/sci/ - Science & Math

>>8165779
He's always trolling.

Master ruseman.

>>8165779
No

>>8165779

He smirks in every video. He's definitely trolling.

>>8166064
Delete this

>>8165779
almost finished watching it
I'm going to get up late because of this

>>8165779
>"'"""''so called real numbers"'"'""'"

>>8166064
>>8166189
>>8166244
>>8166215
>>8166276
http://web.maths.unsw.edu.au/~norman/

I don't think he's trolling. He was accepted into Yale's PhD program. I don't think you can bullshit your way into Yale's PhD program.

Or maybe you can. Maybe there's someone here smarter than me that can tell me if there's any substance to his work.

>>8166298
Rational trig is cool, but I don't think I've seen anything new. I'm not exactly sure where he stops taking from Euclid and the Greeks and where he inputs novel work.
The rotor coordinates stuff may be novel, but I don't think it's more useful than other time-variant basis dynamics.

>>8166298
around minute 37 he convinced me

>>8166298
He's not trolling, he's convinced himself that his fringe interpretation of mathematics is the only correct one, and that there is a quasi-conspiracy among mathematicians to hide this. A lot of "smart" people are susceptible to such conspiracy logic. And that's the only reason why he's popular on youtube.

>>8166323
Let's restate this to get around Wildberger's . Draw any two lines of rational length perpendicular to each other. The probability that the length of the diagonal drawn between endpoints of these lines is rational is 0. So does the line not exist?

>>8166338
> length
> forcing a linear measure this hard

>>8166340
So length does not exist? This seems like a hilariously unintuitive result from a supposedly intuitionist mathematics.

>>8166329
Wildbergers argument boils down to being really pedantic about mathematical foundations (specifically in set theory) and the conspiracy you mention is just him saying that most mathematicians are informal mathematicians (i.e. they don't give a fuck about foundations, much less being really pedantic about them).

>>8166338
Matter is quantized. Your argument is trivially wrong.

>>8166344
>This seems like a hilariously unintuitive result from a supposedly intuitionist mathematics.
This could be said about many results in intuitionist mathematics. Another term that's commonly thrown around is "totally crazy".

>>8166348
I don't think Wildberger is "pedantic" in any way. So many people claim he is being "rigorous" when his arguments are completely lacking in rigor. The construction of the reals is widely known and rigorous, but Wildberger never attacks it logically. He simply does not like the axioms of standard mathematics, except he phrases his argument as being against axioms in general. In other words, he pretends that his interpretation is not axiomatic, but the only way to do mathematics rigorously. He claims that experienced mathematicians are somehow hiding flaws from students, but he never substantiates this claim. Mathematicians are not hiding the axioms of ZFC from anyone. So his argument is fundamentally political (or philosophical if you want to be generous) and not mathematical.

>Matter is quantized. Your argument is trivially wrong.
That's... unresponsive. You draw lines of rational length. This is completely possible if matter is quantized. So what is the distance between the endpoints of these lines?

>>8166351
I just find it interesting that someone who adopts the mantle of "realist" mathematics that "applies to the real world" rejects length, which is an important part of physics.

>>8166362
>I just find it interesting that someone who adopts the mantle of "realist" mathematics that "applies to the real world" rejects length, which is an important part of physics.

Nothing in physics works.
> no transcendental functions
> no harmonic functions

>>8166371
>Nothing in physics works.
Another brilliant argument from the Wild Burgers...

>>8166362
He attacks things at many different levels. Even though he disagrees with axioms, infinite sets, the axiom of choice, and other things he still humors them and shows why even with them the definitions doesn't work.

He has several arguments specifically against Cauchy sequences and Dedekind cuts. One argument relies on the fact that statements of infinite length aren't allowed in formal logic so under axiomatic set theory you can only define a countably infinite number of sets. This gives the definable reals as a countably infinite set and the indefinable reals as uncountably infinite. As such, the only reals you will ever be able to get your hands on are the nice simple ones like sqrt(2).

