/sci/ - Science & Math

*does he mean

>>7527005
Who is this?

>>7527005
It means he's schizo and has a fucked up fascination with fantasy situations where fractions are actually acceptable.

>>7527009
http://web.maths.unsw.edu.au/~norman/

>>7527009
NJ Wildberger. He's a popular math crank on youtube, but at least some of his videos and lectures are useful.

>What exactly does be mean by not doing math with real numbers?
He means nobody ever did.

You have <span class="math">0, 1, 3, -8, 4/5[/spoiler] etc. and then you have the string of symbols
„4 \sum_{n=M}^\infty \frac {(-1)^n}{2n+1}“
which translates to <span class="math"> 4 \sum_{n=M}^m \frac {(-1)^n} {2n+1} [/spoiler]
and the string of symbols
„2\prod_{n=1}^{\infty} \frac{4n^2}{4n^2-1}“
which translates to <span class="math"> 2 \prod_{n=1}^m \frac {4n^2} {4n^2-1} [/spoiler]

For any m, we know how to translate those symbols to algorithms which put out a number like <span class="math">4/5[/spoiler].
For example, for m=13 the first sum is <span class="math">\frac {15411418072} {5019589575} [/spoiler] or 3.07025…
For m=10000, the number starts out with 3.14 and those three digits don’t change for bigger m.

For larger m, two algorithms spit out numbers that are closer an closer, as far as the standard distance function over the rational numbers is concerned.
This fact little is captured by the string of symbols
“\lim_{m\to \infty} (4 \sum_{n=M}^\infty \frac {(-1)^n}{2n+1}) = \lim_{m\to \infty} (2\prod_{n=1}^{\infty} \frac{4n^2}{4n^2-1})“
Such statements are part of „the theory of analysis“.

You may use the three character string „\pi“ to shorten statements such as two algorithms getting closer, but it’s not like doing so improves the theory of numbers you had previously worked with. All of your computational outputs will be some fractions - you might ask yourself what „\pi“ does for you. The answer is probably "A symbol that refers to a particular finite number of digits and as well as a bunch of procedures to spit out more digits."

(cont.)

You can map the naturals N to the strings you can produce on your keyboard
a <— „1“
b <— „2“
c <— „3“
z <— „26“
{ <— „27“
\ <— „50“
aa <— „134“
ab <— „135“
Evidently, in any language there are countable concepts you can talk about.

Whenever any human talked with other humans about numbers, they really talked about either
-rational numbers,
-more generally, algorithms to compute numbers
-statements about algorithms to compute numbers
And if they talked about collections of numbers, those included things that encode one of the above three concepts.

Then 100 years ago, when those collections were formalized in set theory, what happened is that this „set“ formalization introduced new inaccessible objects:
Say you’ve come up with such a procedure that spits out a symbol for all numbers.
‚Set‘ theory is a framework where e.g. the character „N“ refers to what is understood the set of all natural numbers and the symbol „P“ (power set) is reserved for what is understood a short for an operation that makes PN to the set of all subsets of N.
But the theory also says that the elements of PN can’t all be numerated.

You might want to think of all limits like „\pi“ as numbers, ,the real numbers R’, because they are associated with some sequences like 3.14….
This „R“ is then about individual things like „\pi“, but it’s also about things that you can’t even assign a symbol of strings because R is uncountable. It contains non-constructive phantom objects that no physicists or would ever ponder about because you can’t actually give them a name.

Wildbergers says that set theory is not about „sets“, the power set concept fails to capture what was intended and we actually gain nothing in introducing the real numbers.
In fact he claims there is not even a satisfactory string of symbols that characterizes what „R“ together with it’s field features +,·,> stands for.

>>7527228
The upper bounds "\infty" were supposed to be "m"s.

>What exactly does be mean by not doing math with real numbers?
He means nobody ever did.

You have <span class="math">0, 1, 3, -8, 4/5[/spoiler] etc. and then you have the string of symbols
„4 \sum_{n=0}^m \frac {(-1)^n}{2n+1}“
which translates to <span class="math"> 4 \sum_{n=0}^m \frac {(-1)^n} {2n+1} [/spoiler]
and the string of symbols
„2\prod_{n=1}^m \frac{4n^2}{4n^2-1}“
which translates to <span class="math"> 2 \prod_{n=1}^m \frac {4n^2} {4n^2-1} [/spoiler]

For any m, we know how to translate those symbols to algorithms which put out a number like <span class="math">4/5[/spoiler].
For example, for m=13 the first sum is <span class="math">\frac {15411418072} {5019589575} [/spoiler] or 3.07025…
For m=10000, the number starts out with 3.14 and those three digits don’t change for bigger m.

For larger m, two algorithms spit out numbers that are closer an closer, as far as the standard distance function over the rational numbers is concerned.
This fact little is captured by the string of symbols
“\lim_{m\to \infty} (4 \sum_{n=0}^m \frac {(-1)^n}{2n+1}) = \lim_{m\to \infty} (2\prod_{n=1}^m \frac{4n^2}{4n^2-1})“
Such statements are part of „the theory of analysis“.

You may use the three character string „\pi“ to shorten statements such as two algorithms getting closer, but it’s not like doing so improves the theory of numbers you had previously worked with. All of your computational outputs will be some fractions - you might ask yourself what „\pi“ does for you.

Now you can map the natural numbers N to the strings you can produce on your keyboard
a <— „1“
b <— „2“
c <— „3“
z <— „26“
{ <— „27“
} <— „28“
\ <— „50“
aa <— „134“
ab <— „135“

Evidently, in any language there are countable concepts you can talk about.

REMINDER THAT HE LIKES THE p-adic APPROACH

(cont.)

Whenever any human talked with other humans about numbers, they really talked about either
-rational numbers,
-more generally, algorithms to compute numbers
-statements about algorithms to compute numbers
And if they talked about collections of numbers, those included things that encode one of the above three concepts.

But then 100 years ago, when those collections were formalized in set theory, what happened is that this „set“ formalization introduced new unaccessible objects and people decided to call them numbers too:
So say you’ve come up with such a procedure that spits out a symbol for all numbers.
The theory of ‚sets‘ is a logical framework where e.g. the character „N“ is referred to what is understood the set of all natural numbers and the symbol „P“ (power set) is reserved for what is understood a short for an operation that makes P(N) to the set of all subsets of N.
But the theory also says that the elements of P(N) can’t all be numerated.

