New evidence that Cantor plagiarized Dedekind? • Hacker News Board

Re: New evidence that Cantor plagiarized Dedekind?

#31

Post by JasonADrury » Sat, Feb 28, 2026, 6:44 PM UTC

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.

> 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.

Re: New evidence that Cantor plagiarized Dedekind?

#32

Post by dkarl » Sat, Feb 28, 2026, 6:46 PM UTC

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.…

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.

Re: New evidence that Cantor plagiarized Dedekind?

#33

Post by antasvara » Sat, Feb 28, 2026, 6:47 PM UTC

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…

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.

Re: New evidence that Cantor plagiarized Dedekind?

#34

Post by sandslides » Sat, Feb 28, 2026, 6:52 PM UTC

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…

could I just leave my favourite thing ever here? thanks :)

https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Gra...

Re: New evidence that Cantor plagiarized Dedekind?

#35

Post by AndrewKemendo » Sat, Feb 28, 2026, 7:14 PM UTC

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…

> 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

Re: New evidence that Cantor plagiarized Dedekind?

#36

Post by lupire » Sat, Feb 28, 2026, 7:14 PM UTC

antasvara wrote: post #33

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…

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 (s…

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.

Re: New evidence that Cantor plagiarized Dedekind?

#37

Post by buttermeup » Sat, Feb 28, 2026, 7:15 PM UTC

Chinjut wrote: post #24

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…

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).

In school, we talked about “Dedekind cuts” but we never formalized the definition. Kind of disappointed now because your explanation is very simple and elegant.

Re: New evidence that Cantor plagiarized Dedekind?

#38

Post by lupire » Sat, Feb 28, 2026, 7:17 PM UTC

zeroonetwothree wrote: post #15

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…

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.

Critically, "Complete" also means that the supposed limit necessarily exists.

Re: New evidence that Cantor plagiarized Dedekind?

#39

Post by antasvara » Sat, Feb 28, 2026, 7:23 PM UTC

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…

"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.

Re: New evidence that Cantor plagiarized Dedekind?

#40

Post by [deleted] » Sat, Feb 28, 2026, 7:27 PM UTC

wizzwizz4 wrote: post #27

leephillips wrote: post #22

Earlier quoted context omitted.

It won’t be the last time.

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.

[deleted]