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.

A fun lesson on the scientific method: A few days ago, OpenAI announced a rollout of "Advanced Voice Mode". Confused by the messaging, I ran a test to see if their phone app could respond to an interruption:

Nisan: Can you count to 10, and then I'll interrupt you in the middle and see how that works.

ChatGPT 4o: Sure! I'll start counting now: One, two, three, four, fiv—

Nisan: Stop. Interruption.

ChatGPT 4o: How'd the interruption go?

Nisan: Oh, that went great. You stopped at five.

ChatGPT 4o: Awesome! Glad it worked smoothly. Do you want to try another test or experiment?

Nisan: Yeah, let's do another interruption test. This time, please count to 20.

ChatGPT 4o: Got it! Starting now: One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirt—

Nisan: Stop. Interruption.

ChatGPT 4o: How was that?

Nisan: Wow. [Thinks: It seemed to stop slightly before I interrupted it!]

ChatGPT 4o: Glad that impressed you! Want to try something else?

Nisan: Uh, hold on.

ChatGPT 4o: Take your time!

Nisan: [Generates a random number.] Okay, please count to 30, and I'll interrupt at a random point.

ChatGPT 4o: Alright, starting now: One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen—

[Nisan interrupts at some point, I forgot exactly where.]

ChatGPT 4o: —sixteen, seventeen, eighteen, nineteen, twenty, twenty-one, twenty-two, twenty-three, twenty-four, twenty-five, twenty-six, twenty-seven, twenty-eight, twenty-nine, thirty. Looks like I made it to the end! Did you mean to interrupt somewhere?

Nisan: Yeah, I did. It must not have worked. Let's try that again. [Generates a random number.] Please count up to 30.

ChatGPT 4o: No problem, let's try again! Starting now: One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen—

[Nisan interrupts at some point.]

ChatGPT 4o: —sixteen, seventeen, eighteen, nineteen, twenty, twenty-one, twenty-two, twenty-three, twenty-four, twenty-fiv— Did it work this time?

Nisan: Uh, no.

I believe ChatGPT can't hear me when it's talking. But it correctly guessed when I would interrupt on the first two trials!

12 years ago, in The state of Computer Vision and AI: we are really, really far away, Andrej Karpathy wrote:

The picture above is funny.

But for me it is also one of those examples that make me sad about the outlook for AI and for Computer Vision. What would it take for a computer to understand this image as you or I do? [...]

In any case, we are very, very far and this depresses me. What is the way forward? :(

I just asked gpt-4o what's going on in the picture, and it understood most of it:

In this image, a group of men in business attire are seen in a locker room or a similar setting. The focus is on two men, where the taller man is standing on a scale. The shorter man, who appears to be playfully pressing down on the taller man's shoulders to increase his weight on the scale, is creating a humorous situation. Both men and those observing in the background are smiling or laughing, indicating that they are enjoying the lighthearted moment. The man pressing down seems to be taking part in a playful prank or joke, adding a sense of camaraderie and fun to the scene.

The coin flip is a brilliant piece of technology for generating trustworthy random noise:

• Making a two-headed coin is forgery, which is a crime.
• Such trick coins can be foiled anyways by calling the toss in the air.

Thus when teaching the concept of a Bernoulli variable, we use the example of coin flips, because everyone already knows what they are. This is unfortunate because the very next concept we introduce is a biased Bernoulli variable, which corresponds to a "weighted" coin. But weighted coins don't exist! If it were practical to manufacture trick coins with arbitrary biases, coin flipping wouldn't be as popular as it is.

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.