You might want to think of all limits like „\pi“ as numbers, ,the real numbers R’, because they are associated with some sequences like 3.14….
This „R“ is then about individual things like „\pi“, but it’s also about things that you can’t even assign a symbol of strings because R is uncountable. It contains non-constructive phantom objects that no physicists or would ever ponder about because you can’t actually give them a name.

Wildbergers says that set theory is not about „sets“, the power set concept fails to capture what was intended and we actually gain nothing in introducing the real numbers.
In fact he claims there is not even a satisfactory string of symbols that characterizes what „R“ together with it’s field properties (+,·,>) stands for.

>>7527009
https://www.youtube.com/user/njwildberger/videos

He always claims the modern math theory is not rigorous because he is too stupid to understand real numbers and he doesn’t believe in infinite sets (but he does believe in <span class="math">\mathbb N[/spoiler], don’t ask me why) eventhough the few alternatives he has are terribly lame and show that either he has no mathematical background or he wants to kill the subject.

>Let’s do trigonometry!
>We will use a vector space approach.
>No abelian group, no field and no norm, just some random bullshit with fractions that I tell you is the fundamental building block of mathematics.

>Defining natural numbers as the equivalence classes between finite sets for the existence of a bijection is unrigorous. Let’s define it as strokes on a board instead.

>>7527021
funny how a guy that got he's PhD at Yale, taught at Standford and it's currently a math prof gets called a "math crank" by some undergrad at University of Shit

>>7527348
This

>>7527348
Funny also how some people are brainwashed enough to believe that a person's credentials outweigh their ideas. Regardless of where he graduated, his work is not substantiated and his arguments against noncomputability and large cardinals are in no way validated by anything but his opinions. His work lacks vision and scope.

Stop licking the shoes of the fools that the capitalistic academia tells you have worth.

>>7527009

>>7527005
In my Measure Theory course, the professor actually mentioned this guy when discussing the continuum hypothesis and construction of the reals. He said he's a bit of a constructivist himself and that such sentiments aren't uncommon in math departments but rarely go to the extent Wildburger does.

>Stop licking the shoes of the fools that the capitalistic academia tells you have worth.
>capitalistic academia

Wow.

>>7527426
It is literally a capitalistic academia
t. not a commie

Wildberger's work is weird, artificial, rhetorical etc. His videos are very insufficient compared to good math books, his status here comes from being a meme and started with some drop out's emotions. That post is right, NJ is nothing but a rebel with a degree.

>>7527424
This. There are plenty of mathematicians who prefer concrete mathematics even if they don't bother trying to reformulate the 'standard' theory of mathematics and just work more constructively within it, including probably half the professors at any given math dept, and names such as Shreeram Abhyankar, V.I. Arnold, etc.

The problem with Wildberger is that's he's one of the most annoying, over the top strident, yet extremely low level constructivists out there. While most concrete/constructive leaning mathematicians are happy to work constructively, i.e. finding effective proofs rather than existence ones, and let the classical people do their thing, Hamburger insists everyone else's work is "logically weak", without ever producing a logical inconsistency, and that literally everyone is doing everything wrong and needs to do things over Rat instead of R.

Curiously enough, I never understood the obsession with constructionists with Analysis, which is not even hard to reformulate in various constructive/computable settings. What is infinitely worse is algebra (Algebraic closures? maximal ideals and Noetherian rings anyone?)

>>7527424
>aren't uncommon in math departments
yes, mathematicians are not courageous. it is all about a social pressure from a lack of curiosity from the teachers [when the mathematician is formed].
Mathematicians are humans and the older they get, the less inclined they are towards a change of formalism and even worse towards a change of perspective on what is acceptable and not. A constructivist who assumes his stance is nearly all alone when it comes to teaching, and it is rare to find a lab who specialty is constructivism. Today, this constructivism is in CS department, because the mathematicians dislike the logic. Another illustration of the detrimental physical discontinuity of the departments

>>7527458
>What is infinitely worse is algebra (Algebraic closures? maximal ideals and Noetherian rings anyone?)
Hi

Henri lombardi and thierry coquand, plus the author of the sole book on constructive abstract algebra....

Coquand is the one doing a few things, in abstract algebra, along with some other french math guy, Lombardi, Henri.

http://www.cse.chalmers.se/~coquand/algebra.html

he has a lecture note on constructive algebra, and the page
http://hlombardi.free.fr/publis/NotesDeCours.html
List of people involved in constructive fields
http://hlombardi.free.fr/liens/constr.html

>>7527005
fractions are easier to visualize, for example, 1/8^2 = 1/64 is easier to visualize than .125^2=0.015625

but i think his main beef is with infinite decimals

>>7527239
>>7527243
>>7527343
>>7527424
>>7527458
>>7527592

Thanks guys

>>7527343
He prefers to think of ℕ, a and Q not as sets, but as properties a number can have

Bump for more answers

>>7527373

The thing is this. He is questioning the status quo. The very concepts he is questioning are those which have been taken for granted for literally thousand of years. He raises some excellent points. This all makes people like you very uneasy, you know, like when people began questioning the tenets of long held beliefs, for example that the Sun revolved around the Earth, or say, that Humans and Apes are descended from a common ancestor.

It's just too much for your narrow, blinkered, nasty little mind to even consider that the principles of our mathematical systems may just be very flawed.You don't even want to think about it.

>>7528802
>The very concepts he is questioning are those which have been taken for granted for literally thousand of years.
You have no clue what you're talking about.

>It's just too much for your narrow, blinkered, nasty little mind to even consider that the principles of our mathematical systems may just be very flawed.You don't even want to think about it.
There is no such thing as a flawed axiom. There are useful axioms and useless axioms. That's it. That you don't understand what Wildburger actually disagrees with just proves that he is insufficiently explaining his finitism in order to appeal to naive students and idiot laymen such as yourself who like to think they have are different from everyone else because they found a "flaw" in what everyone takes for granted or secret knowledge. Except you clearly don't understand a thing about what mathematicians know about foundational mathematics and are just parroting what you heard on youtube. You are a fool.

>>7528802
While >>7528872 has already addressed most of what I would like to say, there are some things to add. You are saying that I am narrow-minded because I don't want to consider what Wildberger has to say; this is rubbish, considering that I have watched hours worth of his videos with an open mind to his ideas. The fact of the matter is, as much as you would like to believe that you are some radical, free-thinking young student, you have not actually thought about or bothered checking what he spouts. You have lapped up the purple Kool-Aid, and it seems your mathematical reasoning has suffered for it.

