[HN Gopher] New evidence that Cantor plagiarized Dedekind?
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New evidence that Cantor plagiarized Dedekind?
Author : rbanffy
Score : 93 points
Date : 2026-02-25 17:28 UTC (3 days ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| leephillips wrote:
| "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.
|
| Noether's real story is recounted in https://amzn.to/3YZZB4W.
| mymacbook wrote:
| 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!
|
| Scott's Blog on Hit Piece: https://theshamblog.com/an-ai-agent-
| published-a-hit-piece-on... Ars Editor Note:
| https://arstechnica.com/staff/2026/02/editors-note-retractio...
| Ars Retraction: https://arstechnica.com/ai/2026/02/after-a-
| routine-code-reje...
| zeroonetwothree wrote:
| It's not like journalists were very accurate before AI.
| Classic Gell-Mann amnesia
| moefh wrote:
| Being "not very inaccurate" is very different from
| publishing outright fabricated quotes, which is what Ars
| Technica did and later admitted to:
| https://arstechnica.com/staff/2026/02/editors-note-
| retractio...
| shermantanktop wrote:
| I see what you did there. Turning the page of time, I
| guess.
| lich_king wrote:
| 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.
| ajkjk wrote:
| I think your opinion is popular here. Quanta is, while better
| than nothing, universally disappointing. It seems like it
| would be much easier for them to do a better job -- write
| less vaguely, fact-check more, assume the reader is a bit
| more intelligent.
| johngossman wrote:
| I find they are good at identifying interesting topics and
| writing articles that don't deliver. They remind me of Omni
| magazine (which I subscribed to at one point). The articles
| aren't even wrong.
| rvba wrote:
| The articles are unreadable fluff
| QuesnayJr wrote:
| Are you citing your own book?
| leephillips wrote:
| It won't be the last time.
| wizzwizz4 wrote:
| It's best practice to say something like "Noether's real
| story is recounted in my book [link]". This both
| establishes you as a subject matter expert, and stops your
| comments looking like disingenuous grift.
| dvt wrote:
| It's literally cited in his bio, and he's using his real
| name on HN. It's about as far as grift as it could be. If
| he's being curt, he's probably (rightfully) frustrated
| that "journalists" are getting such bottom-of-the-barrel
| facts wrong.
| tadfisher wrote:
| But surely you would agree that "my book" just wastes
| less time all around, and doesn't harm the author's
| message?
| 2b3a51 wrote:
| https://en.wikipedia.org/wiki/Emmy_Noether
|
| 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)
| dkarl wrote:
| > 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 don't get what "suddenly" became apparent.
| terminalbraid wrote:
| The density does not dictate cardinality which is what this
| article is about.
| pfortuny wrote:
| The continuum. Connectedness.
| zeroonetwothree wrote:
| 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.
| lupire wrote:
| Critically, "Complete" also means that the supposed limit
| _necessarily exists_.
| markisus wrote:
| > 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.
| dkarl wrote:
| We've known since Zeno that all of our ways of visualizing
| infinity in finite terms are incomplete and provably
| incorrect, despite being unavoidable in human thinking. In
| other words, we knew the "gaps" reflected incomplete
| reasoning, not real emptiness between "consecutive" numbers.
| If Dedekind and Cantor only changed how we visualize
| infinity, I don't understand why it would cause a stir.
|
| > This method created a new sort of infinity that
| mathematicians were unfamiliar with, and it was vastly larger
|
| I understand that the construction of the reals paved the way
| for the later revolutionary (and possibly disturbing, for
| people with strongly held philosophical beliefs about
| infinity) discovery that one infinity could be larger than
| another. But in the narrative laid out by the article, that
| comes later, and to me it's clear (unless I misread it) that
| the part I quoted is about the construction of the reals,
| before they worked out ways to compare the cardinality of the
| reals to the cardinality of the integers and the rationals.
