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The Unknotting Number Is Not Additive

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Re: The Unknotting Number Is Not Additive

#61

Post by steve_gh »

Timwi wrote: post #29
robinhouston wrote: post #26

Earlier quoted context omitted.

I think this is one of those language barrier things. Non-mathematicians sometimes say ‘obvious’ when what they mean is ‘vaguely plausible’.

A math professor at my uni said that a statement in mathematics is “obvious” if and only if a proof springs directly to mind. If that is indeed the standard, then it's easy to see how something that is vaguely plausible to an outsider can be obvious to someone fully immersed in the field.

Not quite 'obviously', but mathematical folklore has it that 'clearly' is used to mark the difficult conceptual step in a proof.

Re: The Unknotting Number Is Not Additive

#62

Post by Antinumeric »

gcanyon wrote: post #41
iainmerrick wrote: post #28

Earlier quoted context omitted.

You’re getting a lot of pushback here, but I have to say, your intuition makes sense to me too. When you’re connecting those two knots, it seems like you have the option of flipping one before you join them. It does seem very plausible that that extra choice would give you the freedom to potentially reduce the knotting number by 1 in the combined knot. (Intuitively plausible even if the math is very, very complex and…

But this implies that a simple 1-knot might completely undo itself if you join it to its mirror. Which I assume people have tried, and doesn't work. Likewise with 2's, 3's etc. It seems intuitively obvious that there is something deeper going on here that makes these two knots work, where (presumably) many others have failed. Or more interestingly to me, maybe there's something special about the technique they use, a…

(non-mathematical) Implication doesn't mean certainty, which is where I stand with that. But I would posit that it (mathematically) implies that joining two knots with under to over will never decrease the unknotting number from the sum.

Re: The Unknotting Number Is Not Additive

#63

Post by kannanvijayan »

schiffern wrote: post #52
kannanvijayan wrote: post #47

Earlier quoted context omitted.

Hah, nice find :)

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 reals.

The other thing that came to mind when you mentioned root-2 is a similar realization as with pi. That somehow a diagonal is not well defined in discrete terms with respect to two orthogonal vectors. So here once again, you have this weird impedance mismatch between orthogonality (a rotational concept) and diagonals (a linear concept).

I don't have the formalisms to explore these thoughts much further than this.. so it's hard to say whether this is just some trivial numerological-like observation or if there's something more to it. But it's kinda pleasant to think about sometimes.

Re: The Unknotting Number Is Not Additive

#64

Post by amiga386 »

notfed wrote: post #59
amiga386 wrote: post #44

Earlier quoted context omitted.

I'm fairly confident that most mathematics are real, i.e. they have real world analogues. Pi is just an increasingly close look at the ratio between a circle's diameter and circumference. I'm willing to believe elecromagnetic fields are real - you can see the effects magnets (and electromagnets) have on ferrous material. You can really broadcast electromagnetic waves, induce currents in metals, all that. I'm willing…

> I'm willing to believe elecromagnetic fields are real No shade intended, but a philosophical conversation is unconstructive when it centers around highly ambiguous and undefined words. The word "real" does not actually have a general meaning until you give it a definition in support of your comment. (And surely you will find that if you had a definition, you would not need so much "belief" to back up your argument.…

I was mostly going down a sciencey path, but "real" is a fairly well understood word (part of reality; not imaginary).

In terms of philosophy I'm mostly of an empirical bent. Things which are observable are real, and things which aren't observable directly, but have a observable effect that can be repeatedly demonstrated on demand, are real too (though they may not be exactly as hypothesised if all we can see are their effects). This is how electromagnetism and quantum tunnelling can be real at the same time faeries aren't.

Re: The Unknotting Number Is Not Additive

#65

Post by IIAOPSW »

yujzgzc wrote: post #9
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…

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.

Re: The Unknotting Number Is Not Additive

#66

Post by deadfoxygrandpa »

ealexhudson wrote: post #24
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

Surely the example can be "obvious" because it's simple/clear. I don't think they're commenting on whether _finding_ the example is obvious...

did you look at the example? it's incredibly complicated

Re: The Unknotting Number Is Not Additive

#67

Post by CJefferson »

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.

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 natural number given enough time. But then, I'm a constructionist at heart, so I would like that.

Re: The Unknotting Number Is Not Additive

#68

Post by ctenb »

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 discovering universal truths: Given a set of axioms and following theorems, the theorems will apply in any scenario where the axioms are true. So that makes maths both invented (the axioms) and discovered (the theorems) and real in any situation where it applies.

Re: The Unknotting Number Is Not Additive

#69

Post by kcexn »

brap wrote: post #39
twiceaday wrote: post #20

Earlier quoted context omitted.

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.

Mathematics is a philosophy that focuses on the study of logic. It's a bit of an exaggeration to conflate mathematics with 'truth' in an absolute, universal sense.

Mathematical 'truths' are themselves only true in the sense that they can be derived from axioms.

The fact that mathematics can be used to understand the world around us is nothing short of a mystery (or a miracle).

Re: The Unknotting Number Is Not Additive

#70

Post by kcexn »

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…

The natural numbers are 'natural' because they are definite quantities that can be used for counting.

Taylor expansions about a point of a function requires that the function has a derivative defined at that point.

The derivative itself is the point at which an infinite sequence (say, of incrementally closer approximations) converges.

So derivatives and Taylor series are really more of an arbitrary precision approximation of a value rather than a concrete exact quantity.

Arbitrary precision approximation just happens to be a very elegant way to model the physical world around us.

For truly exact solutions, you still have to work with the naturals (and rationals, etc.)

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