Wildberger's claims are neither new nor revolutionary; extreme constructivists have been around for a long time, and the argument has always been the same. Wildberger, however, is fairly unique in believing, AGAINST rigorous proof, that nonconstructive theories are consistent and nontrivial. That is not to say that constructive mathematics has no place with humanity, though. Intuitionistic type theory and now homotopy type theory are extremely important to the theory of computation, which will likely be a facet of human knowledge for millennia to come at the least. However, we can make valid inferences about reality using nonconstructive theories, and even the incessant raving of the mad cannot change that. R exists just as much as any other mathematical theory.

>>7527348
It's one of the great truths of academia that being brilliant and well-educated doesn't protect you from being a crank.- it just means that you're less likely to be fatally crankish in your particular area of expertise.

Nobel physics prizewinners have preached parapsychology and cold fusion; Nobels in biology have been handed out to scientists who also claim that DNA is telepathic and that viruses communicate through radio.

I have no doubt that Wildberger is intelligent, rational, expert in his field, respected, and a crank.

>>7529009
Wildbooger is retarded

>>7527343
>he has no mathematical background

he is an A.Prof at one of the best unis in australia and is 100% better at maths than you, retard.

>>7529009

The fucktardedness is strong within this one.

People call him a crank because he dislikes certain strings of symbols in first-order logic that he, along with plenty of other intelligent people, thinks we can't interpret in any meaningful or useful way.

>>7528802
Flat earthers also question the status quo, but we all know what they are.

>>7529212
He's absolutely right

>>7529046
I hope you're a mathematician.

>>7529598
I'm taking calc 1 at my community college :^)

>>7527458
he's not a constructivist brah

>>7527005
>What exactly does be mean by not doing math with real numbers?
The same thing everyone else means about doing math with real numbers: nothing. All the statements are vacuous because humans can't do an infinite amount of work.

>>7527373
>Funny also how some people are brainwashed enough to believe that a person's credentials outweigh their ideas.

Amen.

>Stop licking the shoes of the fools that the capitalistic academia tells you have worth.

Double Amen.

>>7529351
It's empirically wrong that they can't be interpreted in a meaningful or useful way. They already have been.

>>7527343
Don't know about him, but constructivist math is a valid school, and late Borel wrote a lot about unnameable reals.

>>7527009
and btw I'm gay.

>>7527005
If our computers were designed to manipulate fractions, then we wouldn't have to normalize 3D models?

>>7529009
Spoken like a true skeptic.

>>7528802
high school in da house

Has anyone actually heard of wilborger outside of /sci/?

>>7530305
I just posted this on /b/ so now they know

Just accept the Wildsperger math, you infidels.

How can one man be so based?

>>7527443
>His videos are very insufficient compared to good math books

I think it's pretty rough to compare the work of 1 guy against 100 years of orthodoxy like this

Judging by some of the vitriolic remarks made against the guy, that "he's a crank" etc, I became quite interested in what he has to say.

I've now watched a few of his videos. They are very interesting to say the least. The thing I like is that he goes right to the heart of the matter in plain English, without hiding behind behind academic gobbledygook ( which is something I see a lot of his critics in this thread seeking refuge in ).

I think its good. He is bringing attention to some of the things that irked me as an undergraduate, but which I brushed aside as too troublesome to think about. I like people who challenge the orthodoxy, from them, and those they inspire, come the truly great breakthroughs in human thought. Not saying that he is necessarily one of those people, certainly he has not proposed any radical ideas to replace the existing orthodoxy, merely pointing out their inconsistencies, but he may well inspire someone else to pursue a line of reasoning that leads to the discovery of entirely new, and entirely consistent, form of mathematics.

Certainly one thing I know for sure, the great advances in human knowledge and understanding do not ever come from the people who use ad hominems and insults against those who think differently from them.

>>7527005
Fractions are real numbers.

No, OP, you are the memes.

>>7528802
>The very concepts he is questioning are those which have been taken for granted for literally thousand of years.
if you had actually understood even a single one of his videos you'd know that's literally the opposite of what he questions

soon the movie starring Tom Fruise

>>7530500
The problem is that he has yet to point out even one inconsistency. He merely has stated that he doesn't like that we associate physical values to real numbers.

>>7530459
Who do you think writes math textbooks? A committee? It's usually one guy.

>>7530500
I like how people like you just ignore the content of the criticism (probably because you don't really understand foundational math) and just go right to "well if people are criticizing him, he must be saying something important that people want to cover up." This tinfoil logic sums up Wildberger's appeal. Without using such rhetoric, Wildberger would have no notoriety.

>>7530662
it is really time that there is sociology oh mathematics

>>7530500
This dude called Gödel proved that there can never be any meaningful form of "entirely consistent" math. So yeah.... not gonna happen.

Also i myslef don't really love rationals. Tbh i find most of the hatred towards the reals could be just as easily translated to the rationals. But then we're stuck with natural numbers and fuck them faggets.

>>7530719
Have you actually studies Godel's work in an academic setting? Because you are butchering him. It is perfectly possible to have an entirely consistent math. Modern math is entirely consistent.

>>7530723
tbh i don't even know what entirely consistent means.

>>7530725
Consistency simply means containing no contradiction.

>>7530728
So consistent = entirely consistent?
If so then i don't feel like i butcher him, or do you mean that zfc for instance is consistent?

>>7530734
Godel showed that any nontrivial proof system cannot prove its own consistency. However this does not mean that all proof systems are inconsistent or that consistent math is not meaningful. Anyway, this is all irrelevant to the topic of discussion. Wildburger has not shown anything to be inconsistent.

>>7530750
>Wildburger has not shown anything to be inconsistent.
he does not need to

>>7529088
>beſt unis in auſtralia
No kidding, I manage the top hedge fund in Ethiopia. Gonna listen when I tell you to long oil?

ITT butthurt ultrafinitist cuckshits and decent folk

Nonstandard analysis master race reporting in

>>7530667

Yup, you sound exactly like the sort of person who would have served on a medieval religious council, rooting out the heresy.

Thank God the likes of you arseholes have no power anymore.

>>7530929
Is this bait? This has to be bait.

