The Unknotting Number Is Not Additive(divisbyzero.com) |
The Unknotting Number Is Not Additive(divisbyzero.com) |
It's interesting that you have to first weave the knot around itself, which adds many more crossings. Only then do you get a the special unknotting that falsifies the conjecture.
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 circles are natural and numbers are natural, then why does their relationship seem so unnatural and arbitrary?
You could imagine some advanced alien civilization, maybe in a completely different universe, that isn’t even aware of these concepts. But does it make them any less real?
Sorry for rambling off topic like a meth addict, just hoping someone can enlighten me.
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 directly experience it. It's a useful abstraction but there is, according to that abstraction, an infinity of "natural" numbers that mankind will not be able to ever write down, either as a number or as a formula. So infinity will always escape our immediate perception and remain fundamentally an abstraction.
Real numbers are some of the least real of the numbers we deal with in math. They turn out to be a very useful abstraction but we can only observe things that approximate them. A physical circle isn't exactly pi times its diameter up to infinity decimals, if only because there is a limit to the precision of our measurements.
To me the relationship between pi and numbers is not so unnatural but I have to look at a broader set of abstractions to make more sense of it, adding exponentials and complex numbers - in my opinion the fact that e^i.pi = 1 is a profound relationship between pi and natural numbers.
But abstractions get changed all the time. Math as an academic discipline hasn't been around for more than 10,000 years and in that course of time abstractions have changed. It's very likely that the concept of infinity wouldn't have made sense to anyone 5,000 years ago when numbers were primarily used for accounting. Even 500 years ago the concept of a number that is a square root of -1 wouldn't have made sense. Forget aliens from another planet, I'm pretty sure we wouldn't be able to comprehend 100th century math if somehow a textbook time-traveled to us.
> 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 difference. The translation of mathematical concepts to reality is highly critical, I would say. You cannot just translate it directly, because this leads to such strange questions like "what would happen if we take the law of gravitation by old Newton and let r^2 go to zero?". Well, you can't! Because Heisenberg is standing down there.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.
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
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.
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.
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.
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.
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.
Make the areas between the circle and the square equal, and the infinite precision moves into the ratio between their lower order dimensional measures (circumfence, surface area, etc.).
You can't describe a system that expresses the one, in terms of a system that expresses the other, without requiring infinite precision (and thus infinite information).
Furthermore, it really seems like a bunch of the really fundamental reals (pi, e), have a pretty deep connection to algebras of rotations (both pi and e relate strongly to rotations)
What that seems to suggest to me is that if the universe is discrete, then the discreteness must be biased towards one of these modes or the other - i.e. it is natively one and approximates the other. You can have a discrete universe where you have natural rotational relationships, or natural linear relationships, but not both at the same time.
>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.
Our logical approximation of the universe might need to be, assuming that we don't add more axioms to our system of logical reasoning.
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 to believe atoms, quarks, electrons, photons, etc. are real. Forces (electrical charge, weak and strong nuclear force, gravity) are real.
What I'm not willing to believe is that quantum fields in general are real, that physical components are not real and don't literally move, they're just "interactions" with and "fluctuations" in the different quantum fields. I refuse to believe that matter doesn't exist and it's merely numbers or vectors arranged a grid. That's a step too far. That's surely just a mathematical abstraction. And yet, the numbers these abstractions produce match so well with physical observations. What's going on?
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.)
If one thinks about it, electromagnetism is really bizarre.
How can two electrons actually repel each other? Sure, they do, but it’s practically witchcraft.
Magnetism is even more weird.
Compared to EM it's just weird as hell and tbh I don't like it.
If a function is one-to-one, it has a (right? left? keep forgetting which one)-inverse. But if Moshe the imaginary forgot the milk, his wife may or may not shout at him, whichever way the story teller decides to take the story... So a function being one-to-one is real, but Moshe the imaginary forgetting the milk isn't.
I like this view when I'm being befuddled by a result, especially some ad absurdum argument. I tell myself: this thing is true, so if it wasn't we'd just need to look hard enough to find somewhere where two effects clash.
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 experience of strugling with untying knots in strings.
One issue I sometimes witnessed myself was that Mathematicians sometimes form Groups that behave like pathological examples from Sociology. E.g. there was the Monty-Hall problem, where societies of mathematicians had a meltdown. Sadly I've seen this a few times when Sociology/Mass psychology simply trumped Math in Power.
The counterexample has 7 crossings. Try to explain why the equivalent knot with only 5 crossing is not an counterexample and you may realize why it's not obvious.
And that is why the mirror image had to be taken - you need to make sure that when you join it is over to over and under to under.
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.)
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.
The calculus of scaled rotation is so beautiful. The sacrificial lamb is the unique ordering relation.
I like to imagine they're somehow just an observational error, otherwise the https://en.wikipedia.org/wiki/One-electron_universe is real and we get a universe-sized '—All You Zombies—'
> How can two electrons actually repel each other
Indeed. I think it's something we can only intuit, I don't think we've really gotten to the bottom of it. Trying to push two electrons together feels like trying to push a car up a hill, or pressing on springs. The force you fight against is just there and you feel its resistance
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.
Socrates made a whole career out of it.
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).
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.
Yes, the article is about it ... which has no bearing on my point, and just repeats the logic error.
It is frequently the case that a counterexample is obviously (or readily seen to be) a counterexample to a conjecture. That has no bearing on how long it takes to find the counterexample. e.g., in 1756 Euler conjectured that there are no integers that satisfy a^4+b^4+c^4=d^4 It took 213 years to show that 95800^4+217519^4+414560^4=422481^4
satifies it ... "obviously".
Saying that the counterexample is a posteriori obvious is not saying that the conjecture is a priori obviously false.
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 overloaded to output an exact representation when it detects one of the exact trigonometric values[1] eg cos(60°) = 1/2. It will now give output values as "X + Y PI", or you can optionally collapse that to an inexact decimal with an eval[] function.
That's the closest I got to containing the "messiness" of pi. Eventually I hit a wall because Frink doesn't support exact square roots, so most exact values would be decimals anyway.
Still, I can dream!
[1] https://en.wikipedia.org/wiki/Exact_trigonometric_values
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.
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...
No, the integers that we can count (or build machines to count) are not nearly as fictional.
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".
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.
Non-constructive arguments are things like proof by contradiction i.e., the absence of the negative implies the existence of the positive.
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 intractable, of course.)
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, and it might be possible to use this technique on any/many pairs of knots to reduce the sum of their unknotting numbers.
It could be done symbolically, by generalizing from their rational representation:
X/Y
(X/Y)^(A/B)
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
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.