God is a FRACTRAN program.
God created the real numbers
81–90 of 226 posts
Re: God created the real numbers
#82When I truly grokked complex numbers, I felt as though real numbers were a lie - though I would now say that it was a convenient omission. There are many things that are more naturally described using complex numbers - waves (which much of reality boils down to) immediately come to mind. Even if something does align better with real numbers, it's still just x+0i. Maybe I'll change my mind ~when~ if I finally grok qua…
I think one quaternion contains all complex numbers
I thought this when watching the 3b1b + ben eater collaboration on quaternion visualizers.
von Neumann would say you'll never grok quaternions. but merely get used to them.
Re: God created the real numbers
#83Can't say that I'm completely in the headspace to follow the argument, but wanted to add my 2 cents from a few years ago. Integers come into existence long before god - as the only presumption required is a difference between one thing and another (or nothing). The integers also create infinite gaps. The primes. So no - I do not think reals are closer to the divine. They require we import infinity twice to be defined…
I find primes spooky. They seem to be a concept that exists regardless of reality or universe. How does such a incontrovertible structure arise? ps. Various numerology phenomena have a similar vibe, and no wonder so many people who go off the deep end tend to get trapped by them. Maybe I will be one of them as I become old and senile :-D
> How does such a incontrovertible structure arise?
yes, I also enjoy trying to answer this question.
what is such an structure even mean? how could it be that simply defining numbers, obersving addition, and generalizing it away into multiplication would yield this natural structure?
It all begins with zero. the predecessor of One, the best known number.
zero can be assumed by anyone. the surprise is how all zeros are the same zero. (by uniqueness of emptyset; but as I hope you can see, I'm a crank. a nutjob. I'll stop
Re: God created the real numbers
#84Earlier quoted context omitted.
> Addressing your issue directly, the Axiom of Choice is actively debated: The axiom of choice is not required to prove Cantor’s theorem, that any set has strictly smaller cardinality than its powerset. Actually, I can recount the proof here: Suppose there is an injection f: Powerset(A) ↪ A from the powerset of a set A to the set A. Now consider the set S = {x ∈ A | ∃ s ⊆ A, f(s) = x and x ∉ s}, i.e. the subset of A…
Perhaps this is an ignorant question, but wouldn't you need AC to select the s ⊆ A whose existence the contradiction depends on? A constructive proof, at least the ones I'm trying to build in my head, stumbles when needing to produce that s to use in the following arguments.
No, because you only have to choose _one_ s for the proof to work, and a finite number of choices is valid in intuitionistic and constructive mathematics.
Re: God created the real numbers
#85Earlier quoted context omitted.
> They're unphysical, and yet the very physical human mind can work with them just fine Nah, you're likely thinking of the rationals, which are basically just two integers in a halloween costume. Ooh a third, big deal. The overwhelming majority of the reals are completely batshit and you're not working with them "just fine" except in some very hand wavy sense.
the rationals are 3 naturals with in a "2,1" structure. the first 2 naturals form an integer. that integer and a 3rd natural constitute a real (but this 3rd natural best be bigger than zero, else we're in trouble) what I choose to focus after observing the "unphysical" nature of numbers. is the sense of natural opposition (bordering on alternation) between "mathematical true" and "physical true". both are claiming to…
Huh?
Re: God created the real numbers
#86Earlier quoted context omitted.
What do you mean by "the computable subset of the reals" formally? Is sqrt(2) computable? Is BB(777) computable? Is [the integer that happens to be equal to BB(777), not that I can prove it, written out in normal decimal notation] computable?
A computable real number is a real number for which a Turing Machine exists that can compute it to any arbitrary precision. So yes sqrt(2) is computable. Every BB(n) is computable since every every natutal number can be computed. It's the BB function itself that is not computable in general, not the specific output of that function for a given input.
That doesn’t sound right to me. What about the machines that don’t halt? You can’t compute whether or not to skip them directly.
> A busy beaver hunter who goes by Racheline has shown that the question of whether Antihydra halts is closely related to a famous unsolved problem in mathematics called the Collatz conjecture. Since then, the team has discovered many other six-rule machines with similar characteristics. Slaying the Antihydra and its brethren will require conceptual breakthroughs in pure mathematics.
https://www.quantamagazine.org/busy-beaver-hunters-reach-num...
Re: God created the real numbers
#87Earlier quoted context omitted.
A computable real number is a real number for which a Turing Machine exists that can compute it to any arbitrary precision. So yes sqrt(2) is computable. Every BB(n) is computable since every every natutal number can be computed. It's the BB function itself that is not computable in general, not the specific output of that function for a given input.
That doesn’t sound right to me. What about the machines that don’t halt? You can’t compute whether or not to skip them directly. > A busy beaver hunter who goes by Racheline has shown that the question of whether Antihydra halts is closely related to a famous unsolved problem in mathematics called the Collatz conjecture. Since then, the team has discovered many other six-rule machines with similar characteristics. Sl…
What specifically doesn't sound right?
Re: God created the real numbers
#88Previously yesterday: https://news.ycombinator.com/item?id=45053007
Re: God created the real numbers
#89All math is just a system of ideas, specifically rules that people made up and follow because it's useful. I'm so used to thinking this way that I don't understand what all the fuss is about, mathematical objects being "real". Ideas are real but they're not real in the way that rocks are. Whenever there's a mysterious pattern in nature, people have felt the need to assert that some immaterial "thing" makes it so. But…
Ideas are real in the way rocks are if we are concerned with their informational being. They are real informationally - ideas and math participate in forming the world. Nowadays, LLMs, Search and other apps probably affect the world even more than any common rock. Which is more real?
I don't know what is meant by the informational being of a rock.
Re: God created the real numbers
#90Earlier quoted context omitted.
What do you mean by "the computable subset of the reals" formally? Is sqrt(2) computable? Is BB(777) computable? Is [the integer that happens to be equal to BB(777), not that I can prove it, written out in normal decimal notation] computable?
A computable real number is a real number for which a Turing Machine exists that can compute it to any arbitrary precision. So yes sqrt(2) is computable. Every BB(n) is computable since every every natutal number can be computed. It's the BB function itself that is not computable in general, not the specific output of that function for a given input.
Interesting point about BB(n)... Is it known that BB(n) is finite for every n?