serious-discussion

[fair estuary]

It wouldn't though

[mint patio]

If you think about it that way then everything is arbitrary, don't you think?

[modest rune]

nothing about mathematics would be different except notation

[fair estuary]

I think natural numbers and fractions aren't arbitrary

[mint patio]

Yeah whoops that's what I meant lol

Everything would be the same, just written a bit different

[modest rune]

the radius and diameter are both natural measurements of a circle

[mint patio]

(Maybe a bit more than a bit)

[modest rune]

and the ratio of circumference to both is natural

and the constant you get differs by a factor of 2

[mint patio]

I'll let Max take over, he's more knowledgeable

[modest rune]

in the long run this doesnt matter at all, and while some formulas might look nicer with tau, others look nicer with pi.

At the end of the day its kind of just whatever

but I should sleep and you were all doing a fine job haha

[fair estuary]

It matters for the transcendental integer though

[modest rune]

huh

both are transcendental

[fair estuary]

Get some rest max bro, I'll tell you about it another time

[torn willow]

This should work. If you define this,you can find the area of a circle using integral methods

Now all you need to do is define sin(π/2)=1

And then compute
$4\int_{0}^{r}\sqrt{r^2-x^2} dx$

[fathom swallow]

Drake

[viral dew]

guys im desperate for help

if someone asked you what your major was, but you're a double major, would you tell them just one major or both?

[candid cypress]

just tell them both?

[ancient flame]

lmao

that is such a non-issue

@viral dew what's the point of hiding that fact

[viral dew]

it's a social issue

[ancient flame]

???

elaborate

[arctic grove]

i would tell them both

[viral dew]

based on my own experiences at events, shared rides, etc., telling others you study two STEM subjects when asked makes people uncomfortable and/or stop talking to you. was wondering if i should just deal with it bc it's my own work or if i should try to avoid that type of situation as much as possible

[arctic grove]

[fathom flicker]

Hi I am trying to use set theory to show matching a dictionary key with another dictionary key and using the value of one dictionary as the key and the other value as the value.

[ancient flame]

who tf gets uncomfortable about that? if they would stop talking to you over that then perhaps it's best not to talk to those people anyway

[arctic grove]

yeah

ur choice of studying something shouldnt really be an issue to other people

[ancient flame]

^^^

[fathom flicker]

say i have 2 dictionaries. I want to make a new dictionary using proper set notation where machines are the keys and c is the values.

[ancient flame]

people suck bro

[viral dew]

lol I mean yeah I agree

[fathom flicker]

Anyone here good at set theory?

[arctic grove]

and tbf, i dont think most people would mind it much

"oh cool u study that? nice nice, anyways, look at this cool lego piece i made"

or something

lol

[viral dew]

oh geez that's so true

haha yeah I guess I shouldn't worry about it

[arctic grove]

indeed

[fathom flicker]

Please friends help me haha

How do I do key value pairs in set theory?

[neat frost]

If they’re the kind of people to stop talking to you because they’re uncomfortable that you’re studying two STEM subjects, do you really even want to talk with them? Like who gets uncomfortable about that?

Ah Gmod said that already

Gmod and I are just on the same wavelength

[fervent pebble]

facts lol

imagine getting weirded out bc someone's taking a double major

[neat frost]

Yeah

[fervent pebble]

sah, slurp?

[neat frost]

I mean maybe it’s not the double major part that weirds them out

Maybe one of their majors is like bio 🤢

I’d defo stop talking to someone who’s double majoring in bio

[fervent pebble]

slurp you are the kind of person who would double major in bio and ochem

just kidding

[neat frost]

Wow

[fervent pebble]

sorry

[neat frost]

That’s crossing a line valley

[fervent pebble]

i know that was too far

it's just 2:40 am and i have a headache

[charred mortar]

I'd be more weirded out that they're weirded out in the first place tbh

[fervent pebble]

^

[fervent pebble]

ignored

[neat frost]

Oh sorry. What does sah mean

Still at home?

Stupid ass ho?

[arctic grove]

[neat frost]

Good morning shyshy!

[fervent pebble]

it means what's up

dumdum

[neat frost]

How

How does it mean what’s up

[arctic grove]

Good morning slurp!

[fervent pebble]

what's up --> what's ah --> what sah --> sah

[arctic grove]

i suppose its condensing whats up a lot
so like, u know how people say "sup"

its doing that

but even worse

[fervent pebble]

even worse is my specialty

[neat frost]

That is vile

[arctic grove]

very vile

[neat frost]

Anyway I just got back from the doctor

Sah valley?

[fervent pebble]

hmmm

im very tired

just stopped playing val w/ friends

and today was my last day of school

[arctic grove]

Ooo

[fervent pebble]

so that was amazing

[arctic grove]

thats nice

[fervent pebble]

yes

[neat frost]

Ugh

So lucky

[fervent pebble]

very nice

when does ur school end

[neat frost]

Still have a whole year left

[arctic grove]

im watching stranger things

wait slurp really?

[neat frost]

😭

[arctic grove]

u have a whole year left?

wtf

[fervent pebble]

okay no i still have a year left dum

[neat frost]

Im in 11th

[arctic grove]

thats actually sad

damn

[neat frost]

What

[fervent pebble]

slurp we are the same age

[neat frost]

What

No

Stfu

[fervent pebble]

i meant last day of junior year

[neat frost]

You’re older than me

[arctic grove]

Oo

[fervent pebble]

im calling it school jc

[neat frost]

Stfu

[arctic grove]

i have had my last day of school, period

[fervent pebble]

slurp you are literally older than me in spirit

bitch

[neat frost]

I lil wee baby uwu

D:<

[fervent pebble]

!ping

!pin

[arctic grove]

slurp u arent that small

[neat frost]

I am tho

The doc told me I’m underweight

So I lil wee baby uwu

[fervent pebble]

underweight at 5'6"??

boi

you need to eat more

[neat frost]

I’m 5’4

Smh 5’6 what is this shit

[fervent pebble]

thats even worse

literal twig

[neat frost]

I’m not fckin goliath

[fervent pebble]

bitch neither am i

ur just literally a bug

[neat frost]

You’re like 7 foot or whatever

Wow

[fervent pebble]

you need to eat more

[neat frost]

