help-28

[twin wolf]

in order to move forward in life you must look backwards

[knotty grail]

an x direction is to help you locate

[olive saffron]

the "x axis" is simply an infinitely long 'line' that you can define

[covert heath]

btw @olive saffron for the record, OE replied to this and explained stuff really well until he had to go for a lecture

[olive saffron]

in 3D space you can choose any straight line and call this the x-axis

[covert heath]

so i dont know what this is about

i thought we had moved on

[austere cove]

lecture was cancelled, no one showed up but me.

[valid nimbus]

yes, but where would you place it? in which direction?
keep in mind here, that the object a has to decide which direction it should go, to it there's infinitely many directions. and which directions should it consider x, y and z for it to navigate properly

[knotty grail]

holy moly

[covert heath]

Hi OE!

[austere cove]

hi!

[covert heath]

welcome back

[austere cove]

I'm back

[olive saffron]

Any direction. Doesn't matter.

[covert heath]

he just copypasted the same thing again.

holy shit

[wary condor]

this is what everyone has been explaining for the past 5 hours.

[olive saffron]

I see

[valid nimbus]

but if the object A places x and y and z axis at random directions that it wishes to

wouldn't it end up in the wrong place

most likely

[olive saffron]

Well there is little to be gained with a running meta-narrative on what has happened in this channel

[covert heath]

what

[valid nimbus]

when the object b is already placed

[austere cove]

so first off, "x" direction is leading, you don't even need the directions to be orthonormal, just independent. Your space might not have a well defined notation of the "x" direction as we generally think of it.

[olive saffron]

What is object A, B, etc.? Why do you have objects in the first place?

You can create a 3D space without any objects in it whatsoever

[austere cove]

however, for any euclidean space we can choose an orthonormal set of vectors that span it.

[wary condor]

-# here comes the direction of the axes matter.

[valid nimbus]

well an empty space with nothing wouldn't require anything, a line is also an object, a point is also an object. and in a 3d space, for there to be 2d/1d objects... well u need to have construction of objects right?

[olive saffron]

Sure, you can consider a line to be an object in some sense

[austere cove]

Well, I did explain how to construct a line previously.

[olive saffron]

Likewise for a point

[valid nimbus]

okay now how would u construct a line between 2 already existing points/objects?

if u take x, y, z axis to be in any direction

u have have multiples of co ordinates and equations

that lead to the same line

[olive saffron]

Define "construct"

[covert heath]

OE explained this too previously.

[valid nimbus]

a mathematical equation of the line

[covert heath]

it's like a broken record

I thought we had moved on!!

damnit

[valid nimbus]

hmm well

[austere cove]

please be constructive

[olive saffron]

Again, the meta-narrative doesn't help

[austere cove]

ok, so there are two separate things going on here. First, you have a space, and second, you have some set of coordinate systems.

[covert heath]

I'm just pointing out how he either doesn't well formed questions or is just... not in good faith?

Because... it was answered, right?

[austere cove]

I'm assuming good faith plus confusion

[covert heath]

Anyway if you're fine reexplaining I'm not complaining

[austere cove]

Any coordinate system you define can address any point uniquely in the space with a ordered triple

[olive saffron]

One must assume good faith until given good reason to assume otherwise

[austere cove]

And the same point can have different addresses when different coordinate systems are in use.

[covert heath]

but maybe one can just go back and reread and ask follow up questions?

rather than the same thing again

[covert heath]

alrighty

[austere cove]

I understand that you are frustrated. But please don't interrupt, because it just makes everything more busy and confused as a result.

So circling back (no pun intended), we have a space, and we have points in that space, and we have multiple ways of specifying how to get to points in that space (coordinate systems)

[valid nimbus]

currently, my confusion just lies in this;
there's set 2 points, from point a u have to reach point b via a line, but if someone says just any co ordinates such as 5,0,0 for point b and point a to be 0,0,0.
then u can take numerous directions as x, y and z axis, u would have numerous answers, but only one of them is correct.

My question is, what tells the point a which one is the correct representation in relative directions of point a

[austere cove]

any coordinate system is equal to any other, in the sense that if we construct things like lines, etc using the coordinate systems, the lines will always be the same.

even if the numbers we use to represent the points are different between coordinate systems, the points themselves are the same points.

[valid nimbus]

well yes

[austere cove]

congrats by the way, we've argued about this enough that you've gotten active

[valid nimbus]

5,0,0 with x axis being in one direction and so and so, could be the same as 0,10,6384 with another representation of x axis

[austere cove]

exactly.

[valid nimbus]

But

[olive saffron]

There is a mathematical concept of "changing basis" which you seem to be kinda hitting on without necessarily realising it

[valid nimbus]

if we have a situation where we have no other information as to which direction we should represent x, y and z axis towards to get to the right point B

[onyx glen]

active role speedrun strats crankery timeskip

[valid nimbus]

how would we know that? assuming we don't know the set up already, then how would we reach the correct position

[austere cove]

That information is in the coordinate system itself.

[valid nimbus]

hmmmmmmmmmmmmmmmmm

[valid nimbus]

suppose we have 2 humans, one controlling point b's placement, another point a's movement.
their task was to make point a reach point b,
And human 2 who kept the point b's position, said human 1 that it's at 5,0,0 when point a is at 0,0,0

but then their connection broke off

[austere cove]

For instance, you are correct that there are multiple (infinite really!) coordinate systems such that point A is at (5, 0, 0)

[valid nimbus]

before human 2 could specify the x, y and z by a map

now how would human 1 navigate?

[austere cove]

without more information, we can't figure out which coordinate system is in use just from this

[valid nimbus]

I see

so human 1 cannot navigate

[austere cove]

without more information

you need to give the coordinate system. Alternatively, you can give the locations of 4 points

(noncoplanar)

[valid nimbus]

what really bothers me is, how can 3 dimensional objects even realize something so small

i mean

[austere cove]

I don't understand what you mean by "realize" here

[valid nimbus]

point out something that small in area (0) Without error

Also

[austere cove]

we are dealing with math, if I say a point is at (pi, e, -1) then it is precisely there.

error is its own can of worms

[valid nimbus]

what prevents 3 dimensional objects from being infinite

[austere cove]

nothing, 1d objects can be infinite

a line is infinite.

