/sci/ - Science & Math

Here you go
π

>>9806942
I remember 3.14159265358979323846 but I think I'm missing a couple digits on the end there

>>9806942
Could you please write the whole 1/3 for me? Thank you!

>>9806956
That’s not a number

>>9806959
Yes, so it’s not Pi. Remember, Pi is the ratio of the circumference of a circle divided by its diameter.

>>9806942
I would but posts have character limits and images have filesize and dimension limits.
Pi is just too awesome for 4chan.

>>9806942
Some numbers are too long to be written down, ie the number of chicks ur mum had today

>>9806961
[eqn]0.\overline3[/eqn]
there u go

>>9806961
Retarded. It is a finite decimal in base 3 and many other bases. Literally 0.1 in base 3.

>>9806963
>That’s not a number
Yes, it is.

>>9807009
That's not a number

>>9806963
And my number ki is defined as the ratio of the circumference of a circumscribed dodecadomillion to a quadrilateral pentagon.

Fun question: what guarantees that a number defined like this exists?

>>9807083
it is though. all operations on it are precisely defined, same goes for all its digits. go on try me. i bet i can tell you the nth digit of 1/3 in no time

>>9806942
Ratio is C/D but is irrational in the system of reals.
There's a Dedkind cut you can make on the number line.

>>9807019
Retarded, pi is the number 10 in base pi

>>9807110
Before you define base pi, pi must exist. I know that 3 exists, do you?

>>9807129
https://en.wikipedia.org/wiki/Construction_of_the_real_numbers

>>9806942
That guy has a fundamental misunderstanding of mathematics and it's sad. He's attacking an idea which most mathematicians don't actually believe (only undergraduates still think mathematics is meant to be "a tool for describing reality" and that it's "all built on set theory") so he attacks it and nobody actually believes what he thinks everybody does. Mathematicians who acknowledge full well that you CAN represent all natural numbers, and indeed almost everything in mathematics, with some of set construction, do not ACTUALLY refer to the sets which can be used to represent the number "2" when they say the number "2". The model one can construct out of sets which is remarkably similar to the number "2" is still just a model and the number "2" refers to a number defined in the context of its own theory with that theories own axioms, usually the Peano axioms in that particular case.

>>9807129
>3 exists
prove it

>>9807129
>he believes 3 exists
>he believes numbers exist
Yeah and so does the tooth fairy. Show me a three in real life. You can't. Therefore mathematics is a religion.

>>9807140
what does pi*sqrt2 equal?

>>9807129
>I know that 3 exists, do you?
I know that Pi exists. Do you?

>>9806942
If there is such a number as “Two” as you claim, could you please write it down? Thank you.

>>9806942
>it's another crank episode

>>9806942
Half of pic related

Does there exist a number, b, such that b^2 = 2?

>>9807151
1+1/3-1/5-1/7+1/9+1/11-...

>>9807246
Oops, all numerator should be 4 instead of 1.

Oh and if you don't with the ellipse then you should also not accept 0.333... as a decimal represention of 1/3.

>>9806963
you can't write numbers, you can only write representations of numbers. there is no particular reason to value decimal representations above others

>>9807140
>accomplished professional mathematician
>misunderstanding of mathematics
nah, it's YOU who doesn't know what they're talking about

>>9807246
>...
>infinite sum

>>9807268
0.333... is an infinite sum, hypocrite.

3.14159265358979323846264338327950288419

>>9807101
All operations that define pi are well defined too. All you need is the definition of a limit. And for that all you need is the set of rational numbers.
>inb4 MUH INFINITY
The word infinity shows up nowhere in the definition of limit.
So as you see, as you Yourself said, some numbers cannot be written down in their entirety as a decimal, but only in terms of well-defined operations.

>>9807295
Oh, ok. Without invoking infinity, can you write down the limit that defines pi?

>>9807235
b = 1/^

>>9806942
Yeah because numbers are exclusively defined as the quantities you can write.
Fucking retards I swear.

>>9807422
Ok, pi is the equivalence class of the cauchy sequence whose nth element is 4(sum from k=1 to n of (-1)^(k+1)/(2k-1))

>>9806942
>"If R^n space exist, then plot a cube of n dimensions"
>Sir, R^n is a mathematical construct that lets us understand the properties of other plan...
>"YOU CAN'T MAKE THE DRAWING THEREFORE YOUR R^n GARBAGE IS FALSE, LOOOOOOOOL MATHMATITIANS BTFO, WE DID IT REDDIT!!"

>>9807512
You can write pi, you can't write its decimal representation in base 10. There is a big difference.

>>9807518
>sequence

look man if you really wanna prove him wrong you should at least know his arguments. watch his videos
>i can't be bothered to sit through that crap
well then don't bother trying to counter points you're not familiar with

>>9807531
>numbers are lines on a chalk board...

>>9806956
> Here you go
> π

Fail!

If you had asked OP: "If there is such a number as “1” as you claim, could you please write it down?" then he would have answered by just typing a 1. It clearly evades the question altogether.

>>9807531
>>>/reddit/

>>9807535
>write down the limit that defines pi without invoking infinity
>>ok
>REEEEEEEEE SEQUENCES
Nice attempt to move the goalposts.

>>9807548
>>9807546

>>9807535
There is no argument to prove wrong. Infinity = : ( is not an argument.

>>9806942

Sqrt C^um / pi

>>9807542
Both are valid answers as long as you know what 1 and pi mean mathematically. Do you?

>>9807263
>there is no particular reason to value decimal representations above others
being retarded is a reason

>>9807140
https://en.m.wikipedia.org/wiki/Ultrafinitism
Retard.

>>9807518
Hmm, this really makes me think. You originally said Pi was nothing more than the ratio of circumference to diameter. Now you bring up all that algebraic mess. Hmm... I wonder why your new definition is not as clean.

Also, do you even have any fucking idea of what an equivalence class is you absolute moron?

These are very funny.
>"you can't write down the infinite decimal expansion therefor irrational numbers and infinity don't exist!!
lmao imagine unironically believing this.
>inb4 you have no argument
Every time we write a proof that PROVES these things, they just plug their ears - "lalala infinity dont real!!! there is no Dedekind no R no infinity!! They don't even accept PROOFS why the fuck would I make an argument?
I have no interest in dealing with these idiots and neither should anyone who's serious about math.
>inb4 mathematics is a religion!!
lol

>>9807696

>He thinks a proof that has an invalid prerequisite foundation is in any way a 'proof'.