No I’m not a bug I’m a lil wee baby

[fervent pebble]

healthy slurp = better slurp

[neat frost]

No

[arctic grove]

lucky bastard

[fervent pebble]

stfu bugs dont get opinions

[neat frost]

Why

[arctic grove]

coz im overweight

[neat frost]

It’s not an opinion I’m a lil wee baby uwu

[neat frost]

Awh you don’t need to worry about that

[arctic grove]

i definitely do

[neat frost]

Well being underweight isn’t good either

I mean

Because I don’t do anything physical since I never leave the house, it’s not that big of a deal, but I’m really out of shape

[arctic grove]

yeah true

same

i gotta gym after im done with exams

kinda looking forward to that

[neat frost]

I would gym but I literally could not be bothered

I’m lazy

[arctic grove]

well, u dont really need to gym

if u want to increase ur weight u can like

eat nicely

and run a little

[arctic grove]

relatable

[neat frost]

I bike

Running sucks ass

[arctic grove]

thats better

biking is better

lol

[neat frost]

Yesh I’m a good biker

I only hit like 3 people on average per day

[arctic grove]

uhh

good job

[neat frost]

Thank you!

Also my bike is shit

Falling apart

[arctic grove]

thats sad

[gritty flower]

You can technically freeze time but destroy the world with it

[jade inlet]

How do I freeze time

[naive galleon]

Take 2^256^256^256

[jade inlet]

How do I take 2^256^256^256

[waxen lily]

take ||a shower u smell|| ||like roses|| ||stinky roses||

[ripe wasp]

i went through a whole rollercoaster of emotions reading this

[deep mango]

I suck at biking

Have i ever hit someone?

No.

[sage canopy]

Lol

[lunar spear]

I hit a car once

[fathom flicker]

Guys I have an assignment due in 2 hours and just want to confirm ive done this equation right

Basically I have 2 dictionaries, I want to match the keys and create a new dictionary with the key of one and the matching value of another.

I do it in python and now have to express it

The above equation says create a dictionary with key value pairs. Iterate over a dictionary and get key and value. Then use the key to get the value and set that as the key, then set the value as the value

[neat frost]

Ah well with practice you can!

[deep mango]

Lol

[neat frost]

Anyway good morning Reese!

[fathom flicker]

Hey just quick question

i have d(k) in set theory for accessing the value of a ordered pair at k

But it dosent say which ordered pair its accessing

How do I do this

[neat frost]

This is purely notational

[fathom flicker]

so i dont need to?

[neat frost]

Come up with whatever notation you want

[fathom flicker]

Ok i explained it in text above

[neat frost]

Or you don’t need to if it’s clear which tuple you’re talking about

I assume you’re talking about a tuple since an ordered pair is a pair

[neat frost]

Uh

[fathom flicker]

Does this tell you im creating a new ordered pair from 2 others

[neat frost]

Is it an ordered pair then?

What do you mean by d(k)->v?

[fathom flicker]

im trying to merge two dictionaries together by key then create a new dictionary with key of one and value of other

            in result.items()}```

Thats what im trying to represent

[neat frost]

Oh

So you don’t mean d(k)->v as like a function, it’s just notation?

And machines is a tuple?

[fathom flicker]

Thats what i wrote

hopefully its not too shit

All are dictionaries

key value pairs

Kinda like in mysql when you match tables by primary key, then create new table with key of one and value of other

[neat frost]

Yeah I don’t understand what you wrote

I have to do hw rn tho

[fathom flicker]

ok but just quick

do you know what d(k) is in set notation?

[neat frost]

no, what is it?

[fathom flicker]

Ive heard its a way of accessing a dictionary value at the key

Have you worked with ordered pairs?

[neat frost]

ah well d(k) is not standard notation because the concept of a "dictionary" is not a standard thing in set theory i dont think
in any case i assume youre just defining it as a set of order pairs

[neat frost]

of course, ive done a course in basic set theory

[fathom flicker]

Well basically I have 2 ordered pairs and want to create a third

I want to match keys of them and create new one with key of one and value of other?

Do you understand what I am trying to do?

[deep mango]

Good morning

[brittle socket]

Hi Chalk bird

[deep mango]

Hello grass

[neat frost]

,tex \def\set#1[#2]{\left{#1;\middle|;#2\right}}

$\set{(m,v)}[(k,v)\in R, (k,m)\in M]$

[neat frost]

This is the set you want @fathom flicker

?

[fathom flicker]

Yeah possibly, not sure about the comma tho

should it be a and?

[neat frost]

its not supposed to be, its just another way of writing and

[fathom flicker]

ok so i can use and symbol there?

[neat frost]

yeah

[fathom flicker]

Also do i need to talk in terms of ordered pairs still?

[neat frost]

wdym do you have to

[fathom flicker]

Well i thought a dictionary is a ordered pair in set theory

[neat frost]

a dictionary is not a term that ive heard used in set theory, but yes you could think of a dictionary as a set of ordered pairs

[fathom flicker]

machines[key]

And dont need to use some function access at the key in M?

d(k)

[neat frost]

wdym do you need to?

you could make a function if you want

[fathom flicker]

But works without?

[neat frost]

yes you can do whatever you want without the function

maybe itll be useful to define one in some cases

[fathom flicker]

Ok I understand, thanks for tha

that

We often uses such that terminology but is see no such that

Does this work?

Let me know please it due very soon

Sorry

@neat frost just a yes or no pls

[neat frost]

What does (k,v):(kv) in R mean?

Are you trying to say that R[(k,m)]=(km)?

[neat lintel]

Anybody here good with Discrete / Concrete Mathematics and / or understand flexbox?

I'm trying to master front-end programming for full stack but its not really sinking in.

[cyan goblet]

programming questions would better fit programming/CS dedicated servers

#discrete-math is the topic channel for discrete math itself

[cinder zephyr]

https://twitter.com/neil_calkin/status/1530288990431195136?s=20&t=7RyKNIU4-kAZ4jYBSiJg8w

Neil Calkin (@neil_calkin)

If this holds up --- and both of them are fabulous mathematicians, then it is huge. David Jackson announced just now that he and Bruce Richmond have a 5 page proof of the 4 color theorem.

