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I have a graduate degree in mathematics. AI models can absolutely help you do research math in 2026. I recently asked chatGPT to prove a result which I know to have been published in Advances in Mathematics (a pretty good, but not top tier, journal) this year, and it gave a correct proof which was completely distinct from the one that was published.

I got my mortgage at 2.5% interest. The mortgage was about equal to all of the cash I had before buying, and I'm pretty sure I could (and should) have gotten a much larger mortgage before buying. Mortgages are essentially the only way for ordinary people to access that kind of credit. That's why buying a house is such a good deal. No one's lending an ordinary 30-year-old hundreds of thousands of dollars at 2.5% interest to plow into the S&P 500.

Of course, this kind of credit is _incredibly_ risky. People don't think of it this way, but a mortgage is essentially a massively leveraged bet on your local housing market. Which is already gives your risk profile an incredible lack of diversity. The most valuable thing you "own" in the time of your life when you buy a house is usually your future earnings potential, and your house will be the second most valuable thing you own. The thing is, both of these things are highly correlated with your local economy, so buying a house doubles down on your already high exposure to local economic conditions. People think of owning a home as a kind of boring, safe, low risk/low reward investment, but it's really exactly the opposite.

That's not a nit. There's not much left of the articles point once you take this on board.

Rigor is not the whole point of math. Understanding is. Rigor is a tool for producing understanding. For a further articulation of this point, see

https://arxiv.org/abs/math/9404236

But Houston Street is older than Sam Houston, and was always pronounced that way.

Machine generated text.

I actually have worked in this space and it, uh, has not shaken my belief that powerpoint talks are bad.

I like the idea that various communications media have implicit social contracts that can be broken. In my opinion, power point presentations break an implicit social contract that is held in handwritten talks: if it's worth you displaying a piece of information, so that I the listener feel the need to take it in or even copy it down, it has to be worth your time to actually physically write it on the board. With power point talks this is not honored, and the average power point talk is much, much worse than the average chalk talk. I bet there are lots of other examples.

Last time I flew Delta they no longer had this bot, which made me sad. One of my favorite parts of flying was getting absolutely crushed into a tiny cube by the airplane seat's easy chess bot, and then again by the airplane seat itself when the person in front of me reclines their seat.

Good negative feedback is a public service, a gift from the critic to you, and a severely undersupplied one in this world we live in.

Do you think that the average person can get a gold on the IMO?

In 2019, NPR's planet money did a segment on the Hong Kong protests that heavily featured Jimmy Lai. This segment from the end has always stuck with me.

GOLDSTEIN: China has not allowed more freedom of speech. Publications can still be shut down for criticizing the government. And yet, China has gotten richer. It started to develop its own financial center in Shanghai. Foreign money can now flow into China without going through Hong Kong, so the Chinese Communist Party doesn't need Hong Kong as much as it used to.

This has led to more and more tension between people in Hong Kong and the Chinese Communist Party, the Chinese government. In 2014, there was a fight over how to choose the government official who runs Hong Kong, and a million people in Hong Kong took to the streets to protest. Just last month, the government official who runs Hong Kong wanted to pass a new law that would allow people in Hong Kong to be extradited to China to stand trial. The people in Hong Kong said, we don't trust your mainland courts. Two million people protested in the streets, including, by the way, Jimmy Lai, who is now in his 70s.

What was it like? What was it like walking that day?

LAI: I was very excited - when you see so many people, you know, is fighting for a moral issue. We don't have guns. We don't have tanks. We don't have anything. The only thing we have the Chinese government don't is the moral authority we have, the moral courage we have.

GOLDSTEIN: The moral authority and courage, yeah.

LAI: Yes.

GOLDSTEIN: A few weeks later, on July 1, on the anniversary of Hong Kong's return to Chinese rule, protesters broke into the Hong Kong legislature buildings, smashed glass walls and spray-painted graffiti. Chinese leaders see these protesters and Jimmy Lai, for that matter, as agents for foreign influence - as, you know, basically latter-day colonialists. His house has been firebombed, and there was an assassination plot against him.

LAI: I stopped thinking about this because if I let the fear frighten me, I cannot go on, you know, because with what I have taken up, I have to sustain it. I will be the last to leave. That is like a captain who cannot jump the ship.