Even with length, in order to define that probability you need measure theory and as such are assuming the reals exist. In other words your argument is
The reals exist => the reals exist

>>8166383
>He has several arguments specifically against Cauchy sequences and Dedekind cuts. One argument relies on the fact that statements of infinite length aren't allowed in formal logic so under axiomatic set theory you can only define a countably infinite number of sets. This gives the definable reals as a countably infinite set and the indefinable reals as uncountably infinite. As such, the only reals you will ever be able to get your hands on are the nice simple ones like sqrt(2).
I don't see how any of that follows. The construction of the reals is a finite statement. I'd like to see the actual argument, because it does not make sense as you've described it.

>>8166362
>he fell for the physical distance meme
A rigorous definition of physical distance would involve measuring rulers ad infinitum.

Logic is intuitionist, it works on the everyday scale but doesn't work at other scales.

>I can apply my ideals to the real world rigorously
no you can't, all we have are approximations.

Wildberger shouldn't mention god in his videos, he hate on the square root of 2, but casually mentions god.

>>8166383
>Even with length, in order to define that probability you need measure theory and as such are assuming the reals exist. In other words your argument is
I'm simply saying that the distance between two points can be irrational. Wildburger does not deny this, he denies the entire concept of distance in order to avoid that statement.

>The reals exist => the reals exist
That's not my argument though. I am simply showing how unintuitive Wildberger's argument is. The reals exist because they can be constructed from the axioms of standard mathematics. That is clear and if Wildberger could prove it wrong he would publish the proof. But instead he chooses different axioms while telling gullible students that the construction "doesn't work". He's being dishonest.

>>8166410
>A rigorous definition of physical distance would involve measuring rulers ad infinitum.
You don't know what rigorous means.

>Logic is intuitionist, it works on the everyday scale but doesn't work at other scales.
So Wildberger is illogical?

>>I can apply my ideals to the real world rigorously
>no you can't, all we have are approximations.
Math is rigorous. Physics is the application of mathematics to the real world in order to approximate empirical findings. You are positing a false dilemma.

>>8166383
>As such, the only reals you will ever be able to get your hands on are the nice simple ones like sqrt(2).
Separating the "nice" reals from the "bad ones" requires an arbitrary cutoff and is infinitely more messy and less rigorous than simply accepting reals and accepting that some would be impractical to attempt to use.

Why is it any more of a problem that some real numbers are not calculable, when you already are incapable of listing the set of all integers, or all rationals within a finite range? These facts do not seem to form an impasse. I could go on to point of there are both integers and rationals containing so many decimal places that they too will never be calculated or represented.

So should we abandon the integers and rationals? Should we abandon numbers? Should we suggest that math can only be formalized in a set of integers up to a some large N? Rationals formed with numerators and denominators up to some large N? Where does N come from? What if we change N? What are the possible N's? Are we incapable of speaking of that because it's too difficult of a question? Because a small subset of mathematicians are uncomfortable with infinite sets? What a fucking stupid basis to throw out every bit of mathematics that exists.

>>8166383
I'm aware this is not an argument, that is, I'm not engaging you in discussion or anything else, so don't go sperging out calling 'Fallacy!', this is just a comment.

You sound stupid as fuck, mate. The whole way you word things is lackluster, it comes across as you having no idea what the fuck you're talking about. It seems that you're just repeating misconnected thoughts and sentences.

>>8166405
It's simple:
>The construction of the reals is a process by which we can define sets and set operations such that: Every real number has a corresponding set and arithmetic on these sets is given by set operations.
>The set of real numbers is uncountably infinite.
>Only a countably infinite number of sets can be defined in axiomatic set theory.
>Therefore any construction of the real numbers in axiomatic set theory will fail to give a set for every real number. Only a countably infinite subset of the reals (dubbed the definable reals) can be constructed.

>>8166415
It is unintuitive.

Even if your argument gives you since irrationals, it doesn't necessarily give you all irrationals. That is the second thing wildberger points out is wrong in that picture.

