Limits To Growth

By Robin Hanson · September 21, 2009 10:30 pm · Comments (79) · ·

Me:

Current growth rates simply cannot continue at familiar levels for ten thousand more years.  We’ll eventually learn everything worth knowing about how to arrange atoms, and growth in available atoms will be limited by the speed of light.

Bryan Caplan:

I’m baffled.  You don’t have to be a sci-fi guy to think that in the next century we’ll get working virtual reality.  And once we have that, why couldn’t economic growth of 1% (or 10%) continue forever in simulations?  In the real world, we can’t all be emperor of an infinite universe.  But I don’t see why every one of us couldn’t preside over our own simulated utopias?

As for all this talk of atoms: Economics is about value, not matter.  As long as people regard vivid virtual goods as acceptable substitutes for actual goods, how can the scarcity of atoms stop everyone from having everything he desires?  Robin might respond that computing power will be too scarce to simulate a big virtual world in great detail.  But who really cares if the simulation is accurate down to the microscopic level anyway?

Let’s try some numbers.  Today we have about ten billion people with an average income about twenty times subsistence level, and the world economy doubles roughly every fifteen years. If that growth rate continued for ten thousand years the total growth factor would be 10200.

There are roughly 1057 atoms in our solar system, and about 1070 atoms in our galaxy, which holds most of the mass within a million light years.  So even if we had access to all the matter within a million light years, to grow by a factor of 10200, each atom would on average have to support an economy equivalent to 10140 people at today’s standard of living, or one person with a standard of living 10140 times higher, or some mix of these.

An economy that doubled every century for a million years would grow by a factor of 103010.  To support this using the 1070 atoms found within a million light years, each atom would have to support an average of 102950 people at our living standard, one person with a standard 102950 times higher, or some mix of those extremes.

It seems just physically impossible to create 10140 or more lives we would value like ours per atom, even considering quantum computing and black hole negentropy.  But could individual living standards be that high?

To say that someone had a standard of living 10140 times a subsistence level means that 10-140 of their income could buy a subsistence level standard of living.  Someone with a subsistence level living standard, and a square root type risk-aversion, would reject an offer to jump to today’s world average living standard, twenty times higher, if they could instead roll 69 ten-sided dice, and only get to jump to this 10140 higher standard if all the dice came up 1.  (Someone with a fourth root risk aversion would prefer to roll 35 dies.)  That is just how incredibly fantastic this 10140 higher living standard would be.  A living standard 102950 higher is far far more fantastic.

While I’m sure virtual reality will have its appeals, I just can’t see it offering such lightyears-beyond-astronomical living standards to creatures like us. Sure it would be fun to be “emperor of an infinite universe”, but not that fun.  Perhaps one could design creatures with such values, but we humans just don’t seem capable of such heights of ecstasy.  There is simply no experience so enjoyable that we’d prefer such a tiny chance of it over a big improvement for sure.  Video games and movies today already offer much of the gains that virtual reality have to offer. Sure higher resolution and better crafted story lines will raise the bar, but not by a factor of 10^3000.

Within the next million years, economic growth rates must fall below feasible population growth rates.

Added 9:30a 22Sept: Bryan responded:

Suppose the universe contains one person and one really good virtual reality machine.  In this virtual reality, the universe’s sole inhabitant is a god.  Whatever he thinks, happens.  Question for Robin: As long as this person regards virtual reality as a good substitute for actual reality, why shouldn’t we say that his standard of living is not just 10140 times greater than ours, but infinitely greater than ours?

Is this one person’s willingness to pay for playing god in this sim infinitely greater than his willingness to pay for more familiar lifestyles?  Imagine he starts out with something closer to our standard of living and chooses if to gamble half his income for a tiny chance to play god in this sim, or chooses between having half his income for most of his life, and then spending some tiny fraction of his life in this god sim.  Would he really accept arbitrarily low chances or fractions in such choices?

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64 Comments

  1. Chris Hibbert
    September 21, 2009 at 11:32 pm | Reply

    Well argued, Robin. A little math, used judiciously can be an amazing thing.

    In a million years, we might be able to travel pretty far, pretty fast, but we aren’t likely to be able to travel faster than lightspeed, so your million light-year radius is a generous allowance.

  2. Nathan Cook
    September 21, 2009 at 11:36 pm | Reply

    So one answer to your original question isn’t that we’re uniquely rich but uniquely inefficient. Almost all our descendants will gain close to theoretically optimal value from their free energy. Hopefully we won’t make so many offspring that their share is painfully small.

