serious-discussion

[ember mulch]

It's not that it's hard

It's that it's boring, and therefore I struggle to keep focus

[surreal sapphire]

hm

i dont think i have any helpful tips other than "try to find it interesting"

those ideas are important not just in number theory but for mathematics as a whole, so maybe that helps but 🤷

[ember mulch]

Haha I kinda figured ^^'
I just needed to rant a little bit

I usually like Math, a lot. Compared to my other CS peers, but I just can't with this specific lesson x)

[torn willow]

Understandable

Number theory for undergrad cs folks is just super lame

[bronze pelican]

Thats fine, do you already know this stuff?

[ember mulch]

I mean so far some of it, divisibility was easy, especially considering it's a concept we use a bit in programming
and the congruence stuff is also stuff that kinda makes sense but having to prove it just seems like a chore

[torn willow]

What kind of "proving congruence" are you talking

Stuff like Chinese remainder theorem

Or
"13=7 mod 6"

[ember mulch]

Here are some screenshots I took during the lecture

[torn willow]

Ok ,7=13 mod 6 kind

[ember mulch]

Once again, it's not that it's necessarly hard, but it's just that I find it boring and uselss compared to other stuff in Math

[torn willow]

Yea it's something that should be covered in like 5 minutes

And then you move onto the more interesting stuff

[ember mulch]

My prof spent 1h20 on that x)

And there's more next time apparently

[surreal sapphire]

this is just basic definitions so there isnt any "meat" to it yet

but its important ideas that might be useful in the future if you do more math

[torn willow]

Well idk how the heck you spend 1hr 20 minutes without any content

[restive bough]

modular arithmetic is incredibly important and helpful to do mental math and make basic arithmetic easier.

[ember mulch]

Well, I hope it gets more intersting, so far it's just not engaging me

[surreal sapphire]

eh, this takes more time than 5 minutes

there are quite a few theorems to prove about this

unless you already have ring theory available and just appeal to quotients

[torn willow]

Well I mean 1 hr 20 minutes on literally the definition of congruence

[ember mulch]

I mean I guess, you're right. That is one use for it

[surreal sapphire]

well

[restive bough]

it's also helpful for bit-level wizardry iirc

[ember mulch]

I mean there were a bit more slides but I wasn't able to take all of these since my laptop powered off lol

[surreal sapphire]

if you want some kind of motivation and since you are a CS student (?) you can look at RSA

[torn willow]

Yea you have somewhat non trivial theorems like the remainder on dividing with a number is unique

[surreal sapphire]

this will give you a practical application of where this stuff is used

maybe show you it can be useful and thus raise interest

but i dunno

[ember mulch]

Oh like cryptography ? It is one of the bullet points we're also getting to later, which is the one thing I'm actually looking forward to

[surreal sapphire]

yes

[restive bough]

multiplying is easy, factoring is hard. boom, you know cryptography

[torn willow]

Have you proved divison lemma

[restive bough]

jk

[surreal sapphire]

this is very important in all kinds of cryptography and coding theory

[torn willow]

It's more complicated than what you would expect

[surreal sapphire]

RSA also happens to not need much more than this

and its such a good crypto system that it is still in wide use today

[ember mulch]

divisiom lemma ?

[torn willow]

https://simple.m.wikipedia.org/wiki/Euclid's_lemma

Division theorem here
https://en.m.wikipedia.org/wiki/Euclidean_division

[ember mulch]

We haven't proved that, no

[torn willow]

You need to use well ordering for this

[ember mulch]

Hm, well we might see that later on in the chapter, but I doubt it since we did the "divisibility" section

[torn willow]

Yea, undergrad cs math isn't very deep usually

[ember mulch]

True, but I'm okay with that honestly, I like math, but not enough for it to completely overshadow the programming courses x)

[finite nimbus]

I was trying to make run as slow as possible

[neat lintel]

Hey my understanding of the derivative is pretty superficial

When we say the derivative of a function is blah, what does that say, concretely, about the rate of change of the function?

Like the derivative is what happens when you take looking at the rate of change to its limit basically

So smaller intervals?

So we look at rate of change ‘at a point’ when we look at derivatives but what does really say?

It just confuses me

Is the rate of change of the function really its derivative exactly anywhere

[restive bough]

Yes

Technically the derivative describes the rate of change of a straight line that's tangent to the function at the given point

[neat lintel]

Oh yeah so

Wait but

So we’re not even looking at changes in the function itself? Im just so whizzed out by this concept

I mean im looking into learning about it really but im just asking

[restive bough]

Yes and no. Imagine looking at the function really up close. You can approximate that with a straight line. As you "zoom in" farther and farther, that approximation gets better and better, so we just zoom "infinitely" to get an "infinitely good" approximation

[neat lintel]

Ohhh

Im saying ohh but still whizzed out but the sentences make sense

[restive bough]

Technically it's "arbitrarily close" but that's however close we want, and so we choose to make it so good that it's exact

[neat lintel]

So when we say the derivative is x, we’re saying that as we get arbitrarily precise about the rate of change, we’re getting arbitrarily close to x

Like idk

[restive bough]

That's a correct way of talking about it. Have you used desmos before?

[neat lintel]

No

[restive bough]

Oof, it's a really good tool for demonstrating this.

[undone moon]

Guys how do u do this i need help

[neat lintel]

Imma download it

If its still the same in mobile

[restive bough]

It's a really powerful graphing calculator, the mobile app is basically just the computer site optimized for mobile from what I know

[neat lintel]

Also thank you for engaging w me @restive bough

[restive bough]

So you know the limit definition of the derivative?

[neat lintel]

Uhh smth like lim x—>0 blah blah right

[restive bough]

Yes

[tawny current]

Note that the base is $(h + 7)\text{ cm}$, as described in the problem.
Then use Pythagorean theorem.

[fathom swallow]

pi over four

[restive bough]

lim {h→0} (f(x+h)-f(x))/h

Notice how this looks kinda like ∆y/∆x

[neat lintel]

Yeah

F(x+h) is delta y

No

f(x+h) -f(x) is just delta y

And the h is delta x

And then limit as h goes to 0

[idle copper]

http://txnor.com/mathchallenge

??

[neat lintel]

So as you said thats the infinitely good approximation

Ig

[next swan]

14

[idle copper]

how? its 28?

[next swan]

28?

how tho?

[neat lintel]

I mean its a matter of how u interpret the notation

[idle copper]

its a tricky one?

[next swan]

elaborate?

[neat lintel]

Tbh i think u should treat 24/4 as a fraction and then add 22

Some people say everything after / is the denominator

[idle copper]

what do you mean sorry

[neat frost]

Don’t troll.

The “joke” is that the image is different for different people.

[idle copper]

its not

[next swan]

yeah

[neat frost]

What’s the expression for you then?

[idle copper]

24/4+22

[next swan]

i did VBODMAS and got 14

[idle copper]

its 28

[tawny current]

It is 16/2 + 6 for me

[idle copper]

its still 28

not different for everyone see

[tawny current]

16/2 + 6 = 8 + 6 = 14

[neat frost]

As if we don’t have enough shitposts already

[idle copper]

add 10 its 28

[next swan]

could you elaborate or explain?? please

[idle copper]

if you do the calculous its 28.

[tawny current]

okay sure 9 + 10 = 21

[idle copper]

wrong

[neat lintel]

Im just waiting to see what golden Phoenix was typing

[next swan]

calculus?

