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Comment by DylanDmitri | Hacker News Reader

*speculation warning*

We're still not comfortable checking if rationals are equal in any major language... there's a long way to go.

I could see a future high-level programming language (think Python design goals) using the rigorous foundation of surreals as the underlying numeric system. Sure 95% of the time it'll look the same to the programmer, and the performance will be worse, but you get inherent stability around infinities, irrationals, NaNs, floating-point comparisons, etc

It's reasonable to think that ~200 years from now, highschoolers will learn that numbers are all "really" from this structure of surreal numbers, and that most computers and physics processing etc uses them behind the scenes.

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> We're still not comfortable checking if rationals are equal in any major language...

This seems to be confusing the issue to me. Checking for rational equality is trivial (just cross multiply). What we don't have is major languages deciding the the type rational numbers is useful enough to get first class support. And even that is overstating the issue, some languages do care and make representing rationals easy. In Julia, rational numbers can just be written like 2//3.

If people try to use floats to represent rationals and run into errors, the problem isn't that languages are incapable of representing rationals.

I don't see any upsides to using surreal numbers as a datatype. If you're restricting it to a finite size you're still going to have analagous problems that float or double do and I don't see any advantages in stability or whatever. If you're allowing arbitrary size (but obviously still finite) representation, you can still only represent the dyadic rationals, so you can't even express 1/3. The infinitesimals and infinities would require infinite size, so they're never practical.

Yes, it's instructive to recall that the Pythagorean Cult believed all numbers were rational (and this was a religious sort of belief). There's a legend (albeit probably not historically accurate) that when a member of the cult proved that sqrt(2) was irrational, they put him to death... https://en.wikipedia.org/wiki/Hippasus

I wonder how they would have reacted to a proof that the reals are a larger infinity.

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