Wildberger has already approached what fellow constructivists like Bishop did like 50 years ago. It was called "on-sequences". He was literally in one step from discovering the concept real number = Cauchy sequence. But for some reason, he decided no to go further.

every concern wildberger possesses has been better articulated and addressed by constructivists for over a century

the reason he is so popular on /sci/ is the same reason that .999..., -1/12, and 6/2*3 threads are all popular on /sci/

https://en.wikipedia.org/wiki/Alexander_Esenin-Volpin


>I have seen some ultrafinitists go so far as to challenge the existence of 2100 as a natural number, in the sense of there being a series of “points” of that length. There is the obvious “draw the line” objection, asking where in 21, 22, 23, … , 2100 do we stop having “Platonistic reality”? Here this … is totally innocent, in that it can be easily be replaced by 100 items (names) separated by commas. I raised just this objection with the (extreme) ultrafinitist Yessenin-Volpin during a lecture of his. He asked me to be more specific. I then proceeded to start with 21 and asked him whether this is “real” or something to that effect. He virtually immediately said yes. Then I asked about 22, and he again said yes, but with a perceptible delay. Then 23, and yes, but with more delay. This continued for a couple of more times, till it was obvious how he was handling this objection. Sure, he was prepared to always answer yes, but he was going to take 2100 times as long to answer yes to 2100 then he would to answering 21. There is no way that I could get very far with this.
Harvey M. Friedman "Philosophical Problems in Logic"

>>7530989
holy lol

>>7530989
Ultra finautism.

>>7530989
You fucked up that quote.
It should be <span class="math">2^100, 2^1, 2^2, 2^3[/spoiler], etc, not 2100, 21, 22, 23.

>>7531103
That should be <span class="math">2^{100}[/spoiler], of course.

>>7531103
as if it does not hold for 21, 22 and so on

>>7530975
People find out about him because his videos come up when you search about math on youtube.

>>7530802
>he does not need to back up his misleading rhetoric with actual math
OK...

>>7530929
Nice persecution complex you have there.

I don't understand why people are so upset about Wildburger. He's actually seems to become more and more knowledgable about constructive mathematics.

>>7530975
He's popular because he's explaining foundational issues at a layman level.

>>7531311
the point is that he does not start form classical math, but form the beginning. when he mentions classical math, he only needs to say that he does not like them, the way he has done so far.

>>7530305
I have actually met him irl. He's exactly like you would expect from watching the videos.

>>7531367
which means?

>>7530929
Once again you fail to reply with anything of substance. Maybe you should try learning about foundational math instead of fooling yourself into thinking you know something about it.

>>7531356
>the point is that he does not start form classical math, but form the beginning. when he mentions classical math, he only needs to say that he does not like them, the way he has done so far.
He does not simply say that he does not like classical math, he says that classical math is inconsistent and flawed. He tries to imply that foundations don't exist by pointing to some textbooks that don't contain constructions of the reals. He misleads students with rhetoric and by not presenting a full picture.

>>7531377
juvenile

>>7527387
Top kek

>>7531386
I don't even understand how this can be considered a real (pun hypothetically intended) problem. If we consider the first courses for any mathematician, there is very likely a course on analysis. To justify claims in (real)analysis, there must be a book displaying the construction of the set <span class="math">\mathbb{R}[/spoiler], and actually, there are dozens of them. Because these books exist, there is no need for an author to put the construction into their book, or they can give it in the appendices. Any students doubting the existence of reals can then go and check the construction themselves.

Moreover, it would be unwise to start a book meant for freshmen with Dedekind cuts, tbh.

I would be interested in a book on computational analysis. IE a framework where everything proven could be computed. Wildbergs doesn't appear to offer this.

https://www.youtube.com/watch?v=WabHm1QWVCA

>>7531468
Impossible by the second incompleteness theorem.

>>7531493
Why is he attacking axiomatic mathematics?

>>7531511
Because "modern axioms are just a smoke screen to hide how ill defined a set is", or something like this. This is atleast his argument against axiomatic set theory.

>>7531511
I think an argument can be made that "axiomatic people" removed themselves from what people 200 years considered to be math - nevertheless, I think it must be okay for everyone to do what they like.

Can't you rest analysis on type theory and bypass set theory altogether?

>>7531612
Analysis in the modern formal sense uses implicit characterizations of object (limits, in particlar) that may or may not exist.
Type theories let you implement that, but those theories let you define some notion of set just as well.

If your real question is if there is analysis in constructive mathematics, then yes there is a version with fewer theorems - the classical ones. It's arguably the one everyone (except mathematicans proving things in the standard framework) use anyway - whenever an engineer of physicist implements shit, he has finiteness parameters, such as in this example of constructive analysis:

https://en.wikipedia.org/wiki/Constructivism_%28mathematics%29#Example_from_real_analysis

>>7531612
Analysis in the modern formal sense uses implicit characterizations of object (limits, in particlar) that may or may not exist.
Type theories let you implement that, but those theories let you define some notion of set just as well.

If your real question is if there is analysis in constructive mathematics, then yes there is a version with fewer theorems - the classical ones. It's arguably the one everyone (except mathematicians proving things in the standard framework) use anyway - whenever an engineer of physicist implements shit, he has finiteness parameters, such as in this example of constructive analysis:

https://en.wikipedia.org/wiki/Constructive_analysis#Examples

>>7531612
Analysis in the modern formal sense uses implicit characterizations of object (limits, in particlar) that may or may not exist.
Type theories let you implement that, but those theories let you define some notion of set just as well.

If your real question is if there is analysis in constructive mathematics, then yes there is a version with fewer theorems - the classical ones. It's arguably the one everyone (except mathematicians proving things in the standard framework) use anyway - whenever an engineer of physicist implements shit, he has finiteness parameters, such as in this example of constructive analysis:

https://en.wikipedia.org/wiki/Constructive_analysis#Examples

Let me also reference
https://en.wikipedia.org/wiki/Computable_number
in this context.
People generally work with those, but there are also arguments (Specker sequence blubb) why forming a rich framework over only computable reals is problematic.

Unrelated, I came across this today:
http://topology.jdabbs.com/

>>7531379

Once more you totally miss the point. There is no getting through to you because you are unable to conceive of the conceptual problems raised.

This is very fundamental stuff, which I believe is the reason you have a problem tackling it.

I am sure you might be very good at designing a better mousetrap, but you certainly are not the sort of person who would have conceived of creating a mousetrap in the first place.

>>7531459
>displaying the construction of the set R
no, we is necessary is the construction of the set R according to what you believe is right.