| AndrewKemendo wrote:
| > If Dedekind and Cantor only changed how we visualize
| infinity, I don't understand why it would cause a stir.
|
| Because scientific progress is explicitly the process of
| changing the general mental model of how people approach a
| problem with a more broadly capable and repeatable set of
| operations
|
| This is philosophy of science 101
| dkarl wrote:
| I should have been more specific; I understand why it was
| a mathematical breakthrough. What I don't understand is
| why it would have triggered some kind of psychological
| horror or philosophical crisis. It was a new way of
| understanding numbers, but it didn't reveal numbers to be
| acting any differently than we had always assumed.
|
| If anything, it seems like it would have been comforting
| to finally have mathematical constructions of the real
| numbers. It had been disturbing that our previous
| attempts, the rational and algebraic numbers, were known
| to be insufficient. The construction of the reals finally
| succeeded where previous attempts had failed.
| AndrewKemendo wrote:
| History only seems obvious in retrospect
|
| I would invite you to be more open to the idea that
| people don't live in a world where they operate inside a
| theoretical framework with localized test actions
|
| major breakthroughs tend to cause existential crises
| because most people don't have full scope of their work
| in order to understand where it is broken
| Exoristos wrote:
| Because painting those who objected to these definitions
| of mathematical infinity as "horrified" and "disturbed"
| was a form of character assassination, which was not
| uncommon at the time. The high moderns didn't play.
| antasvara wrote:
| "Knowing" something and proving it mathematically are two
| different beasts.
|
| Zeno couldn't prove that there were no gaps; he showed that
| infinity was different from how we understood finite
| things, bit that's not the same as proving there are no
| gaps.
|
| Later, mathematicians proved the existence of irrational
| numbers. These were "gaps" in the rational numbers, but
| they weren't all the "same" of that makes sense? The square
| root of 2 and Euler's number are both irrational, but it's
| not immediately clear how you'd make a set that includes
| all the numbers like that.
| markisus wrote:
| I'm not sure everyone knew that gaps reflected incorrect
| reasoning. It would have been natural to assume that all
| infinite sets were qualitatively the same size, since
| uncountable infinity was not an idea that had been
| discovered yet. Zeno's own resolution wasn't that his
| reasoning wrong, but that our perception of the world
| itself is wrong and the world is static and unchanging.
|
| As for the importance of visualization (of the reals), I
| don't think you can cleanly separate it from formalism (as
| constructed in set theory).
|
| I think we all have built in pre-mathematical notions of
| concepts like number, point, and line. For some, the
| purpose of mathematics is to reify these pre-mathematical
| ideas into concrete formalism. These formalisms clarify our
| mental pictures, so that we can make deeper investigations
| without being led astray by confused intuitions. Zeno could
| not take his analysis further, because his mental imagery
| was not detailed enough.
|
| From clarity we gain the ability to formalize even more of
| our pre-mathematical notions like infinitesimal,
| connectedness, and even computation. And so we have a
| feedback loop of visualization, formalism, visualization.
|
| I think the article was saying that Dedekind and Cantor
| clarified what we should mean when we talk about the number
| line, and dispelled confusions that existed before then.
| bhk wrote:
| Extraordinary claims require extraordinary evidence. Can you
| cite any claims by mathematicians that there were "gaps"? It
| isn't even true for rational numbers that you can identify an
| unoccupied "gap".
| Quekid5 wrote:
| sqrt(2)
| bhk wrote:
| That's not a "gap" that you find by "zooming in". And how
| can it be a gap when it is occupied?
| vessenes wrote:
| You're thinking of this with the benefit of dedekind in
| your schooling - whether or not your calculus class told
| you about him.
|
| Density - a gapless number line - was neither obvious nor
| easy to prove; the construction is usually elided even in
| most undergraduate calculus unless you take actual
| calculus "real analysis" courses.