Likes

368

Twitter · May 27, 2022, 8:45 PM

👀

[bronze pelican]

What the fuck

This is BIG NEWS

[cinder zephyr]

right?

absolutely huge

[bronze pelican]

WHAT I'm losing my marbles rn

[cinder zephyr]

5 pages is so so so so short

I hope this isn't bait 😭

[bronze pelican]

Why would Calkin post such a tease

[cinder zephyr]

RIP my university tho

[bronze pelican]

What if it doesn't work

[cinder zephyr]

the 4 color proof was computed at UIUC

[cinder zephyr]

nooooo it'll work

I believe

I can't wait for paper to be public

[bronze pelican]

Bro

I have tosee the paper

This is like too good to be true

People have worked on this problem for centuries

[cinder zephyr]

yea

[bronze pelican]

And the solution was a eventually achieved through brute force case work checking via computer

[dense belfry]

People need to get over that tbh

Smh

[cinder zephyr]

I mean

the original paper being brute force by a computer is cool and not cool

[sleek wing]

I could colour them all in 3

[cinder zephyr]

it's an important step in mathematics but it didn't really tell us much

[dense belfry]

Getting it to the stage where you can do that is a big achievement

[cinder zephyr]

yea

[bronze pelican]

I'm not saying that it isn't Emma

[dense belfry]

Sure

[split glade]

Goddamn, that is amazing if it works

[cinder zephyr]

yea

I hope this isn't Twitter bait

I can't see why it would be

[dense belfry]

Thinking about the idea that if you take any proof of a mathematical statement and replace any use of LEM with the statement that the objects you are doing LEM to are Computable you get a Computability theory theorem

Like if you just keep track of LEM you can formally get Computability theorems from math theorems

So cool

[cinder zephyr]

interesting

I wish more computability stuff was offered at my university

[dense belfry]

I think this is a well known thing in reverse math

[cinder zephyr]

wdym reverse math

[dense belfry]

htam

[cinder zephyr]

ok googling reverse math, this looks cool

[dense belfry]

So the idea is you study weak forms of theorems over Computable second order arithmetic

And prove equivalences of theorems

And you classify theorems by how much computational strength on top of basic Computability you need to do it

I only vaguely know about it tbh

[cinder zephyr]

sounds interesting

[cyan fossil]

hi.. is anyone doing A level further maths?

[humble current]

Do you need help with smth

[cyan fossil]

so basically do u know what decision 1 is?

[humble current]

No

[cyan fossil]

ah crap

what about the simplex method

[cinder zephyr]

you'd be better off reading #❓how-to-get-help and then asking a specific question

[fervent pebble]

this means ur actually a bad biker

a good biker would hit wayyyy more than three ppl

i hit ten

at the very least

per hour

[deep mango]

Valley already got back active

meanwhile plegasus is still roleless

[cyan goblet]

guess who just woke up

[dire mulch]

bots playing favorites that why.

[frank orchid]

HOw would I start researching the russian ukrainine conflict

i'm on britannica academia where do I look

[deep mango]

The topic is forbidden for discussion the server for now

So

Not the place to ask

[frank orchid]

I'm asking where to find it that's isn't the same as discussing it

[deep mango]

Not up for discussion

[frank orchid]

techinically

[deep mango]

Nah

[arctic grove]

why is the tweet deleted?

[arctic grove]

i havent even got active either!

[uncut socket]

@wide whale Not you, that Ann was referring to. I'm pretty sure Ann was referring to hello_idont

[neat lintel]

people who learn mathematics, does it make any change in their perspective(outlook towards society) if compare to non math learners?

[charred mortar]

I don’t think it’s made any major changes in perspective for me

At most I’ve just become a bit more precise with some of my words

[dense belfry]

https://c.tenor.com/4LvAD8hD5tcAAAAM/charlie-day.gif

[empty stratus]

I'd like to think so.

For one, it definitely increases your ability to parse arguments and organizing your thinking.

[grizzled grove]

You expect others to say more meaningful or interesting things

[untold sapphire]

@dense belfry what's up

[modest rune]

Not a consistent one, no

Like mathematicians are pretty diverse in their viewpoints on most things

I think mathematical thinking gives you a useful tool for thinking about the world. I think a problematic number of mathematicians probably think it’s the only tool.

[dense belfry]

Just been reading more topos theory stuff for a while

Trying to parse stuff

[untold sapphire]

👍

Goldblatt?

[dense belfry]

Yeah and Bell

I think I need to do more concrete examples

Goldblatt has a lot of those

[fervent pebble]

ys'll i have been transformed

!!

i now realize

that $\sin^{-1}{x}$ is actually good notation

[fathom swallow]

valley

[fervent pebble]

!!

[neat frost]

Good morning!

[neat frost]

Yeah it is. The problem isn’t $\sin^{-1}$ it’s $\sin^2$

[arctic grove]

yeah true

[clever knot]

No love for arcsin?

[fervent pebble]

just write it as $\sin{x}^2$

[fathom swallow]

valley

[fervent pebble]

i love asin too

[fervent pebble]

wait

not that

$\sin(x)^2$

[fathom swallow]

valley

[fervent pebble]

thattt

[fervent pebble]

morningggg

well

technically night

for me

but good morning for you!

[neat frost]

Yesh eggzactly

[neat frost]

[prisma swallow]

Yes it totally screws you over. When you have had a taste of the greatness of mathematics, many aspects of society become infuriating and contradictory. One becomes very disappointed when society, compared to math, is a mess(mathematically speaking).

[light needle]

That’s dumb lol

But yes math does fundamentally change how you view the world imo if u do it enough, and it’s different for different ppl

[prisma swallow]

In math the definitions are well defined, this is not the case for society in general. Math is about "how" and "why" which provides a solid understanding. Society generally concerns itself with only "how", often there is no perfect explanation for why something is as it is. Math is simply more perfect.

