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The Unknotting Number Is Not Additive

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Re: The Unknotting Number Is Not Additive

#71

Post by kcexn »

CJefferson wrote: post #67
empath75 wrote: post #31

Earlier quoted context omitted.

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.

I don't agree, but I agree it's an interesting discussion to have. When is the set of all possible subsets of natural numbers worth considering more than the set of all sets which don't contain themselves (which gets us Russell's paradox of course), once we start building infinite sets non-constructively? The naturals to me are a clearly separate category, as I can easily write down an algorithm which will make any n…

You can construct a real number by using an infinite series so it's no less constructive than a rational function on the naturals.

Non-constructive arguments are things like proof by contradiction i.e., the absence of the negative implies the existence of the positive.

Re: The Unknotting Number Is Not Additive

#72

Post by kcexn »

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 universe, being a physical entity not bound by the rules of human logic doesn't have to be either.

Our logical approximation of the universe might need to be, assuming that we don't add more axioms to our system of logical reasoning.

Re: The Unknotting Number Is Not Additive

#74

Post by CJefferson »

kcexn wrote: post #71
CJefferson wrote: post #67

Earlier quoted context omitted.

I don't agree, but I agree it's an interesting discussion to have. When is the set of all possible subsets of natural numbers worth considering more than the set of all sets which don't contain themselves (which gets us Russell's paradox of course), once we start building infinite sets non-constructively? The naturals to me are a clearly separate category, as I can easily write down an algorithm which will make any n…

You can construct a real number by using an infinite series so it's no less constructive than a rational function on the naturals. Non-constructive arguments are things like proof by contradiction i.e., the absence of the negative implies the existence of the positive.

Except we can only describe those infinite series for a countably infinite number of the reals, so there are all these reals expressed by infinite series we don’t have any way to describe. Why do we need those ones? (To be clear, I realise this isn’t the current standard opinion of most mathematicians, I choose to be annoying).

Re: The Unknotting Number Is Not Additive

#75

Post by tocs3 »

IIAOPSW wrote: post #65
yujzgzc wrote: post #9

Earlier quoted context omitted.

Yes these knots are real and can be experienced with a simple piece of rope. The prime property of numbers is also very real, a number N is prime if and only if arranging N items on a rectangular, regular grid can only be done if one of the sides of the rectangle is 1. Multiplication and addition are even more simply realized. The infinity of natural numbers is not as real, if what we mean by that is that we can dire…

I see infinity all the time. Go look at a one point perspective drawing.

I think that is a little like pi. There is a limit to what we can measure. In a real life drawing on paper the "one point" is not dimensionless. There is a limit to what we can draw.

Re: The Unknotting Number Is Not Additive

#76

Post by kcexn »

CJefferson wrote: post #74
kcexn wrote: post #71

Earlier quoted context omitted.

You can construct a real number by using an infinite series so it's no less constructive than a rational function on the naturals. Non-constructive arguments are things like proof by contradiction i.e., the absence of the negative implies the existence of the positive.

Except we can only describe those infinite series for a countably infinite number of the reals, so there are all these reals expressed by infinite series we don’t have any way to describe. Why do we need those ones? (To be clear, I realise this isn’t the current standard opinion of most mathematicians, I choose to be annoying).

It's been a while since I did abstract algebra, but I'm pretty sure that once you have the additive and multiplicative identities, the rest of the reals can be generated. Which is still a constructive process.

Regardless, the existence of the real numbers is not a matter of need. Their existence is a consequence of how mathematics is defined. Over-simplified, it's a case of if addition and multiplication work, then the real numbers must exist.

Usually, maths doesn't require us to overthink about anything metaphysical. Things either are or they aren't, the problem-solving approach taken to demonstrate a result one way or the other is the fascinating part.

Re: The Unknotting Number Is Not Additive

#77

Post by schiffern »

kannanvijayan wrote: post #63
schiffern wrote: post #52

Earlier quoted context omitted.