    • Dan
      September 22, 2009 at 6:31 am | Reply

      I totally agree with Robins theory just not his *scenario*. Exponential growth is dependent on a delicate balance of interdependence, consensus and self-interest…. I just don’t see any body politic allowing exponential growth to continue at their individual detriment…

  3. Autumnal Harvest
    September 21, 2009 at 11:53 pm | Reply

    I count economic growth in terms of dollar bills, not goods and services. I therefore claim that we can continue at current growth levels indefinitely, merely through rampant inflation. Ha!

    • Dan
      September 22, 2009 at 11:39 am | Reply

      LOL,

      Even that is impossible you will run out of matter for your dollar bills. So lets go electronic you say, that to is impossible to the central bank will take over the universe and reduce it to a memory register to contain the number for the existing money supply!

    • Silas Barta
      September 22, 2009 at 1:17 pm | Reply

      Let me guess: you’re a mainstream macroeconomist?

  4. Tomasz Wegrzanowski
    September 22, 2009 at 12:29 am | Reply

    And you still ignore the main problem that there’s absolutely no reason why population growth must be fast.

    It’s just far too easy to incentivize people to have fewer children (either Chinese style by government decree, or Western style by making it cost too much economic status); subgroups with high fertility all have memetic (usually religious) not genetic basis of it – and their fertility is already dropping pretty much universally anyway; and even if there was some subgroup that really insisted on having too many children, and it was all genetic, they could really easily be forced not to, if the rest of the population decided so.

    I cannot think of anyone who expects explosive population growth.

    Here’s the data. Trivial models completely miss it: http://en.wikipedia.org/wiki/Population_growth#Human_population_growth_rate

  5. Thanatos Savehn
    September 22, 2009 at 12:36 am | Reply

    Two things. 1) Premise: our universe is finite. Conclusion: there’s not enough room to fit an actually infinite anything, people included, into our universe. Me: no duh; not news. It’s essentially a straw man argument. The real question is not what happens when an infinitute of motel rooms is full and a new guest arrives. The question is “can we go from an infinitesimally small number to an infinitesimally small number ^20?” I think you’ll get a different answer that way. 2) Playfully: Think about quantum computing. Then calculate all the possible states of say all the atoms in the universe. Then think of us as energy rather than matter. It may make a difference.

    Anyway, I’m sure people have been worrying about this since there were 20 people in the first village. The funny thing is that you don’t really have time to worry until there’s nothing to worry about. Try cutting down a dead tree or baling hay or planting soy beans or worming cows and worrying at the same time. Work drives out worry. Leisure summons it.

  6. Dan
    September 22, 2009 at 1:21 am | Reply

    1. I do not understand how you can assume that our scientific understanding of today will adequately describe how we understand the universe in 1000 years. I do not understand how you can assure that atoms are the last word in physics. 1000 years ago our ancestors also made weird predictions about the world.
    2. I also do not understand how you can assert that the population will continue to grow. Maybe 98% of the population will be destroyed through a virus or an astroid in the year 2145. Maybe humanity becomes infertile as shown in the movie Children of Men.
    3. Overall I do not see how these speculations matter to our own lives in the year 2009.

  7. Norman
    September 22, 2009 at 1:27 am | Reply

    Again, a fun exercise, but as Tomasz points out, it entirely misses the point that when you describe “feasible population growth rates,” there is no good reason or data to justify the claim that as incomes rise, population will grow as fast as possible, and there is a lot of good reason and data to suggest it won’t. The world population growth rate currently has a constraint on it, and it’s not binding.

    Beyond this, I find the idea of the standard errors around a forecast 1 million years out to be amusing. I would imagine that it is at nearly as likely that humanity will die out due to a confluence of unfavorable random mutations (ie, we lose the war on cancer). Even if we don’t, how can we possibly suggest that our current preferences will bear any resemblance to those a million years from now? What makes us so confident that the word ‘human’ can even meaningfully be applied to our genetic and philosophical progeny that far off?

    And again, why the emphasis on atoms? While this unit can provide some fun math, it implies that the relevant material universe is made up of discrete components. I think this is far from a proven position, and once we allow for the possibility that matter is less particle and more continuous field, there’s no particular reason to rule out technology advancing in a fractal manner, essentially never running out of new stuff to arrange (just making the distance between points continually smaller).

    So here is what I’m getting of your argument: 1. Income (subsistence) is the only binding constraint on human population growth. 2. The countably finite material universe is a binding constraint on technology (and thus on income). 3. Therefore, at some point the binding constraint of (2) must trigger the binding constraint of (1): the Cosmic Malthus.

    While the logic seems solid (in that 3 does seem to follow from 1 and 2), I see no reason to accept premises 1 and 2. I don’t believe income is the only constraint on population growth; I don’t believe we can say with any confidence that there isn’t some bliss point population size; I don’t believe matter can be characterized as countably finite; I don’t believe material inputs are necessarily a binding constraint on technology. I don’t know of any empirical evidence to support any of these claims, and I don’t hold to them for any philosophical or religious reasons. So, for me anyway, the argument has no force.