This is some shitty bait.

>>9807704
>successor function is invalid
my oh my

>>9807676
>You originally said Pi was nothing more than the ratio of circumference to diameter.
No I didn't.

>Hmm... I wonder why your new definition is not as clean.
One is a geometric construction and the other an algebraic construction.

>Also, do you even have any fucking idea of what an equivalence class is you absolute moron?
A set of elements that have an equivalence relation. The equivalence relation of cauchy sequences in the construction of real numbers is that their difference tends to 0. Why are you asking such trivial questions?

>>9807749
Tends to 0?
Wow man slow down let me get a pen to take notes. Wew, all of this to define a single circle constant? Fuck.

Ok I got my pen, what does it mean for something to "tend" to 0 and what does it have to do with the circumference and diameter of a circle?

I hope this discussion ends and you finally properly define Pi before we write a fucking book together.

>>9806963
I choose base pi

pi

faggot

>>9807749
Why are you arguing with a literal retard who can't understand something that a 3 year old can?
These guys aren't to be engaged, they are to be mocked. It doesn't matter - finitism will never become mainstream, it's already been destroyed by hundreds of years of math, there's nothing it can do that we can't, and it's not as good at describing reality OR as a pure mathematical system.
These finitist fags are wrong, they have no chance of winning, and they lost hundreds of years ago. Stop engaging, just mock the retards.

>>9807768
>Wew, all of this to define a single circle constant?
So you don't know the difference between a definition and a construction? You should learn basic math before trying to criticize it.

>Ok I got my pen, what does it mean for something to "tend" to 0
Look up the definition of the limit. I'm not here to teach you what you should have learned in school.

>and what does it have to do with the circumference and diameter of a circle?
It is used to construct the real number that is the ratio between circumference and diameter. Another stupid question with no point.

>>9807535
But he did define infinite sequences of rational functions (giving them retarded name "polynumber on-sequence" or something like that) and their limits

We can call any numerical identity i.e. stuff definable by numerical identies a NUMBER! even error division by zero

>>9807768
>Defining pi takes more than five seconds, so let's say it's impossible to construct it and claim circles have no diameters

Daily reminder finitism is branch of computer science, and not mathematics. They reject infinity and non-constructive proofs, want algorithms, which terminate in finite time, for everything because of limitations of their beloved machines. See how their, including Wildberger, arguments are always something like "the machine won't stop doing that in finite time, that would just run without end and never give the result, that number is too big or too small to store it in computer's memory, that number with infinite binary representation can't be stored accurately"?
Let the cs monkeys play with their finite toys and leave the maths for men.

>>9807802
>pure mathematical system.
no, no you got this backwards, reals are very useful in applied mathematics but are not rigorous. pure mathematics is subservient to applied mathematics because having something useful is more important than building a useless rigorous system for autists

>>9807744

YES, I unironically think proofs via induction are not possible, and while this is a minority, opinion there are some very interesting arguments that I found extremely convincing.

I would love to have a long discussion on this. I am a big fan of Wittgenstein/Doron Zeilberger, and many other finitists.

>>9807802
>there's nothing it can do that we can't,

You can also flip this around, as Wildberger argues in his videos there is literally no advantage with using rational numbers or infinity to doing any sort of analysis or geometry. You can get the same answers to any problem without using anything non-discrete. And since this is the case, there is no reason to use wishy-washy imprecise methods in the first place!

>>9807825
>because of limitations of their beloved machines.
Limitations of the universe itself.

>>9807825

This is about like a psychoanalyst saying "muh made up abstractions are just too complex for these plebs, let the neuroscience cucks with their 'precision' and their 'discrete reality' go on playing with their toys while I make more jerkoff papers based on non-concrete abstractions I'm making up as I go" That is literally how you sound right now cuckboy.

>>9807834
Maths doesn't have to have a model in reality.

>>9807839

This is the same argument SJW's use to back up their incorrect views when called out also.

>>9807838
What's your point? Maths is jerking off to an abstract nonsense. It's not meant as a model of reality or a language to describe natural phenomenon. You can use some parts of it as such, and call it applied maths, computer science or whatever, but don't say that's all the maths there is.
The fact you can't physically represent number pi or sqrt 2 doesn't mean we can't make valid operations on them

>>9807842
Go back to pol, you retard. And your "argument" doesn't even make sense

>>9807845
No, this is bullshit, YOU cannot physically do an operation on Pi you can do an operation on a arbitrarily large subset of significant digits of Pi, but you cannot do an operation on the entity of Pi itself. This is bullshit handwaving that wasn't ok until Cantor shitted up all of mathematics.

Your brain/paper/computer/etc are what is doing the operation. All of them are finite entities within the closure of our physical universe. To do a valid operation on the entity of Pi would require going outside of that closure. Which isn't to say that you can't abstractly conceive of something which could go beyond our physical universe and do such a thing, but it is supremely arrogant to argue that you are doing such an operation yourself in your LaTeX editor.

>>9807839
You have a choice to make here
>human brains are magical thingies that are capable of holding information beyond what's containable within the entire universe
>infinity in mathematics is a hand-wavy concept we only pretend we can wield
so which one do you choose

>>9807229
underrated kek

>>9807829
They are very rigorous, fuck off.

>>9807869
Great argument you have here, "I think they are so fuck you!", it is pretty easy to have rigour about things like infinity or rationals where you just make up shit up as you go until it fits, then in hindsight you say "oh look how wll this fits!". Stay cucked infinifag.

>>9807832
>there is literally no advantage with using rational numbers or infinity to doing any sort of analysis or geometry. You can get the same answers to any problem without using anything non-discrete.
Proof?

>wishy-washy imprecise methods
This is why no one takes Wildberger seriously, he's a polemical liar.

>>9807881

Pick any topic, Trig/Calculus/ODE/PDE/Analysis, doesn't matter, wildberger has a video showing how to do it without Reals or Infinity.

The only thing I haven't seen him do is a Finitist take on Category theory, and I emailed him about it and it sounds like it is in the works.

>>9807857
>No, this is bullshit, YOU cannot physically do an operation on Pi
Wrong, it's called analysis. You can even have a computer do it. You are bullshitting way outside your area of knowledge.

>>9807859
Neither. The amazing thing about analysis such as the calculus is that it allows us to manipulate "infinite information" in a finite way. A calculated integral is not a finite approximation of an infinite sum, it is the value itself. You are arguing from ignorance.

