133 points by rbanffy 3 days ago | 77 comments on HN
| Mild positive Editorial · v3.7· 2026-03-01 07:28:02 0
Summary Academic Integrity Acknowledges
The article is an investigative historical piece examining claims that mathematician Georg Cantor plagiarized his groundbreaking work on infinity. It engages with intellectual property, freedom of expression, and education rights through its examination of academic attribution and scientific discovery. The evaluation shows mild positive signals in Articles 11, 19, 26, and 27, with most other provisions neutral due to the specialized subject matter.
“Noether, who was Jewish, fled from Germany to the U.S., where she died two years later from cancer”
It wasn’t two years, and it wasn’t cancer. These details are unimportant to the (quite interesting) story, but the error is a sign that the author copies information from unreliable secondary sources, which puts the other facts in the article in doubt.
I wrote to him about the error when the article first appeared, but received no reply.
> In their 1872 papers, though, Cantor and Dedekind had found a way to construct a number line that was complete. No matter how much you zoomed in on any given stretch of it, it remained an unbroken expanse of infinitely many real numbers, continuously linked.
> Suddenly, the monstrosity of infinity, long feared by mathematicians, could no longer be relegated to some unreachable part of the number line. It hid within its every crevice.
I'm vaguely familiar with some of the mathematics, but I have no idea what this is trying to say. The infinity of the rational numbers had been known a thousand years prior by the Greeks, including by Zeno whom the article already mentioned. The Greeks also knew that some quantities could not be expressed as rational numbers.
I would assume the density of irrational numbers was already known as well? Give x < y, it's easy to construct x + (y-x)(sqrt(2))/2.
I think we can do without the baity title since most HN readers should know who Cantor and Dedekind are. Edit: okay, maybe not Dedekind.
If someone wants to suggest a better title (i.e. more accurate and neutral, and preferably using representative language from the article itself), we can change it again.
This whole plagiarism thing is too overwrought these days. People discuss stuff and the idea forms in the discussion between the two. Then one writes it up. Oh he plagiarized the other. I don’t know man.
From the article it's hard to tell if Cantor really did plagiarize (though it seems Dedekind thought he did).
According to the article, Cantor proved the theorem first and sent it to Dedekind. Dedekind suggested a simplification of the proof, which Cantor used when he wrote it up. The story doesn't make Cantor look good, but if the original proof by Cantor is correct, then the credit for the theorem still basically belongs to Cantor.
eh this "plagiarism" framing is overreaching
there were two proofs in the paper: countability of algebraic numbers and uncountability of reals
countability of algebraic numbers is a rather trivial induction on countability of rationals/pairs of numbers, which Cantor already knew about
Cantor himself did prove uncountability of real numbers; Dedekind just helped him clean the proof up
to me it seems like Dedekind's assistance was the kind of thing that might merit an acknowledgement, or possibly even joint authorship if subspecialty norms are generous, but far from a novel contribution on its own; unlike the uncountability of reals which was genuinely important and nontrivial. Dedekind, like Cantor, had other very important contributions, but certainly no claim on what Cantor is known for; and the context with Kronecker meant that this would prevent the work from ever being published. Also, this article doesn't actually show Dedekind was specifically upset by the "plagiarism", there may be any number of other reasons they may have stopped corresponding; and Dedekind's "hope this is useful" comment to Cantor can be read as permission to use it for his purposes
Thank you! After Benj Edwards and Kyle Orland's Ars Technica article they published using AI (while saying they didn't), and all the while their article was about an AI agent publishing a hit piece on Scott Shambaugh (matplotlib maintainer), I feel like I now assume journalists are using AI and things need to be fact-checked just as we do for our AI interactions.
I appreciate hearing about details like this and getting the source directly. I hope Kristina Armitage and Michael Kanyongolo from Quanta Magazine respond and you can update us!
I’ll go out on a limb and say the majority of HN users at this point do not know the context and implications of the impact of Cantor - would probably have only heard the name in the context of mathematics but no deeper
I’d go further and say the majority have not ever heard of the name Dedekind
I have an opinion about the editorial style of Quanta that I don't think it's popular here (judging by how often they get upvoted), but I think it's a symptom of that.
They cover science, but the template they consistently follow is a vague title that oversells the premise and then an article filled with human-interest details and appeals to implications. This makes it easy for everyone to follow along and have an opinion, but I feel like science is a distant backdrop and never the actual subject.
In this article, what's the one tidbit of scientific knowledge that we gain? Dedekind's and Cantor's work is described only in poetic abstractions ("a wedge he could use to pry open the forbidden gates of infinity"). When the focus is writing a gossip column for eloquent people, precision doesn't matter all that much.
Complete just means the limit of every sequence is part of the set. So there’s no way to “escape” merely by going to infinity. Rational numbers do not have this property.
How to construct the real numbers as a set with that property (and the other usual properties) formally and rigorously took quite a long time to figure out.
> Before their papers, mathematicians had assumed that even though the number line might look like a continuous object, if you zoomed in far enough, you’d eventually find gaps.
I'll try to interpret this sentence.
We all have some mental imagery that comes to mind when we think about the number line. Before Cantor and Dedekind, this image was usually a series of infinitely many dots, arranged along a horizontal line. Each dot corresponds to some quantity like sqrt(2), pi, that arises from mathematical manipulation of equations or geometric figures. If we ever find a gap between two dots, we can think of a new dot to place between them (an easy way is to take their average). However, we will also be adding two new gaps. So this mental image also has infinitely many gaps.