  8. TGGP
    September 22, 2009 at 1:58 am | Reply

    Karl Smith responded to Caplan here, and I disputed some of his claims in the comments.

  9. haig
    September 22, 2009 at 2:01 am | Reply

    It seems just physically impossible to create 10^140 or more lives we would value like ours per atom, even considering quantum computing and black hole negentropy.

    Based on Lloyd’s ‘ultimate laptop’ and an assumed 100 petaflops to entirely emulate a human brain (with all relevant features), a quick back-of-the-envelope calculation gives me 10^5 emulated brains per atom(Si). That’s a bit less than 10^140 high-quality lives being lived per atom.

    • Silas Barta
      September 22, 2009 at 1:32 pm | Reply

      Are you accounting for data compression, though? Each brain would not need 100 petaflops; once you have the first, then to emulate additional brains you would just need to compute the difference. Or more realistically, have some base brain algorithm, and then to specify each individual brain requires only a small computation on top of that.

      Of course, even if each “marginal” brain only requires one FLOP (an unrealistically low estimate), that only increases the brains per atom by a factor of 10^17, for a total of 10^22 emulated brains per atom, still far below the 10^140 requirement.

      (And I’ll have to echo the comments about population growth of biological beings tapering off.)

  10. fenn
    September 22, 2009 at 2:43 am | Reply

    smackdown!

    And I’m a little surprised that Caplan, who once said that he could not regret any decisions he’d made up until the conceptions of his children because he would then have different children (which he acknowledged he would probably love as much), would point to holodeck utopias as the salvation of the future. Not a direct contradiction, but strikes me as odd.

  11. MattW
    September 22, 2009 at 3:08 am | Reply

    It seems more reasonable to expect that at some point there will be population limits and the number of humans will grow only when available space increases (new habitable planets become available); thus the number of consumable atoms per person will likely not fall much below what it is today (given the ability to recycle practically everything is invented at some point).

  12. Oliver Steele
    September 22, 2009 at 6:45 am | Reply

    This is nicely argued, but it doesn’t apply if a copy-on-write strategy is applied to the creation of individuals and goods. Currently many bits but no atoms are shared between myself and my progeny, or between my possessions and yours.

  13. Dan
    September 22, 2009 at 6:57 am | Reply

    “Let’s try some numbers. Today we have about ten billion people with an average income about twenty times subsistence level, and the world economy doubles roughly every fifteen years. If that growth rate continued for ten thousand years the total growth factor would be 10200″

    I agree 100 % with your fundamental theory Robin. Just one minor quibble. We don’t use double the natural resources every 15 years. A lot of that growth is “abstract” or fake if you are a pessimist. There is also increases in efficiency but even that will hit physical limits. Especially as you refer to the “world economy”.

    But yes even abstract value can be traced back to the exploitation of a natural resource, regardless of the many levels of abstraction that exist between them. So ultimately even exponential abstract growth is impossible, unless it is fraudulent or inflationary.

  14. Robin Hanson
    September 22, 2009 at 9:29 am | Reply

    I just added to the post, responding to Bryan’s response to me.

  15. Andrew Berman
    September 22, 2009 at 10:14 am | Reply

    As long as you don’t care about real reality, why bother with virtual reality? You can have maximum pleasure right now! All it takes is about 20$ and a trip to your local crack dealer!

    If you want it to last for eternity, bring an extra $100 and ask the dealer to wait until you’re in a drug-induced haze of maximum pleasure and then shoot you in the head. I’m using a definition of eternity that is totally based on your existence, but that’s not unreasonable, considering we’re already using a definition of universe that is totally based on your existence, right?

  16. Carl Shulman
    September 22, 2009 at 10:41 am | Reply

    Per capita wealth could remain very high indeed if the future entities are much larger than human minds.

  17. Toby Ord
    September 22, 2009 at 11:07 am | Reply

    3 points:

    1. We are talking about economic value, not utility, so I’m not sure that your risk calculations work. It is, for example, quite plausible that the diminishing marginal returns on real wealth are logarithmic or lower.

    2. I like your idea of thinking about economic value per atom for finding upper bounds, but I’m not sure it works. The economic value of atoms is clearly not additive (a universe with just one atom has zero economic value — and zero moral value too). It seems plausible that value is limited by arrangements of matter rather than by the sheer quantity of it. This might push things much further.

    For example how many arrangements are there of 10^70 atoms? We can bound this from below as follows: consider a single arrangement where they are all in a line in a particular order. Now imagine removing some of the atoms from the line-up. There are 2^(10^70) ways of doing so, and each gives its own structure. These are just structures of one very constrained type, so the true number of structures is much higher.

