Nisan's Shortform

Recent interviews with Eliezer:

• 2023.02.20 Bankless
• 2023.02.20 Bankless followup
• 2023.03.11 Japan AI Alignment Conference
• 2023.03.30 Lex Fridman
• 2023.04.06 Dwarkesh Patel
• 2023.04.18 TED talk
• 2023.04.19 Center for the Future Mind
• 2023.05.04 Accursed Farms
• 2023.05.06 Logan Bartlett
• 2023.05.06 Fox News
• 2023.05.08 EconTalk
• 2023.07.02 David Pakman
• 2023.07.13 AI IRL
• 2023.07.13 The Spectator (Edited transcript of the full interview)
• 2023.07.13 Dan Crenshaw
• 2023.07.28 Coleman Hughes (with Scott Aaronson and Gary Marcus)
• 2023.08.16 Dwarkesh Patel (with George Hotz)
• 2023.08.23 One Decision
• 2023.09.23 Destiny

Hyperbolic growth

The differential equation $y^{\prime }=b{y}^{1+\alpha }$, for positive $\alpha$ and $b$, has solution

$$y=\frac{1}{(t_0-t{)}^{1/\alpha }}$$

(after changing the units). The Roodman report argues that our economy follows this hyperbolic growth trend, rather than an exponential one.

While exponential growth has a single parameter — the growth rate or interest rate — hyperbolic growth has two parameters: $t_0$ is the time until singularity, and $\alpha$ is the "hardness" of the takeoff.

A value of $\alpha$ close to zero gives a "soft" takeoff where the derivative gets high well in advance of the singularity. A large value of $\alpha$ gives a "hard" takeoff, where explosive growth comes all at once right at the singularity. (Paul Christiano calls these "slow" and "fast" takeoff.)

Paul defines "slow takeoff" as "There will be a complete 4 year interval in which world output doubles, before the first 1 year interval in which world output doubles." This corresponds to $\alpha \le 2$. (At $\alpha =2$, the first four-year doubling starts at $\frac{16}{3}$ and the first one-year doubling starts at $\frac{4}{3}$ years before the singularity.)

So the simple hyperbola $y=1/(t_0-t)$ with $\alpha =1$ counts as "slow takeoff". (This is the "naive model" mentioned in footnote 31 of Intelligence Explosion Microeconomics.)

Roodman's estimates of historical $\alpha$ are closer to $0.5$ (see Table 3).

People are fond of using the neologism "cruxy", but there's already a word for that: "crucial". Apparently this sense of "crucial" can be traced back to Francis Bacon.

Conception is a startup trying to do in vitro gametogenesis for humans!

We can derive Newton's law of cooling from first principles.

Consider an ergodic discrete-time dynamical system and group the microstates into macrostates according to some observable variable $X$. ($X$ might be the temperature of a subsystem.)

Let's assume that if $X=x$, then in the next timestep $X$ can be one of the values $x-dx$, $x$, or $x+dx$.

Let's make the further assumption that the transition probabilities for these three possibilities have the same ratio as the number of microstates.

Then it turns out that the rate of change over time $\frac{dX}{dt}$ is proportional to $\frac{dH}{dX}$, where $H$ is the entropy, which is the logarithm of the number of microstates.

Now suppose our system consists of two interacting subsystems with energies $E_1$ and $E_2$. Total energy is conserved. How fast will energy flow from one system to the other? By the above lemma, $\frac{d{E}_1}{dt}$ is proportional to $\frac{dH}{d{E}_1}=\frac{d{H}_1}{d{E}_1}-\frac{d{H}_2}{d{E}_2}=C_1-C_2=\frac{1}{T_1}-\frac{1}{T_2}$.

Here $C_1$ and $C_2$ are the coldnesses of the subsystems. Coldness is the inverse of temperature, and is more fundamental than temperature.

Note that Newton's law of cooling says that the rate of heat transfer is proportional to $T_2-T_1$. For a narrow temperature range this will approximate our result.

Agents who model each other can be modeled as programs with access to reflective oracles. I used to think the agents have to use the same oracle. But actually the agents can use different oracles, as long as each oracle can predict all the other oracles. This feels more realistic somehow.