[covert heath]

3d space is infinite.

[valid nimbus]

I mean it in actual size

3d objects

Because

we can be infinities but bigger and smaller right?

just like the set of natural no.s

[austere cove]

even "finite" objects tend to be comprised of infinite sets of points. A sphere is every point that is within a certain distance of another.

[valid nimbus]

and rational numbers

[valid nimbus]

well yes 3 is certainly finite

but

[austere cove]

natural numbers and rational numbers are both the same size infinity, for the record. But let's not confuse things by introducing new topics.

[valid nimbus]

i dont see the limitation of 3d masses

being finite

I don't see the limitation that I see for numbers like 3

[austere cove]

I don't understand what you mean by "masses" here.

[valid nimbus]

3d objects

3d objects being, 3 meters square or 5 meters cube

[proud dirge]

Why would there be a limitation

[valid nimbus]

is all in all, just measurement of its surrounding right?

[austere cove]

like the points inside of a sphere are infinite though.

[valid nimbus]

a meter cube that we measured couple also be infinite

[austere cove]

it has a finite volume, but the points themselves are infinite.

[valid nimbus]

but this object is like rational numbers, and more infinite than a meter cube of object

[valid nimbus]

well yes

a sphere could be infinite

[proud dirge]

This doesn't make sense

[austere cove]

every sphere has infinite points inside of them.

[valid nimbus]

right but

how about u see a thought experiment first

[austere cove]

so what do you mean by infinite here? What I'm trying to get you to do is make what you're talking about precise.

[restive geyser]

That is, these words have meanings already; so if you're going to use a differing meaning, it's your job to explain what they are.

[valid nimbus]

imagine that, just like our universe, another exact copy of the universe is present in our 3d space,
But this copy universe, is soo small in diameter.
as in; if we take a scale of one quark to this entire universe, the copy universe would be like that quark, to an actual our quark

[austere cove]

because it is my thought that maybe the reason you're getting so confused over things is that you are being imprecise, and then tripping over yourself when you attempt to use imprecise things to reason.

[valid nimbus]

and we can go on forever, while all of them having same thing as perceptions really

exact same perceptions

but different sizes

wouldn't this be able to be possible

and we be infinite, really

just like set of natural numbers

and rational numbers

we differ in sizes

but are infinite

how about this?

[restive geyser]

stop talking

like this

for each phrase

[valid nimbus]

because I don't see something quantifing 3d objects

alright

[proud dirge]

They don't

[austere cove]

If you shrink the entire universe down to a point, and then ask "what happens if I explore within this point"? No. If you shrink it to a point, then it ceases to be 3d. Points are definitionally 0d.

[restive geyser]

It's really difficult to read and parse through

[austere cove]

anything larger than a point, and it wouldn't matter.

[valid nimbus]

right right but, the copy universe in the thought experiment, wouldn't be an actual point, but would be so small that we can never notice

[austere cove]

we just have a different set of coordinates to deal with the small scales involved.

[valid nimbus]

and in that copy universe, there would be another copy universe

same exact idea

and so and so

making us infinite

so mass could be infinite right

[austere cove]

you're confusing several concepts here

[valid nimbus]

and only be measurable on similar infinites like our scales

[valid nimbus]

like I don't push my idea too far

[austere cove]

we're not talking about the "universe" or "mass" or anything do to with physics. We are still talking purely mathematics, yes?

[valid nimbus]

but I just wanna know why mass isn't infinite like this

[valid nimbus]

yeah purely maths I know

[jade pebble]

what is the original question?

[valid nimbus]

but consider this a thought experiment of maths

why would mass be limited here

I mean.. 3d objects

[jade pebble]

well, what was.

[valid nimbus]

whyd they be finite here

[austere cove]

mathematically speaking what is "mass"? Mass is typically a physical quantity of objects in physics. Not in mathematics.

[valid nimbus]

right sorry

3d objects, whyd they be finite here

[maiden vapor]

NORTH! Get ouf of here!

[jade pebble]

what??

[austere cove]

What do you mean by finite and infinite?

[proud dirge]

Leave before it's too late

[valid nimbus]

well, u can divide 3d objects infinitely into smaller and actual 3d chunks

if u couldn't do that infinitely without breaking down into 0 of each

[austere cove]

if they are not point objects, then yes.

[valid nimbus]

then that would be finite

[valid nimbus]

and so, why would 3d objects not be infinite then

[austere cove]

any 3d object that consists of infinitely many points, such as lines, circles, discs, squares, balls, etc, would be infinite in this framework

[valid nimbus]

but

it isn't just points tho

it's infinitely many actual 3d objects

not just 0d points

actual 3d objects

so that would make it infinite right

[austere cove]

define what an "actual 3d object" is, please.

[valid nimbus]

length width breadth

[covert heath]

length width breadth is an object?

[austere cove]

a pyramid doesn't have a single length width and breadth?

[valid nimbus]

one that has non zero, positive length width breadth

[austere cove]

would you consider a tetrahedron to be a 3d object?

[valid nimbus]

Yes

but it's non zero and positive

[jade pebble]

A point in an n dimensional space can be described as a list of n values.

[covert heath]

also why are we talking about infinities now

i mean sure we can talk about infinities

but did you figure how lines work, yet

[valid nimbus]

What's a

Tetrahedrpn

[covert heath]

are we gish galloping

[austere cove]

I think what you're saying is roughly, and correct me if I am wrong: "a 3d object is a set of points for which the smallest rectangular prism that contains all of the points is not degenerate (i.e. has a non-zero volume)"

[jade pebble]

@valid nimbus

[austere cove]

is this what you are trying to define?

[valid nimbus]

it's not a set of points

points have no width, no length no breadth

even if u place 2 points together

it would form no length no width

no breadth

[covert heath]

valid question

[valid nimbus]

and

that's why I argue 3d objects essentially have to be infinite

because there's nothing that makes them up finitely

u have infinitely smaller 3d "objects" that make them up

like numbers of a set

[covert heath]

'numbers of a set'?

[valid nimbus]

well, bad analogy maybe

[covert heath]

analogy?

[proud dirge]

It's an infinite set of points.

[jade pebble]

moderators, please don't purge this channel, let this be historic site.