    3. Even the number of arrangements doesn’t provide a clear upper bound for economic or moral value, as it is quite plausible that there are gaps in the value levels: value levels that are well defined but have no physical instantiations.

    We can represent very large numbers with the matter available. We often imagine gapless representation systems, where every number is representable, but we need not do so. We could use a string in binary to represent 2 to the power of that number, giving a maximum representable value of 2^(2^N) with N bits. Ackerman’s function (or the like) gives even larger upper bounds, and if we do not require the representation to be recursive, we could have a binary string code for how long a program represented by that string takes before halting, allowing representation of values on the order of the busy beaver function (or beyond).

    I’m not sure if value (either economic or moral) is bounded by quantity of matter, or by the number of patterns creatable with this matter, or by the values representable with this matter, or something else entirely. I just offer these as some hints as to how we might be able to get a lot more value than you suggest.

  18. Toby Ord
    September 22, 2009 at 11:23 am | Reply

    An example of why it may be number representation that matters:

    We could design an AI that makes its decision according to risk-neutral decision theory, but which compresses its values for different outcomes by coding them as the log base 2 of its values. It also codes probabilities in the same manner. Its decisions thus represent that it values states in which its coded value is 1 million more than 10^70 times as much as it values states in which its coded value is 1 (in fact, 2^1,000,000 times more: roughly 10^300,000).

    Such an agent would have a lot of gaps in its probability and utility valuations, but this could be very much reduced by replacing the above method with a floating point representation. Allow 100 bits of accuracy for the mantissa and, say, 1 million bits for the exponent. This still has gaps, but has numbers accurate to about 30 significant figures.

  20. Eliezer Yudkowsky
    September 22, 2009 at 12:29 pm | Reply

    Perhaps if technology continues to increase the size of quantum superpositions that can be perfectly maintained at a linear rate, the population can increase at an exponential rate.

    (Note that you and I are now arguing the exact opposite positions from our last discussion on discount rates…)

    • Robin Hanson
      September 22, 2009 at 1:16 pm | Reply

      Scott Aaronson has persuaded me that there just is no exponential gain possible from quantum computing. The typical gain is square, as in the quantum search algorithm.

      Here we are talking total growth rates, last time we were talking interest rates. Interest rates can exceed growth rates if none try hard enough, or are allowed to, to invest for the long run.

      • Eliezer Yudkowsky
        September 22, 2009 at 2:51 pm |

        That’s if you have to collapse the computation afterward. I’m rather proposing that future technological improvements will allow in-place expansion of an ever-larger, never-collapsing quantum superposition (which may even be purifiable via quantum error correction). It may even be possible to keep some parts superposed while collapsing information summaries that can then be communicated to the still-superposed parts – though here I exceed my grasp of physics. Trade between branches would only go as the square of classical productivity, but the branches themselves could grow in population exponentially – you won’t hear from your children very often, but they’ll have a nice space to live in, in the other branch. In fact, this is just the situation we appear to be in now, except that communication is literally zero.

        Actually, this makes for quite an interesting topic – assuming it’s possible at all, and I haven’t made a mistake somewhere – the question of what sort of internal communication is possible in a quantum coherent civilization.

        Problem is, I can’t run out and ask most physicists due to their single-world non-realism brain damage. Any sane physicists want to chime in?

        Interest rates can exceed growth rates if none try hard enough, or are allowed to, to invest for the long run.

        I don’t see how this is possible except as a pure function of utility functions with intrinsic discounts in them, but that gets back to our old argument…

      • Jess Riedel
        September 23, 2009 at 8:44 am |

        Are you referring to public material of Scott Aaronson, or to private conversation? Could you provide a link?

  21. Ken
    September 22, 2009 at 12:55 pm | Reply

    The key phrase in Bryan Caplan’s argument is “As long as people regard vivid virtual goods as acceptable substitutes for actual goods, how can the scarcity of atoms stop everyone from having everything he desires?” But why limit the argument in that way? The phrase “vivid virtual goods” can be replaced with any phrase, provided that it can be made the object of some individual’s desire.

    A person who found the arrangement and contemplation of sand grains utterly engrossing would be wealthy, even living in a cell on a cup of boiled rice a day, provided they had a table and a spoonful of sand. Our entire species could easily be brought to this incredible wealth, although of course by our current standards everyone would seem quite poor (and perhaps even a little autistic).

    This does not, of course, address the issue of growth and limits. In the world of the sand-contemplators, the number of individuals is limited by the amount of rice and number of cells – that is, their physical needs. The wealth of the society, measured by that metric, is also limited. But every individual has everything they want, so how do you attach a value to that, or compare that value to the measure of physical wealth?

