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The Unknotting Number Is Not Additive

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Re: The Unknotting Number Is Not Additive

#11

Post by CJefferson »

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.

Re: The Unknotting Number Is Not Additive

#12

Post by Byamarro »

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…

Math is about creating mental models.

Sometimes we want to model something in real life and try to use math for this - this is physics.

But even then, the model is not real, it's a model (not even a 1:1 one on top of that). It usually tries to capture some cherry picked traits of reality i.e. when will a planet be in 60 days ignoring all its "atoms"[1]. That's because we want to have some predictive power and we can't simulate whole reality. Wolfram calls these selective traits that can be calculated without calculating everything else "pockets of reducability". Do they exist? Imho no, planets don't fundamentally exist, they're mental constructs we've created for a group of particles so that our brains won't explode. If planets don't exist, so do their position etc.

The things about models is that they're usually simplifications of the thing they model, with only the parts of it that interest us.

Modeling is so natural for us that we often fail to realize that we're projecting. We're projecting content of our minds onto reality and then we start to ask questions out of confusion such as "does my mind concept exist". Your mind concept is a neutral pattern in your mind, that's it.

[1] atoms are mental concepts as well ofc

Re: The Unknotting Number Is Not Additive

#13

Post by jrowen »

fjfaase wrote: post #6
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…

What is real? There are strong indications that what we experience as reality is an ilusion generated by what is usually refered to as the subconscious. One could argue that knots are more real than numbers. It is hard to find two equal looking apples and talk about two apples, because it requires the abstraction that the apples are equal, while it is obvious that they are not. While, I guess, we all have had the exp…

It’s more than strong indications. What any individual life form perceives is a unique subset or projection of reality. To the extent that “one true reality” exists, we are each viewing part of it through a different window.

Re: The Unknotting Number Is Not Additive

#14

Post by JdeBP »

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…

More usually, people imagine the reverse of the advanced alien civilizations: that the thing that we and they are most likely to have in common is the concept of obtaining the ratio between a circle's circumference and its diameter, whereas the things that they possibly aren't even aware of are going to be concepts like economics or poetry.

Re: The Unknotting Number Is Not Additive

#15

Post by movpasd »

Byamarro wrote: post #12
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…

Math is about creating mental models. Sometimes we want to model something in real life and try to use math for this - this is physics. But even then, the model is not real, it's a model (not even a 1:1 one on top of that). It usually tries to capture some cherry picked traits of reality i.e. when will a planet be in 60 days ignoring all its "atoms"[1]. That's because we want to have some predictive power and we can'…

I believe this is called epistemic pragmatism in philosophy: https://en.wikipedia.org/wiki/Pragmatism

Re: The Unknotting Number Is Not Additive

#16

Post by kurlberg »

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…

Fun historical fact: knot theory got a big boost when lord Kelvin (yeah, that one) proposed understanding atoms by thinking of them as "knotted vortices in the ether".

Re: The Unknotting Number Is Not Additive

#17

Post by JdeBP »

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.

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 infinite series in order to express physical reality, from Feynman sums to Taylor expansions.

Re: The Unknotting Number Is Not Additive

#18

Post by rini17 »

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 see it like natural sciences strive to do replicable experiments in outside world, while math strives to do replicable experiments in mind. Not everything is transferable from one domain to the other but we keep finding many parallels between these two, which is surprising. But that's all we have, no foundational truths, no clear natural/unnatural divide here.

Re: The Unknotting Number Is Not Additive

#19

Post by rini17 »

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…

But you can't really have chemistry without working with natural numbers of atoms, measured in moles. Recently they decided to explicitly fix a mole (Avogadro's constant) to be exactly 6.02214076×10^23 which is a natural number.

Semiconductor manufacturing on nanometer scales deals with individual atoms and electrons too. Yes, modeling their behavior needs complex numbers, but their amounts are natural numbers.

Re: The Unknotting Number Is Not Additive

#20

Post by twiceaday »

lqet wrote: post #7
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…

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 concepts directly into reality." It's backwards. You must have first chosen some math to model reality. If you get "bad" numbers it has nothing to do with translating math to reality. It has to do with how you translated reality into math.

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