[covert heath]

why bring up analogies

[split hatch]

Why must a "3d object" be infinite? Take any 5 vertices that form a square pyramid, then this set of 5 points is surely 3-dimensional?

[valid nimbus]

it's not an infinite set of points

[covert heath]

it definitely is

[valid nimbus]

a point has 0 width

[proud dirge]

So what

[austere cove]

@valid nimbus take a cube, and consider any vertex. This vertex is joined to 3 others. Now remove the entire cube except for the three lines that join these 4 vertices. Is this object a "3d object"?

[valid nimbus]

if u have 5 points ,let alone infinite points

it would be 0 width

[proud dirge]

No

[valid nimbus]

wouldn't be able to touch positive

[jade pebble]

A point in an n dimensional space can be described as a list of n values.

[valid nimbus]

hoe so

0 times n is 0

[proud dirge]

You have infinite points

[jade pebble]

A point in an n dimensional space can be described as a list of n values. @valid nimbus

[proud dirge]

You can't just add 0 infinite times

A course in measure theory would help here

[austere cove]

or topology yeah

[valid nimbus]

hold on

[covert heath]

welp

[valid nimbus]

I think ur onto something here

well

the 3 lines

would be 2d

[covert heath]

this would end so much quicker if he heeded any of our advice to 'just pick up a book'

[valid nimbus]

but the intersection

should have

width length breadth

sheesh

[austere cove]

3 lines are each 1d separately

[valid nimbus]

yeah same same

hmmmm

but

[proud dirge]

this

[valid nimbus]

something doesn't feel right

[jade pebble]

A POINT IN AN N DIMENSIONAL SPACE IS A LIST OF N VALUES, AN OBJECT IN AN N DIMENSIONAL SPACE IS A COLLECTION OF SUCH POINTS, OR LISTS.

[valid nimbus]

how can u have

infinite points

at one point

[covert heath]

wha

[proud dirge]

what

[austere cove]

sorry, but that question is nonsensical

[valid nimbus]

well u see here

those 3 lines

are one dimensional

but the intersection, if it is 3 dimensional, that means it's made up of infinite points

but it should be just one point

of really small lwb

[austere cove]

so I would define an "object" in 3d space as I defined it before. it's a set of points for which the smallest rectangular prism that contains all the points has a non-zero volume. For some precise definition of smallest (infinum)

[valid nimbus]

hmm

[austere cove]

by that definition, not all 3d objects have infinite numbers of points.

but most common 3d objects that you care about do

[proud dirge]

This whole conversation is the reason why definitions are important in mathematics

[austere cove]

note that you shouldn't conflate the notion of infinite amounts of points with infinite volume

[valid nimbus]

sorry I don't understand something here

[jade pebble]

finally a revolution

[austere cove]

please be nice

[valid nimbus]

why is it that there's not be able to be more than 1 points at a point?

[jade pebble]

i am not even nice to myself how am i supposed to be nice to people.

[valid nimbus]

I'd understand why not for something like a 1/2/3d object

but not a 0d object

that shares no space

[jade pebble]

i am trying my best ok

[proud dirge]

Because it's a point

And a point is a point

[jade pebble]

why did this get 4 sobs

[austere cove]

How would you distinguish two points from each other, if they are located in the same spot?

two objects in math are the same if every feature that we care about is the same, right?

the only features a point has is its location

[proud dirge]

Tfw the space is not T1

[valid nimbus]

How can you distinguish a point at all with respect to 1/2/3d space/objects

[covert heath]

what?

[austere cove]

I can define a point by simply taking my coordinate system, and then identifying it using the coordinate system

My point is at (0, 0, 5) for instance

now you say that I have "infinitely many" points at (0, 0, 5), but how can you tell?

there certainly is a point at (0, 0, 5)

If there are infinitely many points at some coordinates but not others how would you distinguish between the two cases?

(heck if there are 2 points at a point instead of just 1 how would you tell?)

@valid nimbus you still with us?

[jade pebble]

archive this channel as a historical site and create a new help 28 channel.

[full forum]

@valid nimbus Has your question been resolved?

[valid nimbus]

Hmm

[jade pebble]

he's back

[covert heath]

he has!

[valid nimbus]

well you'd be right kind of

[valid nimbus]

but where would the NEXT point to it be?

[austere cove]

there is no next point

[valid nimbus]

5.00000...1?, 0,0

[valid nimbus]

well just like 5,0,0

why can't u

make

another one

Beside it

[austere cove]

this is a feature not of our coordinate system, but of any numbering system that is "dense"

The rationals, for instance, are dense.

Let's say I define a number, let's call it x

and then I say that the next number is y

but then I can construct a new number between them

[valid nimbus]

Hm

[austere cove]

we have y > x, so y-x is positive

then I consider the number x + (y-x)/2

this number is strictly between x and y

so there is no "next" number

[proud dirge]

I don't think bringing more concepts into the conversation is a good idea 😭

[restive geyser]

-# Unfortunately, there's looking to be little alternative

[valid nimbus]

but in a 3 dimensional world, wouldn't there be another point beside it

[proud dirge]

No

[covert heath]

no

[austere cove]

no, why would there be

[valid nimbus]

hmmm

well

okay, how do u categorize masses then? if they're made up of infinitely small points, how many infinitely small points is 1 unit?

[austere cove]

So you're asking a lot of questions that require more learning on your part to understand the answers.

Specifically, as mentioned I think half a dozen times so far, you're looking for "measure theory"

[valid nimbus]

Hmm

[jade pebble]

"pre-university math"

[austere cove]

I know

it's a bit of a trek to get from where you're at to where you need to go

[proud dirge]

Learning measure theory in 9th grade would be wild

[azure onyx]

if the helper is asking questions that demand knowledge of particular fields, I think it's better to be honest than to tell them, "you'll get it later".

[austere cove]

but I'll try to boil it down

[azure onyx]

helpee*, sorry.

[austere cove]

so let's ignore 3d for a moment

[jade pebble]

editing messages is better

[austere cove]

let's instead take an easier case of 1d.

[valid nimbus]

hmm

[austere cove]

now we can define a measure of 1 as all of the real numbers on the interval of (0, 1)

so if we have a line segment from (-2, 3) then that line segment has a measure of 5

we start asking questions about what happens if we start removing points, how does this change the measure?

