly small number of universes that would be "simulated" in the sense of "not being as real as they seem." Simplicity and Occam's razor would further suggest that what we see is what we get; that there may be simplifications in this universe vs. that of our creators' universe, but that ours is indeed the physical atoms & forces we perceive. Their's may, however, have elements beyond our ken. (Like if one were to create a viably independently thinking simulation in software: It would believe that the universe it inhabits, and the laws thereof, are the laws of its universe. And they would be. If theirs lacked a concept of gravity, they simply wouldn't think of it. Just as in Flatland, you can't miss what you have no concept of[8].)

There are, in fact, already known limits beyond which we cannot look: We face the Heisenberg uncertainty principle in observing at the particle scale, and we cannot observe the universe beyond what the speed of light allows. We cannot observe forward in time faster than the clock ticks, nor backward except along strict relativistic paths or via fixed recordings (e.g. film or the imprecision of human memory). The unit of simulation in our case, may simply be the universe.

Likewise, probabilistically, most created universes would be exactly as real as they appear to be.

Whew.

The author now sleeps easier, not having to worry whether there are actually atoms in the earth's core or real stars out there our progeny can reach.

What does Bostrom's Equation prove then?

So far, Bostrom's logic has not been disputed per se -- only the definition of "simulated universe." Substituting "real but created universes" for "simulation" in his argument one arrives at a similar conclusion: That we may likely be living in a constructed, rather than "original" universe.

This formulation would give a sort of probabilistic argument for the existence of "God," – or "a creator" to be more specific – perhaps allowing Pascal to rest easy with odds showing he should have won his wager. (Though the real question remains what the nature of this God is, their intent, whether we have a purpose in this thing or if like in the Hitchhiker's Guide to the Galaxy series, the earth is a giant computer; or whether it's an experiment of the "see where it goes" variety; or a more intent-directed purpose a la George R. R. Martin's "Sandkings," with creatures in their universe who are bred to war and worship. Not to mention it remains an open question whether any of the above are answerable or are as inherent and inscrutable as the Planck constant is to the nature of the universe; and so on.)

However, there are other possible interpretations, and there remains one overlooked element in Bostrom's Equation.

What's at the End of the Rainbow?

The next question about Bostrom's line of thought is the assumption that transhumans would pull the plug on an experiment nearing the transhuman threshold itself. Bostrum assumes this – that each simulation would get terminated (because the simulation completes its purpose, or out of fear it would become a threat to its creators, or what have you). If each universe is a real, true, WYSIWYG universe, then transcension might well be welcomed, not avoided. We might someday be welcomed as newborns. (Arguably even ones then trying to transcend to their next level. Either there's a plateau of advancement, where all entities – indeed, universes – are citizens of some sort, or else it really is turtles all the way down...)

Alternatively, even in a minimalist scenario of rather ordinary computers running rather ordinary software simulations, simulated humans may not be so different from "real" ones. We are already approaching a juncture where our (perceived) biological bodies may meld with our (perceived) electronic inventions (or other computational inventions, such as quantum computing). If being transhuman means, e.g., having software/hardware components to our existence, with some of our memories and thought processes done via software, where our physical bodies may ultimately become simply tools for manipulating atoms (and thus our "selves" may lie outside any specific body, but in a distributed fashion in a networked "computer" [electronic, quantum, or whatnot]) – then whether we are simulated or not right now may be irrelevant if once we meld with extra-human components we join the ranks of other transhumans who started us on the path. If we conduct an experiment with a human egg and human sperm in a test tube, we may well get a human baby: We don’t pull the plug just because we got a baby. It thus seems unwise to conclude that any transhuman Creators out there have an inescapable desire to pull our plug.

Hope for the Atheist

Now, there is hope for the atheist, or in this case, the believers in a universe not created for any purpose relating to humans. Bostrom's Equation assumes humans are the objects of simulation. More generically, if the universe is a simulation, there are, let's call them "entities" that are the object of the simulation or otherwise the purpose of creating a given universe (e.g., perhaps some universes are created for aesthetic reasons).

Thus it may be somewhat arrogant to call the term N in Bostrom's Equation "ancestor-simulations run by a posthuman civilization": The simulation creators may not be running ancestor simulations (but, perhaps, e.g. simulations of what happens with various values for the curvature of a circle, or using universes to generate heat to warm their plants, or bouncy balls for their children, or whatnot) or they may not be "posthuman" in the sense of "related to humans" at all, or, generally their reason for creating the universe may have nothing to do with humans in any way.

Clearly Bostrom's f_sim function ("what are the odds I live in a simulation?") can be stated alternatively as f_sim = H_sim / H_total for

U_sim = number of simulated universes created to study humans,

U_total = total number of universes with humans in them (Universes without humans in them at any time are irrelevant.)

