> 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.
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.
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.
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
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.
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
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.