Good show, and I appreciate your sentiment about the "messiness" of pi. There's a unit-converting calculator[0] that supports exact rational numbers and will carry undefined variables through algebraically. With a little hacking, you can redefine degrees in terms in an exact rational multiple of pi radians. Pi is effectively being defined as a new fundamental unit dimension, like distance. Trig functions can be overl…

I suppose you could have added root two as a fundamental as well. I suppose that's another problem with the irrationals: two irrationals that aren't linearly related by a rational are effectively two fundamentals from each others perspective. It's a sad conclusion - though. Computation exists in the countable space. So there is no computationally representable symbolic model that can ever algebraically capture the re…

Yeah, once I got to "all I need to do is add a root for every prime! And cube roots! And..." I realized this is a path of madness. ;)

It could be done symbolically, by generalizing from their rational representation:

  X/Y
To

  (X/Y)^(A/B)
Again this is tantalizingly close to being workable in Frink -- it supports 'dangling' (unevaluated) rational exponents on units, but not simple numbers.

The problem of course is that I'm trying to twist a (powerful!) calculator into something like a computer algebra system. I really should just use an actual CAS.

But like you say, I'd be happy if I could "just" have an exact representation of (if not the reals because that's impossible, then at least) any number I can describe in finite terms with normal math operators.

Cheers and good day

Re: The Unknotting Number Is Not Additive

#78

Post by IIAOPSW »

tocs3 wrote: post #75
IIAOPSW wrote: post #65

Earlier quoted context omitted.

I see infinity all the time. Go look at a one point perspective drawing.

I think that is a little like pi. There is a limit to what we can measure. In a real life drawing on paper the "one point" is not dimensionless. There is a limit to what we can draw.

The "one point" in "one point perspective" isn't drawn at all, rather it is the point where all lines going into the page perpendicular to the viewing plane eventually converge to. Eg if you were to stand on a set of straight train tracks (don't do this) you would see both rails (and any roads or whatever else is parallel to them) converge to a point somewhere on the horizon line. The artists call it the "vanishing point", the mathematicians call it "the point at infinity".

Indeed with the point at infinity you can simplify geometry by dispensing with Euclid's 5th postulate. There are no parallel lines, any two lines intersect at a single point just the same way as any two points are intersected by a single line, and the intersection points of the lines we call "parallel" simply happen to be "at infinity" (outside the set of ordinary finite coordinates).

The vanishing point in a perspective drawing is a point with a value that is literally beyond the finite coordinates of any object. And you don't need to be looking at a drawing to see it.

In a certain regard its an accounting trick. Saying parallel lines meet at infinity is literally like saying "lets schedule this meeting for never", except the mathematicians added an actual box to the calendar for a date called "never" as an accounting hack, but the hack works so well you really have to wonder if it might actually be a real date or if its just an incredibly useful fiction.

Aren't all numbers just incredibly useful fictions?

Why is a date called never / a point at infinity any different?

https://i.pinimg.com/originals/20/7b/ae/207bae64d2488373fd4a...

Re: The Unknotting Number Is Not Additive

#79

Post by nrds »

empath75 wrote: post #31
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.

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.

Only countably many of those subsets can be distinguished from other ones. So "anything I could ever imagine caring about" is surely still a countable set.

I think the argument you are trying to make rests on a pretty serious fallacy generalizing "I care about some subsets of natural numbers" (and maybe " I care about subsets of natural numbers, in general") to "I care about all subsets of natural numbers, including undefinable ones".

Re: The Unknotting Number Is Not Additive

#80

Post by yujzgzc »

IIAOPSW wrote: post #78
tocs3 wrote: post #75

Earlier quoted context omitted.

I think that is a little like pi. There is a limit to what we can measure. In a real life drawing on paper the "one point" is not dimensionless. There is a limit to what we can draw.

The "one point" in "one point perspective" isn't drawn at all, rather it is the point where all lines going into the page perpendicular to the viewing plane eventually converge to. Eg if you were to stand on a set of straight train tracks (don't do this) you would see both rails (and any roads or whatever else is parallel to them) converge to a point somewhere on the horizon line. The artists call it the "vanishing p…

> Aren't all numbers just incredibly useful fictions?

No, the integers that we can count (or build machines to count) are not nearly as fictional.

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