New evidence that Cantor plagiarized Dedekind?
quantamagazine.org
quantamagazine.org
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.
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.
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.
"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.
I can't say I'm fully comfortable with that characterization of the algebraic numbers. The definition itself does suggest a proof that they are countable:
1. The number of symbols that can appear in a well-defined algebra problem is finite. (For example, if we define algebra problems as being posed in written English, we can use an inventory of no more than 50 symbols to define them all. If we define "algebra problems" in some other way, the definition will specify how many symbols are available.)
2. The number of possible strings describing algebra problems, created from this finite symbolic alphabet, is necessarily countable, because the strings have finite length.
3. Each algebraic number is the solution to one of those strings, and therefore the algebraic numbers are countable.
But I don't really feel like it's possible to learn anything about the numbers from that proof.
How so?
Using the definition of computable numbers where you provide input of n and the output is every digit of the number up to n places past the decimal point, we can rephrase that definition like so:
A computable number c is one with the following property:
A Turing machine exists which, provided with a tolerance δ, will exhibit a rational number q < c such that c - q < δ
Clearly, a suitable rational will always exist, since rationals can be found within any distance of any real.
But for some particular real, we might not be able to find that rational through the use of a fixed Turing machine, in which case the real would be noncomputable. This suggests to me that there is a wider gap between the computables and the reals, where the approximation of a real number is limited by the need to describe it with a Turing machine, than there is between the rationals and the reals, where we can use the same approximation, but without that limitation.
(Obviously the rationals are a subset of the computables, but if we're considering a relationship to the real numbers, the rationals seem to have one that is closer and more direct...? The relationship of a computable number to a real number is defined through intermediary rational numbers.)
I didn't say "wider". I said more subtle. It doesn't take much mathematical intuition and training to understand the rationals versus the reals. Understanding the computables versus the reals is a lot more tricky and takes a lot more thought. The simple arguments that show the difference between the rationals and the reals require a lot of very careful adjustment if you want to translate them to the computables versus the reals.
I agree the gap up to the reals is still bigger than rational -> computable. The reals are weird. This is a meme but there's a lot of truth in it: https://www.reddit.com/media?url=https%3A%2F%2Fpreview.redd....
>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.
[0] https://www.scribd.com/document/977967855/Phlogiston-33#page...
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.
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...
Now they just need to do something about all of the other writers! With the exception of the science lady, the security guy, and the british car guy, it's indistinguishable from the kind of PR-copy-paste blogspam 'coverage' you'd see from a place that will never have the reputation Ars used to.
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.
For learning about actual discoveries, YouTube is much better (Veritasium, Numberphile, 3Blue1Brown, ...).
Indeed. I don't generally give grifters tips on how to disguise themselves.
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)
> However, by October 1933, the issue was straightened out and she was aboard the Bremen, sailing for the United States.
Since she died on 14 April 1935, it was 18 months rather than 2 years.
That sounds like a rather pedantic correction on your part.
That pedanticism is a bad sign and puts your "correction" about the cancer in doubt.
Exageration, the quite mayority of mathematician live very well with infinity, just handle with care.
As you mention: Dedekind stopped corresponding with him after the publication, but also began keeping a copy of every letter he sent to Cantor.
Sure it's circumstantial, but it's exactly what you would do when you're the victim of a plagiarist.
In my eyes the burden of proof has been met.
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.
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
Show up with your hands here if you didn’t know either Cantor or Dedekind.
Cantor's Continuity Credentials Cancelled: Clear Cut Copy Cat Case!
There really is an xkcd for everything
This is a top tier troll, good job.
I think "Cantor: The Man Who Stole Infinity?" would strike a good balance.
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.
> 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.
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.
You can still have sequences with no limits (a_n:=n, going to infinity, where all successive terns differ by 1 and which does not have a limit in the usual metric), as well as sequences with multiple limit points (in which case, subsequences can be considered).
Btw this is "Cauchy completeness", so it is a bit different (but equivalent) way to approach the construction of the real numbers from Dedekind's, but it is also one that can apply to more general metric spaces.
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.
> 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.
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
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.
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
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.
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.
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.
The linguistic sleight-of-hand is what I challenge. What is this "gap" in which there are no numbers?
- A reader would naturally assume the word refers to a range. But if that is the meaning, then mathematicians never believed there were gaps between numbers.
- Or could "gap" refer to a single number, like sqrt(2)? If so, it obviously is not a gap without a number.
- Or does it refer to gaps between rational numbers? In other words, not all numbers are rational? Mathematicians did in fact believe this, from antiquity even ... but that remains true!
Regarding this naive construction you are referring to: did it precede set theory? What definition of "gap" would explain the article's treatment of it?
A couple comments, though - first, all mathematics is linguistics and arguably it is all sleight of hand - that said the word “gaps” that you’ve rightly pointed out is vague is a journalists word standing in for a variety of concepts at different times.
The existence of the irrationals themselves were a secret in ancient greece - and hence known for thousands of years, but the structure of the irrationals has not been well understood until quite recently.
To talk precisely about these gaps, if you’re not a mathematical historian, you have to borrow terminology from the tools that were used to describe and formalize the irrationals -> if former concepts about the lines sound hand-wavy to you, it is because they WERE handwavy. And this handwaviness is about infinity as well, the two are intimately connected. In modern terms, the measure of the rationals across any subset of the (real) number line is zero - that is the meaning of the “gaps”. There is, between any two rationals, a great unending sea where if you were to choose a point completely at random, the odds of that point being another rational is zero.
EDIT: for a light but engaging read about topics like this, David Foster Wallace’s Everything and More is excellent.
I think you will agree that the bulk of your comment employs a post-set-theory nomenclature.
Regarding "if you were to choose a point completely at random, the odds of that point being another rational is zero", I ponder the question of how one might casually "choose" a value with infinite entropy.
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
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?)
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).
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.
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.
https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Gra...
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.
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
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.