  22. Robin Hanson
    September 22, 2009 at 1:23 pm | Reply

    Toby, Ken, I agree it is possible to create creatures with utility functions that go off the scale only for very particular arrangements of atoms. I doubt that we will create many such creatures, or that they will naturally evolve from a competitive scenario.

  23. Karl Smith
    September 22, 2009 at 1:25 pm | Reply

    The issue seems more basic to me. Presumably economic bundles have to be represented by the physical structure of the brain in order to be perferred.

    The structure of the brain only has so many permutations. Thus the number of economic bundles is finite. We cannot create infinite growth with finite bundles and we cannot grow without bound when there is a finite cap.

  24. Sigivald
    September 22, 2009 at 2:11 pm | Reply

    It seems more important to note that “current rates of growth” don’t even matter.

    What I mean is, if in a thousand or two years we’ve managed to effectively remove scarcity such that the significant limit on people’s happiness and experience is other people rather than lack of energy or materials, we’ve gone past the point where worrying about rates of economic growth really matters.

    When everyone is not just rich but rich beyond imagining, so rich that material concerns of acquisition simply don’t matter anymore, it’s okay for growth rates to lower a bit, or even effectively stagnate.

    • loqi
      September 22, 2009 at 4:14 pm | Reply

      What I mean is, if in a thousand or two years we’ve managed to effectively remove scarcity such that the significant limit on people’s happiness and experience is other people rather than lack of energy or materials, we’ve gone past the point where worrying about rates of economic growth really matters.

      This seems to just reinforce Robin’s point. You’re effectively postulating a future where “wealth” is measured by the size of the population you have available to interact with. That doesn’t sound like a recipe for stagnation to me.

      When everyone is not just rich but rich beyond imagining, so rich that material concerns of acquisition simply don’t matter anymore, it’s okay for growth rates to lower a bit, or even effectively stagnate.

      Material concerns of acquisition (of e.g., negentropy) can always be made to matter again with high enough population growth.

  25. Ken
    September 22, 2009 at 2:12 pm | Reply

    Robin, I agree with you about the difficulty with strange utility functions. I would even go further, and see it as a direct counter-argument to Bryan’s claim. After all, what qualitative difference is there between someone who finds fascination in patterns of sand grains on a tabletop, and someone who finds equal fascination in patterns of pixels on a laptop? Or even patterns of pigment molecules on canvas fabric?

    The problem, however, is that we don’t have any sort of consistent way of measuring the relative value of those arrangements. GDP, for example, is measured by exchanges made in markets. Thus, it can’t put any value at all on sand-contemplation; considers the value of “World of Warcraft” to be the membership fee, but completely ignores the hours of play time; and apparently thinks that the transfer of a canvas-and-pigment combination can contribute tens of millions of dollars to GDP – at least, when the pigments were arranged by Van Gogh.

    Until that issue gets straightened out, it’s really difficult to talk about economic growth based on the value of virtual items. Perhaps a clue can be derived from the way we measure the contributions of the financial industry. After all, a large chunk of that industry consists of people doing what, to outsiders, looks a heck of a lot like “World of Warcraft” gold mining – you push the buttons and acquire, or sell, entities that often exist only in the form of computer records of legal contracts. Yet we are quite comfortable in saying that these actions contribute to the GDP.

  26. Doug
    September 22, 2009 at 2:14 pm | Reply

    “There is simply no experience so enjoyable that we’d prefer such a tiny chance of it over a big improvement for sure.”

    Not true. If the people in the future are assured–or even have a small chance at–immortality, one roll at almost any odds would be worth giving up certain shot at any marginal improvement.

    Within the next million years, economic growth rates must fall below feasible population growth rates.

    Why? Negative and zero population growth rates are feasible, and seem to me to be almost inevitible if humans ever learn how to extend their lives beyond a couple hundred years. Plus, if you have everything you ever wanted, and access to everything you could ever imagine, why would you even care about growth rates?

  27. Doug
    September 22, 2009 at 2:28 pm | Reply

    Also, a major fallacy in your argument is that you use the number of atoms as a limit on economic growth, but do not consider its impact on population growth. Population growth, even more than economic growth, will be bound by the number of atoms we have access to. Accordingly, with “feasible population growth” will never be so large as to preclude even larger economic growth. If economic growth is eventually forced to 0, population growth will almost certainly be forced to 0 first.

    Moreover, both population growth and economic growth could taper off and approach 0 as t approaches infinity, such that neither ever reaches 0, economic growth never dips below population growth, and nothing ever reaches or even approaches the bounds imposed by the limited number of atoms.

  28. Noah Yetter
    September 22, 2009 at 2:59 pm | Reply

    You should not rely so heavily on expected value arguments, e.g. dice rolling. Expected value is an extremely poor description of how humans make decisions.