[deep mango]

Well

We still have to pick the definitions in math out of a lot of possible definitions

Mayve you're right about learning math

But the math discovery process is a lot less concrete and the explanations feel like they come after a lot of confusing and nonlinear work

[charred mortar]

I feel like it’s more “perfect” (or structured might be a better word) because stuff we may study in math is simpler than reality

[deep mango]

That too

[prisma swallow]

true

[charred mortar]

Even elementary physical systems can carry an enormous amount of complexity

[spare crater]

aye can yall solve this rq

so som1 is dying right, every 20 deaths he gets 100k, he has 2100 deaths, how much cash has he made?

[deep mango]

Math gives us a load of models that help us to approximate real systems, but real systems are just infinitely deep and detailed (they have no reason to be conceivable by human knowledge) and the thing I like most about math is that it's a language where we can build increasingly better pure structures and arguments which model the imprecise (or arbitrarily detailed) structures we see IRL.

So while I kind of think you have a point about the appeal of math, I would push you to see the interaction of math with the thing you're saying is messy, instead of leaving the messy thing out to rot

I'm not talking about "applied math" (whatever that means)

Logic in human brains is messy as fuck too

[prisma swallow]

The thing I like about math is that it does not need to concern itself with the physical world. This is good because the physical world is a mess.

[deep mango]

Hmm

I feel like the cleanliness of math is almost something that makes it less interesting to me

And what ill say is

Idk what your level is

But in most modern math research fields

[prisma swallow]

Math does not require you to follow the messy rules of the physical world. In that sense, it is more free.

[torn willow]

Or maybe you do math because you have the mindset instead of math helped you get that mindset?

[deep mango]

Modern math is about wading through a bunch of messy rules too

[deep mango]

I dont know about that. At very least math unlocks that mindset. E.g. when I had done enough proof-based math my crisis management skills shifted and improved a ton.

[prisma swallow]

That might be so, but those rules are either part of an axiomatic system, or its perfectly motivated.

[prisma swallow]

Crisis management?

[deep mango]

Yeah literally haha

My job in uni used to involve responding to emergency night calls

[charred mortar]

Ngl that’s actually pretty cool

[deep mango]

One day I realized that I was using strategies I'd learned from studying math (and had never applied outside of solving math problems) to trying to piece together plans to help deal with scenarios

When you dont have time to like

Plan everything out

Its nice to have a set of instincts already there that you usually use during exams or something hahaha

But its more than just strategies for problem solving

[prisma swallow]

Hmm, I think I understand what you are getting at.

Though, I don't think I have made the same observation myself

[deep mango]

It's about the way that you visualize complex problems and attack them at all angles / break them up effectively and naturally, and the way you more clearly evaluate the accuracy of deductions / predictions in scenarios. To me, math is all about taking those "logical processes" and giving them a space to operate in a vacuum.

Thats not really what math is about to me. To me math is definitely more about the "offering more and more detailed models of reasoning" thing I talked about before

[prisma swallow]

Maybe my perception will change as I study more math

[deep mango]

But I guess what I mean is, my relationship with math is founded on it being a sort of playground for that kind of thought-maneuvering

[prisma swallow]

mhm

[deep mango]

I think you'll also find a lot of people for whom math is about the beauty of the structures they find in math themselves

[prisma swallow]

Like fractals?

[deep mango]

What you perceive as the cleanliness of math and the messiness of reality, I perceive as sterility of math and richness of reality. But I still like math as a way of helping me to get closer to the richness of reality (a way which fits my skillset better than experimentation or something)

[prisma swallow]

I see

[deep mango]

Ehh, maybe, and tbh as someone who really likes some math that produces fractals this might be my first example to someone younger. In fact it basically was when I was talking to someone in HS yesterday.

[prisma swallow]

But when something is messy its harder to understand, and you will probably rely more on memorization.

[deep mango]

But I mean more like broad interconnections between fields of math, unifying themes or motifs

[deep mango]

Well you certainly dont have to if you're not trying to pass a class

Like

Sure I could be a chemist and spend my time doing important shit, and that means I would have to remember a bunch of molecules and how they work

But I could also study why those molecules work that way

Its similar in math. Mathematicians dont understand everything that they use. They work together with others and combine expertise, they blackbox theorems from other papers which utilize foundations that they might have never learned, etc

Ofc they could spend a month going and learning from an intro graduate textbook on x and y, but

Mathematicians care more about thinking about what they care about

Or, a lot of mathematicians do I should say.

[prisma swallow]

Simply the fact that math does provide proofs is good enough for me if I had to blackbox something.

[deep mango]

So an example here is the deep relationship between mathematical probability theory / stochastic processes, and the theory of partial differential equations. There's a ton of different bridges tying them together, and this is cool because sometimes to solve a problem in one field, you have to cross back and forth between the bridges a number of times.

[prisma swallow]

That way, I know its not some obscure crap that might not even be correct.

[deep mango]

Another one would be the ubiquity of the fourier transform across various different domains of math, and trying to find general forms which capture all of them at once

[deep mango]

I understand wanting to know that! For me that sounds kinda fun. But I guess I'll never know since I also just stick with math.

[prisma swallow]

Anyways

I should stop procrastinating

Nice chat

[deep mango]

This was helpful to sort out my perspective a little more too!

[compact tartan]

the unreasonable ubiquity of fourier transforms

[wicked ore]

in the natural mathematics

[modest rune]

some of us find the middle ground of studying pure math that is also somehow incredibly messy

[deep mango]

Definitely

Thats also where I am

[modest rune]

i explained to a buddy about how often there are errors in the homotopy groups of spheres

and he was like

aghast

almost philosophically so

[deep mango]

Yeah. I wanted to make this point but i didnt make it super clearly

The perception that math is super objective and pure is at best culturally false and at worst objectively false

There's a ton besides just "what mathematical facts are solidly proven" that goes into doing math

[modest rune]

I also think that like, regardless of whether the results are "objective and pure" the process is certainly not as well

Like the process involves bursts of creativity, half-baked intuitions, and a lot of ridiculously wrong ideas

[deep mango]

Yeah maybe thats what I mean

I dont really separate the process of doing math from the content of it I guess

[modest rune]

I guess I kind of do? Only in the sense that I do think that a bunch of formal results follow from the axioms and that we are doing out best to figure out what those are

but I don't think math papers can be separate from the process

[deep mango]

Definitely

Agreed

I just dont care that much about the formal results following from axioms part

It's less fun than the process

And like, maybe I have the leeway to not care about that since the math im doing is a little closer to being physically applicable