>>8166424
>Only a countably infinite number of sets can be defined in axiomatic set theory.
This is a strawman. We are not defining each number, which we already know is impossible. There are undefinable reals. The construction of the reals does not mean defining each real. so either Wildburger does not understand what a construction of the reals means, or you do not understand his argument.

Is this the real life? Is this just fantasy?

>>8166427
>Even if your argument gives you since irrationals, it doesn't necessarily give you all irrationals. That is the second thing wildberger points out is wrong in that picture.
This is hypocritical. Wildberger has not defined all rationals or even integers.

>>8166435
There are undefinable reals. In fact if you were to choose an arbitrary real number at random then the probability of choosing a definable real is 0.

I'm not saying you're wrong but what does construction mean to you?

>>8166441
>Someone humors you to show you the flaw in your reasoning.
>No! You're a hypocrite! You don't believe in those things, that means you can't use them!!

>>8166423

>>8166444
>There are undefinable reals.
That's what I just said.

>I'm not saying you're wrong but what does construction mean to you?
It's not what it means to me, it's what mathematicians mean when they write about it. What they mean is a way of defining the real number system as an ordered field. Not only are they proven to be rigorous, they are also proven to be isomorphic to each other. So does Wildberger really not understand what mathematicians are saying or is he just playing word games to fool people on youtube with strawman argument?

>>8166445
Huh? The "flaw" is only a flaw if you believe that all numbers must be defined. so how is this person "humoring" me? I would be humoring them by accepting this standard that they don't even apply to themselves.

>>8166451
>That's what I just said.
Sorry, I wasn't arguing. I was exclaiming. I guess I should have written it like this
>There [math]are[/math] undefinable reals!! Thank you!

This seems to at least in our case be a semantic issue. The version of construction used in intuitionist mathematics as well as logic and computer science is of actually giving an encoding for some type of data. This concept doesn't just come up in ZFC but it comes up in other axiomatic systems as well such as lambda calculus (see church numbers).

In this context it is typically always possible to construct integers, rationals, and other countably infinite sets. However, when it comes to constructing real numbers the techniques and approaches tend to be weird and exotic (look up some variants of computable reals, the guy who does abstract stone duality also has a ridiculous way of constructing reals and doing calculus over them using lambda calculus. I don't understand a word of it.).

>>8166454
Suppose we agreed that not all real numbers need be explicitly defined under your construction.

How would you respond to the claim that the undefinable reals don't exist?

>>8166460
The don't exist until you define them.

Also, if it's just defineable reals he has a problem with, why no sqrt(2) and pi?

>>8166464
Wildberger has many different arguments against the reals. I don't agree with all of them, or rather I don't really think there is a problem with practicing intuitionist mathematicis and finitist mathematics as separate but interesting branches of mathematics.

I believe these nice definable reals aren't allowed for other reasons. It could also just be that he stubbornly doesn't believe we really even "need" the reals and wants to see just how much mathematics he can create without them. If that's the case then I'm curious to see how far he can get as that subset of mathematics may be easier to port over to other theories and axiomatic systems.

>>8166470
>I believe these nice definable reals aren't allowed for other reasons.
Sorry, let me rephrase that. I believe that Wildberger may have other arguments for why he doesn't allow these nice definable reals.

>>8166460
If they don't exist then every real number is definable. We know they must exist.

>2016
>not learning the most exciting math of the past 100 years

Plebs need to leave

>>8166338
define "probability". Can you even attach "the" in front of it?

>>8166362
he doesn't reject the concept of length.
that's what quadrance is for anon

http://web.maths.unsw.edu.au/~norman/papers.htm

some of his papers if anyone is interested

>>8166420
can you measure something less than a planck length?

>>8165779
can't someone just bring up the squeeze theorem and get Wildberger to shut up?

If we know a rational that is less than the suare root of 2 and one that is more than the square toot of two and we can get those rationals closer to each other isn't it okay to say the square root of 2 lies in between those two rationals?

>>8167008
it's okay to say that square root of 2 exists.
But it doesn't.