[valid nimbus]

Hm

[austere cove]

we can actually remove infinite sets points without changing the measure

but formalizing this is the key

we can also have finite sets of points with 0 measure.

but again, which sets matter, and formalizing it is the key

what I'm trying to get at is you're asking good questions, but the answers to your questions are complicated and depend on mathematics that you probably won't encounter for a good 5+ years.

[valid nimbus]

hmm

[austere cove]

we generally deal with "nice" sets

so if your set of points is "nice" for some precise definition of nice

then we can say that the measure (which in 3d space we typically refer to as a volume) is a particular limiting process that involves counting up tiny cube volumes and defining the volume of each cube to the its side length cubed.

[valid nimbus]

i see

[austere cove]

but there are stranger spaces we can work in than simple 3d euclidean space, and different notions of volume we can use, and the key is all of this needs to be defined.

nothing is "for free" in math, we have to be very careful how we import our intuitions

[valid nimbus]

hmm all of this needs to be defined

wait for some reason this brings me back to that idea

if you wanna form a line between two points, you have to specify the x axis, y axis and z axis direction, which is usually given by the co ordinate system

But if direction is a line

Like axis would be depicted as,

Then you need to define that x axis, y axis and z axis's directions

in the form of a line

but to generate that line

you need to specify the axis you're using

while doing so, you form lines of direction

and it continues

[austere cove]

so imagine I have a bag of points.

they are unlabeled.

[valid nimbus]

yeah

[austere cove]

but I have a method of grabbing points out of the bag.

just at random

[valid nimbus]

Ok

[austere cove]

I grab a point, I don't know where it is in this space, but I say "Ok, this point, I will name O, and I will say this point is (0, 0, 0)"

[valid nimbus]

Ok

I can agree upon that

[austere cove]

I grab a second point, I don't know where it is in this space, but I will say, "This point is A, and I will say this point is at (1, 0, 0)"

[valid nimbus]

Okay

that's cool too

[austere cove]

I grab a third point, I don't know where it is, but I will say "This point is B, and I will say this point is at (0, 1, 0)" but only after I test to make sure this point isn't in on the line OA

if it is on that line, I just grab a different point at random until it's not on that line

I can make the line OA because I already have O and A

[valid nimbus]

Hmm

you can

but

u need an equation

to do so

us drawing it isn't perfect like us saying oa

an equation, that shows all points of a set is what we need

right?

[austere cove]

What I'm saying is I can identify if two points are the same point by seeing if they're the same point in the space.

so if I have B and I have the line OA then these are both sets, and I can take the set intersection

if this intersection is empty then B is a valid third point, if it's not then I choose a different point

[valid nimbus]

but with this equation, we need to know about which direction x, y and z axis that the equation operates from, but the problem is, this equation literally would be for showing which direction x, y and z axis for our current line is

[valid nimbus]

well yes

[valid nimbus]

well yes

u can get a point where it's not in oa line

[valid nimbus]

well yes

[valid nimbus]

but, how can u really construct an oa line

it seems self contradictary

or paradoxical

[covert heath]

why foes it seeem self contradictory

or paradoxical

[valid nimbus]

well, u need a y = mx + b equation to depict the oa line right? let's suppose for simple case, o is 0,0,0

right now

you need to define the directions of x, y and z axis

[covert heath]

no you dont

[austere cove]

you're thinking about lines in terms of equations

[valid nimbus]

yes that's how they're perfectly formed

[austere cove]

but you don't need to think about lines in this way, you can think about this in terms of sets of points that satisfy some property

[valid nimbus]

sets of points

[restive geyser]

But that's not the only way you can define a line

[valid nimbus]

well yes that property is the equation

[austere cove]

that's one way to define it, but not the only way

[valid nimbus]

hmm

what else can u do

other than an equation

that shows an actual line

[austere cove]

I could just say, "here is an infinite bag of points" and it just so happens that each point in this infinite bag has that property

[valid nimbus]

mathematically how would you define that property

[austere cove]

I just did

[covert heath]

he has

[austere cove]

Formally, this would be something like ${ P \mid P \in \overline{OA} }$

[glossy valve]

OmnipotentEntity

[austere cove]

this is perfectly valid

[valid nimbus]

I can say too that, in my mirror universe, there is a bag that has infinite points and follows the same property that omnipo entity described for his bag, but it follows a property that doesn't count as an actual property yet is a property with all mathematical rigour that exists and disturbs his logic but it would count as the same bag for some odd reason, that follows all axioms properly but disturbs it and contradicts it while being rigorously accepted by actual mathematics

[covert heath]

but whats OA /gen

[valid nimbus]

well, that's only perfectly valid until u imagine logics/properties that disturb your thesis while not having to be considered as a different property bag, but still disturbing it

I'm not sure how or when or where

but in English, I can just say like that

[valid nimbus]

, that could be a property too

disturbing your logic, but still having to get counted as the same bag with properties

[austere cove]

the unique line that passes through both O and A

[valid nimbus]

how would u say this then?

[austere cove]

(we are assuming euclidean space, so there is a unique line)

[valid nimbus]

but

u need to speak in terms of an equation

[covert heath]

isnt that what were defining in the first place

(im fine thinking of other definitions)

[austere cove]

Let's talk about the Greeks. The Greeks did geometry of the plane without coordinates

we can construct a line via two points and a straight edge.

[valid nimbus]

the greeks drew pencil lines that mathematically wouldn't be counted as perfect lines

[austere cove]

no equation necessary

[covert heath]

what are logics

[valid nimbus]

because it wasn't rigorous enough to define what the actual lines are and what are not the actual lines

[restive geyser]

So? The maths still holds, because an equation of a line is not how we mathematically define a line.

[austere cove]

they certainly didn't have pencils.

[valid nimbus]

yes, I agree that, those set of dots can also be lines

but isn't mathematically rigorous enough

to differentiate if my nose

is different than those

why can't my nose, 3000 years later

[restive geyser]

Write a full sentence, THEN press enter, ffs

[valid nimbus]

be counted as a part of that line?

alright

[covert heath]

(ftr i still wanna know wht logics are)

[valid nimbus]

forget logics, I was talking about properties back then

[austere cove]

So in order to define a line, we need to define a few things about our space and about distance

[valid nimbus]

alright

[austere cove]

which means we need to define our metric

[covert heath]

no you were doing word salad

[austere cove]

and now we're getting kind of into the weeds a bit.