Whether or not that's true in practice

[modest rune]

i mean i barely even care about like

mathematicians as a whole being right lol

like its good to catch mistakes and so on

but like, if some big foundational result ended up being false

i would not be emotionally invested at all

(fwiw, in case my future employers are on this discord, i care very much about me being right)

math feels sociologically like a dog chasing its own tail

if we ever proved all the theorems (caught the tail) it would be sad because then what would we do

or like, if you are building a puzzle, its like who cares about the puzzle the second you put the final piece in

luckily most of the time even when you finish puzzles in math it generates new puzzles

which is maybe why its so addicting

[deep mango]

Yeah. That makes sense to me

I guess I view it like

If people are coming to math looking for an escape from the imprecisions and complexities of the real world

They're probably only going to find a shoddy facsimile of what they're looking for at any level beyond undergraduate coursework, and they're gonna have a lot more fun if they allow themselves to be open to experimentation, open ended projects, and even some ambiguities (which do get resolved eventually, hopefully, but we don't usually feel stuff based on what is going to happen eventually)

[untold sapphire]

I wish definitions were better motivated

[velvet dagger]

To be fair I'm not sure what "allow themselves to be open to" means here, since I don't think this preference is a matter of will.

But yeah people who like math for the relative perfection are gonna be sad.

Maybe take the Daniel Litt strat of just never black boxing lol

[steep mountain]

is there a reason why some theorems are only found on the french wikipedia and not the english one

[untold sapphire]

This is hidden knowledge

It's not meant for non Francophone eyes

[steep mountain]

[neat lintel]

Hey im not even an amateur since i havent finished school yet but still can someone explain

Y

What kind of maths/approaches are used to tackle the collatz conjecture by most people?

Im just curious thats all

I swear

[sleek wing]

no one tackles the collatz conjecture

[neat lintel]

Didnt terence tao tackle it

[torn willow]

@empty stratus

[neat lintel]

Yeah ive read that no one really does it

And ive seen the videos ofc

But what techniques from what field yknow

And i reckon its oughta be some weird blend

Ig

[sleek wing]

CV is the only person I know above first year undergrad who cares for the collatz conjecture, talk to him if you want to know more

he's usually more than happy to explain

[neat lintel]

Ahh aw

[torn willow]

He did his PhD on the collatz conjecture

[neat lintel]

Are they active on this server?

He

[sleek wing]

fairly active yeah

[neat lintel]

Ah i’ll just wait then for him to see this and hopefully reply

Thanks for pinging him

[fresh oriole]

Here's Tao's 2019 paper on it https://arxiv.org/abs/1909.03562

He in some sense comes as close as possible to proving it without proving it

[neat lintel]

Thanks for this

[fresh oriole]

Sure. You may also like this much more informal slideshow he did

https://www.google.com/url?sa=t&source=web&rct=j&url=https://terrytao.files.wordpress.com/2020/02/collatz.pdf&ved=2ahUKEwjgubKq3YL4AhVlVTABHXrhC8sQFnoECAMQAQ&usg=AOvVaw00y-sv6RIZRzezdpnvPvKz

[neat lintel]

What did i read just now which is now deleted? Something with he said his aim is to...?

[torn willow]

mb thought it was unnecessary

My goal is to lift Collatz studies out of the realm of crankdom and establish it as a worthwhile field of study in its own right.

[neat lintel]

I’ll like

[neat lintel]

Personally that’s pretty ‘noble’. Why should studying any corner of maths be discouraged

[torn willow]

Well, Because Collatz is associated with cranks

[neat lintel]

Who knows what a surge in research might bring about

Yeah lol ive seen it around on yt i do admit

Itd be neat to meet this CV by the way and their bio and silly little pfp do help

[mellow moon]

Just to flex on non-French speakers

[neat lintel]

Hmmmmmm so from what ive read terence proved the conjecture for a whole lot of numbers but not all

[neat lintel]

Oh thats funny lol

[fierce eagle]

computers have been used to check a lot of numbers

[neat lintel]

Yeah

Reading the slideshow i see a lot of probability stuff

Ofc idk how its incorporated exactly in the proof but i assume theres not much use of say algebra?

Am i correct to think that?

[fresh oriole]

probably right

it's a lot of analysis and probabilistic methods

[empty stratus]

My PFP is not silly. It is a wizard with a dragon doing science.

That’s like... the best combination

[neat lintel]

Bet

Do u have a fav beethoven piece?

[empty stratus]

Oh god.

That's tough.

Playing the Große Fuge usually reduces me to tears.

As does the 32nd Piano Sonata, or the finale to the 30th Piano Sonata.

[grizzled grove]

I wish I had good ears to enjoy classical music

[empty stratus]

And Perlman's rendition of the finale of the Violin Concerto is perfection.

[empty stratus]

You don't need good ears, you just need good recordings. 😉

[neat lintel]

I have some things to check out then

[grizzled grove]

Na i think one requires to develop certain tastes

Like my tastes are different

[empty stratus]

Like, I love the Fuge so much that I borrowed its title for my epic fantasy series.

[neat lintel]

What r ur tastes ssk?

[empty stratus]

(The Great Fugue)

[neat lintel]

Thats epic

[grizzled grove]

It's chill oriental instrumental with electric funk etc

[empty stratus]

It will take over a decade before I'm even ready to start working on it.

[grizzled grove]

It's complicated

[fresh oriole]

Anyone can get into classical music

[neat lintel]

Triceratops or t-rex?

[fresh oriole]

or any genre for that matter

[empty stratus]

A difficult decision.

But my fantasy world has both.

[grizzled grove]

Na some genre are hard for some people to get into

[empty stratus]

Indeed, there's a flaming skeletal T-rex in chapter 1.

[neat lintel]

Cant miss it

[fresh oriole]

it only takes one positive experience to get the ball rolling

[grizzled grove]

You guys interested about dinosaurs?

[empty stratus]

YES

They're so much better than horses, for starters.

SO MUCH BETTER

[grizzled grove]

Lol what

You can ride a horse

[empty stratus]

My fantasy world has no horses. It has dinosaurs instead.

