/sci/ - Science & Math

[Anonymous No.7897827]

What's /sci/'s opinion on the wild burger? Any research mathematicians here that can shed some light on whether what this guy says is bullshit or not? Is modern mathematics fundamentally flawed? Was set theory the worst thing to happen to math?

[Anonymous No.7897847]

Yes most of what he says is bullshit. He's the equivalent of that 0.001% of climate scientists who think man-made climate change isn't real.

[Anonymous No.7897848]

He is full of shit. Rationals don’t exist. Only natural numbers (without 0) exist and are axiomatized as numbers of apples in baskets. This is the true purity of purest of pure intellectual mathematics. There are physical implications as well: a length is always a multiple of the Planck length just like a time is always a multiple of the Planck time and a temperature is always a multiple of the Planck temperature.

[Anonymous No.7897863]

>>7897827

[Anonymous No.7897897]

>>7897848
Yes, but the Planck length is defined to be 1.

Also the biggest number is the number of Planck volumes in the observable universe, which is about [math]10^{180} [/math].

These are the only numbers: the natural numbers up to [math]10^{180}[/math], and no so-called "rationals" (which are actually irrational to believe in).

[Scientist !!ThFjnJh4EkH No.7899097]

>>7897827
Most of the technical details of his math teaching is right as far as I can tell. However, I - and the rest of the mainstream mathematical community - do not share his particular dislike of set theory, especially infinite sets, and all that entails, such as Real Numbers.

[Anonymous No.7899191]

>>7899097
>the rest of the mainstream mathematical community

[Anonymous No.7899194]

>>7897827
https://www.youtube.com/watch?v=tXhtYsljEvY

[Scientist !!ThFjnJh4EkH No.7899195]

>>7899191
Ok. How about this?
>Me, some fool online, in addition to the mainstream mathematical community

[Anonymous No.7899223]

>>7897827
He's awesome. I started from scratch and watched through his Math Foundations series and it's how shit should have been taught.

The undergrads and highschoolers that fill up /sci/ get pissy because he's not what their textbooks say, but nothing he says is wrong, and his concepts are well articulated.

[Anonymous No.7899230]

>>7899097
I share his dislike of set theory. It's convoluted, poor, not rigorous.

What if physics is limited by the way we develop mathematics?
We say quantum physics is about random phenomena, but that's using our mathematical tools. What if in a different mathematical system, it was entirely predictible?

[Anonymous No.7899231]

His ethnicity baffles me. Some videos he looks Indian, others he looks white European. Shitty camera I guess.

I assume he's Dutch descent.

[Anonymous No.7899250]

He's an ultrafinitist so most of what he says is junk.

>>7899231
He's Jewish.

[Scientist !!ThFjnJh4EkH No.7899337]

>>7899230
>It's convoluted,
personal taste
>poor,
personal taste
>not rigorous.
No idea what you're talking about. It's quite rigorous.

The dude once said that math is not just playing games with symbols on paper. The dude said that math is something something about real reality. The guy is absolutely wrong. Math is just playing games with symbols on paper.

[Scientist !!ThFjnJh4EkH No.7899340]

>>7899337
>Math is just playing games with symbols on paper.
And in the context of playing the game of symbols on paper, there are very well defined rules of what moves can be done, and what rules cannot be done. Ergo, rigorous.

[Anonymous No.7899498]

>>7899230
>I share his dislike of set theory. It's convoluted, poor, not rigorous.
How is set theory not rigorous? It's literally the most rigorous part of maths.

[Anonymous No.7899529]

>>7899498
I hate to burst your bubble, but category theory is the hot new thing. It avoids things like Russel's paradox and can do everything much better than set theory can.

[Anonymous No.7899540]

post your rare wildbergers
haven't watched the dude's channel for a while. i kinda lost it when he claimed his new rational approach will give birth to a spectacular operator theory over finite fields.
>JUST

[Anonymous No.7899549]

he's legit mathematician
you shouldn't use his videos to learn stuff though, because he does things differently than they're usually done

[Anonymous No.7899550]

>>7899540

[Anonymous No.7899552]

>>7899340
If it is so rigorous, then how is it possible to construct paradoxes such as 0.999... and 1 being indistinguishable?