[restive bough]

Yep, obviously we can't divide by 0, but we can "pretend" to divide by 0, and use our fancy algebra tricks to eliminate that division by 0 so that we don't have to break the rules

[next swan]

d/dx?

[neat lintel]

Hmmmm

Ive seen a video about that

Lemme get it

By algebra do u mean like

When you have the quotient like ((x+h)^2 - x^2)/h that works out to be something and when you divide by h youre no longer left with h in the denominator?

Thats what ive seen in professor dave explains video

https://youtu.be/x3iEEDxrhyE

Or smth else

[restive bough]

Well, if we input f(x)=x², for example, we can manipulate the numbers and make the h go away before we take it to 0.

((X+h)²-x²)/h
↓ expand the square
(x²+2xh+h²-x²)/h
↓ combine like terms
(2xh+h²)/h
↓ simplify
2x+h
↓ substitute 0 for h
f'(x)=2x

[next swan]

h=12.38238017

[restive bough]

Exactly

[neat lintel]

So f’(x) equals 2x then the approximations for rate of change will approach 2x

?

The derivative is the slope of that tangent line right? What is actually changing with respect to what?

[restive bough]

We can't send h to 0 while there's an h on the bottom, so we just play with it until he isn't on the bottom anywhere anymore, then send it to 0.

H is the difference between the two points that our straight line approximation intersect, so we say there's no difference between them, making our approximation tangent, therefore infinitely good of an approximation

[neat lintel]

Huh

Nice

[neat lintel]

Or do we think of it like well its just in the most literal sense the best approximation out there and so we use it

Or i mean

Can the rate of change ever even be the derivative?

No right? Cause the derivative is what happens as you go smaller intervals indefinitely

So you cant say here in the graph its changing by the derivative

Is the graph even changing at a rate equal to its derivative?

I mean its as h approaches 0 but then how does that give us anything tangible about the function’s changing

Idk im just saying stuff tho

Like what comes to mind

Im really appreciating the engagement here btw but if i kinda tire u out lol its ok if u dont answer yknow

Ultimately its my lack of research that permit these holes in my understanding

But ive just been gathering stuff from here and there and it all just comes down to those questions

Bread

[restive bough]

What's the difference between an exact entity and an infinitely close approximation? Well, basically nothing, and the only thing we really say is different is calling one "real" and the other "approximate." If they end up being the same in all values, why make a distinction?

That's how I see it, at least

It's like the difference between a Riemann sum and an integral. If we use infinitely many tiny rectangles, our error becomes not just negligible, but infinitesimally so, and we can't distinguish between that and no error, so why say it's there at all anymore? It just poofs out of existence by that point

A good way to think of the derivative is in terms of physics: we fall towards earth at about 10 m/s², right? So our position is changing by bigger and bigger intervals every second. The amount of change in our position is constantly getting bigger, but we can measure how much bigger it's getting over time. That's what a derivative is, a measurement of how change itself changes.

[neat lintel]

Hm

Hmmmmmmmmmm

Hmm

I see. Thank you for your time good sir/woman

Very digestible but telling paragraph

[restive bough]

Math is about understanding our universe. Sometimes we can get so abstract that we lose sight of what we're really talking about. It can be helpful to come back to how it impacts us as beings in a beautiful, mathy existence

[neat lintel]

Good to hear that again. . . Getting tired of uneducated Youtubers trying to tell me that math is subjective. . . .

[neat lintel]

People do that?

What

Meanwhile

[prisma swallow]

bruh

Math is not about understanding the universe

You are confusing math and physics

[neat lintel]

Does anyone know anything similiar to fourierseries - a way to represent some function f(x) without the usage of trig functions?

[torn willow]

Taylor series?

[neat lintel]

I have never touched them before

But... I guess they are similiar?

[neat lintel]

"In mathematics, the Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point." according to wikipedia

What if you have a piecewise function?

Pretty sure you can't calculate the derivative of it (maybe one can calculate the derivative within the set boundaries?)

forgive me if my language is wrong, translating from English to Swedish

[neat lintel]

They're both about understanding the universe.

[prisma swallow]

Maybe, but math, unlike physics, is about much more.

The only reason math is pure is that we have distanced ourselves from the physical world.

[neat lintel]

??? In what way have we distanced ourselves from the physical world?

[prisma swallow]

Math lives in the world of ideas

[neat lintel]

Does it?

[prisma swallow]

Yeah

[neat lintel]

Does it really?

[prisma swallow]

Yeah, its just that god used some of those ideas and used them for physical reality.

[neat lintel]

Scumbag Ante0417 : Claims Math lives in the world of ideas. brings up how God used it for physical reality right after that.

[prisma swallow]

uh?

[neat lintel]

@prisma swallow Here, have a burberry hat 🤣

[prisma swallow]

God can have ideas too, so why wouldn't there be any overlap between math and physics?

Math is about the exploration of ideas/definitions

[neat lintel]

Until I can experience god with the sensory tools I have, its just an idea and has no bearing on the conversation of math and physics.

Math isn't about an exploration of ideas, it elucidates truths and axioms that are observable and replicable in reality.

[prisma swallow]

That is not how it works

Math transcends physical reality

It is therefore the ultimate science discipline

[neat lintel]

See this rock? It's not round, at all. It's a bunch of straight lines organized at the decimal point : Put a smaller round rock under a microscope and you can see that.

[restive bough]

oof, ontological arguments of the placement of math in the hierarchy?

[neat lintel]

Dude, what....? Transcends physical reality. . . ?

The hell. . . .?

[prisma swallow]

Dude

[neat lintel]

Nope, explain your point and don't confidently come at me with the knowing : "Dude" remark.

[prisma swallow]

There is absolutely no reason why plancks constant is the value that it is.

[neat lintel]

Okay. . . .Why?

[prisma swallow]

It might aswell have been 0.00000001 off

[neat lintel]

For margin of error in fractional numbers, you'd best explore numerical analysis.

[neat lintel]

fractals releated to the universe?

Set theory?

[restive bough]

coastline problem

[neat lintel]

fr?

[prisma swallow]

Our physical reality might just be one of many variations

[neat lintel]

damn

[neat lintel]

Don't know what those are? Elaborate?

[prisma swallow]

There is only one variation of math

[neat lintel]

https://en.wikipedia.org/wiki/Fractal In short, patterns that are endless

[prisma swallow]

Math is absolute

[restive bough]

might be, but if all those realities can exist, then modally they do, therefore mathematics is the reality, and physics is a measurement of this portion.

[neat lintel]

Wouldn't it be more apt to suggest that we are of a certain type of make that isn't compatible with other forms of reality and the idea that we suggest that the reality we are compatible with is "physical reality" is a fallacy?

[prisma swallow]

hmm, interesting thought.

[neat lintel]

And we're just assuming that because we can walk and talk and live and dream in this reality, it must be the only physical reality?

That's just an idea.

I'm NOT an expert.

I'm just curious.

and aggressive.

[prisma swallow]

That is what we are suggesting

Our physical reality is one of many variations

Math transcends all those variations

[neat lintel]

You honestly should have said that, gotta realize I'm still a laymen level math-wise and I'm willing to gamble most people are. XD

[restive bough]

but it is exemplified in our existence everywhere, since it is the invariant, thereby still tying math and physical reality fundamentally together, regardless of which is supervening on the other

[neat lintel]

Because it's the invariant? The axiom? The constant?

