serious-discussion

[quick bronze]

Gun for high grades

[crystal stone]

Some do, some don't

[quick bronze]

But the ones who do would irritate me as a prof

[crystal stone]

So then it's on you as a professor to create incentives to learn over to nitpick for grades

[quick bronze]

Also the amount of people who cheat

Especially during covid is high

And rampant

[crystal stone]

They eventually get caught

It's hard to cheat your way through a 9-5 job

[quick bronze]

Like in HS that was everyone

[crystal stone]

If you don't actually know what you're doing

[quick bronze]

True

[crystal stone]

People cheating always drives me up a wall

[quick bronze]

Y

Lol

[crystal stone]

If you're graded on a curve, then you are hurting someone else's grade for your selfish unearned gain

Although, most of that fault lies with the incentive brought on by the prof./education system

The other issue is that there are standards to be upheld for a reason. Nurses, Doctors, Engineers, etc. need to have an education, and need to have an honest education

If they cheat their way through the system, then that can literally kill someone

[quick bronze]

Yeah

[pure sun]

grades are the worst thing to happen to education

[crystal stone]

Have you found a way to mitigate their impact in the classroom?

[pure sun]

no

haha

not on a large scale

[crystal stone]

It's very difficult to do so, even when the grades are essentially meaningless

I taught at Russian School of Math for 5th, 6th, and 7th grade - which is an after school program for ultra math nerd kids

Our internal grading and recommendation system meant almost nothing in reality

But the kids took it super seriously if they couldn't solve a quiz problem or something

I kept trying to tell them that this doesn't matter, but they kept getting stressed over it

[gentle bay]

Woah!

Russia is soo cool at that stuff.

[crystal stone]

Russian school of math is an american education company rooted in soviet union curriculum

[gentle bay]

So what was being taught for say 7th grade?

[crystal stone]

Just algebra & word problems

[gentle bay]

What level Algebra?

[crystal stone]

7th grade

[vivid halo]

some people in my department have been experimenting with doing mastery based grading the last few semesters

it's been interesting

[gentle bay]

Wait so how is it ultra math nerd?

[pure sun]

im doing that right now

[vivid halo]

how's that going?

[pure sun]

in my precalculus classes

I think it works really well

[crystal stone]

They are paying to attend extra math classes and enjoying it

[vivid halo]

yea that's the impression I get too

[crystal stone]

Kids that don't enjoy it leave the program quickly

[vivid halo]

I kinda want to work it into my classes

[pure sun]

idk how to scale it up to like several hundred studenst though

[gentle bay]

I have heard that happened to the Chinese Math Olympiad team or something? Did it really?

[pure sun]

which is what our calc classes are

[vivid halo]

oh god

yea that sounds not really possible lol

[crystal stone]

How did you implement the mastery based stuff?

Did you give an oral exam? More open ended questions?

[pure sun]

no i think oral exams are great though

thought about doing that as well but havent tried it yet, but mi ght do it next semester in my intro to proofs class

for me it was just weekly quizzes with one question per learning objective. if you get it right you get a checkmark, if you get it wrong, try again next week. no partial credit

students pick and choose which questions to attempt each week based on what skills they've been practicing

your goal is two checkmarks per objective

(i had some other components to the grade as well, like "application problems" which were multi-part applied problems that combined lots of learning objectives, but the weekly quizzes were the biggest component)

[gentle bay]

Buncho Bananas you are a professor?!!? .

[pure sun]

maybe

[gentle bay]

That's super cool .

[fair mural]

more people here than you’d expect seem to be professors

[gentle bay]

?

[pure sun]

I mean

it depends on what you mean by "professor"

[vivid halo]

Buncho has tenure

[pure sun]

do you mean "someone who teaches collegiate math classes"? or "someone whose job title is Professor"?

hahahaha

no i do not

[pure sun]

if you mean the former, then yes, basically everyone who is a grad student in their 3rd year or above falls in this category. if you mean the latter, then there's maybe like 1 or 2. there's also a middle category of "faculty at a university" in which case there are a couple more

[gentle bay]

1st.

[gentle bay]

I see.

[pure sun]

yeah

[gentle bay]

That's cool!

[pure sun]

so there aren't that many people here of that level

[gentle bay]

Oo.

[pure sun]

but youre right that there are a handful

of us

[gentle bay]

Which books does the Uni use for proofs?

[pure sun]

if you're like 15 years old and assume that discord is only used by high schoolers then it might be surprising lol

but there are twenty-somethings on here as well :P

you mean like intro to proofs?

depends on the person teaching the class

[gentle bay]

I see.

I will soon start proofs so I was asking for proofs books yesterday here lol.

So asked for your recommendations lol 😅.

[pure sun]

i dont have any, sorry :/

[gentle bay]

Okie.

Hey Nami.

[leaden torrent]

the book my uni uses for proofs is a sentence that says "it's recommended that math spec students learn proofs in the summer before first year"

[gentle bay]

Wow.

[fair mural]

[hearty gate]

Apparently the help channels weren't the right place for my question, so hopefully this one is?
It's about a user review system and I'm looking for a better solution than the x/10 rating system.
Like having relative comparison and let the maths/system do the ranking and possibly assigning x/10 values.

Here is the link to the detailed problem, along with the few follow up posts.
#help-17 message
I can move it also to one of the specific channels - if I know which one ...

[vast surge]

This really isn't a math problem. I feel like "out-of-10" questions on surveys come up a lot in social sciences; maybe a social sciences server would be helpful to you?

[deep mango]

I got an extremely good education at berkeley but it was in large part due to getting myself out of my comfort zone to find resources, support systems, and basically never taking anyone else's advice to take it easy (always doing the most amount of work humanly possible for me at that time). It got kinda nightmarishly awful in my 3rd year for a number of compounding reasons. I still loved my experience there and it definitely is the reason I feel very prepared for my grad program now, but I think if I were less willing to take on a lot of anxiety and just deal with it, then I would have come away from it with a more sour perspective.
Berkeley is a school where you can do pretty much anything you want, but you might have to fight tooth and nail to make it happen.
Also as an out of state student it was definitely not cheap for me. I managed to reduce the price pretty significantly by working part time and by endlessly appealing for fin aid, but normally that stuff is very hard to make work.

I feel like I'm making it sound like a nightmare, it was actually a lot of fun, the math community is incredibly supportive and lovely, there are so many groups outside math that are also wonderful, the staff in the math dept were lovely, almost every professor I had was super friendly and loved teaching, etc. It's just easy to get caught up in a whirlwind of taking on too much work at Cal cause nothing is gonna stop you.

