twelve interesting things I learned studying the discovery of nature's laws,

Data-gathering was a crucial step in discovering both gravity and probability theory. One rich dude had a whole island and set it up to have lenses on lots of parts of it, and for like a year he'd go around each day and note down the positions of the stars. Then this data was worked on by others who turned it into equations of motion.

Relatedly, looking at the celestial bodies was a big deal. It was almost the whole game in gravity, but also a little helpful for probability theory (specifically the normal distribution was developed in part by noting that systematic errors in celestial measuring equipment followed a simple distribution).

It hadn't struck me before, but putting a ton of geometry problems on the ceiling for the entire civilization led a lot of people to try to answer questions about it. I'm tempted in a munchkin way to find other ways to do this, like to write a math problem on the surface of the moon, or petition Google to put a prediction market on its home page, or something more elegant than those two.

Probability theory was substantially developed around real-world problems! I thought math was all magical and ivory tower, but it was much more grounded than I expected.

After a few small things like accounting and insurance and doing permutations of the alphabet, games of chance (gambling) was what really kicked it off, with Fermat and Pascal trying to figure out the expected value of games.

Often people discovered more in this combination of "looking directly at nature" and "being the sort of person who was interested in developing a formal calculus to model what was going on".

Thought experiments about the world were a big deal too! Thomas Bayes did most of his math this way. He had a thought experiment that went something like this: his assistant would throw a ball on a table that Thomas wasn't looking at. Then his assistant would throw more balls on the table, each time saying whether it ended up to the right or the left of the original ball.

He had this sense that each time he was told the next left-or-right, he should be able to give a new probability that the ball was in any particular given region. He used this thought experiment a lot when coming up with Bayes' theorem.

Lots of people involved were full-time inventors, rich people who did serious study into a lot of different areas, including mathematics. This is a weird class to me.

Here's a quote I enjoyed from one of Pascal's letters to Fermat when they founded the theory of probability.

"I have not time to send you the demonstration of a difficulty which greatly astonished M. de Mere, for he has a very good mind, but he is not a geometer (this is, as you know, a great defect)…"

In Laplace's seminal work putting probability theory on a formal footing, he has a historical section at the end praising all the people who did work, how great they were and how beautiful their work was. Then he has one line on Bayes where he calls his work "a little perplexing".

"Bayes, in the Transactions Philosophiques of the year 1763, sought directly the probability that the possibilities indicated by past experiences are comprised within given limits; and he has arrived at this in a refined and very ingenious manner, although a little perplexing."

I watched a talk by Pearl about his causal models, and I was struck by the extent to which he had a "philosophy" of counterfactual inference. It had seemed pretty possible to me he would have said "here was a problem, and here is my solution", but instead he had a lot to say about counterfactuals and how he thought about them conceptually that wasn't in the math.

I think in the past I could have found myself unable to justify my interest in the philosophy of something as more than a personal interest. Now I have a practical justification, which is that it helps me come up with guesses about how nature works!

Pearl himself says that he has discovered two laws, and once you have them, you can fire him, because the rest is just algebra! And he calls it a calculus of counterfactuals, just like Newton and Bayes and everyone did. Fascinating.

I updated against expecting to resolve scientific disagreements at the time when the correct theory is known.

In the discovery of gravity, there were a lot of anomalies that didn't fit the data. For instance, Jupiter didn't follow the law: its orbit was a more elongated ellipse when it was further away. Uranus's orbit would jiggle a bit sometimes. Also there were two stars who didn't orbit their collective center of gravity, but instead some other point within the ellipse.

Want to know what they said at the time? For the stars, they said that we were probably just looking at them at a funny angle and that's why it didn't work. For Uranus, they said there was an invisible planet that was knocking it off-course. And for Jupiter, they said the light was moving too slowly for the measurements to work out.

Anyway… they were all right.

Feynman has a wonderful quote on the art of guessing nature's laws that includes at least two paths not discussed above. That said I don't understand them, in particular the ways that quantum mechanics was discovered. I'm tempted to dig into that some.

One confusion I wrote down in advance was "I still don't quite know how to predict that there will not be a simple mathematical apparatus that explains something. Why the motion of the planets, why the game of chance, why not the color of houses in England or the number of hairs on a man's head?"

Looking back on this, I don't know whether I got a direct answer, but I now feel that my answer is something like "look for the places where Nature will show herself directly". Obviously that's not a very well-specified answer, but I feel like it points to a real distinction.

I also made an advance prediction: "I guess I also make the advance prediction that most of the rest of the probability math was developed by people who liked symbol manipulation more than people doing real-world problem solving. But I would be interested to be surprised here."

This prediction was false! It took both! All the probability math was developed by people who liked using math to reason rigorously about the world, and who were interested in understanding the real world!