>it's okay to say that square root of 2 exists
IF sqrt(2) exists

Oh, the square root of 2 is zero

>>8167008
>Hey guise, I figured it out. Here is one irrational number. Therefore all the reals exist!!
Go back to kindergarten.

>>8167039
Well you people are seemingly incapable of following an actual argument constructing the reals in full.

>>8167123
see
>>8166424

>>8166348
How can he be called pedantic or rigorous when he defines naturals as strokes on whiteboard?

>>8166424
>>8167125
>>Therefore any construction of the real numbers in axiomatic set theory will fail to give a set for every real number. Only a countably infinite subset of the reals (dubbed the definable reals) can be constructed.

Right, we can't define every single real number. But we can define the set of all Cauchy sequences of rational numbers (up to some equivalence relation) and addition/multiplication on this set. Or is taking the power set of the rationals somehow not allowed now either because you wind up with uncountably infinitely many sets?

>>8165779
he's trying his best to be relevant

>says parabola
>draws ellipse
https://www.youtube.com/watch?v=zIRlz4apbZA

√2 = X + (X^2 - 2) in Q[X]/(X^2-2)
kikeberger stumped :^)

>>8167163
that equation has no solution.

>>8167163
did you even watch the video? he's completely ok with algebraic notion of sqrt (2) just not the analytic notion (infinite decimal expansion)

>>8165779
He is breaking foundations of bs by pure logic and common sense.
Something that will never be accepted in academia in any branch.

>>8167252
So we can create a solution to this equation. We can define a sequence of rational numbers which converges to this solution. Ergo we have found an infinite decimal expansion of sqrt 2.

>>8167265
>Ergo we have found an infinite decimal expansion of sqrt 2.
no you haven't.
If you have, what is it? Please post it here (or anywhere really)

>>8167273
https://en.wikipedia.org/wiki/Square_root_of_2#Series_and_product_representations

The bottom should allow you to compute arbitrary digits of a base 2 expansion.

>>8167265
You're on the right track, although using solutions to equations won't give you the real numbers. You can construct all the real numbers as Cauchy sequences. This is one of the standard constructions and Wildburger has already rejected it out of hand

>>8167281
not decimal expansion but still cool

>>8167132
You can define the whole set yes, but in order to show that this set contains an equivalence relation corresponding to an indefinable real then you have to demonstrate it.

I conjecture that such a set contains at least one undefinable real (as an equivalence relation of Cauchy sequences). Now prove my conjecture (pretty sure it's impossible).

>>8168186
Pretty sure it's easy, since such a set is uncountable and there are countably many computable reals.

>>8167222
That's a definition. (X^2 - 2) is the principal ideal generated by the polynomial X^2 - 2.

>>8168276
Well that approach uses a double negative but sure, I'll let you have it.

How is defining an "extension field" different from writing start(2) and calling it "irrational?" It sounds like his problem is with the terminology, not acknowledging that it can't be expressed exactly as a decimal.

>>8170306
His issue is that the decimal expansion of sqrt(2) or any other irrational doesn't 'exist', and that irrationals can't (or haven't yet) been realised as 'numbers' akin to rationals. Basically he doesn't acknowledge any accepted construction of the reals and objects to sqrt(2) being defined this way.

He doesn't seem to have a problem with the (admittedly much more pleasant) algebraic constructions of roots and other algebraic numbers. What really seems to frustrate him is appeals to things like the least upper bound property to demonstrate the existence of a number

>>8166298
He did work in Lie theory before starting his ultrafinitist attack on foundations

>>8166362
Space is quantized so length need not be rejected in that context

>>8167127
He actually defines them as multisets containing only one kind of generic object (that can be called a "mark" or "something" or whatever). Which is practically equivalent to saying they're strokes on a board.

>>8166338
>>8166344
<3

>>8166383
>he still humors them and shows why even with them the definitions doesn't work.
Got a direct link to a particular video that does this? First, he probably doesn't, but I'm in the mood to watch some expert level trolling / delusion.