>>9807874
>this is unrigorous because Wildberger said so
>>no it's not
>HOW DARE YOU ASSERT SOMETHING WITHOUT AN ARGUMENT
Get some perspective on yourself, time to grow up.

>>9807908
Nah I've read Rudin, Real analysis is just fundamentally based on bullshit and you can do all of real analysis discretely. This is my point. If you are just going to make ad hominems, instead of making an argument, go ahead, just makes my case stronger. Anyone that is interested should look to Wildbergers videos on the history of calculus/analysis, and watch his methodology on doing all of it without infinity to be convinced.

>>9807902
I asked for a proof, not examples. Typical novice math student mistake.

>>9807943

I'm not just going to type up meticulous LaTeX posts of wildbergers whiteboards from his hrs of videos just to please some random autist person on the internet. The proof is already up there by someone who is a great educator, if you don't care to look at it, why should I retype what is already clearly presented?

>>9807926
>Real analysis is just fundamentally based on bullshit and you can do all of real analysis discretely.
Proof?

>If you are just going to make ad hominems, instead of making an argument, go ahead, just makes my case stronger.
You have not made an argument to reply to. You're just calling things unrigorous even after I've shown them to be rigorous. Show me what specifically is unrigorous. You won't because you are just like Wildberger, pretending to have a valid criticism when all you really have is "infinity = sad face." Pretending that he has found a fault in modern mathematics is why Wildberger is treated as a crackpot and not just another ultrafinitist. This battle was lost over a hundred years ago by much smarter men than he. Instead of putting up a real fight he has opted for building a sycophantic army of naive students such as yourself. It's pathetic.

>be Wildberger
>win the nobel prize in Mathematics
>the nobel prize is set to π Million Dollars

>>9806942
Why do you do this to yourself /sci/?

>>9807957
>I'm not just going to
That's because you can't, because the proof does not exist. Finitism will always be weaker.

>>9806942
>hurr I can't abstract concepts therefore I'll claim that everybody who can is crazy
What phony piece of shit.

>you can't operate on pi because it's infinite
This is what happens when people learn math but never learn how to think first. Go back to logic you dumb undergraduate fucks and stop wasting your time with stupid conceptual masturbation.

>>9808019
What kind of logic?

>>9808033
Good old traditional, intuitive, common sense, simple, aristotelian, first order, predicate logic.

>>9807803
This is a discussion about properly defining pi. It is standard to go over the details. You just admitted you don't know any better but I already knew that from your butchering of the definition of reals as equivalence classes of cauchy sequences.

>>9808105
He got that from wildberger, it is actually a weird agrument from his his videos with Cauchy in the title, there's like three. I'm not entirely convinced, but it is definitely a fun watch.

>>9807814
No, it does take more than five seconds. I am just proving that the people who claim that the reals exist have actually no fucking idea why.

Newsflash, I am an expert on analysis, my field is analytic number theory, and if you know his research, you know that Wildberger is an incredible analyst. What Wildberger argues is that the mechanisms of set theories and infinities that are introduced only to justify the manipulations of calculus are ugly and go against the direction pre-Cantor mathematicians wanted to take math. You are proving his point by showing you have absolutely no idea how any of this works. Heck, you probably think that there is some magic to our current theory when in the reality there are infinitely many ways to set up a symbolic calculus that can justify the manipulations of analysis. Wildberger simply claims to have a better one.

I genuinely feel bad for the worshippers of infinity in this thread that do not know any better.

>>9806942
[eqn] 10_{\pi} [/eqn]

>>9808105
>This is a discussion about properly defining pi
c/d is a proper definition of pi. You don't seem to understand that there are multiple ways to define the same thing and that cauchy sequences are used to construct the real numbers, not simply define them.

>You just admitted you don't know any better
Why do you need to lie about what I'm saying if you actually believe you're right?

>but I already knew that from your butchering of the definition of reals as equivalence classes of cauchy sequences.
Which part is wrong? This should be good.

>>9808119
>Newsflash, I am an expert on analysis
>says the guy who doesn't know that the reals are constructed as equivalence classes of cauchy sequences
Nice LARP, try harder next time

>>9808152
Wrong, they're constructed as Dedekind cuts.

>>9808156
>they're constructed as Dedekind cuts.
That doesn't contradict the fact they are constructed as equivalence classes of cauchy sequences.

>>9808119
>I am just proving that the people who claim that the reals exist have actually no fucking idea why.
Where did you prove that?

>What Wildberger argues is that the mechanisms of set theories and infinities that are introduced only to justify the manipulations of calculus are ugly and go against the direction pre-Cantor mathematicians wanted to take math.
Ah so now they're just "ugly?" Funny how this whole thread Wildberger's sycophants have been crowing about how modern math is "unrigorous" and "flawed." Good to see you've ignored the crackpot polemics and gone straight to the heart of the issue: "infinity = sad face." Too bad this is not an argument, and that all the mathematicians that matter today disagree. But do tell me when you finish that time machine so that you can go back and fix everything!

>>9808167
You don't know what you are talking about. Ugly, unrigorous, flawed, those words are all equivalent because math is supposed to be beautiful and perfect.

I have confirmed you are not one of those loved by God. Your opinions are worthless.

>>9807518
>"without invoking infinity"
>cauchy sequences

>>9808176
>Ugly, unrigorous, flawed, those words are all equivalent because math is supposed to be beautiful and perfect.
No, you are clearly not familiar with mathematical jargon. Mathematical beauty is essentially elegance, i.e. the most result for the least complexity. Rigour is how strongly something adheres to and derives from axioms (this means rigour can be at odds with beauty). A flaw is an invalidity in a proof.

>>9808192
>the poor brainlet can recite from memory the definition of the words, yet fails to recognize how they are all connected.

>>9808183
>"without invoking infinity"
>natural numbers

>>9808198
So where is the flaw?

>inb4 infinity = sad face is the flaw

>>9806942
3
t. Engineer

>>9808201
There is no flaw. You recited the definitions correctly. But in typical brainlet fashion, you are unable to make conceptual connections.

>>9808214
Good, now where is the lack of rigour?

>>9808218
Stop replying to the idiot. His posts are equivalent to "but you don't feel the spirit of the numbers, dude".