>>7531511
>Why is he attacking axiomatic mathematics?
because he says that the axioms today are arbitrary and can be refuted since they do not seem intuitive

axioms are relevant when they are according to the former sense, that is that they are intuitive and commonly shared.

>>7531499
there are plenty of systems which are complete champ

>>7531459
>Any students doubting the existence of reals can then go and check the construction themselves.
You may recognize this debate tactic as "I don't have time to teach you X," when the fact is that's because one has to be indoctrinated into X, there's no actual instruction possible.

>>7532176
>this debate tactic
>I don't have time to teach you X
>because I don't give a shit about your ignorance
FTFY

>>7532237
>a book meant for instruction at a university
>I don't care about your ignorance
haha really

>>7527239

haha! nice garbage!

>>7532483
What do you mean?

>>7530861
>I manage the top hedge fund in Ethiopia
That is like managing the top McDonald's in Gary, Indiana.
GTFO

>>7527373
>le self-taught 4chan wiz
>le establishment is keeping us down

>>7532546
"Better first in a village than second in a city" Julius Ceasar

>>7527005
Everything you do with a computer does not use any real numbers.

>>7532129
Which axioms are not intuitive and how is your opinion over what is intuitive not arbitrary?

>>7533102
So computers only use imaginary numbers? Wow, I learn something new every day.

>>7533083
I don't agree with this quote at all.

>>7533129
>how is your opinion over what is intuitive not arbitrary?
the question is why does the arbitrariness bother you.

inconvenient truths

https://www.youtube.com/watch?v=REeaT2mWj6Y

>>7533755
Stupid

>>7533469
It doesn't.

>>7533755

DEATH TO THE REAL NUMBERS! DEATH!

>>7534362
Rationals : )
Reals : (

>>7533755
42 minute vid on <span class="math">\sqrt{2}[/spoiler]
fgt pls

>>7533132
>So computers only use imaginary numbers?
No.

Turns out real numbers aren't as real as you thought

Why infinite sets don't exist

https://www.youtube.com/watch?v=XKy_VTBq0yk

does this guy believe in ordered infinite sets mapped to by a sequence?
someone mentioned he believes in N
how is that logically consistent?

>>7535247
nvm
i get it
uncountable vs countable

>>7535247
He doesn't believe that there is an infinite set of numbers called N.

>mfw empiricists and finitists exist

>>7535285
yes, there are a few bright people

>>7535261
So of the set of N is finite then there is biggest number. So if it's natural it has its successor, but it's bigger than the set's greatest element so it doesn't belong to the set N, so if the successor is not natural therefore as the greatest natural number doesn't have natural successor it's not natural itself, so how does that hold up?

>>7535314
He doesn't say it's finite, he says it doesn't exist

>>7535314
go to 2m13s
https://www.youtube.com/watch?v=XKy_VTBq0yk&t=2m13s

How many videos has this guy posted so far?

>>7535394
One too many

>>7535394
About 500. They're on a pretty wide variety of topics, though

>>7527239
But you've seen guys who row crew, right?

wildburger is basically like a neo-kronecker (i.e. the guy who tried to shit all over cantor)

>>7534773
But you just said everything done with computers doesn't use any real numbers. That leaves only imaginary numbers.

>>7530861
>hedge fund in Ethiopia.
LOL so what, you probably manage a couple million then?

>>7535333
This is so retarded I don't even know where to begin. Notice how he just states that there are paradoxes in the theory without mentioning what they actually are. Hmm, seems like that would be the first thing you would describe if you believed infinite sets were flawed... But I guess not.

Also doesn't answer the question about what the natural numbers are. Are they an infinite set? A finite set? Oh I guess not even sets make sense. I can't wait for Wilderger to enlighten us all on what the natural numbers REALLY are, and of course none of this will involve stinky AXIOMS which are just "voodoo". No, of course Wildberger does not use axioms, his math comes straight from God!

>>7535333
There is literally nothing wrong with this.

>>7535452
His argument is: the numbers get so big we can't control them.

>>7535471
What does "control" mean and why is it relevant to mathematics?

>>7527496
Coquand is god.

>>7535452
>his math comes straight from God!

>>7535474
basically it is philosophy: does a number exist if we can't store the information in the universe?

>>7535521
Why should we care about whether a number can be stored in the universe? The universe could have infinite storage space. Or it could not. It doesn't change anything about how we do math.

>>7535627
>being this Platonic

>>7535521
What if there are other universes?

Even normal mathematicians hold a lot of the views people like Wildberger do. They're just quieter about them and aren't alpha enough to release dozens of videos where they ramble about them.

>>7527005
To be fair, the first chapter in his book was actually pretty good, and if someone gave it for me for free, I would gladly accept. I especially liked the fact that he pointed out a method for not using arc functions; he constructed lengths of triangles out of his head using only arithmetic theory behind his "rational" trigonometry. He's autism about "real numbers" was showing though.

Let me see if I understand this, Wildberger doesn't believe for example that the sum of all natural numbers is -1/12, is that correct?

>>7535936
Ya because he doesn't believe in the current definitions of infinity and limits

>>7535675
not even close

>>7535675
>people like Wildberger
... all two of 'em.

>>7535675
historically this might be true. not anymore though.

>>7535936
>>7527005
Man I hate this....The sum is not -1/12....There is an infinity - 1/12....But the infinity can be absorbed in another place...Makes my head hurt....Excitation of a string/particle in 12 dimensions to make it massless....

>>7532168
Yup but the post I replyed too wanted a system encompassing all of mathematics.

>>7531459
>Moreover, it would be unwise to start a book meant for freshmen with Dedekind cuts, tbh.

Do you even rudin?

>>7535521
Who cares about the universe, the point of maths is that we can use it to model behaviour without being bound to physics
>inb4 realists sperging about muh insights

>>7531459
>Moreover, it would be unwise to start a book meant for freshmen with Dedekind cuts, tbh.
Why not?

>>7537016
For the same reason that successor functions are not in many textbooks. Are they useful for the course being taught? No.

>>7535936
That is correct. He may be retarded, but not to this point.

I just kek'd

(I think http://www.irb.hr/users/zskoda/ did this.)

>>7537158
top lel

>>7537023
>For the same reason that successor functions are not in many textbooks.
What? Any introductory logic textbook will have Peano arithmetic and therefore successor functions.

