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The Unknotting Number Is Not Additive

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Re: The Unknotting Number Is Not Additive

#31

Post by empath75 »

CJefferson wrote: post #11
brap wrote: post #3

Whenever I encounter this sort of abstract math (at least “abstract” for me) I start wondering what’s even “real”. Like, what is some foundational truth of reality vs. stuff we just made up and keep exploring. Are these knots real? Are prime numbers real? Multiplication? Addition? Are natural numbers really “natural”? For example, one thing that always seemed bizarre to me for as long as I can remember is Pi. If circ…

To me, the least real thing in maths is, ironically, the real numbers. As you dig through integers, fractions, square roots, solutions to polynomials, things a turing machine can output, you get to increasingly large classes of numbers which are still all countably infinite. At some point I realised I'd covered anything I could ever imagine caring about and was still in a countable set.

how large is the set of all possible subsets of the natural numbers?

edit: Just to clarify -- this is a pretty obvious question to ask about natural numbers, it's no more obviously artificially constructed than any other infinite set. It seems to be that it would be hard to justify accepting the set of natural numbers and not accepting the power set of the natural numbers.

Re: The Unknotting Number Is Not Additive

#32

Post by jibal »

deadfoxygrandpa wrote: post #22
Antinumeric wrote: post #21

This example seems obvious to me - Joining the under to the under, and the over to the over would obviously give more freedom to the knot than the reverse.

you're either lying or you don't understand what you're looking at. theres a reason this conjecture wasnt disproven for almost a hundred years

Logic fail. The example is not the conjecture. Saying the example is obvious is not saying that the conjecture is obvious.

Re: The Unknotting Number Is Not Additive

#33

Post by srean »

JdeBP wrote: post #17
CJefferson wrote: post #11

Earlier quoted context omitted.

To me, the least real thing in maths is, ironically, the real numbers. As you dig through integers, fractions, square roots, solutions to polynomials, things a turing machine can output, you get to increasingly large classes of numbers which are still all countably infinite. At some point I realised I'd covered anything I could ever imagine caring about and was still in a countable set.

The entirely opposite perspective is quite interesting: The "natural numbers" are the biggest mis-nomer in mathematics. They are the most un-Natural ones. The numbers that occur in Nature are almost always complex, and are neither integers nor rationals (nor even algebraics). When you approach reality through the lens of mathematics that concentrates the most upon these countable sets, you very often end up with infi…

I agree. Had humanity made turning the more fundamental operation than counting that would have sped up our mathematical journey. The Naturals would have fallen off from it as an exercise of counting turns.

The calculus of scaled rotation is so beautiful. The sacrificial lamb is the unique ordering relation.

Re: The Unknotting Number Is Not Additive

#34

Post by srean »

slickytail wrote: post #4
brap wrote: post #3

Whenever I encounter this sort of abstract math (at least “abstract” for me) I start wondering what’s even “real”. Like, what is some foundational truth of reality vs. stuff we just made up and keep exploring. Are these knots real? Are prime numbers real? Multiplication? Addition? Are natural numbers really “natural”? For example, one thing that always seemed bizarre to me for as long as I can remember is Pi. If circ…

In the words of Kronecker: "God created the integers, all else is the work of man."

Had I been god I would have created scaled turns and left the rest for humans.

Re: The Unknotting Number Is Not Additive

#35

Post by pfortuny »

Antinumeric wrote: post #21

This example seems obvious to me - Joining the under to the under, and the over to the over would obviously give more freedom to the knot than the reverse.

It happens: once you see the example, it may be trivial to understand. The hard thing is to find it.

Re: The Unknotting Number Is Not Additive

#36

Post by jibal »

brap wrote: post #3

Whenever I encounter this sort of abstract math (at least “abstract” for me) I start wondering what’s even “real”. Like, what is some foundational truth of reality vs. stuff we just made up and keep exploring. Are these knots real? Are prime numbers real? Multiplication? Addition? Are natural numbers really “natural”? For example, one thing that always seemed bizarre to me for as long as I can remember is Pi. If circ…

> If circles are natural and numbers are natural, then why does their relationship seem so unnatural and arbitrary?

It is not in any way unnatural or arbitrary.

However, there are no circles in nature.

> You could imagine some advanced alien civilization, maybe in a completely different universe, that isn’t even aware of these concepts.

I can't actually imagine that ... advancement in the physical world requires at least mastery of the most basic facts of arithmetic.

> just hoping someone can enlighten me

I suggest that you first need some basic grounding in math and philosophy.

Re: The Unknotting Number Is Not Additive

#38

Post by gcanyon »

jibal wrote: post #32
deadfoxygrandpa wrote: post #22

Earlier quoted context omitted.

you're either lying or you don't understand what you're looking at. theres a reason this conjecture wasnt disproven for almost a hundred years

Logic fail. The example is not the conjecture. Saying the example is obvious is not saying that the conjecture is obvious.

The example isn't an example -- it's a proposed simplicity of a counterexample. Which is exactly what the article is about and the post you responded to is therefore objecting to.

Re: The Unknotting Number Is Not Additive

#39

Post by brap »

twiceaday wrote: post #20
lqet wrote: post #7

Earlier quoted context omitted.

Philosophical problems regarding the fundamental nature of reality aside, this short clip is relevant to your question: > https://www.youtube.com/watch?v=tCUK2zRTcOc Translated transcript: Physics is a "Real Science". It deals with reality. Math is a structural science. It deals with the structure of thinking. These structures do not have to exist. They can exist, but they don't have to. That's a fundamental differen…

Math is a purely logical tool. None of it "exists." That makes no sense. Some of it can be used to model reality. We call such math "physics." And I think physics is significantly closer to math than to reality. It's just a collection of math that models some measurements on some scales with some precision. We have no idea how close we are to actual reality. I do not understand the framing of "translating math concep…

I think maybe I didn’t really explain myself properly. I didn’t mean that math is real in the sense that atoms are real. Perhaps “true” would be a better word. We know these things are true to us, but are they universally true? If that’s even a thing? Hope that makes more sense.

Re: The Unknotting Number Is Not Additive

#40

Post by schiffern »

kannanvijayan wrote: post #30
brap wrote: post #3

Whenever I encounter this sort of abstract math (at least “abstract” for me) I start wondering what’s even “real”. Like, what is some foundational truth of reality vs. stuff we just made up and keep exploring. Are these knots real? Are prime numbers real? Multiplication? Addition? Are natural numbers really “natural”? For example, one thing that always seemed bizarre to me for as long as I can remember is Pi. If circ…

I don't have an answer to your questions, but I think these thoughts are not uncommon for people who get into these topics. The relationship between the reals, including Pi, and the countables such as the naturals/integers/rationals is suggestive of some deeper truth. The ratio between the areas of a unit circle (or hypersphere in whatever dimension you choose) and a unit square (or hypercube in that dimension) in an…

  >The ratio between the areas of a unit circle (or hypersphere in whatever dimension you choose) and a unit square (or hypercube in that dimension) in any system will always require infinite precision to describe.
Easily fixed! I choose 1 dimension. :)
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