>>9806956

Tautology

>>9808167

You get a "weakening of axioms" when you talk about things like infinite groups and infinite sets. You can't always represent them precisely, or atleast in the same way as finite aets and you lose the pigeon hole principle. In practical terms, this is why you avoid infinity. It does equal "sad face".

It seems that it's possible to do anything in math without talking about infinite sets if we're more careful about definitions than past authors have been. Why shouldn't we?

>>9807834
Mathematics is more fundamental than this universe

>>9807798
>base pi
then it would be "10"

PSA/Links for the lazy

Real numbers as Cauchy sequences don't work!
https://www.youtube.com/watch?v=3cI7sFr707s&index=35&list=PLIljB45xT85DpiADQOPth56AVC48SrPLc

Difficulties with Dedekind cuts
https://www.youtube.com/watch?v=jlnBo3APRlU&index=37&list=PLIljB45xT85DpiADQOPth56AVC48SrPLc

>>9807825
This makes a lot of sense.

>>9807110
OTFB

>>9808331
All bases are base 10.

>>9806942
Get on my level.

>>9808338
No, it doesn't. Modern mathematics is a total shitshow in terms of rigor Go watch Wildberger's videos.>>9808332

>>9808277
Op said to write it down, not define it. >>9806956 wrote it down. Sorry if this makes your brain hurt.

Perhaps this will explain it to you:
https://www.youtube.com/watch?v=M68GeL8PafE

>>9808349

>>9808330
>math is magic

>>9808351

> problem: find x.
> [circles x]
> huurrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr

I can answer the question "write down 4."

{0,{0,{0,{0}}}} where "0" is the null set.

Is this a definition? No

>>9808351
[math]\pi=\pi[/math]
is the best any only effective way to write pi, and there is no way to do any calculations as such. Approximations to an arbitrary level of precision can be made, but it is impossible to do calculations with an infinite level of precision.

Go watch Wildberger's videos as in >>9808332

>>9808358
That's not what I said, and they aren't equivalent statements

>>9808295
>You get a "weakening of axioms" when you talk about things like infinite groups and infinite sets.
How so?

>You can't always represent them precisely, or atleast in the same way as finite aets
Why would we expect to represent them the same way? You can't represent rationals the same way as integers.

>and you lose the pigeon hole principle.
You don't lose the pigeonhole principle. The pigeonhole principle only applies when there are more objects than containers. So saying that you "lose" the pigeonhole principle if you allow infinite objects and infinite containers is as non-problematic as saying you "lose" it when you allow 5 objects and 5 containers.

>It seems that it's possible to do anything in math without talking about infinite sets if we're more careful about definitions than past authors have been.
Wrong.

>>9806942
fuck off faggot its \pi no go kill your self

>>9808385
>>You get a "weakening of axioms" when you talk about things like infinite groups and infinite sets.
>How so?
Check out Wildberger's 3-video group on axioms starting with part one here:
Axiomatics and the least upper bound property https://www.youtube.com/watch?v=rCDRCGjmaO8&list=PLIljB45xT85DpiADQOPth56AVC48SrPLc&index=41

>>9808387
{\pi}

>>9808332
>>9808361
Wildberger's argument against the reals (and basically all of his arguments) rests upon his doubts against the existence of infinity. Once you accept that an infinity exists, you need not restrict yourself to finitely computable sequences. Modern mathematics is built upon the foundations of set theory. One of the most basic (viz., Zermelo) axioms of set theory is the axiom of infinity. Axioms such as this one have been rigorously explored by inner model theorists. Doing mathematics without infinity is perfectly valid, but you get a theory that is needlessly weak and restrictive.

Wildberger math is like doing math with a Turing machine that has to terminate in finite time. Modern math is like doing math with a hypercomputer that can run in very large infinite time. The latter may seem fantastical, but we can imagine it and make true statements about it.

Modern mathematics is more powerful than Wildberger mathematics, i.e., you can encode Wildberger mathematics within the framework of modern mathematics. This inclusion is, in fact, strict. There are certain statements in modern mathematics that you just can't make in the ultrafinite setting.

t. an inner model theorist who used to work with an ultrafinitist

>>9808391
my bad; seems theres only two parts

>>9808391
No, not watching a half hour long video. It shouldn't take that long to answer my question if you actually know the answer and are not just parroting Wildberger.

>>9808360
Read what OP fucking asked, you maroon.

>>9808395
You're missing the big thing that sets Wildberger apart from all the other mediocre ultrafinitists, what makes him so popular with a certain group of special children on the internet. It's not the ultrafinitism, which the children barely understand, it's all the polemics and strawmen arguments attacking modern math as flawed that makes him so popular among idiots and completely ignored by mathematicians.

>>9808397
>No, not watching a half hour long video.

Then try https://www.youtube.com/watch?v=M68GeL8PafE

>I want to understand something complex without sitting through the actual explanation.

>>9808400

> write down pi

I don't see what you're disconnect is you 10th grader

>>9808397
Essentially, there is a lot of ambiguity, imprecision, and straight-up avoidance of defining real numbers, particularly using infinite sets. This is primarily a result of ambiguous axioms. Go watch the videos here >>9808332
You can watch on 1.5x speed or more if you feel like they are too long like this guy>>9808397

>>9806942
[math]\int_{-\infty}^\infty \frac{1}{x^2+1}dx[/math]

>>9808401
Yes, you're right. This does get on my nerves quite a bit.

>>9808332
40 minutes video to say "Achilles never catches the turtle".

>>9808412
As an illustration of why the theory of 'real numbers' doesn't work -- you cant do an infinite number of computations so infinite decimals, sequences, sets, etc. don't make sense.

Short clip discussing the lack of rigor in modern axiomatic theory - https://youtu.be/ncIgoIBLPqQ?t=21m23s

>>9808418
please read >>9808395

>>9808406
This is simply a lie. Wildberger does this all the time. He lies about foundational mathematics and then attacks the lie. Construction of the real is very clear and ZFC is very precise.

>>9808402
There is nothing complex in the video, Wildberger is just lying to naive students who don't know any better. Hopefully they pick up a textbook that goes over ZFC and realize this. But most won't and will become brainwashed youtube cranks.

>>9808417
You can't do an infinite amount of calculations, but luckily you don't have to since you can use analysis, such as the calculus, to transform infinite operations into a finite amount of operations. Which is why you can calculate 1/2+1/4+... = 1

>>9808402
>I want to understand something complex without sitting through the actual explanation.
Can't you just give _one specific_ example that I can read on its own terms? That saves us all a lot of time over me having to attack all the bullshit in 40 minutes of video.