[restive bough]

if this world is a subset of all possible worlds according to mathematics, then it still must abide by the rules of mathematics fundamentally, making the transcendence of math a moot point

[prisma swallow]

So really, math is the only truth.

[neat lintel]

So even if physical reality DID supervene math which is one possibility you suggest, it would still be tied to Math and therefore Math would still supervene physical reality?

[restive bough]

no, but the supervenience relation need not be understood properly to recognize its existence (sorry to any instrumentalist fictionalists out here)

[neat lintel]

(I just learned that word from you, so pardon me if I use it wrong.)

[neat lintel]

Oh, that makes sense.

[restive bough]

dw, it's a slippery one that I've learned through immersion, not rote definition, so I wouldn't be able to tell if you were lol

also it may be helpful to realize that math-as-understood and math-as-existent are two very different things. Our silly scribbles on paper are representative of the nature of mathematics and the universe around us, but we know frighteningly little about the real relations that we're representing. Math-as-language is about understanding reality, which is, at least partially, math-as-fundamental

it's very rare, if ever, that we don't mean something by the mathematical equations we make. It might be very abstract, but even in pure math, it comes down to the application of concepts we have seen demonstrated in reality somewhere. What does it mean to have the half triangle number? that doesn't make any sense in the original definition, but we have seen that all triangle numbers are of the form (n^2+n)/2, so we can then extend that form to make an extension to our understanding and maybe even learn something new about how this pattern applies to other areas of mathematics, which by extension means other areas of reality

[fast ivy]

I am sorry, but "patterns that are endless" doesn't tell absolutely anything.

It is a phrase

But is meaningless

Like, what exactly do you mean by that?

[neat lintel]

I've never studied them before, only seen the visulization of them, which is why that was a useless info. In the context of the convo, I just thought about how something in nature can't, no matter how small the intervall is, have an endless repeating patter.

Hence the linking to the article

But you are absolutely correct, I do not know much, if anything, other than that

[grand axle]

Does anybody know of any completing the square methods for higher degree polynomials?

[grand quartz]

Sorry I was in exams, but yeah, ours just tells us "Programmable calculators forbidden" but we can use them in exam mode so... yeah stupid exams, btw I'm in France and it is superior studies in Economics (I don't know if it is called like that in America)

[neat lintel]

vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv

[leaden kestrel]

my life is a lie, I didn't know the rational root theorem

[wicked ore]

@wide fog the reason vectors always start at the origin is because it would make it very complicated if they didn’t. Usually if you want to think of a vector with its tail at a particular point, you would consider a vector-valued function f: R^2 -> R^2. The input is where the vector starts, and the output is the coordinates of the vector relative to the point.

[restive bough]

it's a good one

[leaden kestrel]

I really didn't know that. I heard, that there is no general way to find roots using "*+/..." took it for granted and never questioned it. it was quite a surprise today.

[wide fog]

yea, I think it would be visually more pleasing to make vectors able to start from other points than origin, but as you say, arithmetically would be messy probably

[restive bough]

most of our equations for describing objects or figures on a coordinate plane also have mathematical provisions in them that effectively build them at the origin first, then translate them to the position they're meant to be at. Take Point-slope form, for example. point-slope form is the equation y=mx where you shift the y coordinate by b units and the x coordinate by a units for some point (a,b). This basically builds the equation where (a,b) is the origin, and then translates that origin point through a shift of ... bases is the wrong word, but I can't think of the right one.

[wide fog]

yea, would be nice to do the same with vectors

[restive bough]

you could, if you like, depending how you construct the vectors, you just replace all instances of x with x-a, and y with y-b

with polar coordinates idk how to do it (yet)

[wide fog]

yea, with polar coordinates maybe, but I'm not that well read on that as well

[restive bough]

the big advantage is that with vectors starting at the origin, they only need be n dimensional, whereas they would need to be at least 2n dimensional if they start elsewhere (magnitude, direction, and vector from origin to the tail of the desired vector)

[wide fog]

visually so clean though

I think I'm just too dumb to understand why it wouldn't work, I got pretty good grade in linear algebra lol

[leaden kestrel]

not only that, adding two vectors feels like adding two points. and I think of vectors like points. it makes sense to start them at 0 because, 0 + v = v. our definition of addition still holds.

[restive bough]

yes, but you could probably do that by forming a vector field instead of singular vectors. something like V(x,y) mapping from R^2 to R^2 (re vectors on a spinning ball or w/e)

[wide fog]

I got the theoretical foundation but still don't get it intuitively why not

[wide fog]

the picture is just me visualising optimizing a function on a sphere, those are vectors pointing at min/max values

studying multivariable calc atm, vector fields are in the last part of course, hope I'm able to have time to learn them before exam in couple of days

[restive bough]

it really depends on application ig

[wide fog]

is it hard to learn how to determine if vector fields are conservative or not and potentials?

[restive bough]

dunno, I'm not super familiar with vector fields

[wide fog]

okay

[restive bough]

you're probably ahead of where I am in mathematics, I only took up to integral calc

[wide fog]

ah ok, you here for fun?

[restive bough]

hobby interest in mathematics, and originally to see if there were any slide rule buddies I could find to learn how to better use my vintage calculators

[wide fog]

ah okay, damn forgot those existed

https://media.wired.com/photos/5935c500f061de0423ccfbf9/master/w_2560%2Cc_limit/alleman-sliderule_f.jpg

slide rule to rule them all

[restive bough]

that's a big boy

aaahhhh bad exposure, oh well

my versalog II, russian k13(?), and soroban

[wide fog]

I'm the opposite in a way, I'm not that interested in the theoretical part of mathematics, I'm of the typical engineer mindset, only interested if it has a practical application

once could argue almost everything in math has a practical application, to be more precise, I mean practical applications relevant to what I'm interested in

[restive bough]

I really like applied math, and slide rules are great at that, but analog solutions don't get you the accuracy of digital calculation

most cases

all math is related to itself somewhere else, so everything matters somehow

[wide fog]

what those bead calculators called again, something scarabus something somthineg?

[tawny current]

abacus?

[wide fog]

yea that's it haha

[restive bough]

soroban, for this one specifically, it's a japanese abacus

I'm hoping for a Facit TK or TKE eventually, or a Curta but that's hard to find in my neck of the woods

[wide fog]

you into some real niche interests here haha

where you from?

[restive bough]

I mean, kinda? old mechanical computers, typewriters, number theory, music composition, it's a big mix.

FL

[wide fog]

florida?

[restive bough]

aye

[wide fog]

ah sorry, european here, was not sure

If I was interested in math as a hobby, I would take psychedelics often to increase the interest/satisfaction

noticed it increased my interest for the theoretical/abstract A LOT

[deep mango]

[wide fog]

then faded away over the years

[cold needle]

florida

[wide fog]

now after couple years I'm back to baseline

[restive bough]

the ability to know is so powerful that I can't go back to not knowing

[wide fog]

but I think there's a trade off between the practical and theoretical most often, it all depends on ones goals, I'm mid 20s now and need to get things done, feel like increasing the interest of the theoretical is somewhat of a distraction when the goal is just to get things done

but I miss those days, right after a period of intense psychonaut exploration, would watch physics/math videos when smoking a J or eating right before going to bed lol

[restive bough]

it's all connected. Knowing the theory makes the process of knowing the practical simpler in many cases

[wide fog]

true, but it's either you learning or you doing in the moment

it increases your capacity to do, but not the output in that moment if it makes sense

[restive bough]

yep, and the more tools in the toolbox, the more likely I have the right spanner for the job

[wide fog]

exactly, been studying for 5 years now, feel like getting tired of preparing, want to start doing, using what one has learned

nice meeting you, have a good one, going to bed now

[wicked ore]

Intuitively you could. It’s just that the theory is more convenient when you define them starting at the origin.