> One argument relies on the fact that statements of infinite length aren't allowed in formal logic so under axiomatic set theory you can only define a countably infinite number of sets. This gives the definable reals as a countably infinite set and the indefinable reals as uncountably infinite. As such, the only reals you will ever be able to get your hands on are the nice simple ones like sqrt(2).

A well known fact. What's your point? I don't see how this is an argument that "the definition [of Reals" doesn't work".

>>8166460
>How would you respond to the claim that the undefinable reals don't exist?
I'd ask you to define your terms.

The number "1" does not exist in terms of a physical object. Numbers do not exist in the same sense that a basketball exists.

Again, thus, I would ask you to define your terms. Exist in what sense?

>>8170539
In short, his actual problems are:

- Strongly objects to using quantification over an infinite set, even Naturals, and using infinite sets at all.

- Strongly objects to using mathematical concepts that are not wholly definable and constructible, such as Reals.

>>8170760
>le unspeakable numbers meme

God I hate this so much

>>8170790
You don't like it. I get it. However, it's entirely rigorous in the sense that:
- It's useful
- It's self consistent
- There are rigorous rules that govern what is and what is not an allowed derivation according to finite rules operating on finite pieces of text.

>>8170793
Fuck, I just realize I replied to a namefag. Nevermind.

>>8170798
No no, not just a namefag, but a tripfag. Also, probably the longest posting tripfag on /sci/. Started with this name probably over a decade ago.

>>8170802
>Ive been on /sci/ for a decade
so ten years of
>0.9999..!=1
>if evolution be true, then how come there be monkeys? checkmate atheists
>current year >believing global warming
>hey guys I found a way to build a perpetuum mobile check it out
>space elevators
>aliens

why would you be proud to have been in this shithole for this long?

>>8165779
>""""""""""""""""""""""""""""""""""""""""intuitive geometry""""""""""""""""""""""""""""""""""""""""

>>8170835
I don't think I am "proud". I was just saying. Meh. I'm bored right now. Just watching a VOD review by Monte.

>>8170802
>been here since /sci/ was started
> tripfag informs me he's been here for a decade
> fuck i've been here for a decade
> wait I started being a chantard in 2008
> close call, pride saved

>>8170802
>>8170841
How do you maintain sanity? Faith in humanity? Especially responding to repeated meme threads like this one? I've had to take weeks, months, once three years off.

>>8170856
I'm a masochist in some ways. I'm a sucker for arguing with idiots and trolls online. It makes me upset, but I also live for it. It's so fun. I also delude myself sometimes into thinking that I'm making the world into a better place by spreading knowledge (when I post at other places besides 4chan).

>>8170784
In what sense, if any, do they exist?

>>8170860
Define "exist".

They exist according to the existential operator in ZF. In particular, the axiom of infinity of ZF is the assertion that an infinite set exists, specifically the Natural Numbers. IIRC, the axiom is usually expressed in symbolic form with the existential operator. In English, this would be rendered as "there exists...".

>>8170867
Okay, so now how do you prove an existential statement?

Let U be a set of undefinable reals. Prove that there exists a u in U.

Bonus question, is U a definable set?

>>8170937
>Okay, so now how do you prove an existential statement?
In this case, you don't. It's an axiom.

> Let U be a set of undefinable reals. Prove that there exists a u in U.

It's pretty straightforward. I'll just resort to the bane of intuitionists and ultra-finitists: A proof by contradiction, and specifically a derivation from "not not X" to "X". Obviously, I'm going to skip over some steps, and I'll phrase it in English.

I'll take as given that the set of "definable Reals" is countably infinite. I'll take as given that the set of Reals is uncountably infinite.

Assume: There does not exist an element u in the set "undefinable Reals". Therefore the set of undefinable Reals is empty. Therefore the set of definable Reals equals the set of Reals. Therefore the set of definable reals has the same "size" as the set of Reals. Therefore countably infinite is the same "size" as uncountably infinite. False. End assumption.

By the preceding proof by contradiction, I therefore conclude that it is false that there does not exist an element u in the set "undefinable Reals". Therefore, there exists an element u in the set "undefinable Reals".

> Bonus question, is U a definable set?