>>9808406
You to. Can you give a specific concrete example of
>ambiguity, imprecision, and straight-up avoidance of defining real numbers
? Take the usual construction of the reals as equivalence classes of Cauchy sequences -- what exactly do you find ambiguous, imprecise, or avoiding of that definition, and/or what axiom do you find ambiguous? You just need to name one thing.

If you can't point out such a thing, odds are you don't actually understand the argument. Which I'd argue makes sense because there is nothing here to be understood; but please prove me wrong.

>>9808432
>Which is why you can calculate 1/2+1/4+... = 1
Rigorously define this process

>>9808441
>specific concrete example
It presupposes the ability to preform an infinite amount of operations, which is not possible.
See https://www.youtube.com/watch?v=I0XA7zFF7EM&list=PLIljB45xT85DpiADQOPth56AVC48SrPLc&index=34

>>9808448
Consider the sequence t(n) = (1/2)^(n+1) for natural n. Informally, t = [1/2, 1/4, 1/8, ...].

For a sequence t, define the sequence s_t(n) as s_t(0) = t(0), s_t(n + 1) = s_t(n) + t(n + 1). Informally, s_t = [t(0), t(0) + t(1), t(0) + t(1) + t(2), ...]. Informally, for this particular t defined above,
s_t = [1/2, 1/2 + 1/4, 1/2 + 1/4 + 1/8, ...]. Call this the sequence of partial sums of t.

For a sequence t, there may be a rational number L, such that for all rational epsilon > 0, there exists a natural N, such that for all n > N, |s_t(n) - L| < epsilon. If that is the case, we call L the sum of the sequence t.

For the particular sequence t defined above, s_t(n) = 1 - (1/2)^(n+1) for all n; this can be proven by induction, and I presume is not contested. For the particular sequence t defined above, for any rational epsilon > 0, let N be the smallest natural number such that (1/2)^(N + 1) < epsilon. Then for any n > N, |s_t(n) - 1| = |1 - (1/2)^(n+1) - 1| = |-(1/2)^(n+1)| = (1/2)^(n+1) < (1/2)^(N + 1) < epsilon. Thus, per the above definition, the sum of the sequence t is 1. Informally, 1+2 + 1/4 + 1/8 + ... = 1.

Please elaborate what exactly about this process you find unrigorous.

>>9807084
Have you tested your number?

>>9808452
>It presupposes the ability to preform an infinite amount of operations, which is not possible.
It does not suppose such a thing. It only supposes the ability to *define* an infinite amount of operations, which is very possible.

I don't think that video provides any justification as to why it presupposes the ability to preform an infinite amount of operations. It does state it, but I don't see a justification for it. If you see any, can you point it out to me?

>>9808417
Computations aren't fundamental, computer science is a subcategory of math.

>>9806942
Everything begins with the primitive notion of a set, for each pair of which we can define a relation called membership. Then apply the von Neumann definition of ordinals. You can restrict to the naturals by adding a clause or taking intersections of inductive sets. Take the Cartesian square of this set and identify pairs whose differences are equal. This gives you the integers. Then take the Cartesian square of this set. Remove pairs whose second coordinate is zero and identify the remaining pairs in the obvious way to give you the rationals. Endow the rationals with the canonical linear order. Consider partitions of the rationals into two nonempty sets, one closed downwards and the other closed upwards. Identify pi with the partition whose constituent sets contain those rationals with values weakly less than or greater than those with the desired property.

>>9808395
>but you get a theory that is needlessly weak and restrictive.
there is nothing wrong with that

>>9808460
>for all rational epsilon > 0, there exists a natural N
> I presume is not contested
This theory appears to work, but still requires an infinite amount of work to check that there is indeed an N for all epsilons>0.
See https://www.youtube.com/watch?v=PYeiNCz0xu8&index=30&list=PLIljB45xT85DpiADQOPth56AVC48SrPLc

>>9808467
This is exactly the problem Wildberger aims to resolve with a basis in Natural number arithmetic coinciding with Comp Sci, rather than set theory.
https://www.youtube.com/watch?v=U75S_ZvnWNk

>>9808474

>>9808479
It's not a "problem to solve", it's simply that computation is not fundamental. Computer science is a subcategory of math.

>>9808490
>It's not a "problem to solve"
With the present objections and ambiguity around the current 'fundamental set theory', I would think any mathematician, or scientist in general for that matter, would welcome the opportunity to consider a logical alternative.

>>9808476
Finitism and ultrafinitism are valid theories. But the fact that they are weak means that there are certain statements of truth that they are unable to convey. I guess you can be a finitist if you do combinatorics and certain types of algebra, but statements made in the analytical hierarchy demand a stronger theory. It all boils down to what a foundation should be. Pragmatically, a foundation should be expressive enough to deal with the major areas of mathematics we use today. Ideally, a foundation should be capable of handling all truths, resolve consistency problems, etc. Analysis alone suggests that there are realms of truth to explore outside the finitist bubble. And these realms of mathematics are useful, predictive, and consistent.

TLDR: The fact that they are weak and restrictive means that they can't serve as good mathematical foundations.

>>9808497
>ambiguity
>t. not a mathematician
lol
t. a math post-doc

>>9806942
3.14 should do, my man. Let me know if your application needs more accuracy, and I will provide some more digits.

t. Engineer

>>9808501
How would you describe the debate over which axioms, e.g. the axiom of choice, should be declared fundamental?

>>9808516
>I will provide some more digits.
Approximation is sufficient for applied fields, but pure math strives for exact answers, which in this case can only be achieved through an infinite amount of work.

>>9808518
three words: inner model theory

>>9808522
pure math deals with things that don't necessarily happen in reality, so infinite amounts of work shouldn't be a problem

>>9808477
>This theory appears to work, but still requires an infinite amount of work to check that there is indeed an N for all epsilons>0.
And yet I proved it for this particular sequence t. Or do you disagree with that?

Suppose that I want to prove that for all natural N and M, if N and M are both odd, then N + M is even. And suppose I were to offer one of the obvious proofs for this as accepted by most mathematicians. Would you have a problem with such a proof in principle? If so, I think there is a problem here more fundamental than ultrafinitism.

Note that in >>9808460 I am not defining the sum of an infinite sequence as a computable arithmetic operation. I am defining it as an algebraic property that you may or may not be able to prove for any particular sequence.