>Are they useful for the course being taught? No.
That makes no sense. You can't do real analysis AT ALL without a definition of real numbers. (Whether that is Dedekind cuts or Cauchy sequences or whatever is immaterial.)

>>7537207
>You can't do real analysis AT ALL without a definition of real numbers. (Whether that is Dedekind cuts or Cauchy sequences or whatever is immaterial.)
It depends of course on your definition of "doing analysis", but I'd say this statement is very untrue.
Evidenced by all the introduction to analysis courses taught to students who are far from understanding Dedekind cuts.

You can do calculus without real numbers

You can do calculus completely without limits.

>>7535431
>That leaves only imaginary numbers.
Wrong.

>>7537520
How am I wrong? The reals describe all numbers on the one dimensional number line. Anything outside it is imaginary.

Choice is really a can of worms

>>7537207
>Any introductory logic textbook will have Peano arithmetic and therefore successor functions.
>logic textbook

I, for one, accept him as my new master.

>>7537491
>>7537502

How?

>>7527348
REKT

>>7537593
>The reals describe all numbers on the one dimensional number line.
Computers cannot represent or calculate with all numbers on the number line.

Nonstandard analysis. You use hyperreals to model infinitesimal change.

>>7538009
And how does that make me wrong? Either:

1. Computers cannot use any reals, which means compters only use imaginaries

or

2. Computers use reals.

>>7537839
The same way your computer does. When doing calculus, you aren't really playing around with limits or anything silly like that, but rather you're manipulating extremely complex algorithms

>>7537839
Guess I should add that you get derivatives by very simple algebraic manipulations of polynomials

FUCKING REALIES

REEEEEEEEEEEEEEEE

>>7538059
But derivatives are based off of the concept of infinity

>>7538056
Taylor series?

>>7535521
The universe is probably infinite.
And no, we can only observe some of it.

>>7530861
why the fuck are you even attempting to type english?

>>7538344
>The universe is probably infinite.
I probably took your mom at least once

Math noob here. I got practically no math education in school on account of going to school in regional Australia (Our math curriculum was 2+2=centerlink payments in 3 more days basically). I recently tried watching some of Wildbergers stuff on youtube at recommendation so I could actually into Algebra, but it was completely mystifying. But I've started watching the tutorials by Maths520 and I'm having no trouble with that at all, though I'm still kind of shaky with equations with no/few known numbers (for example, a(a-b) -a(6+a) had me stumped for a while, don't hate too much, I really can't overstate how little education I got on this stuff in school).

Is Wildberger just shit at explaining or am I just jumping in over my head? From the looks of it, you guys don't consider him all that good at what he does, but that doesn't seem like it should matter for just the basics, right?

>>7538344
Why do so many people believe [insert here] being infinite?

Is the idea of being confined within finite possibilities that uncomfortable?

>>7538382
who's this?

>>7538383
He's alright at education, probably good at what he is researching but holds controversial views on some concepts you won't need for a while anyway.

>>7535452

You are Indian, right?

>>7538391
Layman here

If the universe isn't infinite, wouldn't it have to be inside something else?

Well, except if "nothing" can exist and when you reach one side of the universe and cross it you end up on the other side of it, but if that's the case where is the universe expanding into?

if the universe is finite how would that work?

>>7538035
False dichotomy.

>>7538333
>But derivatives are based off of the concept of infinity
No they aren't. They're based on the concept of the slope of a function at any given point, which can be found algebraically. It's easiest to do this with polynomials.

>Taylor series?
This is the best way to do it for calculus, since you can turn common transcendental functions into polynomials.

>>7538553
>but if that's the case where is the universe expanding into?
It's easier if you just think of it as if the distance between any two things is being increased, or things are being compressed.

>>7532546
>>7535436
>>7538369
>not understanding the point behind the post

He takes constructionist dogma to its logical limits.
Also he hates ZFC for some reason.

>>7538599
He just doesn't like axiomatic systems because he considers them too counter-intuitive to be considered foundations and he doesn't think they're actually that useful when doing most math

>>7538602
Well they aren't useful for doing real math, except when you're actually using them.
Creating a well-ordering of the reals basically requires the C in ZFC, and I'm fairly sure that is required to some math, somewhere.
Besides, the axioms aren't even anything you need to think about, really.
There exists a union of two sets, Russel's paradox can gtfo, there exists a set with a non-finite number of elements, the Cartesian products of a collection of non-empty sets is non-empty.

https://www.youtube.com/watch?v=Bo2w25fDLwY&t=34m36s

>>7538607
>exists
yea, you understand that existence for abstractions means ''some dude wants to consider them to manipulate them and he will try to persuade others to manipulate them as well''

>>7538624
You're making it sound like considering abstractions is some sort if ethical offense.

>>7538577
The negation of the claim "a computer does not use any real numbers" is "a computer uses real numbers"

>>7539220
No, only considering the abstractions he does not like, aka "non-intuitive" is an ethical offense. The abstractions he likes are empirical facts :^).

>>7539250
He's OK with any "abstraction" that can be written down. It's not an "ethical offense" to write down the string of symbols known as ZFC, but to take, say, the axiom of infinity as defining an infinite set into existence when you can't even show an example of one (which would require that you show every element of the set, which is impossible), is what he considers nonsense and what should not be taught as the foundation of mathematics.

>>7539243
Nope.

>>7527005
How can one man be so smug?

>mfw driving and watch wildberg lectures

>>7539220
>ethical offense.
no not at all

>666 videos
Norman confirmed for Satan

>>7540936
Reached 667 a week ago

>>7538369
>>7535436
>>7532546

//whoosh//

>>7527348
>appeal to authority

people sometimes get senile/stupid as they age

>>7540460
Doing gods work anon

>>7539387
Yes.

>>7539291
"Writing down" an abstraction is meaningless. Writing down "1/3" is a representation of an infinite decimal. Writing down "pi" is a representation of of an irrational. Writing down "N" is a representation of an infinite set. Everything in standard math can be written down. If that is actually Wildberger's objection then it just proves how juvenile his position is.

>>7541816
the point is that saying we can put an infinity of objects into a bag is dubious

and as you note, we can label a finite number of objects.

>>7527005

Haha! you dumb mugs all arguing about the wrong stuff.

None of you monkeys get what Wilderberger is driving at.

The very concept of the counting system is fundamentally flawed.

Go and play with some bananas you chimps.

>>7541845
Every rational number can be represented as an infinite amount of digits. So abandon rationals, or you are putting an infinite amount of objects into a bag.

