The rate of value production per atom can be bounded by physics. But the amount of value ascribed to the thing being produced is only strictly bounded by the size of the number (representing the amount of value) that can be physically encoded, which is exponential in the number of atoms, and not linear.
size of the number (representing the amount of value) that can be physically encoded, which is exponential in the number of atoms
The natural numbers that can be physically encoded are not bounded by an exponent of the number of bits if you don’t have to be able to encode all smaller numbers as well in the same number of bits. If you define a number, you’ve encoded it, and it’s possible to define very large numbers indeed.
The configuration does change, it’s just that the change is not about the amount of matter. If there are configurations absurdly more positively or negatively valuable than others, that just makes the ordinary configurations stop being decision relevant, once discerning the important configurations becomes feasible.
So, you imagine that the rate at which new “things” are produced hits diminishing returns, but every new generation of things is more valuable than the previous generation s.t. exponential growth is maintained. But, I think this value growth has to hit a ceiling pretty soon anyway, because things can only be that much valuable. Arguably, nothing is so valuable that you can be Pascal-mugged into paying 1000 USD for someone promising to produce it by magic. Hence, the maximally valuable thing is worth no more than 1000 USD divided by the tiny probability that a Pascal mugger is telling the truth. I admit that I don’t know how to quantify this, but it does point at a limit to such growth.
you imagine that the rate at which new “things” are produced hits diminishing returns
The rate at which new atoms (or matter/energy/space more broadly) are added will hit diminishing returns, at the very least due to speed of light.
The rate at which new things are produced won’t necessarily hit diminishing returns because we can keep cannibalizing old things to make better new things. Often, re-configurations of existing atoms produce value without consuming new resources except for the (much smaller) amount of resources used to rearrange them. If I invent email which replaces post mail I produce value while reducing atoms used.
this value growth has to hit a ceiling pretty soon anyway, because things can only be that much valuable
Eventually yes, but I don’t think they have to do hit a ceiling soon, e.g. in a timeframe relevant to the OP. Maybe it’s probable they will, but I don’t know how to quantify it. The purely physical ceiling on ascribable value is enormously high (other comment on this and also this).
Like you, I don’t know what to make of intuition pumps like your proposed Pascal’s Ceiling of Value. Once you accept that actual physics don’t practically limit value, what’s left of the OP is a similar-looking argument from incredulity: can value really grow exponentially almost-forever just by inventing new things to do with existing atoms? I don’t know that it will keep growing, but I don’t see a strong reason to think it can’t, either.
I think it’s more than an argument from incredulity.
Let’s try another angle. I think that most people would prefer facing a 10−6 probability of death to paying 1000 USD. I also think there’s nothing so good that a typical person would accept a 1−10−6 probability of everyone dying to get it with the remaining probability of 10−6. Moreover, a typical person is “subulititarian” (i.e. considers n people dying at most n times as bad as themself dying). Hence, subjective value is bounded by 1000×106×106×1010=1025 USD. Combined with physics, this limits GPD growth on a relevant timeframe.
I think that most people would prefer facing a 10e-6 probability of death to paying 1000 USD.
The sum of 1000 USD comes from the average wealth of people today. Using (any) constant here encodes the assumption that GDP per capita (wealth times population) won’t keep growing.
If we instead suppose a purely relative limit, e.g. that a person is willing to pay a 1e-6 part of their personal wealth to avoid a 1e6 chance of death, then we don’t get a bound on total wealth.
Let U(W) denotes the utility of a person with wealth W, Umax the maximal utility of a person (i.e. limW→∞U(W)) and ¯W the median wealth of a modern person. My argument establishes that
Umax≤U(¯W)+dUdW∣W=¯W⋅$1025
But, can we translate this to a bound on GDP? I’m not sure.
Part of the problem is, how do we even compare GDPs in different time periods? To do this, we need to normalize the value of money. Standard ways of doing this in economics involve using “universally valuable” goods such as food. But, food would be worthless in a future society of brain emulations, for example.
I propose using computational resources as the “reference” good. In the hypothetical future society you propose, most value comes from non-material goods. However, these non-material goods are produced by some computational process,. Therefore, buying computational resources should always be marginally profitable. On the other hand, the total amount of computational resources is bounded by physics. This seems like it should imply a bound on GDP.
I propose using computational resources as the “reference” good.
