Limits of Imagination


Our finite universe simply cannot continue our exponential growth rates for a million years. For trillions of years thereafter, possibilities will be known and fixed, and for each person rather limited.

Bryan Caplan:

He’s probably right for physical goods. But why couldn’t the quality of life in virtual reality grow at 4% [per year] for ever? Serious virtual reality wouldn’t be like toothpicks; it would be a vast array of virtual goods and experiences. And since these goods and experiences would be imaginary, there’s no reason they couldn’t grow forever. Laugh if you must: Imagination really is infinite!

Let me try to explain (again).

Imagine that in a million years, our descendants occupy all the 1070 atoms in our galaxy and its surrounding volume, and that it will take another million years to grow that number by a factor of ten, to 1071. They’ve spend a million years searching the space of possible physical devices: signal senders & processors, radiators, nuke & black hole power plants, etc. They’ve found some very good designs, and in another million years of searching don’t expect to find designs that are overall a hundred times more efficient. Even if computational capacity grew as the square of available mass (such as might be possible with black holes), for the next million years they expect their total computational capacity to grow by less than a factor of ten thousand, or 0.001% per year.

Over the last million years they’ve also been searching the space of enjoyable virtual reality designs. From the very beginning they had designs offering people vast galaxies of fascinating exotic places to visit, and vast numbers of subjects to command. (Of course most of that wasn’t computed in much detail until the person interacted with related things.) For a million years they have searched for possible story lines to create engaging and satisfying experiences in such vast places, without requiring more computational resources behind the scenes to manage.

Now in this context, imagine what it means for “imagination” to improve by 4% per year. That is a factor of a billion every 529 years. If we are talking about utility gains, this means that you’d be indifferent between keeping a current virtual reality design, or taking a one in a two billion chance to get a virtual reality design from 529 years later. If you lose this gamble, you have to take a half-utility design, which gives you only half of the utility of the design you started with.

If you spend all your time in virtual reality, and if your utility were your years of life times the virtual reality design quality, then you’d be indifferent between a 310 year life in your current design or a ten second life in the 529 year future design.

And 529 years is tiny on a cosmological scale. Over a million years 4% annual growth produces a factor of 1017,000. Could you really be indifferent between taking that infinitesimally small a chance of moving to a million year future virtual reality, where if you lose the gamble you have to accept a half-utility virtual reality? Would you really keep repeating this gamble as your utility fell to zero? And the universe will survive for many trillions of years — in a trillion years 4% annual growth gives a factor of over 101010.

It may be possible to create creatures who have such strong preferences for subtle differences, differences that can only be found after a million or trillion years of a vast galactic or larger civilization searching the space of possible designs. But humans do not seem remotely like such creatures. We like stories, to be sure, but most of us are pretty satisfied with simple variations on standard story lines – we just don’t get billions of times more value from the very best stories, over pretty good stories.

It is also very hard to see how creatures with such subtle preferences would have adaptive advantages in a competitive future scenario. And in a non-competitive scenario I for one don’t see much point in trying to populate our universe with such extremely picky creatures.

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  • I think the argument from human implausibility is flawed. We don’t know how much our descendants will resemble humans. The better argument is your old one that there are a limited number of configurations of the universe. Eventually we will find the possible configurations that result in the highest utility, and after that there is nothing more to find. Caplan asserts “imagination is infinite” but has no support for such a claim. Imagination is the product of the brain producing signals, there is no magic coming in from anywhere that can give rise to infinite possibilities.

    • Anonymous

      Not to mention: There is no reason to assume that future darwinian forces will be well-aligned with “the possible configurations that result in the highest utility”. I think even net-negative expected utility from a populated high-tech universe is quite realistic (i.e. more suffering than pleasure, counting desire frustrations).

    • I think the argument from human implausibility is flawed. We don’t know how much our descendants will resemble humans.