  29. Eric Falkenstein
    September 22, 2009 at 4:18 pm | Reply

    Considering the Easterlin paradox, where happiness levels are not measurably greater as wealth increases in developed countries, why should we expect any greater happiness at any dimension? Habituation and status concerns imply it doesn’t really matter.

    Depressing, but at the same time liberating.

  30. RJB
    September 22, 2009 at 5:24 pm | Reply

    Well, as long as people are bringing up virtual goods and virtual worlds: today Linden Lab announced that users of the virtual world Second Life have transacted 1 billion US$ worth of virtual goods and services among themselves, over 1 billion hours. Wednesday, Sept 23rd at 4pm there will be a panel discussion at the Virtual Goods track of a conference in San Jose, featuring Philip Rosedale (Linden’s founder and chairman). You can watch and chat comments and questions online in real time. See here for more info on how to join in.

  31. Mark Bahner
    September 22, 2009 at 5:38 pm | Reply

    “Current growth rates simply cannot continue at familiar levels for ten thousand more years. We’ll eventually learn everything worth knowing about how to arrange atoms, and growth in available atoms will be limited by the speed of light.”

    Growth of what?

    Money (aka, wealth)? I predict money/wealth won’t even last this century as a meaningful concept:

    Why everyone in 2100 will be a millionaire, and why we should care

  32. Eric Yu
    September 22, 2009 at 6:14 pm | Reply

    For example how many arrangements are there of 10^70 atoms? We can bound this from below as follows: consider a single arrangement where they are all in a line in a particular order. Now imagine removing some of the atoms from the line-up. There are 2^(10^70) ways of doing so, and each gives its own structure. These are just structures of one very constrained type, so the true number of structures is much higher.

    Is it true that the total utility of any region of space with constant radius and energy is bounded because a) the entropy is bounded, and b) since there are a finite number of distinguishable states, there is one with a utility greater than or equal to that of any other state? This assumes that utility is a function of the state of the universe.

    • Toby Ord
      September 23, 2009 at 7:03 am | Reply

      It is indeed conjectured that there are finitely many distinguishable states in any particular region of space. However:

      (1) this is a conjecture, and may turn out to be false (though most people who know about it think it is probably true).

      (2) all finite sets of finite numbers have a finite bound, but this doesn’t mean that we know what it is, or how fast it can grow as the size of the volume increases. Thus, it is a bound in some sense, but not of the type robin is looking for.

      (3) perhaps some arrangements of matter within a volume of spacetime have infinite value.

  33. Forrest Bennett
    September 22, 2009 at 7:52 pm | Reply

    There are three proposals (that I know of) for doing an unbounded amount of computation, and Eliezer just cooked up a forth one on the fly. Perhaps none of these 4 proposals will work. But given that we have no principle analogous to the speed of light to prohibit this, I wouldn’t bet against us figuring something out in the next 10K years.

    1. Injecting information into basement universes. 2. arXiv:astro-ph/0205279v2 3. Tipler’s proposal 4. Elizer’s quantum coherent civilization.

    • Stephen
      September 23, 2009 at 12:14 am | Reply

      It doesn’t seem to me that superposition of universes yields unbounded computation, or in fact any form of unbounded utility. The whole business of counting branches is a bit of a red herring in the first place. Whether a system is in superposition or not is a basis-dependent statement. A spin can be in a superposition of up and down (two branches), but if you consider its orientation along the left right axis the same state of the spin can correspond to being oriented right only (one branch). “How many branches” is just a wrong question.

      If the universe has finitely many degrees of freedom, then the corresponding space of quantum states is finite dimensional, and the whole system, could in principle be simulated with finite computational power by storing and manipulating a gigantic state vector. If the universe has infinitely many degrees of freedom then it presumably has infinite computational power already classically.

      More generally, even supposing we choose a basis, I don’t believe that more branches gives more utility. It is like saying that if you flip 100 coins then you have created 2^100 parallel universes, and therefore we should all constantly flip coins in order to spawn more descendants. Really in our utility calculations we should weight the universes according to their probability. Similarly, when quantum mechanics is relevant we can weight utilities by the squared amplitude. In both the quantum case and the classical probabilistic case, by weighting this way, we no longer have any generic preference for spawning more branches.

      It is true that quantum mechanics is different than ordinary probability in that amplitudes interfere constructively or destructively before we square them. As a result, even without miscounting utility by forgetting to weight universes by their squared amplitude, there can be advantages to using quantum effects. However, these are finite. People even quantify them; see quantum game theory and quantum computation.