|
| The issue is this: for any given number you choose, I
| claim: you cannot tell me a number "touching" it. I can
| always find a number between your candidate and the first
| number. Ergo - the onus is on you to show that the number
| line is in fact continuous. What it looks like with the
| naive construction is something with an infinite number
| of holes.
| wrsh07 wrote:
| You can construct sequences of rational numbers where the limit
| is not rational (eg it's sqrt 2)
|
| Trivially, the sequence of numbers who are the truncated
| decimal expansion of root 2 (eg 1.4, 1.41. 1.414, ...) although
| I find this somewhat unsatisfying.
|
| With the real numbers there are no gaps. There are no sequences
| of reals where the limit of that sequence is not a real number
| JadeNB wrote:
| > Give x < y, it's easy to construct x + (y-x)(sqrt(2))/2.
|
| That's only obviously irrational if x and y are rational. (But
| maybe you meant that, given an arbitrary interval a < b, you
| first shrink it to a rational interval a < x < y < b?)
| Chinjut wrote:
| 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).
| antasvara wrote:
| 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.
| lupire wrote:
| Right, but that's the opposite of what the Quanta article
| says. The article says that Cantor and Dedekind discovered
| infinity in bounded intervals. What they discovered (really,
| what they concocted) was uncountable infinity.
| littlestymaar wrote:
| Quanta messing things up isn't a particularly rare
| occurrence, unfortunately.
| dkarl wrote:
| I just don't understand why this was disturbing. Prior to the
| construction of the reals, the existence of irrational and
| transcendental numbers was disturbing, because they showed
| that previous constructions (rational numbers and algebraic
| numbers) were incomplete. If those gaps were disturbing, a
| construction without gaps should have been satisfying,
| reassuring, a resolution of tension. Was there some
| philosophical or theological theory that required the
| existence of gaps, that claimed that a complete construction
| of the number line was mathematically impossible, because of
| some attribute of God or the cosmos?
| layer8 wrote:
| I think the issue was that most irrational/transcendental
| numbers aren't finitely representable. This means that they
| are mathematical objects which, each of them individually,
| somehow consist of an infinity (e.g. an infinite decimal
| expansion). They are the result or end point of infinitely
| many steps (e.g. a converging sequence) that you can't
| actually reach the end of in practice, and for most of them
| can't even write down a finite description on what steps to
| perform, and which therefore arguably doesn't "really"
| exist.
|
| Another point of contention was the notion that the
| continuous number line would be formed out of dimensionless
| points. Numbers were thought of as residing _on_ the line,
| but it was hard to grasp how a line could consist _solely_
| of a collection of points, since given any pair of points,
| there would always be a gap between them. "Clearly" they
| can't be forming a contiguous line.
| sandslides wrote:
| could I just leave my favourite thing ever here? thanks :)
|
| https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Gra...
| hearsathought wrote:
| > > 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.
|
| Think of the number line stretching from negative infinity to
| positive infinity and let C represent the
| cardinality/size/count of numbers on that number line. Now just
| take portion of the number line from 0 to 1. Let C1 represent
| the cardinality/size/count numbers from the truncated line from
| 0 to 1. You would assume that C > C1. But in fact they are
| equal. There are just as many infinite real numbers from 0 to 1
| as there are on the entire number line. Even worse, this hold
| true for any portion of the number line, how small or big you
| make the line. Rather than infinity being in a far distance
| place at the edge of the line in either direction, there is
| infinity everywhere along the number line.
|
| > I don't get what "suddenly" became apparent.
|
| It appeared suddenly because prior to cantor/dedekind,
| mathematics only understood the countably infinite ( natural
| numbers, integers, rationals, etc ) . By constructing a
| complete number line, cantor/dedekind showed there is a
| cardinality greater than infinity ( countable ). The continuum.
|
| Cantor also showed that there is an infinite number of
| cardinalities.
| dang wrote:
| 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.
