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.
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.
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.
"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.
> 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.
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.
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.
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.
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.
It is not about better or worse, it is about correcting myths created later on that were intended to paint the Church as epitome of backwardness.
Galileo's affair wasn't about noble scientist going against stupid masses and oppressive institution designed to keep people in dark, while providing strong evidence for revolutionary theory, and being punished for his great genius.
Agreed. I'd also say that I think our habit of canonizing whoever happens to be perceived as the 'good guy' in history, and demonizing the 'bad guy' tends to make history much more difficult to learn from, because the people involved go from being real humans to actors in a very artificial Hollywood style story of good vs evil.
The real story here is one that has played out endlessly in history in various contexts. And is a great example of why The Golden Rule is something valuable to abide, even if you're completely self centered. It also emphasizes that all people, even the Pope, are human - and subject to the same insecurities, pettiness, and other weaknesses as every other human. And more. It's a tale of humanity that has and will continue to repeat indefinitely.
But when you turn it into a story of good vs evil, you lose all of this and instead get a pointless attack on one institution, which is largely incidental to what happened. For instance you can see the Galileo story clearly in the tale of Billy Mitchell [1] who went from suggesting that air forces would dominate the future of warfare (back in 1919!) to getting court martialed and 'retired' for his way of trying to argue for such. His views would go on to be shown to be 100% correct in 1937, the first time a plane downed a capital naval ship. However, he died in 1936.
Galileo is a noble scientist going against a Pope who had his fee-fees hurt, which then banned the truth. It doesn't make the Church any less backward.
Because the Church didn't even have a good theological reason for siding against Galileo. It was a fit of pique.
But people have so completely internalized the idea that truth must bow to power that they think the fact that the Church condemned Galileo's ideas because he was rude somehow exonerates it as an institution.
The patron and professor funds a paper, and it contains claims of proofs that don't exist and ad hominems against the patron. The patron then sabotages the author. Sure, not very professional by the patron, but still understandable.
I asked on LM Arena. I got two models I never heard of, and they split on whether I should walk or drive. Ernie 5.0 said I should walk, and then trash talked me for thinking about driving. Octopodus pointed out I had to drive to get my car to the car wash.
Erdos was an incredibly prolific mathematician, and one of his quirks is that he liked to collect open problems and state new open problems as a challenge to the field. Many of the problems he attached bounties to, from $5 to $10,000.
The problems are a pretty good metric for AI, because the easiest ones at least meet the bar of "a top mathematician didn't know how to solve this off the top of his head" and the hardest ones are major open problems. As AI progresses, we will see it slowly climb the difficulty ladder.
I don't know how Polymarket works. Were people betting against Polymarket, or was Polymarket just making book and someone else is on the other side of the bets.
This term is a bit ambiguous, and there's some nuances that make it different from both sportsbooks and poker.
They don't ever take a nominal cut, their revenue model is in holding USD deposits and making money of interest.
> No, Polymarket is not the house. All trades happen peer-to-peer (p2p).
The documentation is purposefully misleading, but it's true that unlike a sportsbook, they don't take the risk of bets. It's a classic case of a blockchain company exaggerating to what extent they are on the blockchain and to what extent they are centralized and just minimally wrapping the blockchain, like when NFTs were actually a URL to an image.
Trades do NOT happen p2p, polymarket functions as an escrow, payments are sent to polymarket accounts and released by polymarket. Each prediction market does have their own contract, but Polymarket staff rules on each event through off-chain (although they are based on the wording used in the specific event).
New events are solely released by polymarket staff (although users can 'suggest' markets).
I think most questions on polymarket use order books now. But they used to use AMMs (where people bet against polymarket) and their FAQ says some questions still use them
What "tone"? Why is it unreasonable to say these sorts of things about Stack Overflow, or about any community? How is "your questions and answers need to meet our standards to be accepted" any different from "your pull requests need to meet our standards to be accepted"?
It's hard to explain, but immediately clear to enough people that it explains why so many people aren't sad to see SO fall on hard times.
I get that there have to be some rules, but it comes across like you derive some sort of satisfaction in enforcing rules. Successful sites with user moderation start out with a big population of people who will tolerate the rules in order to participate in the goal of the site, but eventually they end up dominated by people who feel that the very act of enforcing rules is an important contribution. All of the talk of "community" comes across as a thinly veiled version of Cartman's "Respect my authority" from South Park.
Man, if this was irl, you'd be punched in the face or ostracized. That's a quick way to assess if your tone is right.
If you don't have a mental capacity to do that (nothing against you, some people are just born that way) — I pity you, but still, try to be 'helpful' over 'correct'. That's how civilization is built.
Wikipedia also have this problem, with moderators using some 'wiki-speak' jargon to 'win the comment battles'.
The Friends of the Wanamaker Organ society is doing work with the new owners towards preservation. I doubt it's going anywhere, but concerts will be sporadic for a while.
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.
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