H_sim = total number of humans living in all U_sim universes

H_total = total number of humans living in all U_total universes

In other words, the odds of my being someone created as part of a study of humans is the number of simulation-specific humans divided by the number of all humans, anywhere.

U_total includes spontaneous generation of universes with humans in them and universes created by any entity but not with the intent to study humans, but in which humans came into being. Bostrom identifies the formula for H_sim, but what about H_total? Similarly, what fraction of U_sim are created by humans to study humans, vs. created by others and in which humans simply appear?

To answer, consider the components of U_total:

U_total = U_created + U_noncreated

where

U_created = U_{Created(byAnyEntity)ToStudyHumans} + U_{CreatedByHumansNotToStudyHumans} + U_{CreatedByNonhumansNotToStudyHumans}

This captures the variety of ways in which universes could conceivably come into being, from spontaneously (as is seemingly widely assumed about our own Big Bang birth) to created by assorted entities with assorted goals. While of course we can't conjecture about the ratios of human-like creators to more alien, or about the motives of any such creators, it would be equally ill advised to assume that the only kinds of universes are those created by humans for humans, as Bostrom does.

Viewed through Bostrom's Equation, this assumption would be viewed as overly specific definitions of his terms. He defines f_I as the "fraction of posthuman civilizations that are interested in running ancestor-simulations" then suggests that this should not be low as it is against human nature. Yet the unexplored option that may hold the greatest probability is that humans may exist in universes for a variety of reasons other than for purposes of ancestor-simulations. We may be frequently occurring side effects (as dew happens when temperature and humidity mix), or part of a standardized "kit universe" purchased by incomprehensible entities, we may be decorative, or fulfill a desired minor symbiotic (or even parasitical) function, or any number of reasons we can't even comprehend.

While Bostrom's mathematical expression may have inexorable logic about which terms must go to zero or one, there is great leeway in the interpretation of the meaning of each variable.

Conclusion

There are three fundamental issues to be considered with Bostrom's conjecture about the likelihood of our living in a simulated universe. The first is the nature of "simulation," and how closely it matches reality; it is here argued that it is more probabilistically likely that among universes created for study, "What You See Is What You Get" or "real" universes would greatly outnumber imperfect simulations, thus we each likely live in a universe that is, in fact, as real as it seems.

The second issue involves what would happen when an "ordinary" human civilization achieves the technology to transcend to the level of the creator of its own universe (if there is such an entity). Bostrom tends toward the gloomy projection that the creating entity would pull the plug; whereas it is here suggested as but one alternative that we might be welcomed as a newly born transcendant entity.

This is also a possible answer to the Fermi paradox: What if advanced entities always evolve and leave this universe, hence they aren't coming to visit. Just like a caterpillar doesn't crawl around in the trees after it becomes a butterfly. Just a random thought.

Finally, the underlying assumptions and definitions of the terms in Bostrom's equation are analyzed to demonstrate that other meanings may have been overlooked, and thus the conclusions drawn about the purpose of human existence in any given universe (i.e. for purposes of ancestor-simulation or not) are too limited in scope, and have repercussions on his conclusion that we live in a simulated universe for the purpose of studying our ancestors: It is entirely possible human civilizations have arisen for many other reasons in universes, thus it is premature to draw conclusions about the probabilities of either our purpose or whether our universe is an imperfect simulation.

As an aside, another reassuring result is that the destruction of our civilization may not be as imminent as Bostom suggests. (Remember his formulation is that, more or less, we’re either in a simulation or The End Is Near.) The End need not be so near.

Whatever the reality, for the record, I’m glad to be alive and living in what appears to be a real universe, and am quite optimistic for our future!

References

[1] Nick Bostrom. "Are You Living In a Computer Simulation?" Philosophical Quarterly, Vol. 53, No. 211 (2003).

[2] S. Blau, E.I. Guendelman and Alan H. Guth, "The Dynamics of False Vacuum Bubbles" Phys. Rev. D 35, 1747 (1985).

[3] K. Ghoroku, "Interacting Universes and Their Quantum Birth from Nothing", Proceedings of the Sixth Marcel Grossman Meeting, Kyoto (June 23-29 1991).

[4] Gurevich L, Mostepanenko V Phys. Lett. A 35 201 (1971).

[5] M. Kaku Hyperspace, Anchor (1995).

[6] S. Hawking, "Baby Universes, Children of Blackholes" in Black Holes and Baby Universes and Other Essays, Bantam Books (1993).

[7] Gordon E. Moore. Cramming More Components Onto Integrated Circuits. Electronics, Vol. 38, No. 8 (April 19, 1965).

[8] Edwin A. Abbot, Flatland (1884).

[9] Vernor Vinge, “Technological Singularity”, VISION-21 Symposium sponsored by NASA Lewis Research Center and the Ohio Aerospace Institute, March 30-31, 1993 (http://www.ugcs.caltech.edu/~phoenix/vinge/vinge-sing.html )