| AndrewKemendo wrote:
| Hard disagree
|
| 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
| freehorse wrote:
| I would assume that at least cantors technique of
| diagonalisation should have found its way into some CS course
| that I assume a good part of the audience here has studied?
| Considering that's what Turing used to prove the
| undecidability of the halting problem.
| AndrewKemendo wrote:
| Having been active on this website for 14 years now ... At
| this point I would venture to say The median hacker news
| commentator does not have aa computer science degree
| tchalla wrote:
| > since most HN readers should know who Cantor and Dedekind
| are.
|
| Show up with your hands here if you didn't know either Cantor
| or Dedekind.
| jacquesm wrote:
| Then you can just skip this submission and nobody will be the
| wiser.
| zenethian wrote:
| I am not a mathematician; I barely knew who Cantor was and had
| never heard of Dedekind. I would have likely not read the
| article without the title being so sensational. Your assumption
| sits upon the tip of your nose.
| tgv wrote:
| I'm here for the 19th century drama. Imagine the head lines!
| Cantor's Continuity Credentials Cancelled: Clear Cut Copy Cat
| Case!
|
| Millions of views for Tiktoks about homomorphisms and aleph
| numbers. Just the news we need right now.
| collabs wrote:
| This comment made me think of this xkcd 2501
|
| https://xkcd.com/2501
|
| There really is an xkcd for everything
| freehorse wrote:
| To be fair, if one does not know who cantor and dedekind
| were, the drama about the former plagiarising the latter is
| probably not that interesting anyway.
| JasonADrury wrote:
| > most HN readers should know who Cantor and Dedekind are.
| Edit: okay, maybe not Dedekind.
|
| This is a top tier troll, good job.
|
| I think "Cantor: The Man Who Stole Infinity?" would strike a
| good balance.
| hearsathought wrote:
| > 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 you think most HN readers would know who Cantor is, let
| alone his ideas on infinity, then you have no understanding of
| the community you are modding...
|
| > 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.
|
| May I suggest changing plagiarized to plagiarised to keep in
| line with the King's english you so favor?
|
| Since you are in the mood for suggestions, can I suggest you
| stop with the passive aggressive comment rate limits? Thanks.
| renewiltord wrote:
| 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.
| readthenotes1 wrote:
| Try reading the article?
|
| And you don't like giving credit to people that help you? You
| may be successful by some measures, but not by the more
| important ones
| renewiltord wrote:
| Haha, the sanctimony is really too much.
| shermantanktop wrote:
| 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.
| QuesnayJr wrote:
| 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.
| cls59 wrote:
| 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.
| QuesnayJr wrote:
| I read it the other way. Here's the quote from the article:
|
| On December 7, 1873, he wrote to Dedekind that he thought
| he'd finally succeeded: "But if I should be deceiving myself,
| I should certainly find no more indulgent judge than you." He
| laid out his proof. But it was unwieldy, convoluted. Dedekind
| replied with a way to simplify Cantor's proof, building a
| clearer argument without losing any rigor or accuracy.
| Meanwhile Cantor, before he'd received Dedekind's letter,
| sent him a similar idea for how to streamline the proof,
| though he hadn't worked out the details the way Dedekind had.
| AdAbsurdum wrote:
| I think the relevant quotes are these:
|
| "Dedekind quickly replied that...he'd worked out a proof
| that the algebraic numbers (the numbers you get as
| solutions to algebra problems) could be counted.
|
| [...]
|
| Weierstrass had been most excited about the proof that
| algebraic numbers are countable. (He would later use that
| result to prove a theorem of his own.) So Cantor chose a
| misleading title [for his paper] that only mentioned
| algebraic numbers.
|
| [...]
|
| Writing his paper, Cantor put the proof about algebraic
| numbers first. Below it, he added his own proof that the
| real numbers cannot be counted -- Dedekind's simplified
| version of it, that is."
|
| So the first proof -- the one the article was titled after
| -- was completely created by Dedekind.
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