[covert heath]

dont do thst.

[valid nimbus]

yeah

[valid nimbus]

hmm I see

[austere cove]

So our space is equipped with a notion of distance

[jade pebble]

if this entire conversation was spent explaining measure theory, @valid nimbus would have a phd by now.

[valid nimbus]

yeah

[austere cove]

for any two points in our space we can define a function d(x, y) to give us back a number that represents the distance between these two points.

[knotty grail]

he should have had very active realistically

[valid nimbus]

something doesn't fit right

I thought distance was relative to human units like a metre, a football field, a kilometre, 17 hamburgers, etc..

how do u define absolute distancw

[covert heath]

units

[valid nimbus]

without taking into consideration of relative units

[valid nimbus]

that's relative tho

[austere cove]

to what?

[covert heath]

no its not

[valid nimbus]

if u take oa as 1 unit of oa

[austere cove]

sure

[valid nimbus]

then it's relative to oa

[covert heath]

no its not

[valid nimbus]

and if u take oa as 17 units of ch

it's relative to ch

[valid nimbus]

how many units is oa then

if u say 10 units why can't I say 10039 units then?

that's why it's relative

[austere cove]

may I ask a favor of you

[valid nimbus]

yeah

[austere cove]

can I get you to read a thing?

[valid nimbus]

yeah

what is it?

[covert heath]

you can

[covert heath]

foesnt make it relative

[austere cove]

give me a moment, I need to locate it online

[valid nimbus]

then how would we define an absolute distance unit that we both agree upon

[covert heath]

a unit is whatever i say it is

[valid nimbus]

I can say a unit is whatever I say it is

[covert heath]

yep

[valid nimbus]

and how do we mathematically, convey the true distance between a and be is

maybe

[knotty grail]

why?

[valid nimbus]

Actually

[covert heath]

'true distance' isnt a thing

its the distance in units

[austere cove]

there is no true distance between a and b

[valid nimbus]

Hmm

[covert heath]

(0,0,0) to (1,0,0) is 1 unit

literally

[austere cove]

I can't find it

but essentially, you're mistaking the map for the place

our space is the place, our units and metrics and so on are just a map

[valid nimbus]

I think the more important question is, why do u need to define oa in any units at all?

[austere cove]

the maps should agree with the place, but they are not themselves the place.

[covert heath]

OA is not in units

[valid nimbus]

I mean, distance between a and b

[covert heath]

distance (a measure) is in units

[valid nimbus]

why do u need to say they're in so and so units?

what next

[covert heath]

actually not sure im using measure correctly

[valid nimbus]

The space between them*

[covert heath]

OE can correct me

(because words have meanings which matter)

[valid nimbus]

Hmm

so u could define a line

by talking about those units

hmmm

[covert heath]

no.

[valid nimbus]

no? why else the necessity of those units then

u just measure the distance for what?

[covert heath]

for MEASURES

again not sure im using the word correctly.

but.

lengths areas volumes allat

[austere cove]

I mean, yes, honestly, we can define and translate between different measures, and we commonly call these units of different sorts to distinguish them

the idea is there is some correspondence to any consistent set of measures and units that you can choose that all agree with each other.

[valid nimbus]

is that not USELESS

[austere cove]

many maps, all describing the same thing.

but they're all just maps

[valid nimbus]

when is measuring a distance between 2 points useful?

[restive geyser]

An entire field of mathematics would like to have a word with you

[valid nimbus]

are u playing "who's bigger"

[covert heath]

i dunno mate

[valid nimbus]

or are u playing "I'm stronger"

[austere cove]

Stop taking pot shots please

[valid nimbus]

Alright

[austere cove]

if you're interested in learning, we're here to help

[covert heath]

i cant imagine why MEASURING STUFF is important

[valid nimbus]

but, what would you do with measuring so and so in x units

I won't criticise doing it

but like why.. what next

[austere cove]

we can measure distances

how we choose to measure them isn't important, the exact units aren't important

[covert heath]

do you want rigorous... anything to do with spaces?

rigorous probability?

[austere cove]

and the reason why it's not important is because any other metric that describe the same space will have to give results that agree

[valid nimbus]

but why would maths need to care for that? I thought maths doesn't care about relative systems like human needs

and this distances seems like human needs to me

[covert heath]

nobody said its relative.

[austere cove]

When I say distance, it has a lot of baggage in your brain that you've imported from the outside world.

[covert heath]

arbitrary \neq relative

[valid nimbus]

well forget the relative word

[austere cove]

but I'm not referring to any of that.

distance as it exists in the world of mathematics is just a function

it takes two input points, returns a positive value, and it has to satisfy the triangle inequality property, symmetry, and the fact that the distance from a point to itself is 0

any function that satisfies these properties is a metric.

https://en.wikipedia.org/wiki/Metric_space

[restive geyser]

(I'd rather something easier to digest than Wikipedia for this, tbh)

[valid nimbus]

but something seems troubling to me that distance is an actual real function, because distance in our real world is just useful for needs, and I don't see it being abstract.

If you do consider it abstract (don't ask me the meaning of abstract pls) and mathematical, then wouldnt u have a lot of usele- I mean, a lot of unsolicited functions just like distance as well?
For example, the number of 3d objects in a system could be a function, the edges of shapes in a 3d/2d/1d system could be a function

[covert heath]

(in case any of this is confusing, well... yes. it is.)

[valid nimbus]

there can be a lot of functions

just like distance

[restive geyser]

Do you know what a "function" MEANS? /genuine

[austere cove]

good point, what does a function mean to you?

[valid nimbus]

perhaps, the number of lines of width x between two objects, where x is the number* of units of the distance

this is something mathematically possible to create a formula of

right?

[restive geyser]

In general, what is a function?

[austere cove]

nope

[covert heath]

hooboy

[austere cove]

this is too restrictive

[valid nimbus]

what makes u think distance is something not restrictive and an abstract concept that maths should make a whole theory of?

[covert heath]

(something something sets)

[covert heath]

nope

[restive geyser]

You're jumping around again

[covert heath]

tell us what a function is

[covert heath]

.

