[traderjane]

Professor N.J. Wildberger is probably among the most well known "ultrafinitist" on YouTube.

https://www.youtube.com/watch?v=WabHm1QWVCA

I mention him because I would think he sympathizes with those who have concern over the meaning of this kind of notation.

[rrmm]

Wildberger is great. His lectures that he teaches at UNSW (i think) are interesting, and he usually keeps a clear dividing line between std math and his own predilections. It threads the line between being a kook and legitimate published mathematician very finely.

I actually have some sympathies with his contention that real numbers (limit points of infinite series) are somehow a different animal than rational numbers. But it might be easier for me to go there because practically all numbers on computers that we work with are rational, floating point values. On the other hand, it seems like a philosophical distinction in the end because you can fully order them both on a number line.

[danharaj]

If I give you two representatives of real numbers, say turing machines that write out on their tape the binary digits of those real numbers, in general you will not be able to order them.

[rrmm]

Certainly not non-computable ones, but presumably they lie somewhere regardless of my inability to do it on a TM. Which presumably gives rise to all the weirdness uncountable infinities give you.

I guess I shouldn't phrase it as "you can fully order it". :D Zermelo's theorem at that point right?

[danharaj]

Even computable reals do not have computable ordering.

> but presumably they lie somewhere regardless of my inability to do it on a TM.

Why?

> Zermelo's theorem at that point right?

It is declared by fiat in standard set theory that infinite sets can be well ordered. This is no real mathematical justification. The real justification is social: that it is convenient for mathematicians to not care about the ontology of these nasty infinite objects so long as results are mostly reasonable for objects that mathematicians actually care about. You don't get into too much trouble pretending the reals are nice so long as you don't look too hard.

[rrmm]

I mean that's Wildberger's whole point isn't it?

[danharaj]

Yes, I'm just trying to emphasize that there is real serious mathematics behind his point, it's not just a matter of philosophical taste.

[kangnkodos]

But on computers, you get things such as

console.log(0.1 + 0.2)

// 0.30000000000000004

A mathematician might say that this shows that you do not really have accurate floating point values and arithmetic in your computer, but instead something close to it.

[traderjane]

That's a coincidence of the particular number system you use, and many programming languages have multiple number implementations.

Racket starts with arbitrary precision rationals.