Limits to singularities

No gaming today; physics and economics. In particular, Robin Hanson at Overcoming Bias posts about singularities, which are transitions where the rate of change suddenly changes drastically. He identifies four: Multicellular animals, human brains, farming, and industry. He also expects another one shortly, which seems to me to be extrapolating from four data points, a dangerous activity at best, but never mind. The point is, each of these innovations reduced doubling time drastically. It is not completely clear to me what is being doubled without the context of a human economy – amount of biomass? Metabolic activity? But at any rate the argument is indisputable for farming and industry: Suddenly the time it took to double GDP was hugely reduced.

Anyway. Dr Hanson expects the next singularity within a decade or two, and also that it will reduce GDP doubling time to a few weeks or months. If that happens it will be a truly amazing change; banks will then have to offer APYs of several hundred percent, or nobody will deposit money in them. And this won’t be because of inflation, it’ll be because 100 dollars (or loaves of bread) now will genuinely be worth several hundred dollars next year, because we will all be that much richer. This seems intuitively unlikely to me, but then again so would doubling times of a few years have seemed to anyone from a farming society – hence Malthus. But I got to wondering, suppose you get singularities every so often, is there a lower bound on the doubling time, beyond which you cannot go?

An obvious bound is formed by lightspeed. To double your GDP, the absolute least you must do is move the information on your amazing new doohickey-widget-creator-thingie to all the points that can make use of it. The radius of the Earth is about 0.02 light-seconds, so if humanity remains Earth-bound, that is a lower limit on our economy’s doubling time, beyond which no singularity can move you. Not very stringent!

(A side note: If you do manage to break the lightspeed limit, you’ve also invented time travel, and it’s no longer clear what it means to talk about ‘doubling time’. So I’m going to stick with c as an upper bound on the speed of information transfer.)

An Earth-bound humanity, however, seems unlikely to me if we are all that rich. If everybody has the equivalent of tens of millions of dollars, invested at a few hundred percent APY, then space tourism ought to seriously take off even if no innovation makes the exploitation of space economically viable, which again is unlikely in these circumstances. If you extend humanity to all of the Solar System, then the radius is about 8 light-hours, so the minimum doubling time becomes 16 light-hours. But this gets tricky. The doubling time for Earth is still 0.02 seconds, and similarly for other planets. What’s going on at Pluto while the latest information is crawling towards it? Are they innovating too, and perhaps getting the same invention independently, or do they need Earth – which, you should note, has a huge head start – to invent the latest and greatest? The latter scenario seems plausible to me, for the following reason. There will exist a time when the Plutonian economy is not expanding from innovation, but just from adding the mass of Pluto into the available resources. While that’s going on, Earth is still innovating; so by the time Pluto is fully integrated, Earth is multiple doublings ahead, and there is no way for Pluto to catch up – they can’t innovate on the basis of what Earth has until 8 hours after Earth invented it, and by that point Earth has got all-new stuff. So Pluto will always be 8 hours behind. Which, with a doubling time of 0.02 seconds, is 1 440 000 doublings. There is no analogy to this in history. Uncontacted tribes in the Amazon are way less than a million doublings behind the US. At this point I’m not even clear on what it means to talk about wealth. A million doublings just from the current world product is, quite literally, beyond the dreams of avarice. How will the (fantastically wealthy) Plutonians feel about this? The Mexico/US border has nothing on this wealth gradient!

The foregoing does assume that we get to the 0.02 second doubling time for Earth before we colonise Pluto, which seems unlikely. It’s more reasonable to assume that Pluto gets colonised well before that limit, so innovation is evenly spread over the Solar System, and the doubling time is 0.02 seconds everywhere – that is, each planet is innovating at the rate that gives this doubling time, probably duplicating each others’ efforts considerably. The analysis does apply to anything colonised after the maximal growth rate (MGR) is reached, though, so colonists for other star systems (I note, at this level of wealth you don’t worry about finding habitable planets; build them out of asteroids using your petty cash) will have to reconcile themselves to being poor compared to the ones who stay behind.

Are there other limits? Suppose we look not at the rate of growth, but the absolute wealth of the economy. There is an upper bound to this, given by the mass of the Solar System. A given mass can contain only so much information. Economically useful stuff, generally, contains more information, or less entropy if you prefer, than useless stuff such as rocks. At the MGR, or even at a doubling time of a few weeks, we will be converting Earth and Solar System mass to economically useful stuff at a really huge rate, and it won’t actually take us very long to run into the upper bounds on the information that can be contained in the mass. At that point our growth goes instantly to zero. (Or asymptotically; it depends on whether the last few kilos are easier to remove because we’re so rich, or harder because they’re in the middle of the Sun. Presumably there’ll be some reason we got around to them last.) Clearly this is a much more stringent bound than lightspeed! At this point the economy grows only as fast as we can find new mass to put into it; the economy will now be a sphere expanding at (at most) lightspeed, and converting all that it comes across to wealth. At large scales mass is evenly distributed across the Universe, so the size of the economy will double every time the radius of our sphere grows by a factor cube-root 2. In other words, doubling time will grow exponentially in time.

One might argue that the available energy is a still more stringent bound, which is reaonable for the next few hundred doublings. In that case we’ll hit the wall when we are using the entire output of the Sun, plus the relatively minor sources internal to the planets, like radioactivity and gravitational contraction. Then we can expand the sphere from which we’re getting energy by colonising other star systems, with the same analysis as above. However, the Sun is not actually very efficient at converting mass to energy; in a few hundred doublings we will certainly be able to do full matter-energy conversion, and then we are again limited only by the amount of mass available. We’ll take apart the Sun and use it for fuel, instead of waiting for the slow and inefficient process of hydrogen fusion – which also has the inconvenient properties of eventually causing the Sun to become a red giant, not to mention tying up all that lovely mass just to produce the pressure needed for fusion. Certainly we can do better than this! So in that case we are back to the case above, with the sphere expanding at lightspeed.

So far physics. What about psychology? Here I’m on much shakier ground. It does seem to me, though, that there are limits to human desires. Once you can have absolutely anything you want, up to and including constant, orgasmic intercourse with the sentient being of your choice for the next ten years, just how much more wealth are you going to be motivated to get? We are not so many hundred doublings from this being available to everyone. Perhaps the eventual limit to growth will be, instead, a limit to desire.

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2 Comments

Filed under Physics

2 responses to “Limits to singularities

  1. Elizabeth

    Pretty heady stuff.

    Concerning psychology, I think you underestimate what desire can do materially. You seem to assume that once people have enough material things to satisfy any whim, the material urge goes away. But then why do you have millionaires who focus on making yet more money?

    Competition.

    Who cares if you can have constant, orgasmic intercourse with the sentient being of your choice for the next ten years? Your neighbor can have it for eleven! And that’s why you’ll try for twelve. (etc, etc)

  2. kingofmen

    “Ten years” was just an example. Put in any number you like.

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