[Anonymous No.7899560]

>>7899552
>0.999=1
>paradox
baitin' this early in the day

[Anonymous No.7899589]

I'm almost done with my PhD and I'll bite

WB's ideas are pretty bad, but mainly because he has some more reasonable opinions which he pads with very soft statements that are actually quite wrong.

Real numbers have lots of formulations (which may or may not be the same depending on choice whatever), but when using Cauchy sequences things become very straightforward. You have a computer that stores a finite number of digits. As time progresses you build better computers that store more digits and can carry out computational processes faster and faster, further and further. If you build a theory of, I don't know, PDEs using real numbers and have existence/uniqueness solutions that also guarantee certain computational methods converge to the solutions (where all of this is going on in the magical realm of your mind or whatever), then you are really saying is that as our computers get better we can build better approximations to problems that are modeled well with mathematics, usually because of symmetries in the physical world. Mathematics is the study of structures and lots of processes in the world have some approximate structure, so this seems like a very good idea.

[Anonymous No.7899594]

Who would win in a battle to the death between the wingdings samurai and wildburger?

[Anonymous No.7899596]

>>7899589
Are there lots of paradoxical things associated with real numbers, of course. Is it odd that "most" real numbers aren't computable, I don't know you tell me. Will you ever need to compute an uncomputable number, of course not.

Mathematics is used in the world to do things, and rigorous foundations of mathematics helps us predict what some of those future things might look like. On the other hand, most research mathematics comes from the fact that people are just interested in the structures that mathematics studies themselves, and so it's not upsetting that stuff looks a little weird because the point isn't "how does this connect to our intuition about the world."

This sounds so corny but sometimes math is sort of like art. Lots of contemporary art is about a dialogue with the past, and is principally about art for other artists. This is what the majority of research mathematics is like. On the other hand, at some point you need to build a fucking building and some architect is responsible for doing that, and they incorporate certain artistic/aesthetic elements of their training in order to do a good job. Similarly, mathematics can actually be used to solve real problems and so people do in fact use it.

[Anonymous No.7899602]

He is a respectable mathematician, wrote lots of proper papers like any other math prof and (in contrast to most profs) he's a really good and clear presenter. And he produces free lecture level videos (if a bit slow going).

In addition to the above, he has some philosophical problems - those stem from issues that have been identified over half century ago. But he also doesn't bother to learn computational theory proper himself.
And he's for varying/non-fixed foundations, but he's also not alone in that, let alone the first one to propose such a position.

>>7899596
>Is it odd that "most" real numbers aren't computable
Most are not even definable, i.e. you can't even set them up / speak about them (as your logical alphabet is countable).

[Anonymous No.7899797]

>>7899596

There are some non-computable (but definable) numbers that would be extremely interesting to know.

https://en.wikipedia.org/wiki/Chaitin%27s_constant

[Anonymous No.7899829]

>>7899337
>personal taste
that's why he said dislike, you fucking moron
do you even think before typing something?
>>7899560
>paradoxes are bait
I hope you're ready to catch some pretty big fish if you go into math.

[Anonymous No.7899830]

>>7899589
In Dedekind cuts it's easier to see the issues because the definition for a cut is impredicative.

You don't have the same exact problem with Cauchy sequences but that's only because Cauchy sequences are sloppier. In fact you have many problems with the formalism. In particular if you want to give a cauchy sequence that converges to an arbitrary real number that cannot be described by a string of finite length then you end up having to define sets either impredicatively (hurr durr it's just the set that works) or using a predicate of infinite length (not allowed in logic).

Furthermore, the computable reals are not the same as the reals. They aren't even nearly as well behaved.

https://en.wikipedia.org/wiki/Computable_number#Can_computable_numbers_be_used_instead_of_the_reals.3F
https://en.wikipedia.org/wiki/Computable_analysis
There are classical and intuitionistic versions of computable analysis.

It is obvious you don't know what you're talking about, anon.

[Anonymous No.7899849]

>>7899596
I'm not saying you're wrong, nor that I disagree with you (because I don't) but I want to point out that your argument can be strengthened to support ultrafinitism as well.