I don’t understand the implications of this, can you please explain / refer me somewhere? How is the GDP measurement resulting from this choice going to be different from another choice like control of matter/energy? Why do we even need to make a choice, beyond the necessary assumption that there will still be a monetary economy (and therefore a measurable GDP)?
In the hypothetical future society you propose, most value comes from non-material goods.
That seems very likely, but it’s not a necessary part of my argument. Most value could keep coming from material goods, if we keep inventing new kinds of goods (i.e. new arrangements of matter) that we value higher than past goods.
However, these non-material goods are produced by some computational process,. Therefore, buying computational resources should always be marginally profitable. On the other hand, the total amount of computational resources is bounded by physics. This seems like it should imply a bound on GDP.
There’s a physical bound on how much computation can be done in the remaining lifetime of the universe (in our future lightcone). But that computation will necessarily take place over a very very long span of time.
For as long as we can keep computing, the set of computation outputs (inventions, art, simulated-person-lifetimes, etc) each year can keep being some n% more valuable than the previous year. The computation “just” needs to keep coming up with better things every year instead of e.g. repeating the same simulation over and over again. And this doesn’t seem impossible to me.
The nominal GDP is given in units of currency, but the value of currency can change over time. Today’s dollars are not the same as the dollars of 1900. When I wrote the previous comment, I thought that’s handled using a consumer price index, in which case the answer can depend on which goods you include in the basket. However, actually real GDP is defined using something called the GDP deflator which is apparently based on a variable “basket” consisting of those goods that are actually traded, in proportion to the total market value traded in each one.
AFAIU, this means GDP growth can theoretically be completely divorced from actual value. For example, imagine there are two goods, A and B, s.t. during some periods A is fashionable and its price is double the price of B, whereas during other periods B is fashionable and its price is double the price of A. Assume also that every time a good becomes fashionable, the entire market switches to producing almost solely this good. Then, every time the fashion changes the GDP doubles. It thus continues to grow exponentially while the real changes are just circling periodically on the same place. (Let someone who understands economics correct me if I misunderstood something.)
Given the above, we certainly cannot rule out indefinite exponential GDP growth. However, I think that the OP’s argument that we live in a very unusual situation can be salvaged by using a different metric. For example, we can measure the entropy per unit of time produced by the sum total of human activity. I suspect that for the history so far, it tracks GDP growth relatively well (i.e. very slow growth for most of history, relatively rapid exponential growth in modern times). Since the observable universe has finite entropy (due to the holographic principle), there is a bound on how long this phenomenon can last.
The rate of value production per atom can be bounded by physics. But the amount of value ascribed to the thing being produced is only strictly bounded by the size of the number (representing the amount of value) that can be physically encoded, which is exponential in the number of atoms, and not linear.
The natural numbers that can be physically encoded are not bounded by an exponent of the number of bits if you don’t have to be able to encode all smaller numbers as well in the same number of bits. If you define a number, you’ve encoded it, and it’s possible to define very large numbers indeed.
Great point, thanks!
To me, just ascribing more value to things without anything material about the situation changing sounds like inflation, not real growth.
The configuration does change, it’s just that the change is not about the amount of matter. If there are configurations absurdly more positively or negatively valuable than others, that just makes the ordinary configurations stop being decision relevant, once discerning the important configurations becomes feasible.
So, you imagine that the rate at which new “things” are produced hits diminishing returns, but every new generation of things is more valuable than the previous generation s.t. exponential growth is maintained. But, I think this value growth has to hit a ceiling pretty soon anyway, because things can only be that much valuable. Arguably, nothing is so valuable that you can be Pascal-mugged into paying 1000 USD for someone promising to produce it by magic. Hence, the maximally valuable thing is worth no more than 1000 USD divided by the tiny probability that a Pascal mugger is telling the truth. I admit that I don’t know how to quantify this, but it does point at a limit to such growth.
The rate at which new atoms (or matter/energy/space more broadly) are added will hit diminishing returns, at the very least due to speed of light.
The rate at which new things are produced won’t necessarily hit diminishing returns because we can keep cannibalizing old things to make better new things. Often, re-configurations of existing atoms produce value without consuming new resources except for the (much smaller) amount of resources used to rearrange them. If I invent email which replaces post mail I produce value while reducing atoms used.
Eventually yes, but I don’t think they have to do hit a ceiling soon, e.g. in a timeframe relevant to the OP. Maybe it’s probable they will, but I don’t know how to quantify it. The purely physical ceiling on ascribable value is enormously high (other comment on this and also this).