>>9808551
>I proved it for this particular sequence t.
I grant that for some restricted cases, it is possible to deduce a correspondence between [math]N\text{ and }\epsilon[/math], but this is not the case in general.
>I am defining it as an algebraic property that you may or may not be able to prove for any particular sequence.
This is exactly the problem Wildberger has with the modern theory of limits.
See https://www.youtube.com/watch?v=Ek0URXLCZCE&index=31&list=PLIljB45xT85DpiADQOPth56AVC48SrPLc

The underlying problem is that there is no definition free of ambiguity of the real numbers and infinite processes on which the very notion of a limiting sequence lies.

>>9808571
>I grant that for some restricted cases, it is possible to deduce a correspondence between N and ϵ, but this is not the case in general.
Agreed. Which is why a limit is an algebraic notion, rather than an arithmetical one, just like (say) the zeroes of a function.

>This is exactly the problem Wildberger has with the modern theory of limits.
But why is it a problem? I don't see any problems in this video. Yes, you can write down sequences whose limits you cannot determine without solving some major unsolved problem. Until you solve that problem, you cannot prove what the limit is, nor can you exhibit a function from epsilon to N. Yes, if a sequence t cannot be specified in any finite way in a particular formalism, odds are you cannot produce a finite proof of its properties either. So what is the problem?

>The underlying problem is that there is no definition free of ambiguity of ... infinite processes
Can you explain that? It seems perfectly unambiguous to me. What part do you have difficulty interpreting unambiguously?

>>9806942
here

>>9806942
22/7 is a reasonable approximation

>>9807902
>Without infinity
So, there are finitely many rationals? Sequences terminate after finitely many terms?

>>9806942
π is 4 times the integral of square root of (1-x^2) for x from 0 to 1

>>9808119
I don't care about reality and don't claim reals or even naturals exist as some physical objects. I think of maths as a formal study of symbols, and whatever we can define without contradictions within our formal system exists. We can define infinite sets, we can define reals. Maybe they contradict your intuition about the real world, but don't contradict anything in our formal system. So whilst claims as "there are no infinite sets in our universe, in nature there's no perfect circle with diameter of 2π or line with length √2, we can't do exact arithmetic with irrationals, and even most of the irrationals on computers, we can't do precise operations or the numbers we can store due to some rounding errors" may be right Wildberger's claims as "there are no infinite sets in formal systems called maths, because I haven't seen one in real life, reals don't exist because we don't have a procedure to generate and represent arbitrary real number in finite time, there are no infinitely many naturals because computer can only store that big a number" are completely retarded.
That's why Wildberger belongs to computer science, and not maths, his arguments may be valid for finite machines, but doesn't apply to maths. Look how he never disproves things, just says me no likey likey, therefore all the mathematicians begining from Cauchy or Leinbiz are wrong

Reminder that Wildberger, who claim modern maths is flawed and unrigorous defined "rigorously" naturals as strokes on whiteboard, and his "proof" that multiplication is commutative was to draw an array of circles and "just flip it" (assuming that's not flawed argument, he only proved it for two numbers, 4 and 5 if I remember correctly, so we only know - assuming his argument is right - 4*5=5*4 and nothing about multiplying any other numbers, be it infinitely many or 10^200 of them)

>>9808198
They are connected, but not equivalent. Something flawed won't be beautiful. Something completely rigorous and correct may not be beautiful

>>9808476
Of course there's nothing wrong with that. What's wrong is saying "I don't like the concept of infinity, therefore my finite model of maths is objectively the best and yours doesn't exist, just because I say so"

pi is irrational which is derived from the construction of the circle. a circle is an ideal shape but is logically sound. does logic hold a lower ground than nature for you? you need to take a moment to reflect on what your intuition garners if you wish to obtain any insight as to what is and isn't a part of nature to you. would a physical manifestation of a numerical value change it's meaning that we can describe?

>>9808690
The original is funnier.

construction of real numbers as equivalence classes of cauchy sequences is ugly. what is beautiful is that if you define the reals axiomatically, you can show that they are unique up to isomorphism. so you can completely ignore the actual constructions and work only with their properties which are all as intuitive as it gets.

>>9807129
>I know that 3 exists
Get off 4chan, Plato

>>9808771

I can do better. I can draw a circle and tell you to imagine it is perfect.

>>9808890

>>9808892

>>9808789
>which are all as intuitive as it gets.
ho sweety

How the fuck do you use LateX here, why doesn't [math] work?

>>9808371
that's what's implied

pi = 8958937768937/2851718461558

You may not like it but this is what 29 decimal digit accuracy looks like.

>>9808497
>With the present objections and ambiguity around the current 'fundamental set theory
Can you give me one example of ambiguity in ZFC? Because it seems like you are just parroting Wildberger without actually checking to see if he is correct.

pi = 2.73

Prove me wrong.
Protip you can't.

Wildfags have repeatedly failed to give a single example of these flaws and ambiguities they keep telling us exist. So we can safely assume they are lying and can move on to their real position: infinity is ugly.

Now let's take a vote on which is uglier:

1. The natural numbers are infinite

2. The natural numbers are bounded by some arbitrary number

>>9809070
>infinite in finite universe

>>9809074
>finite universe
According to what evidence?

Which is uglier?

1. A flat uniform simply-connected universe

2. A universe with arbitrary bounds

>>9809080
>Which is uglier?
your mum

>>9809080
>Which is uglier?
This is a science board

>>9806942
Does this bomfungol even to the brtzt of limit?
Can he actual prove the numbers don't?

>>9809094
And what scienctific evidence tells you the universe is finite?

>>9806963
>Remember, Pi is the ratio of the circumference of a circle divided by its diameter.

I like how there are literally hundreds of posts after this one. And another thread regarding how 98% of the population has an iq below 130

>>9809074
This universe is not fundamental, the platonic realm is.

>>9809170
>literally believing in magic

>>9809156
google visible universe brainlet

>>9809181
That's not magic, and there is no "belief" it's just what it is.

>>9809183
>That's not magic
it is

>>9809182
>visible universe
We're taking about the universe

>>9809195
Nope. Platonism is objectively true. Your metaphysics are false, if they are not idealist.

