Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

New evidence that Cantor plagiarized Dedekind?

quantamagazine.org

Post reply on HN
Print view View original

71–80 of 96 posts

Re: New evidence that Cantor plagiarized Dedekind?

#71

Post by dkarl »

bhk wrote: post #43
markisus wrote: post #18

Earlier quoted context omitted.

> 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.…

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".

Yeah, it took me a second, too. By "gaps" they mean numbers that can't be represented in a given construction. So irrational numbers are "gaps" in the rational numbers, and transcendental numbers are "gaps" in the algebraic numbers. Not the best spatial metaphor.

Re: New evidence that Cantor plagiarized Dedekind?

#72

Post by ChocMontePy »

leephillips wrote: post #2

“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, bu…

I'm at the library so I checked your book. You said in there:

> 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.

Re: New evidence that Cantor plagiarized Dedekind?

#73

Post by bhk »

vessenes wrote: post #62
bhk wrote: post #45

Earlier quoted context omitted.

That's not a "gap" that you find by "zooming in". And how can it be a gap when it is occupied?

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 “t…

I think you are getting away from my point, which pertains to what the article said, which is that mathematicians thought there were "gaps". What mathematician? Can I see the original quote?

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?

Re: New evidence that Cantor plagiarized Dedekind?

#74

Post by monster_truck »

moefh wrote: post #65
zeroonetwothree wrote: post #16

Earlier quoted context omitted.

It’s not like journalists were very accurate before AI. Classic Gell-Mann amnesia

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...

Had no idea they even did anything. Was waiting for this. Nice to see some consequences and something resembling an attempt at integrity.

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.

Re: New evidence that Cantor plagiarized Dedekind?

#75

Post by singlow »

ChocMontePy wrote: post #72
leephillips wrote: post #2

“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, bu…

I'm at the library so I checked your book. You said in there: > 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.

Complications after surgery to remove a cancerous tumor?

Re: New evidence that Cantor plagiarized Dedekind?

#76

Post by MichaelDickens »

lich_king wrote: post #10
leephillips wrote: post #2

“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, bu…

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 op…

In my experience virtually every magazine is like this, not just Quanta. I open an article hoping to learn something about some scientific or mathematical discovery, but instead the article is almost entirely about the discoverer.

For learning about actual discoveries, YouTube is much better (Veritasium, Numberphile, 3Blue1Brown, ...).

Re: New evidence that Cantor plagiarized Dedekind?

#77

Post by thaumasiotes »

AdAbsurdum wrote: post #60
QuesnayJr wrote: post #58

Earlier quoted context omitted.

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. Mea…

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 menti…

> he’d worked out a proof that the algebraic numbers (the numbers you get as solutions to algebra problems) could be counted

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.

Re: New evidence that Cantor plagiarized Dedekind?

#78

Post by aaplok »

QuesnayJr wrote: post #25

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 b…

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.

[0] https://www.scribd.com/document/977967855/Phlogiston-33#page...

Re: New evidence that Cantor plagiarized Dedekind?

#79

Post by jerf »

thaumasiotes wrote: post #77
AdAbsurdum wrote: post #60

Earlier quoted context omitted.

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 menti…

> he’d worked out a proof that the algebraic numbers (the numbers you get as solutions to algebra problems) could be counted 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 p…

You can also get to computable numbers through a similar argument, substituting something Turing-complete for algebra. You definitely do get to learn some interesting things about numbers from computable numbers. The differences between the computables and the full reals are much more subtle than the differences between the rationals and the reals.

Re: New evidence that Cantor plagiarized Dedekind?

#80

Post by ozb »

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
Post reply on HN