      Isn’t this what Robin addresses when he says that “picky” creatures (i.e. creatures that have the such strong preferences for subtle differences) aren’t likely to come about either in a competitive scenario (since they probably wouldn’t be nearly as fit) or in a non-competitive scenario (since we humans don’t have such preferences now). The only scenario where picky creatures emerge is one where our decedents ensure a non-competitive future but for some reason have diverged from current humans to be very picky themselves. Why would this happen? It’s possible, but seems much less likely than the alternatives.

  • mjgeddes

    ‘All Good Things’ have limits you say? 😉 Here is the last scene from the last episode of ‘Star Trek: Next Generation’: Picard has a final chat with the super-intelligence ‘Q’, after being rescued from what seemed like a certain demise:

    All Good Things

  • Ben

    What is the probability that in the next million years, humans:

    a) discover new universes, or discover a way to create universes?
    b) discover new forms of utility that drastically alter any utility calculus?
    c) discover new theories of physics, mathematics, or logic that make this analysis moot / inaccurate?

    These things are hard to do, but we have a million years, presumably with growing population and economy. I’m curious how you’d attach a high-level of certainty to any prediction made on such a long time-frame.

  • Steve

    Robin, you seem to have a hard time grasping infinity.

    It is really, really, big.

    Read more Dyson.

  • John

    We like stories, to be sure, but most of us are pretty satisfied with simple variations on standard story lines – we just don’t get billions of times more value from the very best stories, over pretty good stories.

    This explains the popularity of fanfics. You really don’t need much talent to make an entertaining story; the real challenge is making a story more entertaining than the vast number of competing stories written by more skilled authors who can reproduce their slightly more entertaining stories at no cost. But people entertained themselves with stories before mass publishing, and the talents of the local storytellers were enough to keep people interested.

  • Andy

    Ah, but surely we could continue exponential growth with a smaller base until the end of the universe or so :). No reason it has to be 4%.

    • Vaniver

      Andy: Do you understand how exponents work?

      Robin: The approach I would take with discussing this with Bryan is to repeat that brains are made of atoms, and thus imagination exists in the real world, not some mental universe. If you have a finite number of atoms, then you necessarily have a finite brain, and a finite brain can contain only finite imagination.

      • Configuration space is exponential in the number of components, though. Our speed of acquiring atoms must necessarily become linear because of the speed of light, but configuration space could still grow exponentially. If “imagination” goes like the number of possible brain configurations, then you could maintain exponential growth.

      • Finch

        Assuming things stop at atoms. Or stop at some level below atoms. That there’s no fine structure that matters – it’s all a lattice deep down.

        That might be true, and it might not be true. It’s an empirical question. It sure isn’t _obviously_ true.

      • Daublin

        One example of configuration space that could plausibly never run out is mathematical theorems. It takes relatively few atoms to write down absolutely mind-boggling theorems. As math progresses, these get easier to understand and work with, and presumably we’d work on harder theorems. As we do, it doesn’t take many more atoms of notation to write he harder theorems. The complexity and interest of a theorem seems to be more than exponential in the size of the notation, for good theorems.

        Another possible example is genetic information. Both literal genes, and organizational patterns of a culture. Like mathematical theorems, the complexity hidden in a small amount of information seems to be high, more than exponentially high.

        Maybe these are still not enough to prevent an end of progress over the long haul. They deserve some careful attention, though. And anyway, even if it’s just matter that stagnates, Robin’s argument is still well worth taking to heart. We’re living in a golden age.

  • Michael E Sullivan

    Steve, what evidence do you have that the physical universe is finite, or that information and imagination can expand infinitely?

    I wonder if the limits are higher than robin suggests here, but they must exist if the universe is finite (cosmological evidence points this way strongly), and we do not discover other universes that we can reach.

    The hard upper limit on economic/utility growth is going to involve a factorial of the number of discrete elements in the universe than can represent information. If an atom can represent a piece of information, then the hard limit will be somewhere between 2^(10^70) and 10^70 factorial. Now those are a *huge* number — if we can actually get anywhere near that, then we might actually see 4% growth for a million years or more, but of course there are massive logistical limitations on the organization that will limit this below that hard upper bound.

