“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, but received no reply.
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 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.
I find they are good at identifying interesting topics and writing articles that don't deliver. They remind me of Omni magazine (which I subscribed to at one point). The articles aren't even wrong.
I think your opinion is popular here. Quanta is, while better than nothing, universally disappointing. It seems like it would be much easier for them to do a better job -- write less vaguely, fact-check more, assume the reader is a bit more intelligent.
Thank you! After Benj Edwards and Kyle Orland's Ars Technica article they published using AI (while saying they didn't), and all the while their article was about an AI agent publishing a hit piece on Scott Shambaugh (matplotlib maintainer), I feel like I now assume journalists are using AI and things need to be fact-checked just as we do for our AI interactions.
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!
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)
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.
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.
This whole plagiarism thing is too overwrought these days. People discuss stuff and the idea forms in the discussion between the two. Then one writes it up. Oh he plagiarized the other. I don’t know man.
I’ve been in joint discussions where “the idea forming” was really one party thinking out loud and doing almost all the work, and the other providing approximately the same function as a rubber duck.
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.
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.
To be fair, if one does not know who cantor and dedekind were, the drama about the former plagiarising the latter is probably not that interesting anyway.
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
I would assume that at least cantors technique of diagonalisation should have found its way into some CS course that I assume a good part of the audience here has studied? Considering that’s what Turing used to prove the undecidability of the halting problem.
Having been active on this website for 14 years now … At this point I would venture to say The median hacker news commentator does not have aa computer science degree
> 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.
I am not a mathematician; I barely knew who Cantor was and had never heard of Dedekind. I would have likely not read the article without the title being so sensational. Your assumption sits upon the tip of your nose.
> 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 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.
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.
I just don't understand why this was disturbing. Prior to the construction of the reals, the existence of irrational and transcendental numbers was disturbing, because they showed that previous constructions (rational numbers and algebraic numbers) were incomplete. If those gaps were disturbing, a construction without gaps should have been satisfying, reassuring, a resolution of tension. Was there some philosophical or theological theory that required the existence of gaps, that claimed that a complete construction of the number line was mathematically impossible, because of some attribute of God or the cosmos?
I think the issue was that most irrational/transcendental numbers aren’t finitely representable. This means that they are mathematical objects which, each of them individually, somehow consist of an infinity (e.g. an infinite decimal expansion). They are the result or end point of infinitely many steps (e.g. a converging sequence) that you can’t actually reach the end of in practice, and for most of them can’t even write down a finite description on what steps to perform, and which therefore arguably doesn’t “really” exist.
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.
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.
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).
> 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. 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.
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.
"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.
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.
> 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
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.
Because painting those who objected to these definitions of mathematical infinity as "horrified" and "disturbed" was a form of character assassination, which was not uncommon at the time. The high moderns didn't play.
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
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".
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 “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.
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.
You can construct sequences of rational numbers where the limit is not rational (eg it's sqrt 2)
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
> > 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.
> Give x < y, it's easy to construct x + (y-x)(sqrt(2))/2.
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?)
“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, but received no reply.
Noether’s real story is recounted in https://amzn.to/3YZZB4W.
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 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.
I find they are good at identifying interesting topics and writing articles that don't deliver. They remind me of Omni magazine (which I subscribed to at one point). The articles aren't even wrong.
The articles are unreadable fluff
I think your opinion is popular here. Quanta is, while better than nothing, universally disappointing. It seems like it would be much easier for them to do a better job -- write less vaguely, fact-check more, assume the reader is a bit more intelligent.
Thank you! After Benj Edwards and Kyle Orland's Ars Technica article they published using AI (while saying they didn't), and all the while their article was about an AI agent publishing a hit piece on Scott Shambaugh (matplotlib maintainer), I feel like I now assume journalists are using AI and things need to be fact-checked just as we do for our AI interactions.
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...
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...
I see what you did there. Turning the page of time, I guess.
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)
Are you citing your own book?
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.
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.
But surely you would agree that "my book" just wastes less time all around, and doesn't harm the author's message?
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.
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 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.
This whole plagiarism thing is too overwrought these days. People discuss stuff and the idea forms in the discussion between the two. Then one writes it up. Oh he plagiarized the other. I don’t know man.
I’ve been in joint discussions where “the idea forming” was really one party thinking out loud and doing almost all the work, and the other providing approximately the same function as a rubber duck.
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.
Try reading the article?
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
Haha, the sanctimony is really too much.
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'm here for the 19th century drama. Imagine the head lines!
Millions of views for Tiktoks about homomorphisms and aleph numbers. Just the news we need right now.> 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.
> since most HN readers should know who Cantor and Dedekind are.
Show up with your hands here if you didn’t know either Cantor or Dedekind.
Then you can just skip this submission and nobody will be the wiser.
This comment made me think of this xkcd 2501
https://xkcd.com/2501
There really is an xkcd for everything
To be fair, if one does not know who cantor and dedekind were, the drama about the former plagiarising the latter is probably not that interesting anyway.
Hard disagree
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
I would assume that at least cantors technique of diagonalisation should have found its way into some CS course that I assume a good part of the audience here has studied? Considering that’s what Turing used to prove the undecidability of the halting problem.
Having been active on this website for 14 years now … At this point I would venture to say The median hacker news commentator does not have aa computer science degree
> 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.
I am not a mathematician; I barely knew who Cantor was and had never heard of Dedekind. I would have likely not read the article without the title being so sensational. Your assumption sits upon the tip of your nose.
> 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 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.
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.
I just don't understand why this was disturbing. Prior to the construction of the reals, the existence of irrational and transcendental numbers was disturbing, because they showed that previous constructions (rational numbers and algebraic numbers) were incomplete. If those gaps were disturbing, a construction without gaps should have been satisfying, reassuring, a resolution of tension. Was there some philosophical or theological theory that required the existence of gaps, that claimed that a complete construction of the number line was mathematically impossible, because of some attribute of God or the cosmos?
I think the issue was that most irrational/transcendental numbers aren’t finitely representable. This means that they are mathematical objects which, each of them individually, somehow consist of an infinity (e.g. an infinite decimal expansion). They are the result or end point of infinitely many steps (e.g. a converging sequence) that you can’t actually reach the end of in practice, and for most of them can’t even write down a finite description on what steps to perform, and which therefore arguably doesn’t “really” exist.
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.
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.
Quanta messing things up isn't a particularly rare occurrence, unfortunately.
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).
> 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. 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.
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.
"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.
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.
> 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.
Because painting those who objected to these definitions of mathematical infinity as "horrified" and "disturbed" was a form of character assassination, which was not uncommon at the time. The high moderns didn't play.
History only seems obvious in retrospect
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
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?
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 “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.
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.
The density does not dictate cardinality which is what this article is about.
You can construct sequences of rational numbers where the limit is not rational (eg it's sqrt 2)
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
could I just leave my favourite thing ever here? thanks :)
https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Gra...
> > 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.
The continuum. Connectedness.
> Give x < y, it's easy to construct x + (y-x)(sqrt(2))/2.
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?)