>>7541816
You didn't even respond to my point. The problem is not of the symbols you write down on paper, but of the interpretation of those symbols. For example, you can take "pi" to represent an algorithm that computes a decimal that approximates the ratio of a circle's circumference to its diameter. You can take "N" to be an adjective that refers to a type of number, that is, a natural number, or a counting number. However, you can't take them to represent things that contain an infinite amount of information, as it's impossible to display or even think about an infinite amount of information.

>>7541860
Every rational number contains a finite amount of information.

Newton believed in alchemy. This is why alchemy is 100% true.

>>7541845
If it is so dubious drive it into a contradiction bro. Do what Russel did for naive set theory, then you'll have a point.

>>7527348

He is both better at mathematics than anybody on this board AND many of his ideas are very dubious. The world is complicated sometimes. I still would recommend his youtube channel even with all of the dubious things in it.

>>7543050
None of the mathematics he does is dubious in the least. He's very careful to avoid that.

>>7542834
Wrong

his latest video illustrates how bogus the classical trigonometry is

https://www.youtube.com/watch?v=EiP3Jf5hnAI

>This is the first part of the first lecture of a course on the History of Mathematics, by N J Wildberger, the discoverer of Rational Trigonometry.
> by N J Wildberger, the discoverer of Rational Trigonometry.
>the discoverer of Rational Trigonometry.

>>7542815
Numbers don't contain information, only our representations do. "1/3" is an algorithm containing finite information which repeats forever, just as there are algorithms for pi which contain finite information which repeat forever. If you can't even clarify what your objection means then you aren't arguing mathematically, you're just playing around with rhetoric.

Niks

>>7542811
>>7542811
>You didn't even respond to my point.
I did.

>For example, you can take "pi" to represent an algorithm that computes a decimal that approximates the ratio of a circle's circumference to its diameter.
You can take pi to represent a horse's ass. This is irrelevant to the discussion.

>However, you can't take them to represent things that contain an infinite amount of information, as it's impossible to display or even think about an infinite amount of information.
1/3 can represent an infinite amount of information depending on how you represent it. Pi can represents a finite amount of information depending on how you represent it. There is no significant difference between dividing a remainder infinitely and adding fractions infinitely. It's idiotic to claim that it's "impossible" to think about or do math with pi when mathematicians do it every day. If it was at all problematic, then you would show a flaw. But all Wildberger can say over and over again is that the thing itself is the flaw. You see, he's just as axiomatic as anyone else, except that his axioms are less useful.

>>7543488
He didn't say mathematics, he said "ideas". None of the mathematics mathematicians do are dubious, they are very careful to avoid that.

>>7530305
Yes, I remember studying his rational geometry and algebraic topology years before this became a meme.
In his algebraic topology class, he refused to ever call anything "real" and always used rational and affine. Like instead of R^3 he would say A^3

>>7545330
He also failed to teach much of the important material in a decent algebraic topology course.

>>7545151
Representations don't contain information. Our concepts (IE ideas we have in our finite brains, such as numbers) are what contain information. Mathematical information comes after interpretation.

>"1/3" is an algorithm containing finite information which repeats forever, just as there are algorithms for pi which contain finite information which repeat forever.
They do not repeat forever, as you cannot show them repeating forever.

>>7545167
>There is no significant difference between dividing a remainder infinitely and adding fractions infinitely.
I agree. We can do neither.

>It's idiotic to claim that it's "impossible" to think about or do math with pi when mathematicians do it every day.
Of course we do math with pi, but never with any pi that contains an infinite amount of information. It's always a decimal cut short, or an algorithm that computes a decimal we cut short.

>If it was at all problematic, then you would show a flaw.
The flaw is that mathematicians claim the existence of things that cannot be demonstrated, such as an infinite process.

>You see, he's just as axiomatic as anyone else, except that his axioms are less useful.
Saying that you can't complete an infinite process and therefore can't think about one is not an axiom, but a physical fact. Wildberger is not against the concept of "axioms". He simply dislikes certain axiomatic systems in first-order logic that mathematicians claim are the foundation of mathematics. And it's not obvious that his way of doing mathematics is less useful; vast quantities of work have been done in mathematics for thousands of years without modern axiomatic systems, and indeed one wonders just how necessary they are to most modern mathematics.

>>7545171
NJW disagrees.

>>7535404
Underrated post

Nobody even answered OP's question

>>7545365
>Representations don't contain information.
If representations didn't contain information then they wouldn't represent anything. You have no idea what you're talking about.

>Our concepts (IE ideas we have in our finite brains, such as numbers) are what contain information.
Then what makes rationals "finite" and irrationals "infinite" if "conceptions" contain the information and not the representations? Can you show me a concept independent from its representation? You seem to be contradicting yourself by walking from an empirical view into a Platonist view.

>They do not repeat forever, as you cannot show them repeating forever.
It's trivial to show the periodicity of any repeating decimal. Further, if 1/3 did not repeat forever then 1 =/= 1.

>Of course we do math with pi, but never with any pi that contains an infinite amount of information.
You just claimed the concept itself contains an infinite amount of information. I am only talking about the standard pi, not approximations. You are talking about naive calculation and I am talking about math. When Euler proved that e^(i*π) = -1 this is a calculation involving pi to infinite precision. There is no approximation, no error here. When algorithms of pi are found, they do not have an error, they do not stop at a certain point. They are proven rigorously to be equivalent to pi. The implementation of the algorithm to calculate is not the same thing as, and has no bearing on, the proof. You are essentially putting your hands over your eyes and ignoring most of the achievements of mathematicians throughout history for no reason. It's retarded.

>>7545365
>The flaw is that mathematicians claim the existence of things that cannot be demonstrated, such as an infinite process.
What does it mean to "demonstrate" an abstraction? It means to show how your conclusion follows from your premises. That is what mathematicians have been doing for thousands of years, infinite process or not. Math is an abstraction, get over it.

>Saying that you can't complete an infinite process and therefore can't think about one is not an axiom, but a physical fact.
Of course that's not an axiom, it's not a mathematical statement at all. Again, you are confusing math with calculation. Mathematics is not about completing a process or applying an abstraction, that is computer science. Math is *thinking about* that process or abstraction. Mathematics is about taking logic to the extremes, pushing the boundaries of one domain into another.