[grizzled grove]

Not dinosaur

[empty stratus]

Yes you can.

And velociraptors make for excellent little fluffy pets.

[neat lintel]

Are they pets in your story?

[empty stratus]

Yes.

[neat lintel]

Rad

[empty stratus]

Though they are rivaled by the terrestrial cephalopods.

[grizzled grove]

It's also hard for me to enjoy fantasy story or movie

[empty stratus]

Agreed.

[grizzled grove]

I only like realistic movies

[empty stratus]

The genre is insufferable.

[arctic grove]

[empty stratus]

But it's got the best aesthetics.

[grizzled grove]

For example harry Potter..

I never liked it

[empty stratus]

I approach the fantasy genre in my writing more like a science-fiction author than a traditional fantasy author.

[grizzled grove]

Hmm i see.

[empty stratus]

I'm also much more concerned about interpersonal stuff.

[grizzled grove]

In that context I like splice movie

[empty stratus]

ANYHOW...

@neat lintel I currently have anime to watch. I will be available for mathy talk in a bit.

[grizzled grove]

Only anime I ever watched was deathnote

It was cool till 3/4

[empty stratus]

Currently watching Spy x Family.

[neat lintel]

Have fun

[grizzled grove]

Cool. Have fun CV...

[empty stratus]

Anya is best Anya.

So cute.

[neat lintel]

I think i’ll watch it lol sounds funny

[empty stratus]

IT'S ADORABLE

ADORABLE

[neat lintel]

Oh yeah

[grizzled grove]

Lol how do you enjoy too much fantasy

Such creatures aren't real

Nothing is real

[empty stratus]

Realistic fiction is also fantasy.

[sick burrow]

I only enjoy stories with completely realistic plots

[empty stratus]

Still fantasy.

[sick burrow]

[empty stratus]

All fiction, by definition, is fantasy.

[grizzled grove]

Not at all...anime is like people flying and throwing fire from mouth

And realistic is like drama or biography

[empty stratus]

Not all fantasy is equally fantastic.

[neat frost]

Spy x Family is a BAD anime

[grizzled grove]

Like twelve years a slave

That's great movie

[sick burrow]

Ur a bad anime

[empty stratus]

That's an exception, it's based on a real event.

[sick burrow]

Ughui

[neat frost]

Do not assault my eyes with bad weeb imagery

[sick burrow]

[empty stratus]

The protagonist of my current novel is technically a weeb.

[grizzled grove]

Many good movies are inspired by true events and anime is only inspired by cartoon life

[empty stratus]

But he's in his 40s.

[arctic grove]

based

[empty stratus]

Even then, it's still fictional.

[neat frost]

Ryc

[deep mango]

[neat frost]

[empty stratus]

ANYA

[neat frost]

HAHAHAHA SHYSHU WEEB HAHAHAHA

[arctic grove]

fuck no

[deep mango]

[arctic grove]

its from a server im in

[neat frost]

Shyshu has gone this entire time denying he’s a weeb

[arctic grove]

im anything but a weeb

[deep mango]

Spyxfamily isn't weeb

[empty stratus]

Anya can be appreciated by all.

[grizzled grove]

Shawshank redemption is not fictional especially compared to anime

[deep mango]

I'm not a weeb

[neat frost]

And he literally just showed his true colors

[arctic grove]

based ryc

[empty stratus]

Yes it is.

[arctic grove]

shut up slurp

[neat frost]

HAHAHAH IN DENIAL

FIRST STAGE OF GRIEF

[deep mango]

spyxfamily isn't fucking weeb

[empty stratus]

Those characters never existed in reality.

[neat frost]

GET PSYCHOANALYZED RYC

[deep mango]

Btw I've declared myself not a weeb this whole time

[empty stratus]

They are no more or less real than a dragon who happens to be both a werewolf and a vampire.

[neat frost]

Makes sense since you’re such a fan of analysis

[grizzled grove]

But you should understand that those characters are much real and imaginable to occur in real life

[deep mango]

Why'd you sully cheating anya

[neat frost]

You ARE a weeb

[grizzled grove]

Anime doesn't even satisfy physics laws

[deep mango]

me when slurp posts

[empty stratus]

Imaginable? I find it much easier to imagine werewolf vampire dragons than bankers with criminal issues.

[neat frost]

HAHAHA WEEEEEEBBB

[deep mango]

I've never posted something weeby in my life

[neat frost]

YOURE LITERALLY PROVING MY POINT

[neat frost]

THIS! RYC! THIS!

[empty stratus]

Then where's the QED?

[deep mango]

That's a dog

Are you ok?

[neat frost]

Good point

[grizzled grove]

What...do you ever see newspaper. You see criminals more or vampire more?

[neat frost]

[deep mango]

You don't like dogs now?

[neat frost]

SO? ITS A WEEB POST

[empty stratus]

I live in Los Angeles, so, vampires come up more often than you'd think.

[neat frost]

Don’t even FUCKING a start that wew. I LOVE my dog and I won’t have you insinuate otherwise.

[grizzled grove]

Wth...there is no such thing as vampires or dragons

[neat frost]

He doesn’t know….

[deep mango]

Did you just call me "wew"?

[grizzled grove]

Criminal stories in movies are still very much possible in real life

[empty stratus]

There's also no such thing as the character Morgan Freeman played in the Shawshank Redemption movie.

[neat frost]

I DID!

HAHAHAHA

Wow

[deep mango]

Thank you

[neat frost]

I need to eat

[deep mango]

Yeah, you do

[neat frost]

Oml

[grizzled grove]

I think you're trolling me i guess

[neat frost]

I called you fucking wew

Wtf

[empty stratus]

A little.

[grizzled grove]

Gre

Grr

[empty stratus]

That being said, if you peel away the layers of realistic fiction, you actually do get a lot of fantasticality.

It's just in different dimensions.

Realistic fiction is very creative when it comes to fleshing out characters and their psyches.

For example.

[grizzled grove]

I may enjoy animation talent of movies like avatar...but not cartoons

[empty stratus]

Stereotypical fantasy generally channels its creativity in more superficial / aesthetic matters.

However, there's nothing stopping it from doing more realistic portraits of characters.