[valid nimbus]

I could write 1000 pages on this, all with mathematically profound concepts but they're kinda useless if u ask someone

why care about measure theory and distance then

[valid nimbus]

uh

a function is

[restive geyser]

-# If they were useless [assuming you mean by this that no one can understand what you mean by these], then they weren't profound to begin with

[austere cove]

@valid nimbus a function is any way of mapping values from an input set (called a domain) to an output set (called a codomain). Any method you can define to do this is valid, you don't need a formula.

[covert heath]

a function is a bijective relation between two sets.

okay OE does that better

[covert heath]

well certainly not any way

[austere cove]

doesn't need to be bijective

[covert heath]

no?

alright i guess

[austere cove]

any method that actually defines it is valid

[covert heath]

wait ofc not

[valid nimbus]

what makes u think knowing distance is useful? what makes u think to be able to do something makes something useful? for all we care, we can't do anything about big bang, but it's useful. maybe we can generate some laws with it, but there's somethings that we can't do anything but are considered useful and theres not a mathematical definition for useful

[covert heath]

apologies for being dumb

just any relation that doesnt map one preimage(?) to 2 images

or whatever the fuck the terms are

[valid nimbus]

hmm

[covert heath]

11th grade math is wasted on me

[valid nimbus]

mapping values

From an input set

[covert heath]

(jee nahi hoga omg jaa raha hoon kuye mein koodne)

[restive geyser]

You could write a thousand pages, but if your premise is wrong (like the definition you'd given for a function), then the rest of your paper, I'm sorry to have to tell you this the hard way, would be bullshit.

[valid nimbus]

wait hold on, how is this not a function then?

this seems like a perfectly fine function then

[restive geyser]

Because what are your inputs?

[covert heath]

what set does it map from

[austere cove]

In set theory, we can define a function as a set of ordered pairs (input, output) such that in the set each value of the input set only appears once.

[covert heath]

yeah thats good

why the fuck did i say bijective goddamn

[valid nimbus]

just like how u have inputs for distance, for example, the light rays absent to your eyes between 2 points, the length of those together

[covert heath]

i need to get chocolate to stop being dumb

[valid nimbus]

the inputs are same for this

u get the distance, u get the area of the 2 objects

[covert heath]

the input is a distance?

[steel veldt]

Is this still going on??????

[valid nimbus]

and then u map out the domain or something, by seeing how many lines of width x units can pass through those 2 objects (objects, not points here) in whole numbers/integers if u want

[covert heath]

yep

welcome

[valid nimbus]

I just gave an example

of random functions id consider useless

not necessarily distance that's required here

see my idea

[restive geyser]

You see, that's not what a distance metric is

[valid nimbus]

why do we specialize distance as a function

[austere cove]

distance is required to set up a euclidean space

[valid nimbus]

and not those ideas

[valid nimbus]

Hmm

[restive geyser]

A distance metric is a type of function whose OUTPUT is called a distance

Not the INPUT(S)

[valid nimbus]

yeah

[austere cove]

if we use a different idea of distance then we get spaces with different properties and different notions of a line

[valid nimbus]

If u read what I said properly

[valid nimbus]

.

these are outputs

[austere cove]

only a euclidean space has the same idea of a line that we're used to

[covert heath]

'map out the domain'

[valid nimbus]

Hmm

[covert heath]

please dont use words you dont understand

it doesnt help

[valid nimbus]

I see

well so what do u do with this distance then

u use it to define a line?

Or

u use it to describe the line

a quality of the line

[austere cove]

So let's say we have two points O and A which define a line

[valid nimbus]

yeah

[covert heath]

...no

[covert heath]

...maybe??

[valid nimbus]

a line that starts at O and ends at A

[covert heath]

not the word id use

[valid nimbus]

how would u make this rigorous tho

[austere cove]

then for any point on the line, let's call it S, we have either d(S, O) = d(S, A) + d(A, O) or some permutation of this

[valid nimbus]

I mean

[valid nimbus]

yeah

[austere cove]

and none of these equations are true if S is not on the line

[valid nimbus]

that would give all possible points

and yes

if any point is not holding true

then it's not a part of line

hmm

[austere cove]

so now we have defined a line using only our distance metric

[valid nimbus]

I see

[austere cove]

not appealing to any particular coordinate system

[valid nimbus]

well

it is kinda disappointing, because without co ordinates, you can't specify those 2 points/objects to be in 3 dimensional space or any space at all

or CAN you?

[austere cove]

the idea is we are building a coordinate system from scratch

[valid nimbus]

hmm

from scratch

so, what is our next step?

after determining that, we have a line and 2 points

[austere cove]

to review, we picked O, said it's (0, 0, 0)

[valid nimbus]

or two particles or two objects if I may

[austere cove]

then we picked A, said it's (1, 0, 0)

[valid nimbus]

hmmm

[valid nimbus]

hmm

allllright

[austere cove]

next, we pick B, check if it's on the line OA (which we determine using our metric), if it's not on this line, we call it (0, 1, 0)

[valid nimbus]

alright

[austere cove]

Now we pick C, we determine if it's not on the plane OAB

[valid nimbus]

finally we enter relative directions now

[austere cove]

we can still do this using the metric

[valid nimbus]

never mind I mistook u

[valid nimbus]

hmm ok

[austere cove]

if it's not on the plane, we say it's (0, 0, 1)

[covert heath]

no particles.

[valid nimbus]

hmm okay

[covert heath]

no words that have meanings in the real world.

theyre points.

this is not physics

[austere cove]

now we have a complete definition of the coordinate system

but it's not "nice"

it's not orthonormal

our right angles are not right angles, etc.

[valid nimbus]

so essentially

[austere cove]

d(O, A) could be like a million or an irrational number

[valid nimbus]

you just defined basis vectors

[austere cove]

yes

and once we have a set of basis vectors, we can use them along with our metric to define a set of orthonormal basis vectors.

