Since Stephen Miller is such a horrible person and a Jew, I decided he must be a Yecke. This is a derisive name (reportedly from the Yiddish word for jacket) that Jews from Eastern Europe use for the population of German Jews used for the descendants of the earliest wave of Jewish immigration from the early 19th century, mostly living in the upper east side of NYC.
So I googled, “Is Stephen Miller a Jecke”, not really expecting an answer. To my great surprise the AI responded by first defining Jecke (or Jekke) and then saying that he wasn’t, his family having emigrated from Belarus in 1913.
I was and remain impressed. Still, no AI or other source has correctly answered my test question: “Who discovered the shuffle idempotent?” No I won’t explain. The correct answer is readily determinable from the mathematical literature.
No, it was not Louis Solomon, although he may have named them. But he was not the discoverer. His paper appeared a year later than the one in which they first appeared. I say “they” because there is an infinite series of them, one in each dimension.
This is the answer my ChatGPT gave (no idea if it’s correct or not):
Short answer: no single dramatic “Eureka!” moment, no lone Indiana Jones of shuffling.
Longer, more honest answer: the idea of a shuffle idempotent comes out of formal language theory and algebra, and it emerged rather than being “discovered” by one person.
The math backstory (minus the chalk dust)
The shuffle operation (also called the interleaving of strings) was studied heavily in the 1960s–1970s in automata theory. A language is shuffle-idempotent if shuffling it with itself gives you… itself. Math’s version of “I am what I am.”
The concept was developed primarily through the work of researchers like:
Marcel-Paul Schützenberger
Jean Berstel
Dominique Perrin
They didn’t necessarily coin the catchy phrase “shuffle idempotent” in a single paper, but they laid the foundations for:
shuffle (interleaving) operations on languages
algebraic properties of those operations
and conditions under which idempotence occurs
Translation into plain English
Think of it like this:
If you shuffle a deck with itself, and nothing fundamentally changes, congratulations—you’ve got idempotence.
Mathematicians noticed this behavior while classifying languages and algebraic structures, not while shuffling cards at Vegas (sadly).
Bottom line
Not discovered by one person Emerged from formal language theory Strongly associated with Schützenberger and later Berstel & Perrin Developed across multiple papers, not a single “aha!” moment
If you want, I can:
show a simple example of a shuffle-idempotent language
explain why idempotence matters (spoiler: structure + predictability)
or connect it to card shuffles, because obviously that’s more fun.
In the sense used in homological algebra/Hochschild–Harrison theory, the “shuffle idempotent(s)” are due to Michael Barr: he explicitly constructed these idempotents in the group rings of symmetric groups in his 1962 PhD thesis (and later gave an inductive construction), and they subsequently “came to be called the shuffle idempotents.”
I was also surprised at this at first. It seems like the kind of question AI (both the explicit AI mode and all the AI regular Google uses under the hood) always fails at. It’s an obscure very specific question (“Is Stephen Miller a Jecke”) that is closely related to a more general commonly asked questions (“Is Stephen Miller Jewish”)
But thinking about it, it’s actually just two easily answerable questions “what’s a Jecke” (which has a specific answer, a descendant of the earliest German Jewish immigrants to the US) and “when did Stephen Miller’s family immigrate to the US”
Still fairly impressive that it worked it out. But doesn’t change my opinion that AI makes Google far worse at answering very specific questions, with precise wording, than it used to be.
I don’t think of this as a particularly hard question for AI or a search engine. It’s a very specific obscure technical term but there are a bunch of relevant hits that come up when you search for “shuffle idempotent” (including Michael Barr’s homepage). And there isn’t another closely related more.common term that will get the AI confused.
The fact it was able to summarize it correctly was impressive but that’s a very different functionality. AI is quite impressive at summarizing things its not particularly good at finding things out. As I said in a thread on the subject I think Great Debates is more imperiled by AI than Factual Questions
The fact is that every time I tried to use AI, I asked exactly that question and got all sorts of wrong answers including Louis Solomon, but this is the first time I’ve seen the correct answer.
But I was duly impressed by the answer to the Stephen Miller question, including that it was able to “comprehend” Jecke. (Scare quotes because I really don’t think AI comprehends anything.)
This is similar to the context-hopping that happens in a question like “What is the capital of the state that holds Dallas.” It has to contextualize the second half of the question to answer the first. I broadly understand how that happens, but it’s still an interesting feat.
On a side note, I asked ChatGPT the same Stephen Miller question and then challenged it to explain how it answered. It gave me wrong answers twice, and only after I called it out twice did it land on something reasonable. That’s because, as most of you know, it’s not explaining…it’s predicting. (it also offered me a flowchart on the final answer, and it got that wrong too)
Add something like “think hard about it” to the end of your prompts and that will force it (or should, but I have a paid model) into deep thinking mode where it actually researches on the internet and stuff.
Not discovered by one person
Emerged from formal language theory
Strongly associated with Schützenberger and later Berstel & Perrin
Developed across multiple papers, not a single “aha!” moment