The Unknotting Number Is Not Additive
divisbyzero.com178 points by JohnHammersley 4 days ago
178 points by JohnHammersley 4 days ago
Great video coverage from Stand-up Maths https://www.youtube.com/watch?v=Dx7f-nGohVc
Can't recommend this video highly enough. Matt clearly demonstrates with physical examples both the question and the answer, and why it took so long to find that answer.
I read the Quanta article on this when it came out. They show the knots, and they're simple enough that I was almost surprised that the counterexample hadn't been found before. But seeing the shockingly complicated unknotting procedure here makes it much clearer why it wasn't!
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.
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 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.
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 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.
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...
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 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.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.
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.
The age-old problem of a respondent using different definitions of words than the OP.
Socrates made a whole career out of it.
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).
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
I believe this is called epistemic pragmatism in philosophy: https://en.wikipedia.org/wiki/Pragmatism
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.
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.
One could argue that infinite subsets of the natural numbers are not really interesting unless one can succinctly describe which elements are contained in them. And of course there is only a countable number of such sets.
Some people (not me) would consider only countably many of those subsets to be “possible”.
You might appreciate this video where Matt Parker lays out the various classes of numbers and concludes by describing the normal numbers as being the overwhelmingly vast proportion of numbers and laments "we mathematicians think we know what's what, but so far we have found none of the numbers."
You can encompass them all by talking about numbers that can be described. Since you can trivially enumerate all possible descriptions, this is countably infinite. By definition, it is impossible to describe a number outside that 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.
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.)
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.
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.
> 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.
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 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.
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 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
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.
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
(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
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.
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 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?
> 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.
What about the particles that randomly pop in and out of existence?
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.
> What about the particles that randomly pop in and out of existence?
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
Wait until you hear about the gluon, the mediator of the strong force, which is an excitation in the gluon field, and is also the only other particle that is massless and moves at C. However unlike the photon the excitation has a really short range because gluons interact with gluons and form flux tubes between quarks, the further you pull two quarks apart, the more energy you need to use, eventually the energy is so great that it spawns a new quark from the vacuum.
Compared to EM it's just weird as hell and tbh I don't like it.
My pet philosophy is that math is real because the objects have persistent effects, like with the "if a tree falls in the forest..." riddle. Something that isn't real would be a story, because things do not have effects in 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.
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 experience of strugling with untying knots in strings.
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.
there's a great episode of Mindscape where Max Tegmark takes this idea and runs with it: https://www.preposterousuniverse.com/podcast/2019/12/02/75-m...
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.
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.
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.
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".
Theres always an xkcd : https://xkcd.com/435/
Nice line, but it isn't fully complete. After the Mathematicians there's Logic - and Philosophy - And in the end you complete the circle and go all back to Sociology again.
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.
In the words of Kronecker: "God created the integers, all else is the work of man."
I sometimes think about the same things. As of now, my best bet is that math is one of the disciplines studying exactly these questions.
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.
Yes this is an interesting case where something that seems obvious on first thought also seems like it would be wrong once you try it out, and then after 100 years of trying someone looks hard enough at their plate of spaghetti and realises it was right all along.
It happens: once you see the example, it may be trivial to understand. The hard thing is to find it.
> This example seems obvious to me
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.
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
I'm not saying I could have come up with the example. I'm saying looking at the example, and seeing how the two unders are connected togther, and the two overs connected together, makes it obvious that there is more freedom to move the knot around. And that freedom, at least to me, is intuitively connected to the unknotting number.
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.
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 intractable, of course.)
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, and it might be possible to use this technique on any/many pairs of knots to reduce the sum of their unknotting numbers.
(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.
Please don’t jump straight to “lying”, it’s better to assume good faith. I agree it’s likely much more complex than they’re assuming.
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...
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.
"counterexample: an example that refutes or disproves a proposition or theory"
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".
P.S. To clarify:
Saying that the counterexample is a posteriori obvious is not saying that the conjecture is a priori obviously false.
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.
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