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