[neat lintel]

also Berkeley math tourney

[deep mango]

Yeah I was involved with that a little, it was fun. Great group of people

[honest veldt]

anyone got an example of a homomorphism that's not bijective?

[surreal sapphire]

the zero morphism

[deep mango]

sure. the map from Z to Z which is given by phi(n) = 3n is a homomorphism.

[toxic schooner]

any non isomorphic homomorphism?

[honest veldt]

That's what I was asking for

Thanks folks

[deep mango]

it's injective, but it's not surjective. on the other hand, the map from Z to Z/3Z given by phi(n) = n mod 3 is a homomorphism. it's surjective, but it's not injective.

[honest veldt]

I'll check both of these by hand, but why the ?

[deep mango]

as a 3rd example, the map from Z to Z/6Z given by phi(n) = 2n mod 6 is not only nonsurjective, it's also noninjective.

so both can happen.

slim is mad that his answer to this question was just a rephrasing of the question and that i actually gave an answer

[toxic schooner]

i just did it coz i have an impulsive nature to react

[toxic schooner]

lmao

[honest veldt]

Just so I'm sure I've got this right, these maps preserve the structure of these groups under both addition and multiplication?

[deep mango]

oh

umm

i was giving group homomorphisms

if you want them to be ring homomorphisms

uhhh

[honest veldt]

No, no, I do want groups

I meant like, as separate examples

[deep mango]

ok then it's just preserving addition

[surreal sapphire]

what if you have a bijective homomorphism that is not an isomorphism

[toxic schooner]

but group mapping dont preserve both, no?

[deep mango]

Z is not a group under multiplication

[toxic schooner]

dont tell me that is possible

[honest veldt]

Oh right right, I see

Thank you

[honest veldt]

I was thinking of Z once under multiplication, once under addition

[toxic schooner]

Z cant be under multiplication

4/6

[deep mango]

in terms of ring homomorphisms (respecting both multiplication and addition), only this is does both

[toxic schooner]

not in Z

Z? field?

[honest veldt]

Yeah I just realized lol, my bad

[deep mango]

that's because 3*n * 3*m does not equal 3*(mn)

[surreal sapphire]

well, continuous and bijective does not imply homeomorphism

[deep mango]

trying to decipher if you sent this seriously or if you're baiting me into responding

[toxic schooner]

are u making stuff up

[toxic schooner]

[deep mango]

ok then i will react

😌

[errant furnace]

[toxic schooner]

i should do a lot of algebra soon

[surreal sapphire]

but in groups and rings you have a bijective homomorphism, the inverse map is automatically a homomorphism as well
in more general settings you need homomorphisms in both directions such that their composition is the respective identity

[toxic schooner]

I see

what would be such a general setting?

[surreal sapphire]

topological spaces as i mentioned

[toxic schooner]

ah

[surreal sapphire]

in even more general settings (categories in which the underlying objects arent sets), it gets worse

[toxic schooner]

categories coming to haunt me again

[errant furnace]

Grp

[deep mango]

[flat harbor]

just pretend they dont exist youll be fine

[deep mango]

Oh shit i got an email from courant tech support

[surreal sapphire]

this

[deep mango]

Lets see if they'll change my website url to something not idiotic

[toxic schooner]

Thats what i try to do

[surreal sapphire]

yes i do the same

its fine

[errant furnace]

but you brought them up

[toxic schooner]

i doubt if i can do that for long tho

[errant furnace]

[deep mango]

Nope! Lol

[surreal sapphire]

i just wanted to sound smart

[deep mango]

Alright fuck me I guess

[toxic schooner]

u are smart tho

[surreal sapphire]

debatable

[toxic schooner]

[tiny marten]

Who needs categories anyways

[deep mango]

I'll just keep using my berkeley website for now until someone decides to have respect for me

[toxic schooner]

homology?

[surreal sapphire]

but like the topological spaces example is worth to keep in mind

[toxic schooner]

i see, yeah i will try to keep it in mind

[atomic hornet]

Whats your website

[deep mango]

Not gonna self dox that easy 😇

[fair mural]

ryc i won’t dox you

[bright hill]

You guys have like, secret identities on discord?

[mortal igloo]

I'm secret

[leaden torrent]

you say that as if its abnormal?

has social media normalized posting your personal info publicly nowadays

when i was a kid, we were always told to not share anything personal on the internet

even saying my university here gives me weird vibes even if i know its fine

[bright hill]

I was told that too

But I was never told the reasoning behind it

So ig I just shrugged it off as superstition

[deep mango]

Well if everyone promises not to dox me, I guess I could just tell everyone.

Isn't the recording of my talk enough??

[toxic schooner]

i have doxxed myself quite a lot

i shared my face here, so thats already a lot

[pale orchid]

shyshu of the golden doxx

[ancient flame]

gdoxx

[toxic schooner]

Lol yeah

[honest veldt]

Is there a way to check for myself if this is a homomorphism? Or do I take this for granted

Oh wait I cropped it out

[pale orchid]

D_2n

you are tempting me, strad

[honest veldt]

Eddy don't do it

I know it's unbearably tempting but you are strong

Thank the lord this book uses the D_2n notation and not the other common one

[toxic schooner]

what is the common one?

[honest veldt]

It's exactly what eddy cannot resist

D_n

[toxic schooner]

Ooof

lol

[honest veldt]

Say for example I have an element rr (r^2), how do I know what this is mapped to in D_2k?

[brave hollow]

just check the conditions for all elements of the homormorphism

[brave hollow]

group homomorphism condition so phi(rr)=phi(r)phi(r)=r1r1

[honest veldt]

Ohhh right right, I get it

Merci beaucoup

[steep mountain]

i feel like it doesn't really matter
like i like my privacy and i would not share more than needed but if i tell someone online i'm in uni x or my real name is x
its like...ok? and? how is that different from introducing yourself in public
what am i missing

[icy forge]

Your privacy is gone if you're superstar X, but if you're not, then your privacy has non-zero (and presumably non-negative) worth

Well, not all aspects of privacy are gone for superstars, but having a Wikipedia page detailing aspects of private life (even in line with WP:BLP - that policy also only protects against contentious details - and ~2yrs after death) will make privacy disappear, just about forever

[ancient flame]

im just skeptical about everything

like I share as little as possible

y'know

bc I feel like if people have my info then they will come to my house and murder me irl

which I know won't happen

but if I keep that in the back of my mind then I won't post anything out of stupidity

[deep mango]

yeah it's too late gmod i already doxed you to the whole server

[fair mural]

someone’s already coming to your house to murder you

[rotund steppe]

😳

[icy forge]

Don't grief my MC house

It's built with nether brick

[mint patio]

This seems really useful

Like a really nice way to describe a group

[bronze pelican]

How do you prove this

[honest veldt]

The book proves stuff about presentations at some point, I don't remember what exactly

[bronze pelican]

Oh this is just a definition

I was thinking this was saying any group that is finitely generated has its relations also being finitely generated

[honest veldt]

Wait, how can you generate relations?