      • Jess Riedel
        September 23, 2009 at 9:43 am |

        Whether a system is in superposition or not is a basis-dependent statement. A spin can be in a superposition of up and down (two branches), but if you consider its orientation along the left right axis the same state of the spin can correspond to being oriented right only (one branch). “How many branches” is just a wrong question.

        This is true near the atomic level, where branching is fuzzy and not well-defined. But as you approach the macroscopic level, the branches become well-defined (though not necessarily discrete). Otherwise, we wouldn’t be able to agree on what our current classical world was.

        In other words, it’s not sensible to say whether a single spin is in a superposition or not (since it’s basis dependent), but it is sensible to say that we are in the (discrete) branch where the spin was measured to be spin up.

        Although a single spin doesn’t have a preferred basis, macroscopic systems (i.e. the classical world) do. This basis is preferred througheinselection, and the origins can be traced back to the form of the interaction Hamiltonian. For most classical systems, the Hamiltonian has a momentum-dependent part and a position-dependent part. The einselected basis is then roughly the set of states which approximately diagonalize in both position and momentum: minimum uncertainty wavepackets.

      • Jess Riedel
        September 23, 2009 at 9:52 am |

        If the universe has finitely many degrees of freedom, then the corresponding space of quantum states is finite dimensional, and the whole system, could in principle be simulated with finite computational power by storing and manipulating a gigantic state vector. If the universe has infinitely many degrees of freedom then it presumably has infinite computational power already classically.

        If space is continuous, then a single spin-zero particle has an uncountably infinite dimensional Hilbert space even though it is said to have 3 degrees of freedom.

        I think that classical computational power is still bounded in continuous space because of experimental error, which may have entropy-imposed minimums. I’m not sure about this.

      • Jess Riedel
        September 23, 2009 at 9:56 am |

        It is like saying that if you flip 100 coins then you have created 2^100 parallel universes

        Branching/decoherence happens all the time, at a furious pace, whether we flip coins or not (or measure spins or not). Our measurements do not significantly alter the rate of branching.

        However, you are right that the only sensible way to value future states is through the Born weights.

      • Stephen
        September 23, 2009 at 1:09 pm |

        > If space is continuous, then a single spin-zero particle has an
        > uncountably infinite dimensional Hilbert space even though it > is said to have 3 degrees of freedom.

        Yes, sorry for the sloppy terminology.

        > I think that classical computational power is still bounded in
        > continuous space because of experimental error, which may
        > have entropy-imposed minimums. I’m not sure about this.

        This is a good point. I retract my statement that continuous systems “presumably” have infinite computational power. In fact, Jose Felix Costa and some others have a body of work showing that in a classical universe where higher precision measurements take more time to perform the computational power is reasonable.

        Incidentally, I am currently visiting UCSB, if that is where you are. Email me if you’d like to have lunch or something.

  34. Stephen
    September 22, 2009 at 8:10 pm | Reply

    Eliezer:
    I don’t think the concept of communication between branches really makes sense, because quantum time evolution operators are linear. At best, branches can interfere, but at that point they are not separate branches anymore.

    Robin:
    I doubt that is exactly what Scott said. Probably he was referring to speedups for NP. For the general speedup of NP probably the best we can achieve is quadratic improvement via Grover’s algorithm. However, there are certainly examples of exponential gain from quantum computing. Factoring integers is one*.

    *Technically, the best classical algorithm for factoring has runtime proven to be 2^(c sqrt(n log n)) and conjectured to be 2^(d n^(1/3) (log n)^(2/3)), for some constants c and d, so the speedup achieved by Shor’s polynomial-time factoring algorithm is not quite exponential. However, it is nearly so, and anyway it is far beyond quadratic. If you want to be pedantic about exponential speedups, you can still find examples. They are just less well-known than factoring. See any BQP-complete problem such as estimating Jones polynomials or simulating quantum systems. A comprehensive list is here.

    • Psy-Kosh
      September 22, 2009 at 11:14 pm | Reply

      Stephen: “But QM’s linear!” would have been my reply… except there’s that pesky nonlinear Born statistics business.

    • Psy-Kosh
      September 22, 2009 at 11:25 pm | Reply

      Whoops, to clarify… maybe the intention then could be to set up a very carefully controlled and limited interference that basically effectively shifts just some small aspects, without continuing to influence each other that much afterwards. In terms of Robin’s view, a very carefully controlled and limited mangling. I’m not sure if this would be possible.

  35. Tim Tyler
    September 23, 2009 at 4:33 am | Reply

    Basement universes, yes – but I am not sure those other proposals actually fit the bill of removing limits to genuine growth.

    The other main way is with an enormous quantity of room at the bottom – which we can’t rule out – but which seems even more unlikely.

  36. Pete Murphy
    September 23, 2009 at 9:15 am | Reply

    If world population growth continued at today’s rate, in only 1100 years every drop of water on earth (including oceans) would be locked up in the makeup of human flesh.