>>9809202
yes that's the universe unless you believe in magical phenomena

>>9809204
nope it's ojectively magic

>>9809208
Nope. Essence precedes existence, it is more fundamental. If you refuse this, it makes sense that you'd be a dumb finitist lol

>>9806942
Only if you draw me a perfect circle first

>>9809210
>>9809208

>>9809215
checkmate norman

>>9809220
Not magic, there's no such thing as magic. This universe is not fundamental.

>>9809232
>there are things larger than the universe

>>9809206
Mmmhm so if that's the universe then mass and energy is being created out of nothing at the edge, and that mass is instantly forming structures that should take billions of years to form. And that mass instantly gains momentum from nothing. Oh and the Earth is the center of the universe for no reason (hello geocentrism). Oh and if we send a telescope out suddenly mass is created for no reason to accommodate the telescope. And that reminds me, which universe is the correct one, the telescope's observable universe or our observable universe?

Or we could just recognize that there is a universe outside our observable space and answer all these questions. Retard.

>>9809258
Yes, this universe is not fundamental.

>>9809259
>he doesn't know what observable universe means
the state of /sci/

>>9809261
>>>/x/

>>9809264
lol so this is the final brainlet argument. Sorry dude, you're metaphysics are objectively wrong, there's nothing magical or supernatural about it. Essence precedes existence and this universe is not fundamental.

>>9809278
>>9809181

>>9809282
This universe is not fundamental. No matter how many times you say "that's magic" doesn't matter. There's no such thing as magic.

>>9809285
>This universe is not fundamental.
Why?

>>9809288
Because I can prove things objectively exist, which you will never find in this universe. Thus, this universe does not contain everything that exists, and there is a more fundamental category of objects.
>que THERE IS NO SUCH THING AS INFINITY REAL NUMBERS DON'T EXIST THERE IS NOT ROOT 2 AAAHAHHHGHGHGHGHGGFFFF

>>9809292
do wut

>>9809296
No matter what you do with numbers, I can show that sqrt(2) is irrational, that pi is transcendental, that infinity always exists.
You finitists confuse numerals with numbers. They are not the same.

>>9808360
If there is such a thing as the "null set", can you please write it down?

>>9809262
Please explain to me what I got wrong about the observable universe. This should be good.

>>9806942
>NUMBURS CAN ONLY BE RATIONAL

>>9809307

The null set is different because it is basically an axiom. Pi is not an axiom.

>>9809300
>confuse numerals with numbers. They are not the same.

Nicely summarized. I give you a cat.

>>9808430
My point was not about the veracity of Wildberger, about which I hold no opinion.

It was just to express disdain for the idea that "my unwillingness to invest time in learning the answer means the answer is not valid, so please do my work for me."

>>9808412
>40 minutes video to say "Achilles never catches the turtle".

Turtles are slow, so it is just common sense that it would take a long time to talk about them.

>>9809319
You can get analysis once you have axiom of infinity, choice, and the other axioms of ZFC. CH and GCH may be necessary for some other things, too.

>>9809338
Your point was proven wrong by the video indeed being an utter waste of time and not even relevant to my question. Next time stay out of arguments you don't understand the context of.

>He actually believes in the "real" numbers

I didn't know I was in the company of such sheeple

>>9809361

Given ZFC by itself is almost surely inconsistent, that clusterfuck is definately inconsistent

>>9806942
all numbers are created by humans and imaginary. If you can't accept a number then just don't, worst that will is you wont be able to solve questions made by people that accept it. Would be no different if the number 3, or 8889984875.

>>9809367
>Given ZFC by itself is almost surely inconsistent
Proof?

somebody should tell wildberger about intuitionism, constructivism and HoTT already

>>9809366
They aren't something to "believe" in, they just exist.

>>9809374

I'm waiting for one of you to make an objection to what I wrote here. >>9808395

I'm also going to elaborate on why I chose to talk about the axiom of infinity and not AC. The most common objections to AC involve constructions in ZFC that defy probabilistic intuition. A colleague of mine recently wrote a conference paper (yet to be submitted) showing that similar "paradoxes" can be constructed in theories as weak as Z or even just Z4. The takeaway is that if you reject AC on the basis of probabilistic intuition, you might as well reject axiom of infinity.

>>9807926
being rude is not an ad hominem

>>9809440
Looks like a crackpot. Post proof.

>>9807926
I'm guessing you did baby Rudin. Most of the statements here are Pi^0_1 statements or around there. Finitism cannot handle stronger statements in analysis, particularly some you come across in measure theory and functional analysis.

>>9809468
>>9809440
Just looked up the paper. It's a crackpot. It's so wrong that the same "proof" could be used to show that the empty set doesn't exist.

Pi is not a number for it is the attempt at quantifying the unquantifiable, it is doomed from the get-go since a circle cannot have divisible properties.

Zero is not a number either. It's the complete lack of a number. Zero is currently used like the brackets in set theory, i.e.
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

The zeros simply act as the brackets that contain the set, that also act as the separators between other sets, like 11, 12, etc.

I'm working on a new number system that does not use 0, we fucking need it.

22/7

>>9809580
>we fucking need it.
No we don't, and we already have that, it's just the Peano axioms starting at 1

>>9809580
>I'm working on a new number system
justcrankthings.jpg

Are the infinity deniers here trying to claim that something as concrete and physical as the proof of the cobordism hypothesis is based on "wishy-washy imprecise methods"?

>>9809400
>They aren't something to "believe" in, they just exist
lmao

>>9808395
>The latter may seem fantastical, but we can imagine it and make true statements about it.
no you can't

>>9807926
>you can do all of real analysis discretely
So there is discrete complete field? Intermediate value theorem works for functions on Q?

>>9809440
lmao

>>9809615
Do you think the cobordism hypothesis is convincing evidence for infinity? Sure you make reference to infinity categories, but category theorists tend to encode infinity using colimits. Sure, there's a big difference between a colimit of an infinite diagram and an infinity category. After all, one HoTT is having trouble defining the latter in its native theory.

But wouldn't it be more convincing to the Wildfolk to give more direct statements about infinity, such as those from measure theory, point-set topology, or set theory?

>>9809684
We can imagine what infinity would be like and make statements about it. Proof: there are set theorists who study consistency strength of large cardinal axioms. The fact that we can study these at all means that there is math to be done about them.

Just because you can't build it in reality doesn't mean that you imagine it. Mathematics has all these layers of abstraction anyways. What's a circle? Perfect circles don't exist in nature. Does that mean we should be studying rough approximations to circles, which have really small bumps in them? What does it mean to have two of something? Saying that you have two of something means that you are identifying two objects as the same thing. How do we ensure that these identifications are well-formed? If you demand that the only mathematics is that which exists in nature, the property of two does not exist independently of the objects to which we assign it.

