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New evidence that Cantor plagiarized Dedekind?

quantamagazine.org

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Re: New evidence that Cantor plagiarized Dedekind?

#41

Post by hearsathought »

dkarl wrote: post #4

> 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…

> > 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.

Re: New evidence that Cantor plagiarized Dedekind?

#42

Post by cls59 »

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…

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.

Re: New evidence that Cantor plagiarized Dedekind?

#43

Post by bhk »

markisus wrote: post #18
dkarl wrote: post #4

> 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…

> 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".

Re: New evidence that Cantor plagiarized Dedekind?

#44

Post by Quekid5 »

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".

sqrt(2)

Re: New evidence that Cantor plagiarized Dedekind?

#45

Post by bhk »

Quekid5 wrote: post #44
bhk wrote: post #43

Earlier quoted context omitted.

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".

sqrt(2)

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

Re: New evidence that Cantor plagiarized Dedekind?

#46

Post by 2b3a51 »

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…

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)

Re: New evidence that Cantor plagiarized Dedekind?

#47

Post by hearsathought »

dang wrote: post #5

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.

> 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.

Re: New evidence that Cantor plagiarized Dedekind?

#48

Post by dkarl »

AndrewKemendo wrote: post #35
dkarl wrote: post #32

Earlier quoted context omitted.

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 ne…

> 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

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.

Re: New evidence that Cantor plagiarized Dedekind?

#49

Post by markisus »

dkarl wrote: post #32
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.…

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 ne…

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.

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