I think if you really wanted to, you could define vectors with two points, the tail and the tip, and then consider two vectors to be “equivalent” if tip minus tail is the same for both

Then you’re allowed to have moved vectors, and they’re equivalent to the anchored ones

[fresh oriole]

that works fine but only in R^n

[wicked ore]

Yeah none of this works in a general vector space, eg in function spaces where would the functions “start” and “end”?

[acoustic anvil]

is it weird that i keep forgetting how to do stats and mechanics

like i have to revist the basics quite often

whereas in pure maths i dont really forget anything

like i can do the hardest integrals

but if i have a ladders question in mechanics i cant budge it until ive gone over the general method over and over

[shrewd ocean]

can i get a recommendation for a math practice problems website?

[fair estuary]

If you could get access to an uncomputible number. What would you want to know about it?

[swift sinew]

We already know quite a few. Here’s one
https://en.wikipedia.org/wiki/Chaitin's_constant?wprov=sfti1

[fair estuary]

But what would you want to know about it?

Does it have any kind of intrinsic value?

[fading hull]

The problem with this is that it doesn't actually define a particular number

It's just an algorithm for generating a class of numbers

[dense belfry]

Okay here's an example of a useful noncomputable number

So there's something called a PA degree

A PA degree is a noncomputable number that can compute a path through any Computable tree

The reverse mathematics system WKL_0 is equivalent to existence of a PA degree

And there are a list of equivalent things and consequences

Of that

In particular being able to find convergent subsequences of a Computable sequence of rationals in [0,1]

Is something you can do with a PA degree but not in regular computability

That's one example

@fading hull that's not true, fixing a prefix free universal Turing machine defines a particular number

@fair estuary so the reverse math program is big on using existence axioms for certain classes of non Computable numbers to characterize theorems

[fair estuary]

https://en.m.wikipedia.org/wiki/Reverse_mathematics

Is this the one?

[dense belfry]

Why am I sending you down new rabbit holes

Smh

Yeah

[fair estuary]

Cool, thanks

For zeta, which are the values that are equal to -1/12?

[dense belfry]

Lol

I guess I'll give some exposition for myself

[fair estuary]

So far I've got -1 and -13

Are there any others?

[dense belfry]

About properties and muchnik degrees

So tennenbaums theorem says that there are no Computable models of PA

Ie models of PA with Computable complete theory

However the low basis theorem says there is a low Turing degree which computes a complete theory of a nonstandard model of PA

Finding the complete theory of the standard model of PA takes omega many jumps

So you can't get that

But you can get a nonstandard model which has low complete theory

So a Turing degree that can compute a complete theory of a model of PA actually can compute a path through every Computable tree

This is interesting because you can make a tree so that the paths through it are exactly the complete extensions of PA

ie all of the presentations of omega models of PA

[fair estuary]

What is a math tree?

[dense belfry]

It's a subtree of the complete binary tree

[inner finch]

moochik

[fair estuary]

What is a binary tree?

[torn willow]

Wait it has to be a binary tree?

[fair estuary]

Emma said complete binary tree

Which means there are also incomplete binary trees

[torn willow]

Ok,I think I kinda get it

So It's like a decision tree?

[fair estuary]

I dunno

Not sure what a binary tree is

[torn willow]

Something like
1)you start at root,
If leaf,terminate.
if the query is true, move to left child and repeat 1)
if query is false move to right child and repeat 1)

[dense belfry]

So the complete binary tree is the tree of all finite strings of 0's and 1's

You can make a Computable tree whose set of paths are exactly the complete theories of models of PA with domain the natural numbers as follows

First enumerate all of the sentences in the language of PA. The nth level of the tree will denote the nth sentence in this enumeration

The idea is you can test formulas to see if PA proves their negation, which is when a formula is inconsistent with PA

Given a finite string of 0's and 1's of length n, you take the conjunction of the first n formulas in your list of sentences with a negation in front of the formulas so that the corresponding element of the string is a 0

And you check to see if PA proves the negation of that

You do this by searching all the possible proofs from PA (we have a list of all such proofs)

So the idea is if you find that it proves the negation of a particular string you stop building the tree on all extensions of that string

On the left is a complete finite binary tree, on the right is an incomplete finite binary tree

We are discussing infinite binary trees

Okay so anyway the complete theories of models of PA are called PA degrees and they are the path through this tree we built

Note this tree is very non unique, it depends on 1) our ordering of the sentences and 2) our algorithm we are using to build the tree

[fair estuary]

Are all fractions considered even numbers?

Because you can divide them by 2.

[dense belfry]

Slight variations in those will make slightly different trees, so something you would want to prove is a theorem establishing some sort of computational isomorphism between the trees

So you know they produce the same set of paths from a Computability pov

But okay, so the neat thing is this is actually the most complicated Computable tree

In the sense that any path through this tree can compute a path through any other Computable tree

So this idea actually motivates muchnik reducibility

Muchnik reducibility is the idea that if I can pick a solution to one problem (of which there may be many) then you can pick a solution to a different problem

So in this case the problems are of the form "find a path through this tree"

The solutions are the set of all possible paths through the tree

And the statement that a path through the PA tree can compute a path through any other Computable tree is equivalent to the path problem for any Computable tree being muchnik reducible to the path problem for the PA tree

[chilly coral]

Even numbers are numbers that can be divided by 2 without leaving a fractional part, so no

[dense belfry]

So this says that this is the most complicated Pi^0_1 class (a logic name for a set of paths through a Computable tree) in terms of muchnik degrees

Okay I'm done with this exposition I guess

Although one thing I'll add is a reason why you should care about muchnik degrees is because they're actually a lot more natural than Turing degrees

In the sense that the set of turing degrees that solve a certain problem for a muchnik degree

[devout nacelle]

What does "computable" mean in this context?

[dense belfry]

It means there is an algorithm that can decide if something is true or false

About your structure

[devout nacelle]

[dense belfry]

I said tennenbaums but actually this is just a consequence of Godels incompleteness theorem

Tennenbaum's theorem is a different statement

So the only (known) natural Turing degrees are jumps basically

But there are many distinct known natural muchnik degrees

You can think of muchnik degrees as upwards closed sets of Turing degrees

And the regular Turing degrees are represented in the muchnik degrees by a Turing degree d is represented by the set of turing degrees that compute d

Which looks like a cone with d at the bottom

So the muchnik degrees include all the stuff that aren't just represented by cones

Other examples of muchnik degrees

1. the non low_alpha degrees
2. the non alpha-generic degrees
3. the degrees which compute a random
4. the degrees which compute an isomorphism between any two structures in a given theory

Lol why am I saying technical stuff

[inner finch]

something something oracle

[dense belfry]

sotrue

Okay I'm done now

[devout nacelle]

😵‍💫

[torn willow]

What you described feels true

Although I don't get the specifics I can confirm it's true by intuition

[dense belfry]

Nice

[inner finch]

محکوم کے الہام سے اللہ بچائے

[dense belfry]

I could probably make what I just infodumped about into an hour talk I guess

Maybe longer

Depending on the level of detail

[neat lintel]

Hi emma

[dense belfry]

Hey BritS

Things got resolved with my sister

Very happy about that rn

Also I just gave the final for my class

[neat lintel]

[dense belfry]

So just grading left to do

And then I'm done

[neat lintel]

You're a teacher

?