>>9809817
>We can imagine what infinity would be like
literally can't. your brain is finite and limited by its physical reality. there's no room for infinity in the universe let alone in your tiny head. unless you believe the brain is some transcendental metaphysical object you and your fellow mathematicians can at best pretend to imagine infinity. that's also what gives rise to nonsensical statements such as "some infinities are larger than others" when what you're really saying is "some very large numbers are larger than other extremely large number"

>>9809830
So then there must be some flaw in math like the calculus that uses infinity. Dive we can't possibly understand infinity, we must be getting something wrong when we use it. Can you show me where it is?

>>9809830
Well, there's not enough precision in the universe to have a perfect square either. I guess we should throw away all of geometry, too.

>>9809830
>when what you're really saying is "some very large numbers are larger than other extremely large number"
please pick up a book on set theory

>>9809857
set theory is trash, you cantor whore

>>9809862
Don't you have some shitty youtube comments to make, kid?

>>9809850

>>9809830
>literally can't.
lmao literally that's exactly what mathematicians have been doing for the past century and never in the history of mankind have mathematics flourished so much. bottom line, you CAN describe infinite quantity with finite data. just because you can't comprehend this fact doesn't make it nonsensical.

>>9809879
do you deny that your brain is incapable of infinite processes?

>>9809891
you must admit there's something really special about the human brain that clearly isn't limited by its physicality unlike what many scientists and autists on this board would have you believe

>>9809830
>literally can't
Yes you can, and we do it all the time.
>your brain is finite and limited by its physical reality
And within it's physical reality, you can accurately conceptualize infinity
>that's also what gives rise to nonsensical statements such as "some infinities are larger than others"
The statement isn't nonsensical
>when what you're really saying is "some very large numbers are larger than other extremely large number"
This is not what you are saying with the above statement.
>>9809897
Irrelevant

>>9809908
>we do it all the time.
you pretend all the time

>>9809915
Nope, we do it.

>>9809920
nope you do

>>9809897
You imagine it, as you would an idealized square.

>>9807229
Nicely done.

>>9807429
kek

>>9809929
you only are aware of the imperfections of a square the smaller you go. the idealized square is only possible because we can't go infinitely small.

>>9809879
kek this, saved

3.1

pi = 4 brah

>>9809947
You're confusing what you're imagining with what you're imagining represents in reality.

>>9809830
I'm still waiting for a response to >>9809849

What's wrong?

>>9807140
Undergraduate here; how is math not "a tool for describing reality"?

>>9810080
all approximations, not exact

>>9810156
Ah I see, so what is the error term in the "approximation" 1/2+1/4+1/8+... = 1?

>>9810180
never actually reaches 1, obviously

>>9810191
So how far away is it that it never reaches 1?

>>9810196
do what?

>>9810197
If it never reaches 1 there must be some distance between it and 1. What is that distance?

I'll cut to the chase: for any distance you propose there is a partial sum of a finite number of terms that is an even shorter distance from 1. Thus the distance between the sum and 1 is 0, they are equal.

>>9810202
>What is that distance?
depends how far along you're willing to go with your summation

Is this the most autistic thread on /sci/ right now?

>>9810213
Infinitely far because it is an infinite series.

>>9810217
you can never complete an infinite process by definition

>>9810226
You don't need to, you can transform an infinite process into a finite one. That is what the calculus does. Thus 1/2+1/4+1/8+... = 1 is not an approximation. Pick up a textbook and learn instead of showing your ignorance.

>>9809830
Here's an infinite process you should be able to wrap your brain around. There is a natural inclusion of a complex coordinate space into complex coordinate spaces of higher dimension given by mapping into the left-most coordinates. This induces a natural inclusion of complex projective space into complex projective spaces of higher dimension. The colimit of all such inclusions is given by the space of points generated by the finite data of a point within any one of the constituent spaces. This is called infinite-dimensional complex projective space, and it is a model of the Eilenberg-Maclane space of type [math]K(\mathbb{Z}, 2)[/math].

>>9810213
No, the sum is defined as the limit of that infinite process. The values of any intermediate results are irrelevant.

>>9810253
>>9810267
make belief. you'll never reach 1 from that sum EVER! you'll only get closer

>>9810283
Closer to what? You are describing a limit. A limit is not an approximation. A limit is what an approximation approximates. Thank you for proving yourself wrong.

>>9807429
Kek

>>9810226
So you will never define the difference between the series and 1.

>>9808360
>where "0" is the null set.
nice meme

>>9809777
>Do you think the cobordism hypothesis is convincing evidence for infinity?
I think that merely the existence of \infty-groupoids which are not n-groupoids for any n is convincing evidence for infinity. The cobordism hypothesis is just a physical manifestation of "similar" ideas.
>give more direct statements about infinity, such as those from measure theory, point-set topology, or set theory
I don't consider these fields to be mathematically important enough, as do most other mathematicians. So I don't see any value in doing this.

Define pi as the ratio of the circumfrence to the diameter of the circle with an approximation accuracy to 10^-2

3.14

>>9810703
Approximations are all you need.
Pi = e = 3

>>9810716
An approximation of 3 might be adequate depending your situation. However, imagine you have a cylinder and you need to make three gears to fit inside. These gears need to have evenly spaced teeth or they may jam up. How well do you think 3 will do for pi when working this out?

>>9810753
>Designing based on theory and not trial and error
>Thinking that pi is needed at all for that calculation
>Not using sin(x)=x to make the teeth evenly spaced
>Assuming that there's any way for three gears to not jam up

>>9810785
sin() is just a theory.

>>9810785
>>Not using sin(x)=x to make the teeth evenly spaced
holy shit

>>9809777
Infinity is a law of numbers, a law of mathematics. Numbers can have no true beginning and end, only we assign an arbitrary beginning and end. It's the same with physical reality, there's no true objective beginning and end of a physical object, no truly objective, empirical measurements, because energy is constantly changing and interacting with itself, physical reality is not precise, static or separated cleanly enough to be truly represented by numbers.

Numbers are a representation of infinitely perfect separation and connection - static, precise, objective. They exist in metaphysical reality, outside of the physical, just as consciousness does.

>>9811483
>physical ""reality""
>>>/x/