    On the other hand, that doesn’t change the essential nature of the issue. The real problem is postulating infinite growth in a bounded universe. Of course this is all pretty academic from our current situation, where we are many thousands of years from running into any kind of universal hard limitation. We are more likely to run into a technological limitation in the next few hundred years that stalls such growth at the planetary or solar system level than to be dealing with Robin’s scenario before, say 100,000 years go by.

    • …if the universe is finite (cosmological evidence points this way strongly)…

      This is wrong. Opinions about whether the universe is finite or not vary greatly among cosmologists. Most would probably side with “infinite” given the popularity of inflation models, although opinions are based mostly on a priori arguments. There is basically no useful direct observational evidence one way or the other, except for the fact that we don’t look out and see a wall, or see the universe repeating (suggesting periodicity e.g. a torus), or see closed curvature (in which case you could conclude the universe was finite if you assumed homogeneity).

      The hard upper limit on economic/utility growth is going to involve a factorial of the number of discrete elements in the universe than can represent information.

      Even in an infinite universe, the speed of light limits you to a linear rate of growth in physical goods long term. The only way out of this is to think that the things we want are somehow exponential in physical good. But Robin argues persuasively that neither humans nor creatures that are likely to come about will have the appropriate utility functions.

      • The speed of light limits you to cubic growth, not linear growth.

      • Dániel Varga

        Peter de Blanc: While we are nitpicking, the speed of light, together with the holographic principle limits us to quadratic growth.

  • billswift

    there are only a finite number of different brain states

    Even if the brain is made up of all the particles in the universe, eventually all possible experiences would be had.

    Quote from

  • Michael Wengler

    I think the most important and sticking thing I have learned since reading Robin Hanson is that Malthus is “usually” right, but that living in the most recent century could make you not realize it. Not only ought Robin’s explications of what it really means to compound 4%/year convince you, but we can also look to history. There have been over 100 billion humans by modern estimates. Reading Matt Ridley’s The Rational Optimist, we see a brief history of all the times in human history that Malthusian limits have caught up with humanity.

    We live in an extraordinary period where the human population has not yet overrun our expanded technical prowess. And so we look out on the modern world and see growth and plenty everywhere, just as other humans have looked out to their horzons and declared confidently “The world is FLAT.”

    When I was young, the dominant theory of the universe was “steady state,” it had always been this way. Then some people pointed a strong enough telescope in to the sky and saw something they didn’t expect: the edge of the universe. (This is a poetic but not misleading way of describing the big bang. Think about it.)

    Follow the implications of plenty and 4%/year growth carefully. Don’t make the mistake of thinking that what you have seen your whole life constitutes the way the whole universe is and was and will be.

  • Michael Wengler

    I followed the link to Bryan Caplan’s post about 4%/year growth and saw a Tyler Cowan response there of 0%.

    Any answer except 4%/year asymptotic growth rate for the entire economy strikes me as innumerate. What am I missing?

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  • rapscallion

    But there won’t ever really be an “end of possibility” for individuals if the end stage of development is a game with no stable equilibrium. In that case, things might always possibly get better or worse for you depending on luck and strategic behavior. Just because the pie is fixed, it doesn’t mean that individuals might not always have a shot at getting a bigger chunk of it (or losing some).

  • Instant Karma

    We all need to think bigger and get out of our current GDP paradigm.

    Once we are millions or billions of times wealthier, with broader ranges of better experience, why do we need to create more experiences? We can spend the next few trillion years experiencing and sharing the ones we already created.

    Let me restate: the scarce factor is experience. I think we will have enough to keep us busy for the next few trillion years. After that, well we can ask Robin then.

  • arch1

    I find Instant Karma’s point to be a good one. Why would the eventual leveling off of growth in our technological capabilities alone (or perhaps in our achievable utility per second) necessarily limit a person’s possibilities in a meaningful sense?

    If a person’s Acme9000 VR library has on tap, say, 10^10000 years’ worth of profoundly distinct, absurdly high utility-per-second experiences, do we say that this person’s “possibilities are limited” because that person is not constantly upgrading the experience quality to that achievable using Acme9001, Acme9002, etc?