[static loom]

I think that's a difficult problem in general

I forget the technical phrasing for "how bad" it is lol

[honest veldt]

Oh, I thought it mightve been something I missed in the book, definitely not then

[static loom]

I guess you can get a presentation by starting with the cayley table and mucking around there and boiling down stuff idk

I think this has something to do with the definition of solvable groups, I'm sure someone knows more than me here

[bronze pelican]

So if you have a set of generators of a group, that's essentially specifying a group homomorphism from the free group on that many generators to your group, sending the i'th generator of the free group to the i'th generator of your group. The relations will be the subgroup of this free group which is the kernel of this homomorphism. What i meam by saying the relations are finitely generated is that this kernel is finitely generated .

[honest veldt]

I'll save this and reread it later because I still have a while ahead of me before I even get to free groups lol

[modern drift]

Hello guys, I need to do a research for the noise reduction in acoustics, and need to find a probabilistic method for it. Any recommendation?
I firstly thought of Kalman filtering, but it is way too hard for my actual level

[bronze pelican]

Free group is just a group without any relations. The free group on n generators is the group whose elements are finite strings of letters from an alphabet of size n (you also need the inverses of each letter). The group operation is concatenation.

[tiny marten]

free groups and words are cool

[agile sundial]

Nothing in life is free 😡

[frigid lark]

🤔What about the cosplayers who walk in fairs with a sign that says free hugs.

[fair mural]

https://tenor.com/view/hug-anime-hug-anime-cute-uwu-gif-17264448

of course that’s free

[frigid lark]

No new avatar 😭

[fair mural]

nope i can’t find any

[rapid folio]

Hey all

[alpine kindle]

hi corra

[rapid folio]

Hope you're doing good, thought I'd hang out while I prepare some lessons for when teaching starts up again I find lurking and doing some chatting here gets me motivated

[alpine kindle]

oh

nice

[alpine kindle]

@rapid folio what are your lessons on

[tender musk]

Is Year 7 and Extension 2 the equivalent to a senior in high school

[rapid folio]

Right now I'm taking it a bit easy, I'm planning for Year 7, which is 13 y/o's coming into high school from primary. So after doing some number line stuff, it goes into place value and then comparing numbers with >, < and =

[alpine kindle]

year 7 is 6th grade

[rapid folio]

Year 7 is the youngest year of high school, Extension 2 is a course you can take in Year 12, which is the final year of high school

[alpine kindle]

huh?

what country

[rapid folio]

Australia

[tender musk]

So you teach 6th grade, 9th grade, and 12th grade?

[alpine kindle]

ohhh

nvm

year 7 uk is 6th grade

how is being a maths teacher

[rapid folio]

Primary school: Kindergarten, then Year 1 to Year 6. Students start kindergarten around age 5 and finish Year 6 at age 12.

High School: Year 7 to Year 12. Students start Year 7 are 12/13 years old and finish Year 12 when they're around 18

[alpine kindle]

interesting

[rapid folio]

I really enjoy it, it's got its tough times. But I know the content, so what really interests me is answering the "How do I teach this to you?", which changes year to year depending on what students you get.

I find it fascinating that with my job I can introduce someone to something that's foreign to them, that they've never encountered and get them to a point where they understand and it's familiar to them. It's a good process for them to appreciate.

[alpine kindle]

in the uk we have:

Primary school:
Key Stage 1: Foundation Stage/Year 0-2 (starts at 4, unless your birthday is very early september)
KS2: Year 3-6 (starts at 7 with same caveat)

Secondary School:
KS3: Year 7-9 (starts at 11 with same caveat)
KS4: Year 10-11 (starts at 14 with same caveat) (at the end you do most of your GCSE exams, although some schools do some earlier)
KS5: Year 12-13 (starts at 16 with same caveat) (you don't necessarily need to be in school at this point, just some form of full time education. most people do A Level exams at the end, but others may do BTECs or otherwise)

[alpine kindle]

I'm thinking of being a professor eventually, so it sounds interesting

[rapid folio]

I sometimes peak at your A level tests and other senior ones, I'm always keen on seeing how they compare to what we have. You find the questioning technique is slightly different, it's a nice change from what we do here and I sometimes use some of the questions in my own topic tests. The Irish was another one we looked at in the staff room, they set theirs out slightly different again.

[alpine kindle]

the further maths a level papers are interesting

they give you a lottt of space

I'm currently in year 11

[rapid folio]

I got asked at uni if I wanted to do a PhD and lecture, with the idea of going on to do research, but I have a problem with the lecturing part, because it's not quite teaching. By definition, it's lecturing.

And I can always do any research or learning I want in my own time because my university give alumni access to the library and academic paper

[rapid folio]

Good to hear! If you're in this discord, that's reassuring as a teacher that you're keen 😅 The further maths are good papers from a teaching and testing pov. And yeah, the space is one of those small design things you pick up on. I find questioning techniques and exam design really interesting

[alpine kindle]

I'm very excited for university in a few years

because then I'll be around people who actually know what I'm talking about

[sly thistle]

Sup nerds

Nvm looks like this convo happened a while ago

😢

[rapid folio]

Great to hear! I hope you get some good lecturers when you go, it can make or break a topic I've found. Some calc course I took were a drag because of the person lecturing, other were fantastic! I did medical mathematics and financial calc with the same person and he was by far my favourite!

[rapid folio]

sup

I just didn't respond because I was working 😅

[sly thistle]

Hello

[rapid folio]

How's your part of the world going at the moment?

[sly thistle]

Hmm

In what respect?

I’m Australian so

[alpine kindle]

I'm hoping to go to cambridge

[plush venture]

wait, you're not in college? so are you like a sophomore or something?

[rapid folio]

Ayyye! Same!

