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on Gopher (inofficial)
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COMMENT PAGE FOR:
URI The Unknotting Number Is Not Additive
Antinumeric wrote 6 hours 1 min 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.
James_K wrote 3 hours 40 min 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.
pfortuny wrote 4 hours 23 min ago:
It happens: once you see the example, it may be trivial to
understand. The hard thing is to find it.
deadfoxygrandpa wrote 5 hours 56 min 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 wrote 4 hours 34 min ago:
Logic fail. The example is not the conjecture. Saying the example
is obvious is not saying that the conjecture is obvious.
jibal wrote 1 hour 55 min 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 wrote 4 hours 19 min 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 wrote 2 hours 30 min 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".
iainmerrick wrote 5 hours 22 min ago:
Please donât jump straight to âlyingâ, itâs better to
assume good faith. I agree itâs likely much more complex than
theyâre assuming.
robinhouston wrote 5 hours 26 min ago:
I think this is one of those language barrier things.
Non-mathematicians sometimes say âobviousâ when what they mean
is âvaguely plausibleâ.
Timwi wrote 5 hours 1 min ago:
A math professor at my uni said that a statement in mathematics
is âobviousâ 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.
Antinumeric wrote 5 hours 36 min 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 wrote 5 hours 15 min ago:
Youâre getting a lot of pushback here, but I have to say, your
intuition makes sense to me too.
When youâre 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 wrote 4 hours 11 min 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.
ealexhudson wrote 5 hours 38 min 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...
qnleigh wrote 7 hours 52 min ago:
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 wrote 8 hours 8 min ago:
Whenever I encounter this sort of abstract math (at least
âabstractâ for me) I start wondering whatâs even ârealâ.
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 ânaturalâ?
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 isnât 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.
eprparadox wrote 2 hours 37 min ago:
there's a great episode of Mindscape where Max Tegmark takes this
idea and runs with it:
URI [1]: https://www.preposterousuniverse.com/podcast/2019/12/02/75-m...
amiga386 wrote 3 hours 28 min ago:
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?
pelorat wrote 50 min ago:
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.
BobbyTables2 wrote 2 hours 23 min 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 itâs practically witchcraft.
Magnetism is even more weird.
amiga386 wrote 2 hours 12 min ago:
> 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 [1] is real and we get a universe-sized 'âAll You
Zombiesâ'
> 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
URI [1]: https://en.wikipedia.org/wiki/One-electron_universe
jibal wrote 4 hours 20 min ago:
> 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 isnât 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.
kannanvijayan wrote 4 hours 39 min 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 wrote 4 hours 16 min 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.
Easily fixed! I choose 1 dimension. :)
kannanvijayan wrote 2 hours 24 min ago:
Hah, nice find :)
schiffern wrote 1 hour 35 min ago:
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!
[0] [1]
URI [1]: https://frinklang.org/
URI [2]: https://en.wikipedia.org/wiki/Exact_trigonometric_valu...
rini17 wrote 6 hours 22 min 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.
kurlberg wrote 6 hours 41 min 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".
JdeBP wrote 6 hours 45 min 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.
Byamarro wrote 7 hours 10 min 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.
[1] atoms are mental concepts as well ofc
movpasd wrote 6 hours 43 min ago:
I believe this is called epistemic pragmatism in philosophy:
URI [1]: https://en.wikipedia.org/wiki/Pragmatism
CJefferson wrote 7 hours 14 min 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 wrote 3 hours 53 min 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."
URI [1]: https://www.youtube.com/watch?v=5TkIe60y2GI
empath75 wrote 4 hours 36 min 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.
JdeBP wrote 6 hours 24 min ago:
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 wrote 4 hours 28 min 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 wrote 6 hours 14 min 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
6.02214076Ã10^23 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.
jaffa2 wrote 7 hours 43 min ago:
Theres always an xkcd :
URI [1]: https://xkcd.com/435/
adornKey wrote 5 hours 55 min ago:
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.
yujzgzc wrote 7 hours 43 min 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 wrote 7 hours 55 min ago:
Philosophical problems regarding the fundamental nature of reality
aside, this short clip is relevant to your question:
> [1] 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.
URI [1]: https://www.youtube.com/watch?v=tCUK2zRTcOc
twiceaday wrote 6 hours 12 min 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 wrote 4 hours 16 min ago:
I think maybe I didnât really explain myself properly. I
didnât mean that math is real in the sense that atoms are real.
Perhaps âtrueâ would be a better word. We know these things
are true to us, but are they universally true? If thatâs even a
thing? Hope that makes more sense.
IAmBroom wrote 2 hours 2 min ago:
The age-old problem of a respondent using different definitions
of words than the OP.
Socrates made a whole career out of it.
fjfaase wrote 7 hours 57 min 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 wrote 7 hours 9 min ago:
Itâs more than strong indications. What any individual life form
perceives is a unique subset or projection of reality. To the
extent that âone true realityâ exists, we are each viewing part
of it through a different window.
fedeb95 wrote 7 hours 58 min 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.
slickytail wrote 8 hours 6 min ago:
In the words of Kronecker: "God created the integers, all else is the
work of man."
srean wrote 4 hours 26 min ago:
Had I been god I would have created scaled turns and left the rest
for humans.
ZiiS wrote 8 hours 10 min ago:
Great video coverage from Stand-up Maths
URI [1]: https://www.youtube.com/watch?v=Dx7f-nGohVc
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