Numbers Are Leaves • Hacker News Board

Re: Numbers Are Leaves

#71

Post by lupire » Mon, Dec 30, 2024, 5:06 AM UTC

bubblyworld wrote: post #37

nthingtohide wrote: post #16

I think people will like the following tangent. https://en.m.wikipedia.org/wiki/Benacerraf%27s_identificatio... In the philosophy of mathematics, Benacerraf's identification problem is a philosophical argument developed by Paul Benacerraf against set-theoretic Platonism and published in 1965 in an article entitled "What Numbers Could Not Be". Historically, the work became a significant catalyst in motivating the deve…

Thanks, that was an interesting rabbit hole. Although I can't help but feel there's a philosophical map/territory confusion here. Like, sure, numbers can't possibly just "be" sets because there are many different models of the naturals in (ZF) set theory, even in higher order logics. But I feel like a Platonist would just counter that of course this is the case - we are simply modelling the properties of the "true" n…

It's not a map/territory confusion. It's a warning about map/territory confusion. The problem is that we have only maps, so we cannot fully understand the territory. This is a major concern in mathematics, whose reason for existence is to understand things precisely, not just rough and ready.

Re: Numbers Are Leaves

#72

Post by nthingtohide » Mon, Dec 30, 2024, 5:07 PM UTC

openquery wrote: post #48

some-unique-x wrote: post #46

Despite several negative comments, I thought the author did a great job of explaining and playing with set theory (which, as can be seen by the response, is good fun.) I do take some issue with intepreting set theory's membership relation in terms of the tree child relation, though. First, the child relation is presumably transitive, whilst set membership is not. (The subset relation is transitive. Presumably it is '…

Thanks for the feedback. Honestly sets as trees isn't original. While I was learning about ZFC I came across some lectures[0] by Richard Borcherds which was the seed of insipiration for this project. [0] https://youtu.be/oWN13ktp8gg?t=1154

HACKENBUSH: a window to a new world of math

https://www.youtube.com/watch?v=ZYj4NkeGPdM

Thanks for the blogpost, I think you will like this above SOME3 entry. Owen Maitzen committed suicide which is mentioned in the following video.

The Endless Universe of "Bean and Nothingness"

https://www.youtube.com/watch?v=bRIXgb4UgmY

Re: Numbers Are Leaves

#73

Post by bubblyworld » Mon, Dec 30, 2024, 8:30 PM UTC

lupire wrote: post #71

bubblyworld wrote: post #37

Earlier quoted context omitted.

Thanks, that was an interesting rabbit hole. Although I can't help but feel there's a philosophical map/territory confusion here. Like, sure, numbers can't possibly just "be" sets because there are many different models of the naturals in (ZF) set theory, even in higher order logics. But I feel like a Platonist would just counter that of course this is the case - we are simply modelling the properties of the "true" n…

It's not a map/territory confusion. It's a warning about map/territory confusion. The problem is that we have only maps, so we cannot fully understand the territory. This is a major concern in mathematics, whose reason for existence is to understand things precisely, not just rough and ready.

I'm not convinced there is a coherent "territory" when it comes to mathematics. Sometimes (remarkably!) mathematical concepts neatly line up with some part of the universe, and we gain some insight into how it works. But often that part of the universe lies entirely within somebody's mind - mathematics as an hobby or aesthetic pursuit, for instance, occupied with questions of pure logic.

Different ways of thinking about the same thing can lead to conflicting truths even without set theory getting in the way. The standard model of the real numbers has no infinitesimally small elements, for instance, but the hyperreal numbers do, despite satisfying all of the same first-order properties.

Re: Numbers Are Leaves

#74

Post by nthingtohide » Wed, Jan 08, 2025, 2:16 PM UTC

Viliam1234 wrote: post #70

bubblyworld wrote: post #37

Earlier quoted context omitted.

Thanks, that was an interesting rabbit hole. Although I can't help but feel there's a philosophical map/territory confusion here. Like, sure, numbers can't possibly just "be" sets because there are many different models of the naturals in (ZF) set theory, even in higher order logics. But I feel like a Platonist would just counter that of course this is the case - we are simply modelling the properties of the "true" n…

Could we just say that natural numbers can be represented using sets, and leave it there? Natural numbers can also be represented by even natural numbers (e.g. the easy way where 2n represents n), but that doesn't tempt people to make metaphysical statements about natural numbers somehow fundamentally "being" even. There is no reason why a representation of natural numbers by sets should be any more tempting.

There are multiple represenation of rational numbers . n * p / n * q (for any n). So this problem is not unique to reduction of numbers to sets. Generally people say unique reduced form.