[dense belfry]

Yes

I'm a grad student

And grad students have to teach

That's the rule

[inner finch]

no emma is a logician

[neat lintel]

I see

[inner finch]

or is she

[dense belfry]

I am

[inner finch]

no you're not

[dense belfry]

I am do logic

[inner finch]

you are a grad student 🙏

[dense belfry]

I can wear many hats at once

[inner finch]

many heads

[dense belfry]

Lol I got into an argument over what the meaning of proving consistency is earlier

That's top tier logician shit

Or bottom tier

[inner finch]

whats consistency

[dense belfry]

Depending on who you ask

[light needle]

thats consistent with what i saw yes

[dense belfry]

Lol

[inner finch]

and what does it mean

[neat lintel]

Lol

[light needle]

con(theory) is the statement that theory is consistent

[dense belfry]

I don't want to get into that again today

Lol

[light needle]

goedel did memes where he showed

zfc cant prove con(zfc)

what does prove consistency mean?

[inner finch]

lol

[light needle]

idk ask the logicians

[dense belfry]

It's all about internal proof systems

That you define in a complicated enough structure

So if your structure is complicated enough you can do coding inside of it and define a proof as a finite sequence of formula satisfying certain rules

If you use the Hilbert proof system the rules are you can use your axioms and a preset list of tautologies in your proof

As well as the rule modus ponens

Which is a implies b, a then b

So if you have a on one line of your proof and (a implies b) on another then you can write down b

So once you have a complicated enough structure you can start coding things

So you first code formula and then finite strings of formula

And then proofs are just certain finite strings of formula

[inner finch]

hMMM

hilbert proof system would be a way of interpreting these strings then right

[dense belfry]

Yes, the strings are just syntax until you fix an interpretation method

So then the cool thing is being able to express the statement x is a proof of y

Internal to your language

[torn willow]

You are encoding so that it reduces to a language accepted by some TM?

[dense belfry]

So the Turing machine interpretation came later

Initially it was just model theoretic

[inner finch]

hmmm true

so is this impossible?

[dense belfry]

Yeah you can do that in the language of the natural numbers if you assume a strong enough theory

Let me state some theorems

[inner finch]

realshit

[bronze pelican]

ay boys ima learn hodge theory with my prof in the summer

[dense belfry]

Nice

[light needle]

hot

[dense belfry]

Okay so I am going to be referring to the theory A_E as in enderton

[inner finch]

whats A_E

[dense belfry]

Wow I forgot how much of a hot mess the coding in this chapter is

[bronze pelican]

A_E is arithmetic with exponentiation

[dense belfry]

Yes

Lol

[inner finch]

smexy

[bronze pelican]

Im going to fail logic final tmrw

[dense belfry]

Ohno

What is it on?

[inner finch]

logic i think

[bronze pelican]

Part 2 of the logic qualifying exam

[dense belfry]

Oh okay

I'm glad I'm never going to take an exam again

If anyone tries to make me take an exam I'll bite them

[inner finch]

lol

[dense belfry]

Wow my little sister's attitude is rubbing off on me

She says that sort of thing all the time

[inner finch]

is she 8 or something

[dense belfry]

She's 15

[inner finch]

oh rite

perfect biting people time

[dense belfry]

Yes

Yeah I'm actually not going to do exposition on this tonight

idk why I started even

Lmao

I literally said I didn't want to get into it

And then immediately did

Maybe I'll post constructivism memes later

And do exposition on top of that

I actually both really like Troelstra's and Bauer's exposition

And then I should read Baez's topos theory Rosetta stone thing

[inner finch]

smh smh

yes

go to bed

[dense belfry]

No

I actually should tbh

Like that would be really good

But I won't

[untold sapphire]

@dense belfry yo

[dense belfry]

Hey

I've been doing some reading and I'm down to talk to you tomorrow night about proofs and types

[untold sapphire]

👍

[ancient flame]

yo I've been doing a lot of induction proofs recently and they're like super fun idk why

they feel like cheating tbh

like I get whats going on but damn

[bright hill]

[dense belfry]

Gross

[bright hill]

oh

[smoky cradle]

Question 1
C=
5
9
(F−32)

The equation above shows how temperature F, measured in degrees Fahrenheit, relates to a temperature C, measured in degrees Celsius. Based on the equation, which of the following must be true?

A temperature increase of 1 degree Fahrenheit is equivalent to a temperature increase of
5
9
degree Celsius.
A temperature increase of 1 degree Celsius is equivalent to a temperature increase of 1.8 degrees Fahrenheit.
A temperature increase of
5
9
degree Fahrenheit is equivalent to a temperature increase of 1 degree Celsius.

A) I only
B) II only
C) III only
D) I and II only

EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE

[wintry bramble]

Did you straight up copy that from your exam?

[sleek wing]

you in an exam right now boss? @smoky cradle

[swift sinew]

Dear anyone who is considering applying to the University of Minnesota,
don’t.

• Sincerely, everyone who has graduated from there

[torn willow]

I mean University of Minnesota can't be that special in that regard

[swift sinew]

That’s what I’m wondering. Do most universities just suffer in regard to quality of staff, admin, and structure?

[hexed rover]

Good day gentlemen. I am 14 years old and I have grown a big passion for mathematics. I would love to be introduced to this server and told how to start learning from the basics to more advanced stuff. I joined with my friend from to learn more advanced mathematics in our free time and maybe get prepared for upcoming tests.

[bright hill]

that 13 y/o was from university of minnesota iirc

[arctic grove]

[swift sinew]

He was. I watched him graduate
In fact, other than the dude who dressed up as Captain America. He received the loudest applause when receiving his diploma cover

[surreal sapphire]

what do you already know?
the first step would probably be khan academy up until calculus and then some intro proofs type book

[hexed rover]

Thank you for your tip Lochverstärker I will get right into khan academy. If you do not mind I will ask further questions if I run in to some problems or if I do not know what to do.

[bright hill]

bruh

[arctic grove]

based

[brave hollow]

omg reddit clips irl

I saw the captain america on r/all

[cyan goblet]

how do you denote this using one bigcap

[torn willow]

$\bigcap_{i=1}^{r} \ker({f_i})$

[fathom swallow]

Drake

[cyan goblet]

dope

[neat lintel]

I have a question, What if you have a Good GPA and Credits but failed the SAT , is that alright for college ?

[fervent pebble]

probably fine?

you can choose not to submit sat i think maybe

unsure

[vivid halo]

It can be alright but it definitely kills your chances at top programs outside of exceptional circumstances or unless they make the SAT/ACT optional

[tender tulip]

There's a theorem called the Bourbaki-Witt theorem that states the following:
For any poset (S,≤) such that there is a supremum for every chain, that for any function f(x) < x, then for every x there exists a y > x such that f(y) = y. The proof of this uses induction and transitivity to show {x, f(x), f(f(x)), f(f(f(x))),...f^n(x)} must be a chain and thus must have a supremum, so there exists an n such that f^n(x) = f^n+1 by transfinite induction. But this proof seems really dependent on this induction. I'm wondering if you can prove the theorem by contradicting by assuming x < f(x) is strict, and taking advantage of chain properties via transitivity

(such that the subset of every chain is a chain, adjoining any maximal or minimal elements to a chain is a new chain, etc.)