Note that because the universe is finite then there must exist numbers too large to be computable and much less representable within the universe. Does it make sense to make claims about such numbers? Does it even make sense to claim that such numbers exist?

To take the argument further, one could point out that really fine rationals end up becoming the ratio of huge integers. At some point said integers become to large to exist and so the existence of these rationals becomes dubious.

I think if we took your argument to it's natural conclusion we would say that it only makes sense to talk about a computable, finite but arbitrarily large subset of the real numbers (that also incidentally doesn't contain all integers or all rationals).

Philosophically I agree with this notion. However I must also acknowledge that this appears to be incredibly difficult.

[Anonymous No.7900736]

>>7899830
>>In Dedekind cuts it's easier to see the issues because the definition for a cut is impredicative.
you can define these cuts in predicative maths.

[Anonymous No.7901250]

>>7900736
How do you do that? I'm actually curious. My current understanding is that you need to define least upper bounds and greatest lower bounds which are impredicative by nature.

[Anonymous No.7901299]

>>7897827

He's just a mathematical atheist. Ignore him like all crackpot atheists.

[Anonymous No.7901303]

>>7901250
johnstone himself gave a talk on them
https://www.youtube.com/watch?v=pKWYa9sc5UI

here is the definition
https://ncatlab.org/nlab/show/Dedekind+cut#Definitions

the point is to define Q, then define constructively the ideal completion of Q (Q as a poset).
it is natural to consider subsets of Q which is closed under the order, or, in other words, the subsets L of Q which are a directed supremum of its principal ideals. which means:
L= directed U_{k in L} {q in Q st q < k}

the directed joins are constructive because they correspond to ''the certainty that a program finishes'', once you take the perspective of CS.

[Anonymous No.7901320]

>>7901303
I will take a look at these, thank you. I much consider Dedekind cuts to be the right way to deal with real numbers as compared to Cauchy sequences.

I'm very surprised to hear that there's a way to formalize them in predicative math.

[Anonymous No.7901336]

>>7901320
here is how they are defined, in hott. it is the same as the definition in predicative locale theory.

[Scientist !!ThFjnJh4EkH No.7901473]

>>7899529
>It avoids things like Russel's paradox
Conventional set theory also avoids Russel's paradox.

>and can do everything much better than set theory can.
Explanation / citation please.

[Scientist !!ThFjnJh4EkH No.7901477]

>>7901303
You are an awesome anon.

[Anonymous No.7901616]

I wish he and his proponents would read up on serious constructivism, maybe check out certain type theories as alternatives to set theory instead of meming up the internet with garbage

[Anonymous No.7901869]

yeah

[Anonymous No.7903111]

bump

[Anonymous No.7903177]

>>7899552
So, what's 1-0.999…?

[Anonymous No.7903190]

>>7899250
>Jewish

It all makes sense now. He hates set theory because Cantor was a white Christian

[Anonymous No.7903204]

>>7899589

Listen to this shit.

Christ, you are living proof that they give PhD's away to any fucking monkey.

[Anonymous No.7903221]

>>7899830

That's not being sloppy, that's you rejecting set theory. There is nothing sloppy about Cauchy sequence construction and it's entirely equivalent to the set construction(but makes more sense intuitively). It's construction method is also how you expand any metric space into being a complete metric space.

[Anonymous No.7903286]

>>7903221
It is sloppy in set theory.

Since logic doesn't allow sentences of infinite length then you have at most a countably number of sentences in your set theory. You cannot use a countably infinite number of sentences to define an uncountably infinite number of objects.

At best you get cauchy sequences for a countable number of real numbers.

This problem goes away in Type Theory and Topos Theory as other anon's pointed out.

[Anonymous No.7903295]

>>7903286

Except you can as it's specifying their form, not enumerating them beforehand.

[Anonymous No.7903342]

>>7899829
>an unknowable value can be presumed to be a higher value due to it's predicted motion towards that value and the impossibility of it ever reaching that value while under observation
Good call autist, that is totally a paradox

[Anonymous No.7903391]

>>7903295
This is exactly the problem. You are implying that for every real number you can write a sentence of finite length that uniquely characterizes it. However you can prove that the set of sentences of finite length is countable.