[sly thistle]

What does sophomore mean

[sly thistle]

Nice

[plush venture]

second year studies

in USA

[alpine kindle]

year 11 uk
so yes

[plush venture]

Oh nice, are you hoping to study math>

[alpine kindle]

yes

[rapid folio]

I'm hoping you go to Cambridge too! That's great

[sly thistle]

@rapid folio I’m still pogging about the fact I got a 50 in methods last year

[rapid folio]

What part of the country you in?

[sly thistle]

Vic

[rapid folio]

Thought so, from the word "methods" 😅 I'm NSW

[sly thistle]

Idk why I assumed you were Vic too

[alpine kindle]

they go through content very quickly
which i think I'll enjoy

[plush venture]

for now just enjoy your days in school.

[rapid folio]

See, I'm a slower mathematician, university was a lot of walks and sessions in coffee stores because I had to take time to digest and understand stuff

[rapid folio]

Also, this

[alpine kindle]

i am

[plush venture]

because as soon you step foot on campus, time will go so quickly that you will forget everything in the world

[alpine kindle]

huh?

wdym

[plush venture]

like time flies

[alpine kindle]

no i know what you mean

sorry

just like

[sly thistle]

so does fruit

Sorry that’s a bad pun

[alpine kindle]

I've never heard that before

about university

[deep mango]

I can kind of agree with this

[plush venture]

Yeah just a heads up. I know it sucks. I wished I knew earlier and took high school for granted. Instead of wanting to leave quickly.

[deep mango]

Some of it moved in slow motion at the time but in hindsight it was all lightning fast lol

[sly thistle]

I mean that’s how it is isn’t it?

Everything in hindsight that is

[deep mango]

Idk

[plush venture]

It is but then you wished you had done more things than just study.

[deep mango]

When I think about high school it feels like it dragged on forever for me

[alpine kindle]

you only remember the novel experiences

[deep mango]

I do not really wish that lol

But

I do wish that I had done certain things differently

I mostly wish that id like

Not decided to do the job I decided to do

Cause I think that the networks and social ties I got out of it are all totally worthless now

That might not end up being true

Idk

[alpine kindle]

what job

[deep mango]

But it would have been nice to focus a little more on the math circles

Resident Advisor in the dorms

It was a lot of work, sometimes fun but I had one particularly awful year with a lot of really serious situations I had to respond to

I guess I gained a lot of skills from it but i'm also a very anxious person and it definitely worsened that

In a way which I think is a little irreparable

But probably not totally

[frozen merlin]

I have some friends at cambridge so if you have any questions I can forward them

but it might be a bit early to have uni-related questions seeing as you're in y11

[plush venture]

In college, I would avoid taking an interesting class because it was with a bad professor. I regret my time as a undergrad. I wished I took those interesting classes. Now I tell my friends and peers to take those interesting classes.

[autumn aspen]

Idk, hard to learn from a bad professor. When you have an unbelievably inspiring teacher it is hard to go back.

[plush venture]

Is different for everyone. But i understand your point

[spare kelp]

hiii

[gentle bay]

Nice PFP.

[bright hill]

It just means you treasure your memories there

I wish I would feel the same

Although, I constantly feel like an outlier so I honestly doubt it

[icy forge]

Good advice but you have to be aware it's not always easy. Life in school can be very difficult for people

[autumn aspen]

I totally agree 🙂 it is different for everyone. I've been going to school for a while and have learned the teacher makes the experience. Many classes I've taken I could of just read the book and learned what was needed. Undergrad is about jumping through hoops to get you in the door of the more interesting stuff in the real world or grad classes.

[frigid lark]

My linear algebra prof just copied the script onto the board literally. 🤔My crpto prof barely looks at the script and explains the most complex constructions and the reason of every step in a first intuitiv and then formal approach.

[devout nacelle]

Almost all of my profs just read out the notes on the screen, barely even writing

[surreal sapphire]

imagine having notes and not just winging every lecture

[flat harbor]

why write when you can experience

[devout nacelle]

Imagine not simply skipping every lecture

[neat lintel]

hello guys how can I apply for a role

role that does nothing

[fair mural]

#changelog

scroll up in there

you’ll see a message you can react to to get it

[surreal sapphire]

you are only allowed to do this in your last year

also i cannot skip lectures that i have to teach 😭

[sick burrow]

Not with that attitude at least

[alpine kindle]

just get your stunt double to do it

[toxic schooner]

read #❓how-to-get-help

[steep mountain]

what

[bright hill]

U ok bro?

[devout nacelle]

Just git gud and you can

[alpine kindle]

just be better smh

[dire mulch]

Let me just find someone who is 195 cm tall for my stunt double.

[fair mural]

,w 195cm in inches

[fathom swallow]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by quantum#6320

[fair mural]

wow

that is tall

are you actually that tall

[vagrant kestrel]

6' a billion''

[dire mulch]

Yep.

[fair mural]

my gosh

have you ever hit your head on anything

[dire mulch]

My basement.

[fair mural]

rip

stereotypical tall person stuff

[honest veldt]

Is it fine to visualize basically everything in topology in R^2 in my head? I feel like that helps it stick, but at the same time like I should stop doing that and get used to it as an abstract, general subject. Which one would help more in the long run?

[frozen merlin]

it worked fine for my metric spaces course, but it might be unhelpful as you study topology more

i'm interested to hear answers too

[pure sun]

I think maybe the eventual goal is to gain intuition for when you can rely on R^2 and when you cant

Sometimes drawing a picture in R^2 or R^3 is good enough and sometimes it’s not

[last oxide]

to me, it happens to be enough more often than not

[pure sun]

And being able to look at a problem and recognize “if i just think about R^2 I shouldnt be missing anything important” or “i cant rely on R^2 here because something more subtle is happening”

[last oxide]

with minor adjustments ig

[pure sun]

Is important

[last oxide]

well behaved metric spaces tend to work pretty ok, especially separable ones
res is right. general topology can get kinda fucked, like projective spaces, the sorgenfrey line/square, zariski thingies and so on

[honest veldt]

Alright, understood, thanks folks. I guess I'll stick with it until it's made clear at some point that R^2 just won't cut it

From you what you said, Fractal, it'll probably be pretty clear when that time comes

And got it Bananas, I'll try and start practicing that recognition of whether R^2 suffices, many thanks

[frank orchid]

what are the unusable help channels for

[wild lantern]

The ones in the hidden section?

[frank orchid]

yea

[ancient flame]

they're there so they don't fill up the available help channels category

[frank orchid]

uhh

[deep mango]

gmod is so wise

[frank orchid]

shouldn’t they be available?