WITHOUT using Zorn's Lemma, as that implies choice

this is in ZF

Also a chain union its supremum is either itself or another chain with a new supremum

Just gonna write out properties of this chain-completion

1. The subset of any chain is a chain
   1a. Intersection of two chains is a chain
2. Adjoining a maximum or minimum [unique due to totality] to a chain is another chain
3. Every chain has a unique supremum
4. Every chain union its supremum is either itself or another chain (with a new supremum)
5. For any pair that is a chain, there exists a set of chain of elements between x and y by the order

we assume that a function f exists such that x < f(x) and the set is still chain complete, next is to find how we can define a contradiction without repetitive application

[tender tulip]

Induction might possibly be the only way

[wooden flax]

What does "fail" mean

[vivid halo]

God I got another like

11 hours to go until I can get into a bed

[wooden flax]

dont worry, if you read a math textbook you can probably fall asleep anywhere within 10 minutes

[vivid halo]

[wooden flax]

especially an algebra book!

[vivid halo]

[narrow rock]

[bright hill]

ng must be cringing so hard rn

[wooden flax]

lmao fr

[stark charm]

Reading a topic and not having interest in it can make you sleepy

But sometimes you can develop an interest this way

Not all boredom comes from a bad place

[wooden flax]

that was a joke

[crisp meteor]

hey guys, do you have recommendations for latex apps on android? and also any nice resources to learn latex?

by latex apps, i mean ones that support \usepackaging stuff and etc

[devout nacelle]

Android devices isn't where I would usually expect someone to work with LaTeX

Although I do remember one app that was functional, let me see if I can find it

VerbTeX

About learning: you could use one of the many introductory tutorials on YouTube or the Overleaf website. Most of it is however learnt continuously as you work on LaTeX itself and constantly look for solutions online (on platforms like TeX Stackexchange).

[clever knot]

Plug the ryc video

https://youtu.be/CSjVUbVGlA8

Oh and the Google doc

https://docs.google.com/presentation/d/1HsmNIN9acY6rYQwOYPJ97NS8dzxbdRFT-UKJqLvNYKI/edit?usp=sharing

@crisp meteor ^

[crystal tendon]

j

[odd narwhal]

Where in the proof of bourbaki-witt do they use zorn's lemma. You don't need choice to do transfinite recursion, they're already well-ordered

[fading hull]

You can't fail the SAT

[split glade]

Further to what @odd narwhal said which is correct, you CAN prove Bourbaki-Witt with Zorn's lemma. But the proof would be completely trivial. Bourbaki Witt says: every inflationary function f (i.e. f(x) >= x for all x) on a non-empty chain-complete poset X has a fixed point. Zorn's Lemma implies X itself has a maximal element y, so y <= f(y) implies y = f(y). So done.

But this would be missing the point entirely. Bourbaki-Witt is the 'choice-free' part of Zorn's lemma: it can be proved without AC and it easily implies Zorn with AC

[odd narwhal]

That's a good point greenman, thanks

[crisp meteor]

thank you!

[crisp meteor]

thank you

[tropic isle]

Introduction to Probability, Statistics, and Random Processes https://g.co/kgs/Fs2Urq
opinion on this book? (is it worth reading?, are there better books?)

[tender tulip]

every proof i've seen uses either Zorn's Lemma or Transfinite Induction

[bright hill]

bruh

[olive kayak]

https://www.amazon.com/Brownian-Stochastic-Calculus-Graduate-Mathematics/dp/0387976558

what are prereq for this book?

do I just need measure theory or do I also need course on measure theory based probability and stocahstic processes?

[deep mango]

You should probably know some measure theoretic probability

[olive kayak]

okay

[deep mango]

I also don't think they're going to like

Give you too much background when it comes to discrete time processes

This book starts off pretty strong

[olive kayak]

oh okay

[deep mango]

in terms of intensity

[olive kayak]

maybe I should first take a course on stochastic process

like the syllabus states prereq so vaguely

[deep mango]

i don't know if that's super essential, but knowing measure theoretic probability theory is a must

yeah

[neat frost]

Ryc I just had my final, I finished stupid Hebrew

[deep mango]

sometimes people also teach stochastic calculus in like

[olive kayak]

can you recommend me some good books on that

our school does not offer course on measure theory based probability

except for gradaute ones

[deep mango]

Nice! I hope it went okay, I know you were frustrated about it yesterday

[deep mango]

but they're offering a stochastic calculus class?

or do you just want to read this?

[olive kayak]

actually its pretty strange

the problem is

the probability course our school offers is heavily based on

financial mathematics

so it is fairly applied and lacks contents needed to tackle stochastic calculsu

[deep mango]

cuz what i meant to finish here was "in a way which doesn't rely on measure theoretic probability"

similarly to how you can teach continuous probability without measure theory if you use riemann-stiltjes integration

[olive kayak]

btw stochastic calculus is my ultimate goal during undergraduate

[neat frost]

Me? Frustrated? Never ryc, never.

[deep mango]

i see

[olive kayak]

maybe I will take class on that subject 2 years later

so I have some time

[deep mango]

i didn't learn this stuff until this year (first year graduate)

yeah

i mean the book i read for measure theoretic probability, Varadhan, was super hard

but also really good

I'm sure there are better options out there

[olive kayak]

https://www.amazon.com/Probability-Theory-Courant-Lecture-Notes/dp/0821828525

this?

[deep mango]

I don't know the canonical ranking

yes

[olive kayak]

okay

[deep mango]

i think i found them for free pretty easily

[olive kayak]

plan for me is to take classes on manifolds and measure theory next semester

probaiblity the next semester

[deep mango]

cool

[olive kayak]

and finally stochastic process after im ready

i mean calculus

do you think its realistic/

?

[deep mango]

here are the contents of varadhan

[deep mango]

yes, that sounds realistic to me

[olive kayak]

anyways thnx

[little vine]

Doob

[deep mango]

Doob is all over probability

Funny name

[olive kayak]

btw I know its subjective but how difficult do you think stoch calc is when you learn it first time

[prisma swallow]

Dynamic programming ?!

Makes sense it is the last chapter

[compact tartan]

my introduction to stochastic calculus was very smooth and easy

depends on how deep you wanna go I guess

[deep mango]

well i found it very smooth because i'd just learned a bunch of discrete time stuff and the theorems are pretty similar

[cyan goblet]

good morning chat

[deep mango]

and tbh

none of the definitions are super weird

if you've done enough measure theory / probability / analysis they're all the natural definitions that you'd want to make

e.g. Ito integrals

[olive kayak]

oh okay

[neat frost]

Good morning anemone

[deep mango]

but the results can be strange, a lot of the subject hinges on the fact that brownian motion is almost surely nondifferentiable

"hinges" is maybe a strong word but

there's a lot that comes out of that

[split glade]

Again, the proof using Zorn's lemma is a single sentence. X has a maximal element, so for any inflationary f on X, x <= f(x) implies x = f(x). QED. This is a bit nonsensical and I doubt it's the typical proof you've seen. The typical proof of Bourbaki-Witt (which you can read for yourself on Wikipedia) uses only definition by recursion on the ordinals (maybe what you call transfinite recursion) and Hartog's theorem, neither of which use choice or Zorn's lemma. I'm confused why you initially asked for a proof without Zorn when the most typical proof of this theorem doesn't use Zorn.