    The biggest obstacle we face in changing attitudes toward overpopulation is economists. Since the field of economics was branded “the dismal science” after Malthus’ theory, economists have been adamant that they would never again consider the subject of overpopulation and continue to insist that man is ingenious enough to overcome any obstacle to further growth. Even worse, economists insist that population growth is vital to economic growth. This is why world leaders continue to ignore population growth in the face of mounting challenges like peak oil, global warming and a whole host of other environmental and resource issues.

    But because they are blind to population growth, there’s one obstacle they haven’t considered: the finiteness of space available on earth. The very act of using space more efficiently creates a problem for which there is no solution: it inevitably begins to drive down per capita consumption and, consequently, per capita employment, leading to rising unemployment and poverty.

    If you‘re interested in learning more about this important new economic theory, then I invite you to visit either of my web sites at OpenWindowPublishingCo.com or PeteMurphy.wordpress.com where you can read the preface, join in the blog discussion and, of course, buy the book if you like.

    Pete Murphy
    Author, “Five Short Blasts”

  37. fourcultures
    September 23, 2009 at 10:03 am | Reply

    I’ve written about growth and models at ‘how to spot a model that actually works‘ and at ‘models, reality and the limits to growth‘. From the latter: “Limits – ‘real’ limits – are only ever met in the concrete and the particular, not in the universal and abstract. Therefore, what concrete and particular limits have to tell us about the world in general is always open to interpretation and argument.”

  38. Anders Sandberg
    September 23, 2009 at 1:17 pm | Reply

    While the Bekenstein bound and similar considerations put an upper limit on the number of distinguishable states in a region of space with a certain amount of matter, it does not place any limit on their value (as noted above). However, the value has to be experienced by entities inside this region. There is also a finite bound on the number of distinguishable utility functions. However, this number is very large: even fairly simple computational systems can calculate very complex functions, and here the evaluation system could potentially be as big and complex as the entire space region (leaving little but itself to evaluate).

    I wonder whether one can get more utility by having a lot of agents with evaluation systems that are relatively simple, or having few agents with very complex evaluations?

  39. Anders Sandberg
    September 23, 2009 at 5:19 pm | Reply

    As a spin-off, I did a calculation of the inevitable thermodynamic costs of expanding a civilization: http://tinyurl.com/l55d8b

    It turns out that a “classical” civilization that just converts matter into information storage has no real limit to its expansion speed. However, a civilization close to the Bekenstein bound in terms of information storage has such an enormous entropy cost in converting matter that its expansion is actually heat dissipation limited!

    Still, dealing with the entropy output of an advanced civilization is going to be challenging. Good waste management will just grow in value.

  40. Mark Bahner
    September 23, 2009 at 10:05 pm | Reply

    If world population growth continued at today’s rate, in only 1100 years every drop of water on earth (including oceans) would be locked up in the makeup of human flesh.

    The world is growing at about 78 million people per year. If growth continued at that rate for 1100 years, the world population would be about 90 billion.

    I think you think world population is growing exponentially, but it’s not. It hasn’t been been growing exponentially for several decades.

    The very act of using space more efficiently creates a problem for which there is no solution: it inevitably begins to drive down per capita consumption and, consequently, per capita employment, leading to rising unemployment and poverty.

    Computers have been improving in space utilization more than just about any industry for many decades. Not only is our consumption of computers not slowing down, it’s increasing. This is leading to more employment and less poverty (ask the employees of Microsoft…especially those who bought MS stock).

  41. nick
    September 24, 2009 at 7:33 am | Reply

    If it turns out that baryon number is not conserved, with the result that we can convert neutrons and protons into photons, we should count photons instead of atoms. Given that there are over 20 orders of magnitude worth of frequency to work with, we may be able to store 10^30 bits per photon, with the result that the neutrons and protons in our Sun, converted into photons of average energy blue, could store about 10^97 bits. Of course, we will need at least a few leptons around to actually process that information and recycle photons. That’s classical data storage — entangled photons may be far better still, at least in terms of raw information capacity.

    That gives us some more room at the bottom, but still sub-exponential unless, as mentioned, we rarely collapse qubit superpositions (i.e. if qubit-minds rarely actually think about anything but just keep storing more memories).

  42. fenn
    September 25, 2009 at 12:06 am | Reply

    sounds like warfare’s not going anywhere

  43. Marcio Rocha Pereira
    September 25, 2009 at 11:44 pm | Reply

    Human beings are not driven to maximize pleasure. Instead, pleasure is a tool to drive human beings to maximize other things (i seem to think we are currently unsure exactly what, though). Bryan seems to imply that improve must mean have more pleasure, but we are much more complicated than that…

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