[grizzled grove]

Yeah but then you can write whatever you want and call it creative

[deep mango]

[empty stratus]

Also, one of my FAVORITE aspects of fantasy is its ability to juxtapose the ordinary and the extraordinary.

I absolutely adore stories about confrontations with the unknown (think Star Trek).

[grizzled grove]

The only imagination I like is complex numbers

[empty stratus]

One of my biggest complaints about the way fantasy is done most of the time is that it puts crazy fantastical stuff on the same footing as ordinary interpersonal reality.

And when you do that, I feel it ends up diminishing the sense of wonder that the fantastical elements have.

[grizzled grove]

Lol you want to fantasize emotional bonds now

Invent a new emotion

[empty stratus]

Actually, I had that as an idea for a magic system.

Alien / otherworldly concepts invade reality, and by seeking to understand them, you gain supernatural powers, but at the cost of being gradually transformed and re-made by the very concepts you study.

Also, my first magic system ended up helping me better understand Lie groups and Lie algebras.

[grizzled grove]

What

[empty stratus]

Yes.

[grizzled grove]

That's a weirdest way to learn math

[empty stratus]

I didn't intend it that way.

It just turned out that way.

[grizzled grove]

When I read about Ramanjuan stuff...this all may be true I think

[empty stratus]

That being said, I have borrowed a heck of a lot of functional analysis and harmonic analysis for filling out the background of my grand unified theory of magic.

[grizzled grove]

A very strong complex spiritual way to understand mathematics

[empty stratus]

I've even used the Fourier-analytic manifestation of the uncertainty principle as a plot point.

(Specifically, the fact that a function with compact support cannot have a Fourier transform which is also compactly supported)

@neat lintel Okay, I'm available.

[ancient flame]

@deep mango I swam in my pool again

:p

[neat lintel]

yo only in ur head. is 15.99÷3.99 less than or more than 4?

[ancient flame]

no clue

probably more

oh ye

def more

[neat lintel]

why

[ancient flame]

bc 3.99*4 would be .01*4 less than 4

[neat lintel]

huh?

[deep mango]

Did you swim lengthwise or widthwise

[ancient flame]

what

[deep mango]

Like

Did you swim the long way across

Or the short way across

[ancient flame]

I still don't understand

[deep mango]

Do you get in a pool and just sit there

[ancient flame]

idk how to explain lol

[deep mango]

what's there to understand

(gmod)

[ancient flame]

I swim and sometimes sit and stuff

idk

[arctic grove]

more

[deep mango]

Sigh

ok

[ancient flame]

it's not that big tbh

[arctic grove]

3.99 is 0.1 less than 4

[deep mango]

do you stick around in the deep end

[arctic grove]

so 4 times it would be 0.4 times less than 16

[ancient flame]

sometimes

[deep mango]

I see

Can you stand there

[ancient flame]

I like to sit on the steps too sometimes

[arctic grove]

so it will be 15.6

[deep mango]

Cool

[arctic grove]

which is less than 15.9

[ancient flame]

I am almost tall enough to stand there

[arctic grove]

so the quotient will be larger

[deep mango]

Hahahahaha

Oh ho ho

[ancient flame]

you are getting a lot of information

[deep mango]

I've gotten so much information from this conversation

LOL

[ancient flame]

LMAO

[deep mango]

Wow. I outfoxed you big time

Now I need to draw up my blueprints...

[ancient flame]

nah I was aware

[arctic grove]

Lmfao

[deep mango]

[hushed turret]

the ryconnaissance continues

[arctic grove]

i used to be able to swim nicely

[deep mango]

like hell you were

[arctic grove]

now i havent swam in 3 years properly

fuck u covid

[ancient flame]

I literally said this

[deep mango]

Shyshu doesn't like the public pool

[deep mango]

Only after I laughed maniacally

[ancient flame]

😐

public pools suck

[deep mango]

Ugh

[ancient flame]

I don't need to drink pee

[deep mango]

Not another one

UGH

Ok come on gmod

Who cares about that

We're not 13 anymore

[ancient flame]

but I am

[deep mango]

eww cooties! eww!!!

[arctic grove]

there is no public pool

[deep mango]

ew its a spider!!!

[ancient flame]

ryc has cooties!!!!

[arctic grove]

trust me if there was i would be there all the fucking time

[full folio]

[arctic grove]

i used to swim in a stadium

[ancient flame]

...

[arctic grove]

i got to swim for less than 2 months

[full folio]

what

[deep mango]

Wow

That's cool

[arctic grove]

coz there is no swimming pool here

[deep mango]

I don't have cooties

[ancient flame]

goldfish

[full folio]

yum

[ancient flame]

I stabbed a goldfish to death once

[empty stratus]

Sounds cruel.

[full folio]

bruh

[ancient flame]

he did not smile back

[deep mango]

Goldfish are so gross

(the snacks)

[full folio]

Why

[ancient flame]

im kidding I did not stab a goldfish

[deep mango]

I hate cheese crackers

[ancient flame]

ur weird

[deep mango]

Cheese is so good on its own

cheese on top of crackers is delicious

Why would I want to ruin both by melding them together

[ancient flame]

I used to fish with my friends but I don't much anymore because
a) I lost them lol
b) I once almost killed a fish bc I couldn't get the hook out for so long

[full folio]

Idk but homemade ones are cool

[deep mango]

Hmm

[ancient flame]

damn I never thought of doing that

[proud olive]

Crackers love cheese

[full folio]

4head

[cyan goblet]

chat i hate leg day

[proud olive]

Why can't you get the advanced role gmod?

It's been ages since my last leg day

How much ya squat @cyan goblet ?

[cyan goblet]

i do dumbbell stuff at home

[proud olive]

Oh

[ancient flame]

idk it isn't letting me

even a mod tried to give it to me but it immediately removed it

[proud olive]

Tf

Did you do something wrong?