[tender tulip]

fair

[bright hill]

Why does "an usual person" sound weird?

Despite seemingly abiding by the "a/an" rule?

[restive bough]

because the u in "usual" is not acting as a vowel, it's acting as a semivowel diphthong

[arctic grove]

dipthong?

oh god its a real word?

[primal seal]

Also, it would be extremely confusing in speech, as it sounds close to ‘unusual’

[restive bough]

I'm not spelling it in IPA but in more sound-like spelling it would be "yoozhoowal"

[restive bough]

yep, wait until you hear about triphthongs and tetraphthongs

[arctic grove]

[bright hill]

[swift sinew]

its much more natural to switch from n sound to a vowel sound than it is to switch from n to a "yoo" sound

[restive bough]

monophthong: one vowel sound in one syllable (a in father)
diphthong: two vowel sounds in one syllable (o in hope, or ow in cowl)
triphthong: three vowel sounds (ueue in queue)

[swift sinew]

the former is just opening the mouth and voicing, the latter involves a whole setup and movement of the lips and tongue

[next schooner]

Who wtf pronounces queue with a triphthong

[restive bough]

most people

it's the same as the triphthong "you" with an extra plosive

[swift sinew]

kehyoo

checks out

[bright hill]

OMG JOSH'S RAT IS SO DERPY IN ANDROID

HAHAHA

[restive bough]

there's a shift at the very end of the "oo" sound that becomes more of a "w" sound, making it a triphthong

[next schooner]

Right I guess this is a US English thing

[fervent pebble]

omg that's actually a goated banner lmfao

[restive bough]

I'd need to be on my phone to write it in IPA

[fervent pebble]

i love the rat

[restive bough]

as opposed to? the English do it more than Americans do

[swift sinew]

so if you cut it short so its a cleaner sounding word, then is it just a dipthong?

[bright hill]

Check this shit out

[fervent pebble]

oh wait we're talking abt diff rats

i meant this bad boy

[next schooner]

I mean the UK is a dialectical nightmare. Guess I'll have to listen out for it.

[bright hill]

ikr

What an absolute unit

[restive bough]

could be, a pure "oo" is hard to end on neatly. it is done, but not commonly, and it often gets mutated by the words around it

[fervent pebble]

too?

i mean

would too be an example

[restive bough]

as the end of a sentence it can be. it depends on dialect

[fervent pebble]

hmm

[next schooner]

Yeah that checks out tbh

Neat

Never noticed that

[fervent pebble]

the only other one i can think of off the top of my head is goo

but goo has the same sound as like shoe, or flew, or dew

so not sure it counts

[sleek wing]

poo

[fervent pebble]

...

[restive bough]

this also depends on what your first language is. Native english uses glides EVERYWHERE which makes there be very few true monophthongs in the language, whereas languages like Spanish pronounce their vowels very clearly and purely, and the northern germanics like Norwegian tend to land in between the two

[hushed turret]

josh when's the next rat fact dropping

[next schooner]

soon

[swift sinew]

Ya boy is now officially registered for his first semester of graduate courses

[next schooner]

based

what are you taking?

[swift sinew]

Just a General Algebra course, a Real Analysis course, and a Manifolds and Topology course

the basics

[next schooner]

good stuff

[fervent pebble]

real anal is basics??

wait

for ug?

no way

[tender tulip]

For this major set theory weirdness, I’m assuming ZF

Well, for sets like the Reals, they are axiomatically defined up to isomorphism mostly by the Dedekind-complete total order, and it’s field operations. Well, to specify certain elements from the endless abyss of the continuum, we use the order and operations to specify them.

When we ajoin existence axioms to ZF, most notably the existence of well-ordering/choice-sets [which essentially do the same thing in a way], we say certain structures exist regarding the elements of every set… even one’s like the reals. For the names example, that we can “chose” an element out of each subset in a [specified] partition of R WITHOUT specifying the elements explicitly.

Are most “theorems” in ZFC that say you can form structured over sets WITHOUT specifying its elements explicitly correlated to Choice

[fervent pebble]

OH grad

thank god

gj alex

[tranquil dove]

is anyone here done with their igcse pure maths paper 1

if yes then how was it

[bright hill]

real analysis can be taken first semester of ug if that's what you're asking

he's probably taking something more advanced tho lel

congrats alex!

[deep mango]

oh yeah you're starting grad school this year or something right? sick

[stark charm]

Yeah they are attending UDN

[queen raft]

what do P and Q mean in the cubic formula?

[atomic hornet]

Congrats! Im not officially enrolled in my program yet… waiting for UG transcripts

[sick burrow]

Room assignments are in for my REU and I also now know the other people who are going to be on my project. So I'm debating whether to send out an email suggesting setting up a groupchat before the program starts.

[fervent pebble]

defo do

establish rapport 👍

also gj on getting an reu

[sick burrow]

the upside is possibly getting to know my fellow students before the project starts, the downside is possibly having the other students not like the idea and ending up embarrassing myself before the project starts

[fervent pebble]

the idea being the groupchat?

[sick burrow]

yeah

[fervent pebble]

eh i cant think of a reason any normal person would dislike the idea

[peak tide]

i agree, worst case maybe one or two antisocial people won't want to participate but even they probably wouldn't think it was a bad idea

also agree with the congrats on the reu @sick burrow 👍

[sick burrow]

yes I am very excited

[peak tide]

what's the general area/field of focus?

[sick burrow]

it's fairly broad and I don't know the specifics yet but some sort of discrete stuff I think

[cold needle]

oh wow congrats

[next schooner]

Awesome congrats

[wild lantern]

If I were doing an reu and somebody set up a group chat for it I'd join it.

[cold needle]

Do it, itll save you a bunch of hassle later on

i did it for mine and i learned a lot of stuff that way

plus everyone is probably gonna be nice

[dire mulch]

I hope when I apply for grad school after I graduate they will overlook my first semester at undergrad.

I was not ready for university, had a big wake up call.

[deep mango]

They tend to do that

Usually theres a place to put a note about that

Everyone has the blanket "covid school was weird" excuse ofc

Also welcome back from your permastudy

[dire mulch]

My first year in college was behind a laptop. lol, caught me. It was too boring after I finish studying.

I decided to get rid of discord from my phone. That will definitely help.

[deep mango]

Thats a good idea

I wanted to do that too

Then they asked me to be a mod

I still could I suppose

[dire mulch]

Didn't you want to be mod

[deep mango]

Well yes

I also didnt want to delete discord from my phone

[cyan goblet]

real

[deep mango]

But I was going to for productivity's sake

Turned out fine anyway

[cyan goblet]

i tried to uninstall everything to prepare for midterms

lasted about 2 hours

[deep mango]

Almost didn't though!

Yep

I did remove discord from my taskbar

On my laptop

Now I gave to navigate to the start menu for discord

[rose dock]

I need to get off my league of legends addiction once I get to college

[deep mango]

Yes you do mr grad classes in freshman year at berk

[dire mulch]

called me out.

[deep mango]

Or ms idk

[rose dock]

Mr.

but ye

[dire mulch]

That is the only game I have on my phone.

[deep mango]

[rose dock]

I skipped like 5 days of school to play league

[deep mango]

Simon tathams puzzle collection moment

Thats all i need

[dire mulch]

Ok, lol not as addicated as you.

