Recent discussions with Nick Bostrom (e.g., the Simulation Argument) have prompted this commemoration of some previous thoughts. The magnitudes of the following simple numerical estimates appear nonsensitively to imply that humanity is a digital simulation ("sim"). Detecting the computational error which must surely lurk herein is left as an exercise for the reader.

I. Some Assumptions

• Generously assume 100 teraflops per human brain (~10¹⁵ bits/brain-sec, assuming 8-bit words)
• ~10¹⁰ human brains have ever lived, to date.
• Each human brain has lived ~10⁹ seconds.

è Total bits processed by all human brains, ever, is ~10³⁴ bits.

• Capacity of conventional mechanical nanostorage is ~1 bit/nm³ (equivalent to linear DNA, approximately midway between nanomechanical registers and spooled nanotape; NMI:345).

è Physical volume of nanomechanical computronium to store all human thoughts that have ever been thought, to date, is ~10⁷ m³ (a ~215-meter cube).

• A conventional reversible mechanical nanocomputer processes ~10¹⁷ bits/watt-sec [NS:1] or ~10³² bits/sec-m³ [NMI:344].

è Physical volume of computronium to compute all current human thought is ~10^-7 m³.
è Computronium power required to compute all human thought is ~10⁸ W (vs. ~10¹¹ W for all biological human brains currently alive).

• Maximum human sensory bandwidth (primarily visual) is ~10⁸ bits/sec.
• If ~10¹⁰ human sensorial sets have ever existed, and each sensorial set received sensory data for ~10⁹ sec, then ~10²⁷ bits of sensory data have ever been received by all human sensorial sets that have ever existed.
• Assume each bit of sensory data requires ~1000 bits to simulate.

è ~10³⁰ bits must be processed to simulate all human sensory data ever received by all human sensoria, up to the present.

 

II. Some Arguments

Computation Power arguments:

• Global hypsithermal power limit on planet Earth is ~10¹⁵ W (NMI:175).
• Number of ancestor sims that could be run on Earth at hypsithermal limit is ~10⁷.
• Assume all ~10⁷ possible sims are of equal probability to the one "reality" (as subjective observers, we cannot distinguish with any certainty whether we inhabit reality or a simulation).

è Probability we live in a sim is ~(1 - 10^-7) = 99.999 99%.

• Power available to Type II "natural" civilization is ~10²⁶ W (~1 solar output).
• Number of ancestor sims that could be run by "natural" Type II civilization is ~10¹⁸.
• Assume all 10¹⁸ sims are of equal probability to the one "reality".

è Probability we live in a sim is ~(1 - 10^-18)= 99.999 999 999 999 999 9%.

Storage Volume argument:

• Volume of the terrestrial biosphere is ~10¹³ m³.
• Volume of all planetary mass in solar system, theoretically accessible to a "natural" Type II civilization, is ~10²⁴ m³.
• The number of ancestor sims whose entire mental data production could be stored ranges from ~10⁶ (all terrestrial biosphere) to ~10¹⁷ (for "natural" Type II civilization).
• Assume all possible sims are of equal probability to the one "reality".

è Probability we live in a sim ranges from ~(1 - 10^-6) = 99.999 9% to ~(1 - 10^-17) = 99.999 999 999 999 999%.

Computation Volume argument:

• Assume the same volumes as in the Storage Volume argument, above.
• The number of ancestor sims whose entire mental data production could be computed ranges from ~10¹⁹ (all human artifacts) to ~10³¹ (for "natural" Type II civilization).
• Assume all sims are of equal probability to the one "reality".

è Probability we live in a sim ranges from ~(1 - 10^-19) to ~(1 - 10^-31).

Sensory Simulation argument:

• A computer must process ~10³⁰ bits to simulate everything that all humans everywhere, anywhere, have ever perceived sensorily.
• Conventional reversible mechanical nanocomputer processes ~10¹⁷ bits/J.
• Thus ~10¹³ J of energy are required to process all human sensory data to date.
• The fusion energy available from jovian dismantlement, assuming 0.6% mass-energy conversion efficiency and assuming ~100% jovian mass is fusionable, is ~10⁴² J.
• The number of ancestor sims whose entire sensory experience could be computed by a jovian-burning "natural" Type II civilization is ~10²⁹.
• Assume all 10²⁹ sims are of equal probability to the one "reality".

è Probability we live in a sim is ~(1 - 10^-29) = 99.999 999 999 999 999 999 999 999 999%.

III. A Conservative Conclusion

So even if these estimates err by many orders of magnitude in the contrary direction, perhaps because only a small fraction of the maximum possible number of sims are actually being run, the overall conclusion should remain essentially unchanged. The probability that we live in a simulation, rather than the reality, may range from (1 - 10^-6) to (1 - 10^-31).

Last updated on 2 November 2002