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.
brap 6 hours ago [-]
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.
yujzgzc 6 hours ago [-]
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.
lqet 6 hours ago [-]
Philosophical problems regarding the fundamental nature of reality aside, this short clip is relevant to your question:
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.
twiceaday 4 hours ago [-]
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.
brap 2 hours ago [-]
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.
IAmBroom 13 minutes ago [-]
The age-old problem of a respondent using different definitions of words than the OP.
Socrates made a whole career out of it.
Byamarro 5 hours ago [-]
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.
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?
BobbyTables2 34 minutes ago [-]
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.
amiga386 24 minutes ago [-]
> What about the particles that randomly pop in and out of existence?
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
kannanvijayan 3 hours ago [-]
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.
schiffern 2 hours ago [-]
>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.
> 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.
CJefferson 5 hours ago [-]
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.
gcanyon 2 hours ago [-]
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."
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.
srean 3 hours ago [-]
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.
rini17 4 hours ago [-]
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.
empath75 3 hours ago [-]
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.
rini17 5 hours ago [-]
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.
JdeBP 5 hours ago [-]
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.
fjfaase 6 hours ago [-]
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.
jrowen 5 hours ago [-]
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.
kurlberg 5 hours ago [-]
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".
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.
slickytail 6 hours ago [-]
In the words of Kronecker: "God created the integers, all else is the work of man."
srean 3 hours ago [-]
Had I been god I would have created scaled turns and left the rest for humans.
fedeb95 6 hours ago [-]
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.
Antinumeric 4 hours ago [-]
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.
pfortuny 3 hours ago [-]
It happens: once you see the example, it may be trivial to understand. The hard thing is to find it.
James_K 2 hours ago [-]
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.
deadfoxygrandpa 4 hours ago [-]
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
jibal 3 hours ago [-]
Logic fail. The example is not the conjecture. Saying the example is obvious is not saying that the conjecture is obvious.
jibal 7 minutes ago [-]
P.S. To clarify:
Saying that the counterexample is a posteriori obvious is not saying that the conjecture is a priori obviously false.
gcanyon 2 hours ago [-]
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.
jibal 41 minutes ago [-]
"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".
ealexhudson 4 hours ago [-]
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...
iainmerrick 4 hours ago [-]
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.
Antinumeric 4 hours ago [-]
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.
iainmerrick 3 hours ago [-]
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.)
gcanyon 2 hours ago [-]
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.
robinhouston 4 hours ago [-]
I think this is one of those language barrier things. Non-mathematicians sometimes say ‘obvious’ when what they mean is ‘vaguely plausible’.
Timwi 3 hours ago [-]
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.
Sh4p3Sh1fter 3 hours ago [-]
[flagged]
Rendered at 14:10:34 GMT+0000 (Coordinated Universal Time) with Vercel.
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:
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.
Socrates made a whole career out of it.
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'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?
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.
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
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.
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.
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.
https://www.youtube.com/watch?v=5TkIe60y2GI
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 calculus of scaled rotation is so beautiful. The sacrificial lamb is the unique ordering relation.
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.
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.
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.
Saying that the counterexample is a posteriori obvious is not saying that the conjecture is a priori obviously false.
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".
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.
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.
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.