GOLDSTEIN: I mean, you're rich. You could leave if you wanted.

LAI: Yeah. If I'm rich but an a*hole...

GOLDSTEIN: (Laughter).

LAI: ...What my kids will think about me?

GOLDSTEIN: Yeah.

LAI: You know, being rich, you can be very poor...

GOLDSTEIN: Go on. Say more.

LAI: ...Because if you only have money, you lost the meaning, you lost the dignity, you lost everything as a human being. What else do you have?

Yes, Malthusian society was very boring, that's my point.

But in this world the distribution of social backgrounds in the population had very low entropy. Over 90% of people were just working the land.

If the Canterbury Tales had been actually representative of the time in which they were written, it would not have been the Knight's Tale, the Miller's Tale, the Reeve's Tale, etc. It would have been the Subsistence Farmer's Tale, the Subsistence Farmer's Tale, the Subsistence Farmer's Tale, etc.

Maybe? But you could have written this same thing in 1999 with OpenAI and Google replaced by Google and Yahoo, respectively.

Matrix multiplication is not ugly, but matrices themselves are ugly, mainly because they encode the arbitrary operation of choosing a basis. There's nothing especially nice about the pixel basis for images, or about the token basis for language. But of all the things that make up modern deep learning, matrix multiplication is surely the _least_ ugly. Relu/gelu is not pretty! Batch normalization is vomit-inducing!! Imagenet normalization? JFC!!!

https://people.mpim-bonn.mpg.de/zagier/files/math-mag/63-2/f...

"How often should you beat your kids?", by Don Zagier.

This note is a follow-up to the note "How to beat your kids at their own game" by K. Levasseur, in which the author proposes the following game to be played against one's own children: ... Levasseur analyzes the game and shows that on average you will have a score of n + (sqrt(pi * n) - 1) / 2 + O(n^{-1/2}), while the kid, of course, will have an average score of exactly n.

We maintain, however, that only the most degenerate parent would play against a 2-year-old for money, and that our concern should therefore be not by how much you expect to win, but with probability you will win at all.

The qwen clocks are art.

Are you saying that a non-aphantasic person can recall the eye color of everyone in their office?

I do not, and have never, understood why absolute R&D investment is compared to relative growth rate in this literature. It would make sense to compare R&D spending as a percentage of GDP to relative growth, or to compare absolute R&D spending to absolute per-year growth, but I remain mystified by the comparison of absolute R&D spending with relative growth.

Agree, but I think there is another, more important factor. They are a highly visible part of the internet, and their existence is mainly internet-based. This means that the people assessing them are mainly on the internet, and as we all know, internet discourse tends to the blandly negative (ironically my own comment is a mild example of this).

I think this point is less forceful than it seems due to the high dimensional nature of knowledge. I just don't know that much more than I did when I was younger; there are too many directions to set out in from any given point. I learn important new things all the time.

Do you hate anyone who doesn't recycle? If so, I think you should set a higher bar for the sins necessary to earn your hatred.

Most people don't think hard or carefully about politics, and their political views are a very tiny fraction of what they give to the world (this is true even for most people who do think hard and carefully about politics, by the way). Their vote is never pivotal, and their views do not shape any major institutions.

I don't agree with your first sentence, but I agree with the rest of this post.

In what sense is "Attention is all you need" a pun?

Clickbait titles are something of a tradition in this field by now. Some important paper titles include "One weird trick for parallelizing convolutional neural networks", "Attention is all you need", and "A picture is worth 16x16 words". Personally I still find it kind of irritating, but to each their own I guess.

Totally unrelated, but this reminds me of my favorite title of a math article: "How often should you beat your kids?" (it's about a certain simple combinatorial game)

https://people.mpim-bonn.mpg.de/zagier/files/math-mag/63-2/f...

(My favorite line: "Levasseur analyzes the game and shows that on average you will have a score of n + (sqrt(pn) - 1)/2 + O(1/sqrt(n)) while the kid will have exactly n. We maintain, however, that only the most degenerate parent would play against a two-year-old for money, so the question should be not by how much you expect to win, but with what probability you will win at all.")

My grandfather was a (non-professional) left wing agitator who was brought in front of HUAC when he joined the NIH, and also won the Lasker-Debakey award for his contributions to the development of combination chemotherapy.