[ancient flame]

if not there would be like 20 of those channels up there

there are always a few available

[deep mango]

wow

[ancient flame]

yes

[deep mango]

ctrl a delete

you sniped me

[frank orchid]

why aren’t the hidden ones available

[ancient flame]

oof

[deep mango]

cause it would clutter up space

why would they be available

you only need a few available

[ancient flame]

^^

[wild lantern]

I need at least 20 at once

[ancient flame]

YES.

[deep nebula]

i went thru precal and its basically just algebra 2

[jovial ember]

20$/1 35$/2

19 dollar Fortnite card

Who wants it??

[fair mural]

chmonkey

[jovial ember]

What

[fair mural]

chmonkey

[jovial ember]

Quantum

[fair mural]

[fair mural]

darn you got me

[mint patio]

when proving G is a group do I just have to prove that
operation is associative
identity element exists and ea = ae = a for a in G
every a in G has an inverse a^-1 in G
(so just definition of group)

[jovial ember]

Yes

Or if you’re me you just think “are there inverses” and once I convince myself there are I move on

[mint patio]

okay

how about this

here's context

ignore the words

i like talking and i am bad at this

[mint patio]

so do I always have to show that the inverse is in the group too?

I feel like (zw)^n = 1 is implied since the definition of a bianry operation means the group is closed right?

[brave hollow]

but this is you showing the group is closed

closure is not defined by binary operation

[static loom]

yeah you gotta show for z != 1 that you have w such that wz=1

[jovial ember]

It’s cuz

n ranges in Z

Not in N

[mint patio]

(wz)^n right?

[jovial ember]

No

[mint patio]

n ranges in Z^+

[jovial ember]

wz

Oh okay well it actually doesn’t matter

teehee

[mint patio]

OH an inverse lol

wait I'm confused

[jovial ember]

Wait what am I even talking about

[mint patio]

so would I just add "assume z \neq 1"

[jovial ember]

I am being Hurb

[mint patio]

in the inverse section

[jovial ember]

Well no

If z = 1

It is its own inverse

[mint patio]

okok can we back up I think I'm getting confused

[static loom]

well it doesn't matter just trying to make it clear when it is nontrivial yeah

[mint patio]

what am I missing from my proof?

[static loom]

let's look at it concretely

what's the inverse of i

[mint patio]

1/i? 🥴

(i)(1/i) = 1

(1/i)(i) = 1

[static loom]

well what's 1/i in the form x+iy

[mint patio]

1/(0+1i)

[static loom]

nope

[mint patio]

do you want me to conjugate?

that's not x+iy though

[static loom]

when you showed stuff with associativity earlier you only were using stuff that looked like x+iy

[mint patio]

do I conjugate and divide

[static loom]

x,y are real numbers

well, truthfully the proof I have in mind this doesn't matter but still good to know how to do this stuff

main thing is you can't just say 1/i because we're sort of saying that its existence is in doubt, it's like saying the inverse of 0 is 1/0, I can't just write that symbol down and guarantee it exists

[mint patio]

huh

okay

how would I prove it exists then?

[static loom]

well the proof I have in mind is we know z^n=1 so we can write it as z^(n-1) * z = 1

[mint patio]

Trying to see what we can do with that

[static loom]

we know z^(n-1) is in our group cause (z^(n-1))^n = (z^n)^(n-1) = 1^(n-1) = 1

I think maybe I shouldn't have told you this so soon haha

[mint patio]

I follow this

[jovial ember]

I feel like one thing I should mention

I said you only have to prove the group axioms

Inverse, unit, associative

[mint patio]

Yes, but apparently I'm missing something there

[jovial ember]

But in this case we’re constructing the group as a subset of C

So you need to prove that it’s closed under the operation

[mint patio]

Okay

[jovial ember]

Idk if this was mentioned earlier

But imo this is the hardest part

[mint patio]

Why don't D&F mention closure as a group axiom :(

[jovial ember]

So

It’s because it isn’t one

Let me explain

A group operation is a function G x G -> G

It tells you how to multiply two things

But here we have this intuitive notion of multiplication inside C

And we’re taking that to be the multiplication in the group we constructed

This gives a function G x G -> C

We know the output is a complex number

But does it live in G?

[mint patio]

GOT IT

[jovial ember]

So it’s not an axiom of a group because

An operation has to be closed

[mint patio]

But the group doesn't have to be closed under the operation

[jovial ember]

No it has to

[mint patio]

:zonkies:

[jovial ember]

Since it maps G x G -> G

[mint patio]

That's what I originally thought

[jovial ember]

In this specific case

We don’t know that it maps into G

yet

You can prove it

But a priori all you know is the function goes G x G -> C

So it’s less of “does this set with this operation satisfy the group axioms”

[mint patio]

Okay

[jovial ember]

You don’t even know that this is an operation on G

Since you aren’t sure if it maps into G

Does that make sense?

[mint patio]

I think. We don't know that it maps into G (at least, not "yet") so we must show that it does map into G

[jovial ember]

Yeah

[mint patio]

So I guess I'd take

[jovial ember]

So it’s not an issue of “is G a group with this operation”

Because you aren’t even at the point of being able to say it’s an operation on G

That’s why closure isn’t mentioned as an axiom

[mint patio]

[jovial ember]

Because you’re already at the point you have a function G x G -> G

It doesn’t even make sense for it to not be closed under the operation

[mint patio]

So once we show that it is an operation on G then we're all good?

[jovial ember]

I think so?

[mint patio]

And then after that we know it maps G x G -> G and so it's closed under the operation

[jovial ember]

You’ve shown inverses exist

You’ve shown it has an identity

[mint patio]

I need to show that inverse^n = 1

and that zw = 1 (this isn't true is it?????)

[jovial ember]

(I mean you have to know this to even do inverses lol)

Right

And you know it’s associative

[jovial ember]

It is

w = z^n-1

[mint patio]

wait

[jovial ember]

You assumed z^n = 1

[mint patio]

zw
= z(z^(n-1))
= z^n
= 1

oh

[jovial ember]

Swag

[mint patio]

but how can you just say w = z^(n-1)

[jovial ember]

Cuz why not

It’s an element

And you can show it lives in G

[wicked ore]

How do you know z^n-1 is in G

Oh it’s closed under multiplication

z is an element, so z multiplied by itself n-1 times is an element

Right?

[mint patio]

that's what we're trying to prove though isn't it???