[ancient flame]

no

it's just broken lol

[proud olive]

Could try to rejoin the server

U know, turn the server on n off

[ancient flame]

true

still doesn't work

now I lost all my roles

fuck

at least I screenshotted it :p

@deep mango

hi!!

ily

do u mind uh

giving me these again

thanks :D

[deep mango]

Wat

[ancient flame]

LOL

[deep mango]

Did you leave and rejoin

[ancient flame]

yes

[deep mango]

Oh

[ancient flame]

it didn't fix the advanced role thing

[deep mango]

Ok the bot should give back emeritus

Uhh

Do you have advanced now

[ancient flame]

!talks

OH SHIT

I DO

TYYYY

[deep mango]

Ok hang on

[ancient flame]

epic

[neat lintel]

hi

[empty stratus]

Hello.

[neat lintel]

Hey so you got time to say smart stuff about the thing i talked about now?

[empty stratus]

So, you had Collatz questions?

[neat lintel]

Yes i did

[empty stratus]

Ask away.

[neat lintel]

So what approach people who are serious about it take? Like, what kind of maths, if you had to describe it

[empty stratus]

That's actually part of the problem.

[proud olive]

U have advanced role @ancient flame !

[empty stratus]

There's no unified set of techniques.

[neat lintel]

So people are just kinda looking around to see what works

[proud olive]

Congrats

[empty stratus]

Most of the biggest advances (though that's not saying much) are probabilistic in nature.

With Tao's 2019 paper being the best example of this.

[neat lintel]

Damn probabilistic

[empty stratus]

The first big breakthrough was in the early 1970s, due to Riho Terras.

Do you know what the density of a set of integers is?

[neat lintel]

Do you think further advancement will also look like that?

I dont know that

[empty stratus]

NOPE

[empty stratus]

Then let me explain.

Let V be any set of positive integers.

For any integer N, let V(N) denote the set of all elements of V which are ≤ N.

For example, if V is the set of all perfect squares ({1,4,9,16,25,...}), V(10) = {1,4,9}.

[neat lintel]

Hm

[empty stratus]

For any N, the quotient V(N) / N is then the proportion of the numbers {1,2,...,N} which belong to V.

The density of V, denoted d(V), is the limit of V(N) / N as N —> ∞.

Importantly, not all Vs have a well-defined density. There exist Vs for which d(V) does not exist.

[neat lintel]

So the set of integers does have a density?

[empty stratus]

In that case, we define the upper and lower densities, respectively, as the largest and smallest quantities that V(N) / N gets arbitrarily close to infinitely often.

As a rule, if V is finite, then d(V) = 0.

Meanwhile, the set of all positive integers (or all but at most finitely many positive integers) has density 1.

[neat lintel]

I see

[empty stratus]

Let c be a positive integer.

Then, the set of all positive integer multiples of c ({c,2c,3c,4c,..}) has density 1/c

So, the set of all even numbers has density 1/2.

The set of all numbers which are 1 more than a multiple of 3 has density 1/3

And so on.

This makes sense. When you divide an integer by c, the result is either that c divides the integer evenly, or there is a remainder. The remainder is going to be a number in the set {1,2,3,...,c-1}.

So, when you divide a number by c, there are c possible values for the remainder: 0,1,2,...,c-1.

[neat lintel]

Hm

[empty stratus]

In other words, if you choose a remainder r in {0,1,2,...,c-1}, there is a 1/c chance that a randomly selected integer will have remainder r when divided by c.

In this sense, d(V) (which, if it exists, is always a number in the interval [0,1]) can be thought of telling you the likelihood of selecting an element of V if you choose a random positive integer.

[neat lintel]

Yes

[empty stratus]

However, this isn't a perfect analogy, partly because not all sets V have a well-defined density.

Anyhow, in the early 1970s, Terras used probabilistic methods to prove the following result:

Terras' Theorem: Let S be the set of all positive integers which Collatz does not eventually send to ∞. Then, d(S) = 1.

In other words, if you select a positive integer at random, you are basically guaranteed that it will not be sent to infinity under iteration by 3x+1.

[neat lintel]

Damn

[empty stratus]

However, this does NOT mean that there are NO integers that get iterated to infinity.

It just means that, if they exist, they must be extremely rare.

[neat lintel]

Hmmm

[empty stratus]

This is the achilles' heel of all probabilistic methods.

[ancient flame]

yeah they fixed it

[empty stratus]

Probability theory can be used to make very powerful conclusions about systems. However, these conclusions cannot cover all possibilities unless the system being studied only has finitely many possible states.

And, even then, there are issues.

[neat lintel]

Yeah thats the thing

Damn thanks

Also

Do you think

[empty stratus]

However, probability can be very useful for making predictions.

For example...

Consider the 5x+1 map, which is the same as collatz, except with a 5 instead of a 3.

[chilly coral]

How rare is extremely rare? Does that mean finitely many? Do they get exponentially more spread out?

[empty stratus]

If one exists, infinitely many exist.

Let x be iterated to infinity.

Then, 2^n x is iterated to x.

[neat lintel]

There are so many hard problems about numbers that have to do with elliptic curves, like fermats theorem or the abc conjecture. Do you think collatz is related to those fields of study?

[empty stratus]

Moreover, every number you get by applying Collatz to x will also get sent to infinity.

[empty stratus]

I'll be getting to that in just a bit.

Let me finish talking about the probabilistic methods.

[empty stratus]

When you apply Terras' methods to 5x+1, you get an amazing result:

Terras' Theorem (5x+1): the set S of all positive integers which do not get sent to infinity by 5x+1 has density 0.

[neat lintel]

The 0 is now a 1?

[empty stratus]

Yep.

[neat lintel]

I mean other way around

[empty stratus]

In other words, if you select a random positive integer x, x will almost always not go to infinity under 3x+1, but almost always WILL go to infinity under 5x+1.

And here's the most embarrassing part: we have yet to prove that a single integer gets sent to infinity by 5x+1.

It is believed that 7 gets sent to infinity under 5x+1, but no proof is currently known.

[neat lintel]

Wow so its the same problem but in reverse

Sorta

[empty stratus]

Even though Terras' result does not give specific details, it does tell us what kinds of behaviors to look for.

And that is very, very useful, because it means we aren't going in blind when we start studying these maps.

As for MY research...

[neat lintel]

Hmmmm :)

[empty stratus]

My goal was less about trying to solve these problems and more about trying to find the right context in which to study them.

As others quoted from me earlier this morning, I want to lift this area of mathematics out of the ghetto of crankery.

[neat lintel]

Hmm