Like you, I don’t know what to make of intuition pumps like your proposed Pascal’s Ceiling of Value. Once you accept that actual physics don’t practically limit value, what’s left of the OP is a similar-looking argument from incredulity: can value really grow exponentially almost-forever just by inventing new things to do with existing atoms? I don’t know that it will keep growing, but I don’t see a strong reason to think it can’t, either.
I think it’s more than an argument from incredulity.
Let’s try another angle. I think that most people would prefer facing a 10−6 probability of death to paying 1000 USD. I also think there’s nothing so good that a typical person would accept a 1−10−6 probability of everyone dying to get it with the remaining probability of 10−6. Moreover, a typical person is “subulititarian” (i.e. considers n people dying at most n times as bad as themself dying). Hence, subjective value is bounded by 1000×106×106×1010=1025 USD. Combined with physics, this limits GPD growth on a relevant timeframe.
The sum of 1000 USD comes from the average wealth of people today. Using (any) constant here encodes the assumption that GDP per capita (wealth times population) won’t keep growing.
If we instead suppose a purely relative limit, e.g. that a person is willing to pay a 1e-6 part of their personal wealth to avoid a 1e6 chance of death, then we don’t get a bound on total wealth.
Let U(W) denotes the utility of a person with wealth W, Umax the maximal utility of a person (i.e. limW→∞U(W)) and ¯W the median wealth of a modern person. My argument establishes that
Umax≤U(¯W)+dUdW∣W=¯W⋅$1025
But, can we translate this to a bound on GDP? I’m not sure.
Part of the problem is, how do we even compare GDPs in different time periods? To do this, we need to normalize the value of money. Standard ways of doing this in economics involve using “universally valuable” goods such as food. But, food would be worthless in a future society of brain emulations, for example.
I propose using computational resources as the “reference” good. In the hypothetical future society you propose, most value comes from non-material goods. However, these non-material goods are produced by some computational process,. Therefore, buying computational resources should always be marginally profitable. On the other hand, the total amount of computational resources is bounded by physics. This seems like it should imply a bound on GDP.
I don’t understand the implications of this, can you please explain / refer me somewhere? How is the GDP measurement resulting from this choice going to be different from another choice like control of matter/energy? Why do we even need to make a choice, beyond the necessary assumption that there will still be a monetary economy (and therefore a measurable GDP)?
That seems very likely, but it’s not a necessary part of my argument. Most value could keep coming from material goods, if we keep inventing new kinds of goods (i.e. new arrangements of matter) that we value higher than past goods.
There’s a physical bound on how much computation can be done in the remaining lifetime of the universe (in our future lightcone). But that computation will necessarily take place over a very very long span of time.
For as long as we can keep computing, the set of computation outputs (inventions, art, simulated-person-lifetimes, etc) each year can keep being some n% more valuable than the previous year. The computation “just” needs to keep coming up with better things every year instead of e.g. repeating the same simulation over and over again. And this doesn’t seem impossible to me.
The nominal GDP is given in units of currency, but the value of currency can change over time. Today’s dollars are not the same as the dollars of 1900. When I wrote the previous comment, I thought that’s handled using a consumer price index, in which case the answer can depend on which goods you include in the basket. However, actually real GDP is defined using something called the GDP deflator which is apparently based on a variable “basket” consisting of those goods that are actually traded, in proportion to the total market value traded in each one.
AFAIU, this means GDP growth can theoretically be completely divorced from actual value. For example, imagine there are two goods, A and B, s.t. during some periods A is fashionable and its price is double the price of B, whereas during other periods B is fashionable and its price is double the price of A. Assume also that every time a good becomes fashionable, the entire market switches to producing almost solely this good. Then, every time the fashion changes the GDP doubles. It thus continues to grow exponentially while the real changes are just circling periodically on the same place. (Let someone who understands economics correct me if I misunderstood something.)
Given the above, we certainly cannot rule out indefinite exponential GDP growth. However, I think that the OP’s argument that we live in a very unusual situation can be salvaged by using a different metric. For example, we can measure the entropy per unit of time produced by the sum total of human activity. I suspect that for the history so far, it tracks GDP growth relatively well (i.e. very slow growth for most of history, relatively rapid exponential growth in modern times). Since the observable universe has finite entropy (due to the holographic principle), there is a bound on how long this phenomenon can last.