[Anonymous No.7903419]

>>7903391

I'm not. Cauchy sequences are real numbers and they can be approximated to arbitrary precision but no one claims to be able to uniquely characterize them all nor do you need to.

[Scientist !!ThFjnJh4EkH No.7903510]

>>7903286
>Since logic doesn't allow sentences of infinite length then you have at most a countably number of sentences in your set theory. You cannot use a countably infinite number of sentences to define an uncountably infinite number of objects.
Yes you can. Not constructively, but you can define them.

[Anonymous No.7903547]

>>7903419
>approximated to arbitrary precision.
As soon as you talk about approximating to any precision you lose the ability to distinguish between different reals. You may as well be dealing with rationals.

>>7903510
Only a countable subset. Which is fine if you only care about "muh pragmatic real world" and not about mathematics.

[Scientist !!ThFjnJh4EkH No.7903554]

>>7903547
I don't understand what you're talking about. No: It defines the whole set. It doesn't constructively define the whole set, but you don't need to be able to construct something in order to define it.

[Anonymous No.7903574]

>>7903190
Taking over math is just the next phase of their global banking scam.

[Anonymous No.7903577]

>>7903554
First one defines the set of Cauchy sequences, C.
Then one shows that by definition every element in C is defined through restricted comprehension on the rationals. eg.
[math]c\in C\Rightarrow c=\{q\in\mathbb{Q} | \Phi (q)\}[/math]
where [math]\Phi (q)[/math] is a predicate (where [math]q[/math] is free) in your logic (and thus [math]\Phi (q)[/math] has finite length).
Finally one uses a simple formal language argument to show that the set of all sets defined using restricted comprehension on the set of rationals must be countably infinite because the set of finite predicates over your language is countably infinite.

[Anonymous No.7903597]

this is from the talk by johnstone above

R_d is the set of points of the locale R_f constructively formalizing the real line through the dedekind cuts

[Anonymous No.7903606]

>>7903577

Except it's not of finite length. Your same argument says the power set of the natural numbers is countable when again thats not the case. There is a fundamental flaw in your argument.

[Scientist !!ThFjnJh4EkH No.7903615]

>>7903577
Work with me please. This is slightly above my pay grade, but only slightly. I think I can keep up.

First, forgive me laziness in learning Latex (or however it's capitalized).

First question, that's an unusual set-builder notation. I'm used to it being the other way around:
> {q∈Q|Φ(q)}
Shouldn't that be this instead?
> {Φ(q)|q∈Q}

And this anon named my other complaint:
>>7903606

Let me put it in my own words. From the axiom of infinity, we have the naturals. It's trivial and straightforward to construct the integers, then the rationals. Naively, and it's been a while, I would think that you would define Cauchy sequences as this:
{ f | f is function N -> Q }
equivalently:
{ f | f set-member-of (N x Q) }
Where I mean the Cartesian Product, which is straightforwardly done from standard ZF set theory.

As the other anon said, your problem seems to be your definition of Cauchy sequences.

[Scientist !!ThFjnJh4EkH No.7903618]

>>7903615
>First question, that's an unusual set-builder notation. I'm used to it being the other way around:
>> {q∈Q|Φ(q)}
>Shouldn't that be this instead?
>> {Φ(q)|q∈Q}
Ignore that part. Brain lapse. I haven't done this in years.

[Anonymous No.7903676]

all of these questions of what is right or legitimate hide the question of ''what do you expect from maths, why do you do maths?'' which hide the question ''why do you get up in the morning and do maths (rather than doing something else) ?'' which hides the question ''why do you take your formalizations, abstractions seriously enough to spend time on them''

[Anonymous No.7903703]

classical mathematics bfto


there is a nice course on logic, philosophy and mathematics at Cambridge

http://www.phil.cam.ac.uk/curr-students/II/II-lecture-notes/

for the theorems by Godel,
http://www.phil.cam.ac.uk/curr-students/II/II-lecture-notes/ii-gwt5.pdf

where you increase the number in the name ''gwt5'', to get all the pdfs.

[Anonymous No.7903718]

>>7903703
so which one is correct?