What do you mean? I can simply define U as the set subtraction
U := (Reals) - (definable Reals).

>>8170945
You're assuming that every definable real is identified uniquely with a finite string. Why can't a definition be infinitely long?

>>8171322
Then a human can never define it. So to a human, it is indescribable

>>8171411
Math isn't built around the specifics of humans. What is 1, for example? It is an abstract concept, no matter what you associate it with or how you represent it materially, not a real, solid, replicable object.

>>8171322
Because formal logic doesn't allow logical sentences of infinite length. Such sentences cannot be evaluated as true or false and proofs cannot be written to establish them.

>>8171589
>but how do you "know" that the parallel lines don't go off and intersect somewhere we can't see?

>>8171608
That is a sentence of finite length.

>>8171411

if you have a process to generate it to arbitrary precision, then it is described.

>>8172799
You'd also need a process to prove each of those arbitrary length statements, and if such exists, you'd probably be able to restate the entire problem as a simpler inductive proof.

>>8166323
>the ratio is irrational
kek

>>8166323
This is.... Embarrassing. I hate Wildberger and he's correct, because space is likely quantized and if not the graphite particles certainly are. Math literalism is bad for your health. Reals matter, and this is not why.

>>8170860
Logically speaking, any theory that has a model is "true" in some sense (ie, it is consistent), and anything posited by that theory (like the set of real numbers) exists in the domain of the theory.

The question of whether it models a particular domain of discourse (like the universe we live in, for example) is a separate question. Even people who know model theory and logic seem to be totally clueless about the distinction between these two things, even though they're literally built into how modern model theory works. Then they go saying misleading things like "real numbers don't exist."

>>8172819
Space may be quantitized, but for calculations above the planck scale it is useful to calculations with real numbers rather than rational numbers.

>>8172893
You don't use actual real numbers, though, but approximations generated by algorithms.

>>8173169
Real numbers substitute said approximations during symbolic manipulation, sometimes disappearing from the resulting equation entirely. Your argument is tangential.

>>8173169
Reals are just a very convenient system of working with formula that happen to describe the real world very well. Also imaginary numbers and complex numbers.

>>8173172
They don't substitute them, as you can't write them down. The symbols represent the algorithms/approximations.

>>8173173
Complex numbers with rational coefficients can be written down, so they aren't a problem.

>>8173269
Many real numbers can be written down too. What's your point?

Defining calculus in Reals is very easy and straightforward.

Apparently without Reals, it's incredibly difficult to even talk about the length of the side of some triangles, which I consider to be obscenely ridiculous. See:
>>8166338

>>8173267
>>Real numbers substitute said approximations
>The symbols represent the algorithms/approximations.
Pedantic.

>>8172799
That is the problem. The number of infinite processes for which you can give a finite description is countably infinite.

In order to have a true infinite process with no finite description it would have to be something like

Choose 1, then choose 4, then choose 2, then choose 5, ...
I can argue that this process continues infinitely and gives an infinite string of digits for a real number in my head but the only way to even know it goes forever or to distinguish it from another real is to actually ask me what each value is, all infinite of them. It has no finite description and no closed form.

>>8170553
You keep saying this without thinking about it. The length of a line between two random points with rational coordinates will almost always be irrational. And space is not quantized according to most physics.

>>8175390
Length is properly measured by quanta

>>8170945
>What do you mean? I can simply >define U as the set subtraction
>U := (Reals) - (definable Reals).

Is the set of definable reals a definable set?

>>8175628
What exactly do you mean? Presumably yes. Look at any intro book on Real Analysis.

>>8175704
Real analysis books rarely concern themselves with raw set theory and even the ones that do like Landau's Foundations of Analysis do not ever deal with the issue of definable sets.

You can, by definition, write a finite set theoretic sentence for each definable real. There are countably many such reals. The question is, does there exist a finite set theoretic sentence that gives you the set containing all definable reals and only all definable reals?

If there is then you can use set difference in order to produce a finite sentence that defines the set of all undefinable reals. If you can't then you will need to find another way to define the set.