[static loom]

this was my proof earlier

[mint patio]

oh

yes

[wicked ore]

Oh that’s lit

[mint patio]

fuck

I don't like that

LMAO

okay

well I like the proof

[static loom]

lol

[mint patio]

don't like that I have to do that

So I guess

[wicked ore]

Wait what are we proving

[jovial ember]

Oh I guess it does follow from G being closed under the operation

Group of roots of unity forms a multiplicative group

[wicked ore]

Do we know that multiplication is closed on G

[jovial ember]

Feather still has to prove this

[wicked ore]

How does one normally prove closure

[cold needle]

directly ?

[mint patio]

1. Show associativity, existence of identity, and existence of an inverse
2. Prove that the operation maps to G (and hence G is closed under it) by showing that any product under the operation is in G

is that right?

[wicked ore]

That was stupid

[cold needle]

which parts did u do

[mint patio]

1

[cold needle]

cool

[wicked ore]

What’s the associativity proof

[mint patio]

Oh I just FOIL'd it

[wicked ore]

Oh

[cold needle]

foil

[jovial ember]

Honestly I would’ve just said multiplication in C is associative

[cold needle]

[jovial ember]

Which we know from our complex analysis class

[mint patio]

that we haven't taken yet

[jovial ember]

They don’t know that

[mint patio]

LOL

[wicked ore]

Oh that’s right

[jovial ember]

I mean if you weren’t sure why yourself

[wicked ore]

Not that complicated ok

[jovial ember]

Then it’s good to go through it

[cold needle]

why foil tho you have exponential form and exponent rules n all

[mint patio]

right

[cold needle]

add the args multiply the magnitudes

[mint patio]

I forgot

metal

that's it

i forgor 💀

[cold needle]

thats fine

[mint patio]

are u happy now do u feel superior metal ?

[cold needle]

u still proved it

i never feel superior

there is always a bigger fish

[mint patio]

Is this p[art right

2. Prove that the operation maps to G (and hence G is closed under it) by showing that any product under the operation is in G

Or is there more I have to do to show closure

[cold needle]

nah

[jovial ember]

No that’s it

[cold needle]

just product two roots and thats all

[mint patio]

poggers

[jovial ember]

Just show if you have z,w in G then so is zw

[mint patio]

and we do that by saying let w = z^(n-1)
but then we have to show that such w is even in G
which it is since w^n = (z^n)^(n-1) = 1^(n-1) = 1 (thanks mero)

and we're all set

okay

that's not bad

good thinking

[wicked ore]

This is for closure? Or

[mint patio]

yes

[jovial ember]

No

[mint patio]

no.

[jovial ember]

That’s showing inverses exist

[wicked ore]

Ok ok

[mint patio]

wat

[jovial ember]

That doesn’t show it’s closed under multiplication

This was just showing z has an inverse

You have to let z,w be arbitrary to show it’s closed under multiplication

So z^n = 1 and w^m = 1

Show (zw)^k = 1 for some k

[wicked ore]

Ohhhh

[gentle bay]

Hey abs!

[wicked ore]

Yo

[gentle bay]

Wassup?

[mint patio]

uhhh

[wicked ore]

Not much

[gentle bay]

What you studying these days?

[wicked ore]

Trying to prove closure

Lol

[gentle bay]

Wow lol .

[mint patio]

Let k = nm? lol

[jovial ember]

Yah

[mint patio]

oh

[jovial ember]

Why does that work tho

[wicked ore]

wait can you actually do that

[jovial ember]

Yes lol

[cold needle]

why not

[wicked ore]

Oh I’m stupid

[mint patio]

oh

[wicked ore]

Yeah I though k was some fixed unknown quantity

[gentle bay]

Oh and University started or still 12th grade going on lol?

[wicked ore]

You can sorta choose what k you want

[wicked ore]

12th

[brave hollow]

There's plenty of k just choose one 4head

[jovial ember]

Yeah

[brave hollow]

"I just happen to know..."

[gentle bay]

Cool.

[wicked ore]

Man proving that sets and operations form groups feels like

[mint patio]

what if k is prime?

[wicked ore]

Im crawling at the rock bottom of concrete math

[jovial ember]

k won’t be prime

I mean unless m or n is 1

But prime doesn’t mean anything

[wicked ore]

And algebraic tricks are the only way to climb up

[jovial ember]

Just compute (zw)^mn

First break it apart

z^mn • w^mn

Now what?

[mint patio]

(z^n)^m * (w^m)^n

[jovial ember]

The exponent rules you know from like, algebra still@hold

Yeah

And that becomes?

[mint patio]

1^m * 1^n = 1 * 1 = 1

[cold needle]

ya

[brave hollow]

[jovial ember]

[neat lintel]

cute

[toxic schooner]

[mint patio]

I'm confused why I have to prove that an inverse element exists
but I can just "say" an identity element exists? that it's 1?

[cold needle]

?

[toxic schooner]

no?

thats not obvious

[jovial ember]

Wel you have to prove it

[cold needle]

wait

[jovial ember]

But like

[wicked ore]

Inverse and identity are part of the axioms

[jovial ember]

Take any z

z•1 = z

[cold needle]

what do you mean feather

[jovial ember]

So 1 is the identity

[toxic schooner]

for example, in R, or Q even, there is no inverse for 0

[jovial ember]

You technically have to prove that 1 is the identity

[toxic schooner]

under multiplication

[mint patio]

like I don't see why this doesn't work?

[jovial ember]

We just omitted mentioning it

Oh this technically works

[toxic schooner]

[mint patio]

.

[jovial ember]

But it’s hard to see that z^-1 is in G

And again, you technically either
1:

Need to know that’s the inverse in C

[cold needle]

need to do a bit of multiplication to see it clearer

[mint patio]

whereas it's obvious that 1 is in G

[jovial ember]

Or 2: manually compute what happens when you multiply z and z^-1

I mean yeah

[cold needle]

right cus 1+0i

[jovial ember]

1^n = 1 for any n

Lol

[mint patio]

yeah

[cold needle]

and its fixed

[mint patio]

that wasn't sarcasm

LMAO

[jovial ember]

Oh okay

[brave hollow]

to me it is obvious that inverse is in G

[jovial ember]

Lol

[static loom]

you still need to know how to show 1/(a+bi) in the form x+iy from earlier when you had trouble

[jovial ember]

Well, sort of

[wicked ore]

What is 1/(a + bi)? I’ve not done much complex numbers

[jovial ember]

I mean a lot of this honestly

[static loom]

I mean independent of everything

just need to know how to do that

[mint patio]

why can't you just conjugate it and then divide real parts

wait

[jovial ember]

Depends on how much of C you are allowed to assume you know

[mint patio]

kidding

i lied i lied

stop

i wasnt thinking

[jovial ember]

No that is kind of why it is

What*

Inverse is

[mint patio]

im going to stop breathing now

[static loom]

lol

[jovial ember]

(x - iy)/(x^2 + y^2)

[toxic schooner]

u can multiply and divide by (a-bi) to get a a+bi form for the inverse

[wicked ore]

Isn’t 1/(a+bi) just shorthand for (a+bi)^-1? Then it’s circular

[jovial ember]

But if you know that or not depends on how much you know about C

[toxic schooner]

this

[jovial ember]

See my point above

[wicked ore]

Oh

[mint patio]

I think I need to prove more abstract or unfamiliar sets are groups to really understand

because right now I'm still kinda 4heading some of this

[wild lantern]

Did you do linear algebra?