    I suppose we do if we are selling Acme9001. But it seems to me that any meaningful discussion of the Acme9000 owner’s limitations should only start around year 10^9997 or so – which is to say, an immense span of time after Robin’s exponential growth rates and Bryan’s growth of quality of VR life had ceased.

    (Having said this I admit that I find the whole discussion, in which we are throwing around notions such as utility per second, kind of awkward; it seems to miss what’s important, but I haven’t thought it through well enough to to do more than report my unease:-)

  • Instant and arch1, a lack of growth does not at all imply a bad life. Growth could continue for a while and then stop at a high level and then capacities could stay high forever after. Not such a bad outcome at all. But still, growth does end.

    • Instant Karma

      Robin and Arch,

      This clarifies the issue for me.

      This points out the potential for a phase transition of prosperity measured by growth in experience (Gross Domestic Experience?) to prosperity measured in terms of percent of potential experience actually experienced.

      We may not be creating any new virtual realities, but nobody will have the time to live all the great ones already available. Sounds great to me.

  • arch1

    Yes, growth does end (as long as one person can’t control ever more resources, which seems likely at present). But why does that matter?

  • Ari T

    Since this is a sci-fi topic might aswell take it to another level.

    Our finite universe simply cannot continue our exponential growth rates for a million years. For trillions of years thereafter, possibilities will be known and fixed, and for each person rather limited.

    Some smart economist quipped that there’s a chance we’re living in a virtual reality. If we are in one, why can’t the rules be changed? One of the dangers of making absolute statements is not taking into account exogenous factors. Happens all the time in the market even when people have thought they thought of everything (eg. discrete multivariate distributions applied to real estate).

    If we are not in a virtual reality or something similar (), which “seems” likely, growth will stop at some point, but the real question is what kind of life do the people (or whatever beings) enjoy.

  • It is not clear that the universe is strictly finite. The volume inside a black hole can be infinite, which implies infinite entropy. You can’t get to the infinite entropy from the outside, but maybe you can by entering the black hole. It may appear to be finite from the outside, but it may not be finite from the inside.

  • To follow up, maybe that is the solution to Fermi’s Paradox. We see no aliens because they have all entered black holes to experience an infinite future.

  • arch1

    Robin, you’re implicitly assuming that relative utility (whether measured probabilistically or via relative durations) is multiplicative – so that if someone thinks experience A0 equal in desirability to a 1/1.04 chance at experience A1, or to experiencing A1 for 1/1.04 the duration of A0, and A1 stands in the same relation to A2, etc, then it will necessarily be the case that they would be indifferent to experiencing A0 for a year vs. A440 (if I did my math correctly) for one second, and also vs. having a 1 in 31.6 million chance of experiencing A440 for a year.

    Is there any reason to believe that peoples’ utility really works this way?

  • arch1

    Answering my own question: It seems that relative utility is multiplicative (in the way Robin implicitly assumes) as long as the pairwise equivalences are scale invariant, and indifference is transitive.

    I’m aware of ways in which each of these fails when applied to human preference, but I don’t know how much those failures affect Robin’s argument (that exponential growth must end even if resources grow cubically)

  • Fergus Mackinnon

    This is very similar to what occurred to me when I was in the middle of writing a similar comment to your original post as Mr Caplan.

    Does anyone have any thoughts on how to approach memory archiving in this scenario? If we can’t find ways to somehow expand the amount of matter at our disposal to create more memory storage capacity at an equal or greater rate than we accumulate memories, we would presumably be eventually forced to die or delete old memories. While this might stop us from becoming bored when we run out of new experiences, I’m curious as to what sort of mechanism could be created to deal with this situation.

  • Re: “Our finite universe simply cannot continue our exponential growth rates for a million years. For trillions of years thereafter, possibilities will be known and fixed, and for each person rather limited.”

    If the “person” has a brain the size of a planet then “known”, “fixed” and “limited” would most-likely be a strange and inaccurate way of describing their “possibilities”.

  • As unbelievable as 4% growth for a trillion years is, your argument is really just that: an argument from incredulity. If such incredible growth is so obviously impossible, then it should be easy to explain exactly why.

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