[valid nimbus]

that's cool and all but

how do u make it nice now

[austere cove]

and once we have that, we have just a regular set of x, y, z axis that you're normally dealing with

[valid nimbus]

because our 3d world is kinda nice

if we ignore relativity

and we end up here with a different

[valid nimbus]

hmm

[austere cove]

https://en.wikipedia.org/wiki/Gram–Schmidt_process

This is how ^

[valid nimbus]

is there a way that u convert these

[valid nimbus]

I see

[austere cove]

https://upload.wikimedia.org/wikipedia/commons/e/ee/Gram-Schmidt_orthonormalization_process.gif

[valid nimbus]

Hmm

that is very interesting

[austere cove]

Note, once you do this, you will have a coordinate system that agrees with your metric, but you can have different equivalent metrics for your space

[valid nimbus]

hold on

[austere cove]

This all depends on your notion of distance. And this is a choice you need to make.

[valid nimbus]

something is not hitting me here

you defined basis vectors

by oa

distance, cool, cool everything is cool

but

if u were to take

another object or a point

at any arbitrary location

how do you plot it's location?

you're not just seeing the imaginary graph and saying it's at that location right? I mean.. theres a lot of error with that

is there a function that gives u like the exact position of it

[austere cove]

So we can do so with OABC directly, each point in space is some linear combination of these vectors

[valid nimbus]

of the vector perhaps

yeah

but how do u know

what the combination is

I mean.. it's really hard to guess

[austere cove]

it can be, yeah

[covert heath]

stop talking about relativity!!

[valid nimbus]

then is there a process

that gives the precise exact answer

it seems impossible to guess

[austere cove]

so one thing you seem to need to get comfortable with is a non-constructive proof

[valid nimbus]

non constructive proof hmmm

what is that

[austere cove]

I can prove to you that every point can be given by a unique combination of vectors, but that doesn't necessarily mean that I can give you a recipe for making each point.

just some combination exists, we don't know what it is apriori

[valid nimbus]

well yes I'm not really uhm, destroying your system because u haven't given me a way to locate things

[valid nimbus]

are u saying that there's no correct / rigorous way to locate things?

[austere cove]

ok let's say that you have a nice set of orthonormal axes.

[valid nimbus]

we just have to do so by guessing?

[austere cove]

let's not fuss with how you got them

how would you located the point R using your axes?

This is the first time I mentioned R to you, and you don't know anything else about R.

[valid nimbus]

orthonormal as in, the arbitary, kinda disturbed basis vectors that u took instead of the nice ones that we see on graphs of lab notebooks?

[valid nimbus]

Alright

[austere cove]

ortho meaning perpendicular, normal meaning of length 1.

[valid nimbus]

hmm

well to me as a human

I would

well I mean u have to show me the point and I would numerically try to narrow it down by removing chunks of area around it

I would cut and narrow it down into a point

visually

and calculate it numerically

with a scale

in regards to the basis vectors

[austere cove]

sure, that's a process that you could do, and you could estimate the point.

I could do something similar with my basis vectors.

[valid nimbus]

yeah

it's only estimation tho

and like thats

really approximately rough

[austere cove]

sure, but without knowing more about R, that's as good as you can do

[valid nimbus]

yeah

that is

[austere cove]

you can get an arbitrarily small region around R

[valid nimbus]

yeah

[austere cove]

there is some coordinate, but we might not be able to know precisely what that coordinate is.

[valid nimbus]

yeah

[austere cove]

and that's OK

[valid nimbus]

it kinda seems like a limitation

of the system

[valid nimbus]

WHAT

okay?

huhhh

[austere cove]

in the sense that it's as good as you can do for any coordinate system, if this is your process.

[valid nimbus]

yeah but

every answer is like 99.9999...% innacurate tho

[austere cove]

the key is that you can take the limit of this process.

[valid nimbus]

no matter how small we narrow it down

[austere cove]

to get the exact answer.

[valid nimbus]

oh

I see

wait there's

an actual process to cutting smaller circles?

I can guess it yeah but like

[covert heath]

now we're gonna talk about calculus are we

[valid nimbus]

what if

the object is

not a point

[austere cove]

just like we can say that pi is just pi, or we can try to numerically approximate it by saying it's 3.141592653589793238...

[covert heath]

blud is gonna get very active any time now

[valid nimbus]

and has like a shape

[valid nimbus]

u have to define something in ur limit that would stop when we reach the shape

[cinder thorn]

NO WAY

[valid nimbus]

for a point we can just narrow it down a lot

[covert heath]

no we're not talking about calculus IN THE SAME CHANNEL

[valid nimbus]

for a shape, we have to stop at some point

[covert heath]

at least make a different one

...later

idk unless OE wants to

[valid nimbus]

hmm well yeah

[austere cove]

I actually need to go soon

[covert heath]

in which case sure ig you got mods on your side

ripp

[austere cove]

I mean, it's christmas, I'm alone at home, I have no car and absolutely nothing better to do, sure, I can spend several hours on this I guess.

But I do need to go shower.

[knotty grail]

[covert heath]

just keep it alive for another hour or so until OE comes back

how long has it been already

blud is def getting VA

[austere cove]

more than 3 hours so far

[proud dirge]

hi

[foggy vapor]

What does having a car have to do with this

[valid nimbus]

I think I need to learn how we would make equations/ functions that would reduce it to the exact position of the point/object

[knotty grail]

literally a battle at this point lmfao

[valid nimbus]

with respect to those weird co ordinates

of basis vectors

[covert heath]

equations/functions

[valid nimbus]

well functions

[covert heath]

theyre not the same thing

[valid nimbus]

yeah

take functions

[covert heath]

points and objects are also not the same

[valid nimbus]

well they're kinda same over here

u need to find position of a point

u need to find position of an object

[covert heath]

basis vectors dont have 'coordinates'

etc etc

[valid nimbus]

well yeah

I spoke about the weird shape these basis vectors have

even though they're all 1,0,0 etc..

why do u take everything so literally

[covert heath]

well

its maths mate

[restive geyser]

This is the literal subject where nuance seldom has room to exist

[wary condor]

you now know more math than me @valid nimbus ! and you achevied all of that in one day!

I commend your endurance.

[cinder thorn]

How many times did we tell you that you need to be precise in your definitions

[covert heath]

please stop the word salad.

everything else is fine

[valid nimbus]

well it's kinda fascinating when you go deep into concepts

thank you

[wary condor]

right?! my deepest concept is... time-releated problems. you literally know more math than me 😔

[valid nimbus]

well hey, sometimes I'm not defining and referring to some things

[wary condor]

you might've also set a record for the fastest person to ever get the active role.