[mint patio]

and wait chmonkey you said I need to technically show 1 is an element
yes dootdooter

[static loom]

slightly different perspective on what chmonkey already showed $z\bar z=|z|^2$ then $\frac{\bar z}{|z|^2} = \frac{1}{z}$ and since $|z|$ is real you're fine

[fathom swallow]

Merosity

[wicked ore]

This you mean?

[mint patio]

what's wrong with this chmonkey

isn't that proving it's the identity

[cold needle]

right but is 1 a root of unity

[mint patio]

oh

that is

I have to

man

but that's like...

😭

[cold needle]

Obvious?

[static loom]

how can unity be a root of itself

[mint patio]

you know

that I know

I can't say that

LMFAO

[wild lantern]

Vector spaces are examples of groups when you consider their addition op.

Idk if you proved things were vector spaces or not?

[mint patio]

nope

[wild lantern]

Ah dang :/

[mint patio]

we did approximately zero (plus or minus zero) proofs in my LA class

because it was a lame engineering class

[cold needle]

rip

[wicked ore]

This feels a little sketchy, I don’t know. If I was inventing complex numbers, wouldn’t I have to invent what it means to divide them? I can’t just use variables like z and then say 1/z as if it’s just a real number.

[wild lantern]

Well, you have the right idea as far as proving lots of things are/aren't groups for practice.

[mint patio]

OKAY so let me get all this homophobic shit down

To show a set is a group under some operation

[smoky tangle]

Hey

This is me aman

[static loom]

sure, if you want you could write it as $z \bar z = |z|^2$ then since $|z|$ is real divide it through to get $$z \frac{\bar z}{|z|^2} = 1$$ keep in mind this is all after you've already shown multiplication of numbers of the form x+iy is associative and commutative and stuff

[fathom swallow]

Merosity

[wicked ore]

Ok I see

[mint patio]

we need to show

1. Associativity of the operation
   (Just take three arbitrary elements a, b, c in the set and show (ab)c = a(bc))
2. Existence of an identity e
   (First verify that whatever you declare as the identity is even IN the set, then show that ea = ae = a)
   3, Existence of an inverse
   (Have to take two arbitrary elements in the set and show that one is an inverse of the other, and also we need to show that the inverse is in the set too)
3. Show that the operation maps into the set (which means the set is closed under the operation).
   To do this we just show a product of two arbitrary elements is in the group

[wicked ore]

So the 1/z thing is just applying “real number logic” to complex numbers, more as a notational choice sorta thing?

[static loom]

yeah sorta

if it's a commutative group, fractions are well defined

[wild lantern]

Another way to look at 3 is that you need to pick an arbitraty elt from the set then find its inverse in the set.

[mint patio]

So is my list complete or am I missing anything?

[wicked ore]

I always thought a/b is notational shortcut for a * b^-1

[static loom]

it is yeah

[wild lantern]

The list looks right but I could be forgetting group axioms lmao.

[wicked ore]

Oh so then that complex number thing you showed me is like

How to find an explicit a+bi form

For inverses

[static loom]

yeah, like it's just a way of putting them in a form so that you can prove axioms and is convenient or whatever

[wicked ore]

I think proving closure should come first, because logically the others can’t follow without that, and sometimes proving the others is easier with closure

[jovial ember]

Yeah

Like I was saying before, to even get to asking “is G a group” you need to know you have an operation

So closure should be first

[mint patio]

oKAY

so put 4 at the top of the list

and NOW we're gucci

[wicked ore]

Yes

[mint patio]

[jovial ember]

[wicked ore]

So like

I’ve learned all these basic definitions and theorems and whatnot for abstract algebra

But it seems like it’s all in a void

Who invented these things? Why? What problems were they working on?

[jovial ember]

Arguably Galois was doing stuff to prove Abel-Ruffini

I think

[wicked ore]

I hear the history is kind of messy

[jovial ember]

Insolvabilitu of the quintic

These show up quite naturally in various fields

[wicked ore]

And the clean, simple definitions today were developed indendently of their original problem-context

[mint patio]

quintics are solvable in terms of radicals bro

[jovial ember]

This is a bit afterwards, but people had long realized number theory was related to things other than just Z

[devout nacelle]

The clean formulations are rather recent iirc

[jovial ember]

The Gaussian integers

[mint patio]

x^5 = 1

[jovial ember]

C

[mint patio]

x = 1^(1/5)

[jovial ember]

Various number fields

[wicked ore]

I mean

[jovial ember]

Things like Z[sqrt(n)]

Etc etc

[devout nacelle]

Maybe Noether was the one who formalised a lot of this?

[jovial ember]

You can deal with these algebraically

[wicked ore]

In 1800s did they even have the kind of rigor we have now

[jovial ember]

Not really?

[mint patio]

no lol

[jovial ember]

Set theory wasn’t even founded

But the work done is still largely good

[wicked ore]

So now it sounds ridiculous to talk about groups without the concept of a set, logic, all that foundational stuff

[jovial ember]

Yeah, I mean you can find a book by like uhhh

Hall?

No

It is linked in a reference on Wikipedia

It’s like from 1896?