The only time I ever skipped school was to play swtor in highschool.

Star wars old republic.

[rose dock]

probably a more respectable game choice

[dire mulch]

What your favorite champion you use a lot?

[rose dock]

master yi because I have no brain

[dire mulch]

Ah jungle person, I always use caitlyn and lux.

I can't fight close range no matter how hard I try.

[wild lantern]

I could never skip college classes. What if my professors miss me gracing them with my presence? I could never be so cruel.

[dire mulch]

My last course I had I went to every class. The professor scared me and everyone in that class by saying if you miss 3 classes you fail.

[deep mango]

I dont think I ever skipped a lecture except for if I was sick, if I was traveling for a good reason, or if there was a recording

Hmm

[graceful crag]

Can someone help me with some problems

[deep mango]

Check out #❓how-to-get-help

This isnt a help channel

[arctic grove]

whos taking grad classes in freshman year

[fair estuary]

Does pi have a definition?

[arctic grove]

yes

pi is defined as the ratio of the circumference of a circle to the diameter of the circle

[fair estuary]

Yeah that makes sense.

It seems like there could have been other ways of doing it though.

[fair estuary]

Actually isn't it circumference to radius?

[arctic grove]

no

thats tao

or 2pi

[fair estuary]

Oh ok

I feel like the circumference has to stay the same or be in the ratio

But the diameter or radius part, that length can be defined by a lot of other stuff

[peak tide]

there are other ways. in real analysis pi is defined as the first positive root of sin, or an equivalent definition

where sin is defined in terms of its taylor series or in terms of a differential equation

[fair estuary]

Like we could make something like a quadius which is 1/4 a radius.

[peak tide]

and not in terms of the geometry of a circle

[fair estuary]

Can you also get tau from the positive root?

[peak tide]

by tau do you mean 2*pi?

[fair estuary]

Yes

[arctic grove]

that would be the second root of sin

[peak tide]

then sure, once you have defined pi you can just define tau as two times that

or yeah, second root of sin

[fair estuary]

But say people decided that tau was our circle transcendental number from the very beginning.

Then there would be no pi, just 1/2 tau

[peak tide]

it doesn't matter which one you define first, you can always define the other one once you have the first one

if you have tau defined then define pi := tau/2, done

[fair estuary]

People could define a circle transcendental number with any natural number or fraction, but do you think they could do it with other numbers like negative numbers or with other transcendental numbers?

[fervent pebble]

what

[fair estuary]

Say you have circumference of a circle

C =2(pi)r

You can have a different transcendental circle constant if you change up the 2r part of the formula.

[fervent pebble]

i dont understand what you're saying

[fair estuary]

I'll give you an example

let's use ć as the new transcendental circle constant

And we'll redefine radius to have a different ratio

You could have a formula for circumference of a circle like C=4(ć)r

And ć would be that new transcendental circle constant

It would be the same as pi/2

[fervent pebble]

whats the point then

?

[fair estuary]

It just feels like you can have any transcendental circle constant that you want

Like it is an arbitrary definition

[torn willow]

Well you need to define what a circle is

Defining with a circle is cyclic

[arctic grove]

(x-a)^2+(y-b)^2=r^2

go crazy now

[torn willow]

Locus of all points at a constant distance from a point works

[arctic grove]

ik

[wild lantern]

Part of the point is you get the same constant no matter what circle you use. That seems pretty non-arbitrary?

[arctic grove]

defining with a circle isnt a bad way to introduce pi

[fair estuary]

You don't though :(

[wild lantern]

Like, you can scale it and say "I wanna use pi/2, 4pi, 222233pi" or whatever but you'll still have some value in terms of pi?

[arctic grove]

and u can formalize to make it afterwards with geometry

[wild lantern]

Can you find me a circle where C=2pir doesn't hold? 🤔

[fair estuary]

That's because we already decided to define pi as our circle constant

[fervent pebble]

imagine not introducing pi as the series tbh

[arctic grove]

ofc

[wild lantern]

Okay, so it's just unique up to constant multiples?

[arctic grove]

we should teach kids about taylor series

5th graders will love it!

[wild lantern]

What's wrong with that?

[fair estuary]

I'm trying to figure out if you can use other transcendental numbers in the ratio

[arctic grove]

u can

but why would u want to do that

[fervent pebble]

you can help them memorize it using mnemonics with taylor swift songs

[torn willow]

A convenient definition of pi is ratio of area to square of radius

[fervent pebble]

lets define a to be (pi * e)/e

[fair estuary]

Can you give an example?

[arctic grove]

also works

[fervent pebble]

boom

[arctic grove]

u just did

multiply divide pi with any constant

and then u can change the equation C=2pir

accordingly

like i can define pi as tao/2

[fervent pebble]

shyshu praise my intelligence

[torn willow]

But then you have to use some predefined property of pi wrt sin or cos to prove the property of pi wrt a circlr

[arctic grove]

then i get C=tao*r

[torn willow]

You just can't escape it

[fair estuary]

Can you do it with e another transcendental number?

[arctic grove]

praise be

[arctic grove]

sure

[wild lantern]

Why you gotta do ma boi terry like this shyshu?

[arctic grove]

ur equation will change accordingly

[arctic grove]

[wild lantern]

(Jk)

[arctic grove]

yeah this

u will basically end up with pi all the time

no matter what other number u use

so why use something else

[fair estuary]

Can you show me the version with e in the ratio?

[wild lantern]

C=(2/e)(epi)r

[fair estuary]

Hmm yeah

[arctic grove]

yeah this

or u can do pi/e*2re

same thing

doesnt affect anything

[fair estuary]

I think people would be able to figure out the e though and divide it out

[arctic grove]

thats the thing

it doesnt matter whichever constant u use

[fair estuary]

I think fractions and integers can work though

[arctic grove]

u will end up with the same thing

[fair estuary]

The problem is that I have something where it doesn't end up the same

[fervent pebble]

show us

[fair estuary]

They know the thing. It'll make everyone mad though so I don't think it's a good idea to post it

[fervent pebble]

?

the fuck

[torn willow]

I guess you can define π as "half a rotation" in radians

Then you can show area of a circle is πr^2

And then derive circumference is 2πr

[torn willow]

This is still problematic. You need to define a lot of stuff for this to work

[fair estuary]

It just feels like circle constants themselves are arbitrary

[mint patio]

pi shows up pretty naturally

Ratio of circumference to diameter

[fair estuary]

It's circular reasoning though

[mint patio]

Is it? Make a circle, measure the circumference and diameter using another tool, approximate the ratio

[fair estuary]

If they did it circumference to radius then you have tau

[mint patio]

Sure

[fair estuary]

And then you'd be telling me how tau shows up pretty naturally

[mint patio]

Right

Because it does

They both do

I'm not sure we chose to define everything in terms of pi instead of tau

So perhaps that part is arbitrary (there may be a good reason, I don't know lol)

But the constants themselves show up very organically

[arctic grove]

afaik there isnt

u can define all we have in pi in terms of tau

[mint patio]

Yes

[arctic grove]

it wont change shit

[fair estuary]

If there something like a pentadiameter at that time where pentadiameter was 1/5 the diameter then the circular constant would be something else.

[arctic grove]

coz u know

they are both constants

[mint patio]

Sure it would!

And our entire mathematics system would be different as a result