>And it's not obvious that his way of doing mathematics is less useful; vast quantities of work have been done in mathematics for thousands of years without modern axiomatic systems, and indeed one wonders just how necessary they are to most modern mathematics.
It's obvious that his way of doing mathematics is less useful since it erases most of the work of those thousands of years.

>NJW disagrees.
Disagreement without a mathematical argument is irrelevant. Wildberger claims modern math is wrong but cannot prove it. All he can do is point to an alternate axiomatic system. This is a dishonest rhetoric.

>>7545721
https://www.youtube.com/watch?v=Q3V9UNN4XLE

>>7545878
Wouldn't that mean the area of a circle
would be defined as <span class="math">\lim_{n \rightarrow \infty} \pi_nr^2[/spoiler]? Isn't that much harder to explain than <span class="math">A = \pi r^2[/spoiler]?

>>7527005
Who is this semen demon?

>>7545802
>calculation involving pi to infinite precision
the thing is that your infinite precision only happens to at best repeating sequences of numbers. There is no infinite precision in pi.
infinite precision says that you know what all the numbers in pi are AND you are able to get a process to compute them whenever you want [once that you have forgotten all the numbers in pi] up to some chosen cutoff.

also, you cannot say that the algorithms have no errors, since for being able to say this, you must compare the output of the algorithm with pi, but you do not know pi.

finally, to be equivalent to pi is NOT to know pi.

>>7546171
not if you decide that pi=1

c=pi

pi does not exist outside of the mind. It is a paradox because it is not in the world, but a consequence of how we process the world. The world is, and is indivisible. Yet we can only see a small amout. The rest we put together with memory into a narrative which is only a representation of the world. How we put this together gives rise to paradoxes that are not in the world. but are only in our narrative of the world. The choice of breaking space into three dimensions comes from a world where the forces of motion are orthogonally independent, but we do not know that the world is a metric, only that our mind thinks it is.
Our logic come from the feeling of inference that was trained by living in the world. If our universe were different, our logic would be different. If large things fit inside smaller things, our deduction would be backwards.
Pi doesn't exist. it is the feeling we get from the process loop of thinking about pi. It is part of a shorthand system that allows us to see more of the world than we can sense, and thus, navigate that world better.

It is a tool, and is only as good as what it gets us.

It is useful to believe that pi exists.

>>7545151
>>7545365
Regarding a formalized perspective on information, there are some strong results on how representations don't matter too much

https://en.wikipedia.org/wiki/Kolmogorov_complexity#Invariance_theorem

>>7529088
Wild Burger isnt a full professor? Damn what a chump.

>>7545151
Are you fucking stupid? The information content in bits of a number n is log2(n)

>>7547416
That's only for a decimal integer you retard. And anyway bits are just another representation of numbers.

>>7546498
Kolmogrov complexity applies to computations, I'm not talking about computations.

>>7546171
>the thing is that your infinite precision only happens to at best repeating sequences of numbers.
Wrong.

>There is no infinite precision in pi.
So what is the precision of the expression e^(iπ) ?

>infinite precision says that you know what all the numbers in pi are
Wrong. It's like you didn't even read the post you're replying to. Again, we don't need to know any digits of pi to know certain things about it. Knowing the digits of pi is not fucking interesting at all to mathematicians and has almost nothing to do with the mathematics involving pi. Take off your computer science blinders and try to get on topic.

>finally, to be equivalent to pi is NOT to know pi.
Wrong. To prove a statement about pi is to know something about pi. To know the digits of pi is to know something about pi, but not particularly interesting.

>>7545151

>pi repeats

>>7547462

>I'm an illiterate faggot

>>7547460
all that you say is that you have several numbers/algorithms having various properties and then you claim to manage to link, thanks to a carefully chosen deduction system, these various numbers/algorithms via bundles of shared properties and that you can show, for a each one of a few properties, that only one number/algorithm has the properties in question.
You do not need to go beyond this...
This is nice, but then you claim that ''all these numbers are the same'' because you are too insecure to admit that you conflate extensional equality with intentional equality.

>>7532085
>look at me i'm so much smarter than you

>>7539387
You're dumb. Computers don't use irrational numbers, they use real numbers, not all of them.

>>7527239

>how to translate those symbols to algorithm
>algorithms spit out numbers that are closer an closer
>All of your computational outputs will be some fractions

haha he thinks computable is the same as rational.

>>7547591
>All that you say is that it's proven mathematically these numbers are equivalent
>Then you argue they are equivalent, because you are insecure!
God damn you are retarded. I'm out.

>>7547772
not the poster you're replying to, but if I had to venture a guess as to what they mean, it's that we can only perform finitistic computations, and so any output we receive after a finite amount of time is necessarily rational

even the idea of a computable real is an idealization of a computation after an infinite amount of time

>>7547772
How so?
Computable = rational would restrict the computable numbers severely. Statting that every computed output (results of a terminated computation, say of pi to 10^9 digits) is a rational is much weaker.

>>7547958
You fail so so hard

Who here has learned Rational Trigonometry?

>>7547967
But anon, the real are properties, not sets. Properties are finite.

>>7527021
How the fuck can you be a "math crank"? I have come up with a couple amateur physics and math work in my time and only the math stuff ever got positive recognition because it's plain to see whether it's true or not from the proof.

From what I can gather this guy is not saying the current way is wrong he is just saying there is an alternative way to do it. Why are people attacking him just for being different? He gets the same answer, who cares?

>>7548322
http://www.crank.net/maths.html

bump for norman

wildbump

>>7548337
He is definitely saying the current way is wrong.

Bump

>>7551134
GEOMETRY WITHOUT THE REALS

He's a lot better in his Math history lectures.

Mathematicians are scared of Wildy

>>7533376
"Tis better to reign in Hell than to serve in Hev'n"
~Milton

Better?

Has anyone here actually read his book on "rational" trig?

What the fuck, this thread is still alive

>>7554076
As it should.

>>7547724
Computers only use fixed sized integers.

How many hours a week do you think Norman works?

>>7554338
i picture him on his knees in the kitchen working hard on his sheets

>>7554345
He's a real mathematician, you know.
Which is weird, because he's a crank.

>>7554114
So are integers imaginary or real? Take your time.

>>7538035
Holy shit you're retarded dude. COMPUTERS CANNOT WORK WITH REALS, ONLY WITH REALLY GOOD APPROXIMATIONS WHICH ARE ACTUALLY RATIONALS

fuck

>>7554803
So you think rationals are imaginary?