The terminology is ass

Because they viewed these things in different ways

[wicked ore]

Jeez

Why did the like

[jovial ember]

Oh I think it’s by Burnside

[wicked ore]

Breakthrough in logical rigor occur

[jovial ember]

Uhhh

Idk, I think… Cauchy among others

Cantor

[wicked ore]

Seems like the philosophy of math was completely overturned

[jovial ember]

Okay so like

A lot of assumptions were made

Because people were operating ok vibes

So defekind Cauchy, etc formalized things I think

Limits weren’t like formally defined yet

[wicked ore]

Yeah Gauss and Euler would have laughed at something like Rolle’s theorem

[jovial ember]

A lot of people assumed that continuous things were differentiator

So things like the Weierstrass function

[wicked ore]

Ah

[jovial ember]

The like fractal saw one

Continuous everywhere nowhere differentiator

[wicked ore]

There’s this quote from Bertrand Russell

[jovial ember]

People started constructing these to say “hey guys, these assumptions aren’t true”

[wicked ore]

I think it captures it kinda well with the Weierstrass function type stuff

[jovial ember]

So I think people started poring over the details to put things on more rigorous footing

There’s in algebraic geometry also around the turn of the century and into the start of the 20th

[wicked ore]

Ah

[jovial ember]

The Italian school of algebraic geometry had sort of abandoned rigor to operate just on vibes

And this led to a lot of false things being “proved”

Which brought some people like Weil I think to try to seek to make a rigorous foundation

[wicked ore]

Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner.

In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that.

If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratologic museum.
Henri Poincare, 1899

[jovial ember]

And then Grothendieck said fuck you to them all and invented schemes which put all those other guys stuff into the trash

[wicked ore]

What? Really?

[jovial ember]

Yeah

[wicked ore]

Wait grothendieck invented schemes

Jeez

[jovial ember]

I mean some of the stuff they “proved” we managed to reprove ages later

[wicked ore]

Schemes are so weird

[jovial ember]

But there’s also just false things

Like people made counterexample a and I think some of the ppl of the school were like

So things got messy

Also in the past math was a lot more tied to physics and chemistry and stuff

[wicked ore]

Bro what

[jovial ember]

A lot of the mathematicians also did other sciences

[wicked ore]

Yeah I was thinking math was in the natural sciences right

[jovial ember]

And they used physical intuition, or some things about the laws of certain physical phenomena to justify things

[wicked ore]

Yeah

[jovial ember]

I forget the specific one, but there’s a kind famous example, maybe by Riemann?

Idk

[wicked ore]

Which is kinda like this quote

A practical purpose

And even if it was not practical, they used practical intuition

[jovial ember]

I guess a lot of it was just that people began to question things and then they made counterexamples

So people went back to iron out the foundations

I’m sure there’s a lot more to this, and math history is a studied field

[tiny marten]

for a long time math and physics were deeply associated

[jovial ember]

So I’m sure books have been written on them

[wicked ore]

Ah

[tiny marten]

tons of famous historic mathematicians were also engineers or physicists

[wicked ore]

And the obsession with axiomatic stuff in the 20th century

[tiny marten]

super common

for a long time people used way less than rigorous assumptions about algebraic manipulations and stuff

[jovial ember]

Fluxions

[tiny marten]

and mathematics didnt get away from stuff like that till cauchy's era (often specifically because of cauchy) and then later with formal logic

formal logic sorta helps cause an explosion of math theory that differentiates pure math way more from sciences

there was an assumption called the universality of algebra or something

and it was this pretty much not rigorous at all assumption that allowed them to extend some operations like arithmetic to infinite sums

people like euler used it iirc

i mean effectively many times these assumptions did work but it falls apart at interesting stuff

formal logic was so powerful in mathematics that a bunch of people really did think it could be used to "finish" large swathes of mathematical and philosophical inquest

and it's why the philosophy adjacent to mathematics in the first half of the twentieth century appears to be disrupted and upturned with some results from later in that half

it wasn't really clear how far you could go with logic but it appeared to be really far, advents in logic made massive improvements to a ton of fields of math and the humanities

but some of the more ambitious ideas sorta gave way to reality

[wicked ore]

What’s an example of this?

[tiny marten]

well the really famous one is bertrand russell's logicism

[wicked ore]

Isn’t that a philosophy?

I’ve heard of it

But I don’t really get what it is

“All of math reduces to logic”, or something like that

[tiny marten]

it's been a long time since i took the class where we discussed this but bertrand russell hoped that logic could be used to answer a broad class of arithmetic questions

that is not true by the incompleteness theorems

since the incompleteness theorems are about a logical system that can express a certain level of arithmetic theory

[devout nacelle]

Didn't this concern Hilbert's program as well

[tiny marten]

probably, i remember some stuff was all entangled

logic was very prominent back then and iirc some of hilbert's program is just problems in logic

[devout nacelle]

Right, I believe Hilbert also held the opinion that all math could be formalised inside some sequent calculi so as to reduce all of math to mechanical symbol pushing

But Gödel blew it up with his theorems

[tiny marten]

yeah people had tons of big brain ideas on it

von neumann was also working on completeness

goedel's work was a really big deal and people really saw it as a work of genius immediately

its parallel with the halting problem is really beautiful too

the philosophy of math and math diverge a lot by like the 60s

a lot of philosophers still are trained in math (usually logic) by then but it's not like people betting their philosophical ideas in provable/disprovable notions as much lol

[devout nacelle]

[wicked ore]

Pretty cool that you know all that stuff

It’s interesting how logic was such a big deal back then

I don’t think it’s nearly as large now

[blazing pawn]

I think its more so an increasing segmentation of math from philosophy, the philosophers of math and logicians still exist

This is kind of a general phenomenon though

[random solstice]

Does anyone here have mojang account>

[timid bluff]

You just blew my mind 🤯

[quasi torrent]

wait the number 'n' in base n is represented as "10"?

damn

that's awesome

[toxic schooner]

[tender grotto]

yep lol its interesting

[haughty bear]

what determines how many help channels are available?

[deep mango]

need

[sick burrow]

What do you need ryc

[deep mango]

there's basically always 5 available

unless they're ALL occupied

here i'll find it gamma

#discussion message

[sick burrow]

I see

[deep mango]

also keep in mind that the menu is heavily contingent on me getting full

if we can also avoid that, then i would stack it with a lot more food

[sick burrow]

No that's a very respectable perfect day

[deep mango]

most notably not included (which i mention later): seared scallops and connecticut style lobster rolls

[sick burrow]

My opinion of you has grown slightly

[haughty bear]

oh that's cool

[fathom swallow]

FLAME

[mint patio]

what makes DG notation so cursed

why does everyone hate it

anyone have pics

[leaden torrent]

it doesnt like

visually look bad

but its very inconsistent and technical

like heres the definition of the exterior derivative in terms of differential forms

[livid yacht]

Is ring theory knowledge worth to learn algebraic geometry?

[opaque patrol]

yes

[leaden torrent]

you need it, yes