Dedekind and Cantor figured out a way to fill all the gaps simultaneously instead of dot by dot. This method created a new sort of infinity that mathematicians were unfamiliar with, and it was vastly larger than the gappy sort of infinity they were used to picturing.
I don't like the way it's written, but what they are talking about is completeness in the sense of "Dedekind completeness"; i.e., that given any two sets A and B with everyone in A below everyone in B, there is some number which is simultaneously an upper bound for A and a lower bound for B.
Note that this fails for the rationals: e.g., if we let A be the rationals below sqrt(2) and B be the rationals above sqrt(2).
Take something like the integers (1,2,3,etc.). They are infinite; given an integer, you can always add 1 and get a new integer.
However, there are "gaps" in that number line. Between 1 and 2, there are values that aren't integers. So the integers make a number line that is infinite, but that has gaps.
Then we have something like the rational numbers. That's any number that can be expressed as a ratio of 2 integers (so 1/2, 123/620, etc.). Those ar3 different, because if you take any two rational numbers (say 1/2 and 1/3), we can always find a number in between them (in this case 5/12). So that's an improvement over the integers.
However, this still has "gaps." There is no fraction that can express the square root of 2; that number is not included in the set of rational numbers. So the rational numbers by definition have some gaps.
The problem for mathematicians was that for every infinite set of numbers they were defining, they could always find "gaps." So mathematicians, even though they had plenty of examples of infinite sets, kind of assumed that every set had these sorts of gaps. They couldn't define a set without them.
Cantor (and it seems Dedekind) were the first to be able to formally prove that there are sets without gaps.
If I understand the article correctly, that second proof was published as a rider on a first proof that was entirely Dedekind's. So, there was definitely a credit owed at time of publishing.
I came away with the impression that the biggest villain in this story was Kronecker. Without the need to tiptoe around his ego and gatekeeping, these results may have been published as a paper with joint authorship.
Is the wikipedia page more or less correct or in need of editing in your view?
(Given that you are probably the current world expert on Noether having written the book)
I’ve been in joint discussions where “the idea forming” was really one party thinking out loud and doing almost all the work, and the other providing approximately the same function as a rubber duck.
Sometimes the one doing the heavy lifting is me; sometimes it’s the other person, and I’m happy to make squeaky rubber duck noises that help. And with some people we have switched roles, even during the conversation. And perception will not track with reality because we’re all the hero of our own story.
Very hard to assign credit after the fact without a verbatim transcript, which written letters provide here.
That the credit for the theorem belongs to Cantor is not under question. This is acknolwedged in the article:
>The revelation about Cantor’s result doesn’t undermine his legacy. He was still the first person to prove that there are more real numbers than whole ones, which is what ultimately opened up infinity to study.
What he is alleged to have plagiarised are the proofs, or at least one of the proofs. The original article by Goos [0] contains a lot more details about this, including a partial transcription of the letter by Dedekind that Cantor is accused of plagiarism. The story is complex.
1. Cantor's paper has two theorems: the countability of algebraic numbers and the uncountability of reals.
2. The proof of the former appears in Dedekind's letter, and Cantor acknowledges this in his response to the letter. Dedekind mentions in his letter that he only thought about proving this because of Cantor's prompt and only wrote it with the hope of helping Cantor. Dedekind felt that the proof by Cantor is "word for word" his, although it is quite the case. It is essentially the same proof though.
Cantor also felt that Dedekind's proof that the set of algebraic numbers is countable is essentially the same as his own proof of the countability of tuples. It remains that he didn't think of adapting that proof himself, and that Dedekind was the first to prove the theorem is not under question.
3. Dedekind was not the first to prove the uncountability of real numbers. However, he gave a number of ideas to Cantor in that same letter. Namely, he suggested proving the uncountability of the interval (0,1), and it seems that gave a pointer towards how to build the diagonalisation argument, although how this statement was useful to Cantor (page 76 of Goos' paper) escapes me.
EDIT: it's not a pointer to the diagonalisation argument, it is an argument why proving the theorem on (0,1) is enough.
4. Cantor proved the uncountability of reals shortly afterwards, and shared his proof with Dedekind. Dedekind simplified the proof in his reply, and Cantor seems to have come up with a similar simplification on his own. None of these letters are analysed in Goos' article.
5. Cantor published the two theorems; the first proof is essentially the same as Dedekin's, and the second proof is possibly the one Dedekind's simplified version of Cantor's. Dedekind is not acknowledged at all in that paper, due to academic politics.
Goos' paper is very detailed and quite readable. I recommend it. The site is pretty annoying and you can't download the article without creating an account, but you can read the article online.
Even if the most important theorem of the two is unquestionably creditable to Cantor, the first one should likewise unquestionably be credited to Dedekind, at least partially. This is where the accusation of plagiarism stems from.
Beyond the question on plagiarism, there is no question that Cantor and Dedekind worked together on this. The lack of acknowledgement by Cantor is certainly quite unfortunate.
Describes historical mathematical discovery and claims of intellectual theft; implies value of knowledge advancement and attribution, but not directly addressing human dignity or rights
FW Ratio: 60%
Observable Facts
The article narrates Georg Cantor's proof about different sizes of infinity in 1874.
The article reports claims of plagiarism against Cantor based on newly discovered letters.
The article presents historical academic debates about credit and attribution.
Inferences
The discussion of intellectual theft and proper credit implicitly acknowledges the value of human knowledge and achievement.
The historical analysis of mathematical discovery suggests awareness of progress in human understanding.
build af177b1+4aph · deployed 2026-03-01 06:49 UTC · evaluated 2026-03-01 12:31:25 UTC
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