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Comment by yellowapple | Hacker News Reader

"So for those of you who don't know the result, here it is in simple, non-technical terms:"

proceeds to immediately use a character that can't even be copy/pasted due to needing MathJax to render it, refuses to elaborate further

Thankfully, it ain't hard to find the Banach-Tarski theorem in actual simple, non-technical terms, so for those wondering what in tarnation that character is: it just means three-dimensional space (and is usually rendered as ℝ³ instead of 𝕽³, at least so Wikipedia claims).

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Had a laugh about that too. It’s sometimes hard to tell what’s »non-technical« when you’ve spent a lot of time on something, and even then what it means to be »technical« depends a lot on the audience. Non-technical here means knows basic math but is not familiar with measure theory; it does not mean _anyone_.

I like that they added my favorite math joke in their (linked at the very top) limited audience jokes section [1] (bottom left, starts with »Two mathematicians are in a bar«), which is also a about what basic math knowledge is.

[1]: https://www.solipsys.co.uk/new/LimitedAudienceJokes.html

> Non-technical here means knows basic math but is not familiar with measure theory; it does not mean _anyone_.

I guess "knows basic math" is relative; I tend to draw the "basic" line at "it's required for a high school diploma or GED", but I suppose I can blame my country's education system for that bar being much lower than what mathematicians seem to assume :)

Unrelated to the article in question, but using ℝ over 𝕽 for the reals is more of a modern development. If you read older articles and textbooks, many will use 𝕽 rather than the sleeker ℝ (most in my experience, but results will be heavily affected by your cut-off for 'old').

I don't have direct evidence for my speculations, but I presume the reason [fraktur](https://en.wikipedia.org/wiki/Fraktur) was more common in mathematics back in the days is largely down to articles having to be type-set using movable type. If you insisted on using ℝ over 𝕽, you were likely to make life considerably harder for your printer (which in turn meant higher printing costs), since they would be considerably more likely to have to cast new types. As printing was modernized and movable type was replaced by more flexible printing-technologies, this pragmatic reason for preferring one glyph over the other went away. Another explanation/contributing factor is that the switch seems to have occurred in tandem with the barycenter of mathematics switching away from continental Europe and towards the US in the post-WWII period (at least if we disregard Soviet mathematics which also flourished in this period, but which was largely published in Russian). The average American would probably be less familiar with 𝕽 and other fraktur glyphs than the average German.

I did get a chuckle at that, like the running joke about how a monad is “just a monoid in the category of endofunctors”. To be fair, you could delete the “R^3” part and the explanation can still be understood.

Hell, deleting the 𝕽³ part probably would've made it clearer: "ball" already implies 3D space for me, whereas seeing that qualified with some janky-ass medieval R had me second-guessing about it being one of those math quirks involving non-euclidean space or imaginary-number dimensions or whatever else mathematicians come up with in their efforts to compute the passcode on the Gates of Hell.

Yeah I feel you - I think these things come so naturally to mathematicians that it's hard for them to conceive of the idea that people like you and I exist :)

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