[valid nimbus]

but alright

[valid nimbus]

lol

[restive geyser]

sometimes I'm not defining
is precisely the problem, since you're using words whose actual definitions differ to what you MEAN

[covert heath]

and never say 'logics' again

please

at least use real words

[wary condor]

I'm really impressed how OE was able to do all of this!!!!!!

[covert heath]

well

[valid nimbus]

well I underestimated that y'all would know what logics actually mean

[covert heath]

hes fucking omnipotent

its in the name

[valid nimbus]

instead of the usual idea of what's right

[covert heath]

...what?

[valid nimbus]

Alright

[covert heath]

'Logics' is not a word.

[restive geyser]

It's more that you're using philosophically loaded words that DO NOT MATCH WHAT YOU MEAN

[covert heath]

There's Logic.

[wary condor]

girl 😭

[covert heath]

Which is not related

at all

well at least not really

[valid nimbus]

yeah but usually, when you say "logically and logic", what u mean is what I was referring

I know philosophical logics are something different even though kinda same

[covert heath]

i dont say logically.

[valid nimbus]

but yeah

[covert heath]

LOGICS IS NOT A WORD!!

[valid nimbus]

alright why don't we accept I was at fault and move on!

[covert heath]

it's Logic!!

[valid nimbus]

I need to do something rq

I've been sitting here for 7 hours

arguing here

[wary condor]

PEACE AT LAAAAAST (for you)

[austere cove]

be nice

[covert heath]

clever edit

take my joy emoji

hi OE!

[austere cove]

In most of the US "I don't have a car" means "I literally cannot go anywhere, because every destination is several miles from where I live."

[wary condor]

cars are amazing people. you should get to know one!

[knotty grail]

but really huge respect for standing on your own for a third of a day against 10-20 people

[austere cove]

I know one, they're sick 🙁

[foggy vapor]

Sadye

[covert heath]

hopefully he got something out of this

[austere cove]

I think he did

[proud dirge]

Can this chat be saved as a memorial

[knotty grail]

🫡

[covert heath]

i swear to fucking hell if i hear another 'how do we get coordinates'

[wary condor]

you did an amazing job 😭 I'm honestly so impressed with your patience

[covert heath]

ill just ping OE.

[wary condor]

90% of the helpfuls did crashout like 20 minutes in. (not-not-not discluding myself.)

[austere cove]

oh one thing I didn't mention @valid nimbus sorry about the ping.

I mentioned that the choice of metric is a choice, you can also just change the metric, once you have OABC, you can just define a new one based off of the natural coordinates they represent. In other words, OABC is an orthonormal set of basis vectors in the metric you defined based on the coordinates they give.

[covert heath]

im not crashing out i just dont wanna be asked the same blood thing like i could come up with a different answer this time

also the word salad

[austere cove]

if this seems strange to you, this is because you're not yet comfortable with the idea that space can be deformed by a linear transform and remains exactly the same afterwards.

anyway, I need to go AFK for a bit again

[valid nimbus]

yeah I mean.. just like how u took b or c and decide if it was on the line or not, u could do the same and get a lot more

but

I thought the major problem now is locating something with that astronomically hard and weird basis vectors

that I don't think we would have an intuition of

[austere cove]

no, this isn't actually a problem.

locating anything with any basis vectors (without knowledge of exactly where it is a priori) is the hard part

but once you know exactly where something is, you can easily and rapidly construct coordinates for it in any basis

[valid nimbus]

yeah that's what I'm talking about

[valid nimbus]

if u see here

that mushroom is really hard to know the exact co ordinates of

u said something called functions and limits would get me a good answer

knowing those functions would be hard tho I'm assuming

[austere cove]

Here's a procedure you can take

Consider a bounding box around your point, you located it to a region that has coordinates about (a, b, c) to (a', b', c')

[covert heath]

im sorry whats the mushroom cloud for

[austere cove]

the mushroom is the object lol

[valid nimbus]

an object that u are given to locate

[covert heath]

ah

[valid nimbus]

u have to be really studious to locate it precisely

very hard for me personally

[covert heath]

really disturbing imagery lol

[austere cove]

let a > a', b > b', and c > c'

[valid nimbus]

hmm

[austere cove]

so we have a point in the middle of this box, ((a + a')/2, (b + b')/2, (c + c')/2) right

[valid nimbus]

wait a, b, c and a' b' c' are just points tho

how is it a bounding box

[austere cove]

yes, but they define the opposite corners of a box

[valid nimbus]

Oh

Opposite corners

wait lemme get the imagery of this

so

u have a box

inside that is the object btw

[valid nimbus]

now u have a pink, lime, white and purple corner

[austere cove]

that's a 2d box

I'm talking about a 3d box

[valid nimbus]

ah well

imagine 2 rectanges ok

[austere cove]

or nd in the general case

[valid nimbus]

and tell me for the first rectangle

[austere cove]

anyway

[valid nimbus]

which corner

and second rectangle

which corner

then it's pretty easy to imagine the box

[austere cove]

you have a point in the center of the box

[valid nimbus]

wait

would that be

at the centre of the object as well

[austere cove]

no

your point is somewhere in the box

[valid nimbus]

oh

hmm alright

[austere cove]

your object is composed of many points, so you would need to do this procedure for each individual point

[valid nimbus]

now if that's a, b, c

[valid nimbus]

I mean.. we can ignore the many points

for now

[austere cove]

sure, let's talk about a single point

[valid nimbus]

one point in the box

alright

[austere cove]

we don't know where in the box this point is

[valid nimbus]

Where is a', b ' and c'

[valid nimbus]

yeah

well I mean

[austere cove]

(a, b, c) is the top left front corner of the box, and (a', b', c') is the bottom right back corner

[valid nimbus]

top left and bottom right

ohh yeah

[austere cove]

for relevant definitions of left front etc.

[valid nimbus]

I get it

yeah yeah sure

[austere cove]

so you choose some point in the box, and then you can use this point to make 8 smaller boxes inside of the box.

[valid nimbus]

it's kinda confusing a bit tho cause... the box literally acts as the hidden object, how would u know the boundaries of the box, doesn't it have like infinite points?

[valid nimbus]

yeah

hmmm