/sci/ - Science & Math

[Anonymous No.9784384]

Has Wilderberger finally put the nail in the coffin of modern mathematics?

https://www.youtube.com/watch?v=U75S_ZvnWNk

[Anonymous No.9784395]

>>9784384
>literally named "Wild Burger"
>Canadian
eh?

[Anonymous No.9784411]

Pure autism. Imagine if he spent all this time learning something useful.

[Anonymous No.9784429]

>>9784384
I'm not really familiar with this guy.

Is he a real mathematician? What makes him such a meme?

[Anonymous No.9784533]

>>9784384
>Modern "Set Theory" - is it a religious belief system?

It is neither a belief system nor is it religious. It's not a belief system because different axiomatic systems are known to result in distinct consequences, and no axiomatic system is really "believed" above any other, just employed in different contexts where one wants to see what can be derived from a certain axiomatic system. It is definitely not "religious". It's too bad that he pulls this clickbaity garbage because his History of Math courses and courses on geometry contain interesting information.

[Anonymous No.9784544]

>>9784429

He is an actual, university-employed, PhD mathematician who has dubious arguments purporting to debunk the existence of real numbers and infinite sets.

One problem is that he has such a large volume of material that it is difficult for a normal person to garner a comprehensive framework for his disagreements with mainstream mathematics. I happen to think his arguments for the illogicality of modern mathematics are mostly unconvincing and not rigorous.

[Anonymous No.9784551]

>>9784429
He's a real mathematician who rejects the existence of the real numbers because he doesn't believe in infinity

[Anonymous No.9785108]

>>9784429
he's a finitist and honestly won me over with his arguments.
reals are extremely useful in applied mathematics and, here's the fun part, are only widely accepted in """"pure"""" mathematics precisely because of this. modern day """pure""" mathematics is subjugated to the much more useful applied mathematics

[Anonymous No.9785695]

>>9784429
He's legit, but a bit of an iconoclast. It's definitely worth checking out his videos. He's remarkably lucid and understandable.

I think the meme around him is mainly in jest due to his frequent rejection of real numbers. It's not that he believes the reals don't exist (they do), but rather he argues that they aren't required as a foundation for other mathematics. For example, he demonstrates how trigonometry can be fully developed using only the rationals.

[Anonymous No.9785712]

>>9784384
He's a serious mathematician but has pathological attitudes and beliefs towards infinity which can be safely disregarded.

[Anonymous No.9785877]

>>9785695
>It's not that he believes the reals don't exist (they do)
they don't

[Anonymous No.9785881]

>>9785695
Could you point me in the direction of a "real" number?
I've never seen one before.
A single example will do.

[Anonymous No.9785919]

>>9785881
>1<
Here you go. No need to thank me

[Anonymous No.9785926]

>>9784533
Yeah, I hate overuse of 'religion'. Religion includes things like history, governance, hierarchy, ritual, custom, philosophy, faith, and more, all wrapped up in a distinct grouping. It's not just being dogmatic or close-minded, or simply preferring something over another.

[Anonymous No.9785939]

>>9784384
holy shit this guy's onto something

[Anonymous No.9785943]

>>9784384
no, he still is an autistic nut

[Anonymous No.9785948]

>>9784411
You mean like testing scientific theories

[Anonymous No.9785976]

>>9785919
>the only real numbers are integers
unicorn status confirmed

[Anonymous No.9785988]

>>9785926
of those ritual and faith are the things people would normally think of when using the word religion, and wildberger's obviously emphasizing the faith aspect. i think he's just a skeptic philosophically.

[Anonymous No.9785989]

>>9785108
He is an ultrafinitist. It is essentially different to finitism

[Anonymous No.9785993]

>>9784533
I agree with you - you set the axioms and you look what comes out and many a bit different axiom sets result in different theories which are however on similar footing. E.g. to what sets turn out to be ordable or measurable etc. Or what inclusion properties are allowed impossible. That being said, to be fair, what you say isn't really true. If you read MathOverflow posts of the top set theorists, they indeed consider ZFC in some sense the >muh only real set theory.
E.g. this guy
https://mathoverflow.net/users/1946/joel-david-hamkins
They don't take a pluralistic approach at all, as you may expect.
I haven't really followed their arguments, as they are quite elaborate.

[Anonymous No.9785994]

ultrafinitism is just a metaphysical position (a kind of anti-platonism) and doesn't actually change the math in practice.

[Anonymous No.9785998]

>>9785994
>a kind of anti-platonism
Either that, or the exact opposite. If I don't believe in the existence of non-standard model of the natural numbers (which are however formally constructible), then that doesn't make me a non-Platonist either. If I restrict the existence of things as tightly as an ulta-finitists but believe in existence of the very finite, I'd say I'm a good kind of Platonist

[Anonymous No.9786017]

>>9785998
but if you're going to throw out the infinite then why even bother believing in the independent existence of numbers? you can't actually show that a number exists outside of your construction of it, much like with infinity.

[Anonymous No.9786019]

>>9785919
Looks like a natural number to me. Are you sure we need these "reals"?

[Anonymous No.9786030]

>>9786019
>nah, I'll just keep eating crayons

[Anonymous No.9786137]

>>9784384
Category theory will be taught in high schools in about 50 years.

[Anonymous No.9786150]

>>9786137
No. Too useless

[Anonymous No.9786197]

>>9786137
Haha have you ever talked to a highschool teacher? Living in the basement so long forgot what the real world is like

[Anonymous No.9786214]

>>9784384
YAWN! Wildberger's a broken record here. Boring! OK--real numbers are a spook, but so what? Mathematics is all spooks.

[Anonymous No.9786219]

>>9786017
no the problem goes deeper. you can't even imagine infinity, your little monkey brain is not capable of infinite processes, even computers can't and never will because our universe is ultimately finite

[Anonymous No.9786241]

>>9784429
He's a non-real mathematician

[Anonymous No.9786293]

>>9786241
a rational mathematician if you will

[Anonymous No.9786304]

>>9786137
High school teachers barely know algebra, let alone category theory. High school students (the average student in the US, really) barely has an understanding of algebra by the time they graduate. What makes you think they can learn it?

[Anonymous No.9786313]

>>9786304
High school teachers will be replaced by super-AIs. High school students will be taught by super-AIs from a very early age (and are genetically modified anyway).

[Anonymous No.9786314]

>>9784544
>He is an actual, university-employed, PhD mathematician who has dubious arguments purporting to debunk the existence of real numbers and infinite sets.
The existence of the real numbers has not been proved.
>s. It's not that he believes the reals don't exist (they do)
There exists no complete proof of the existence of the real numbers; they are simply assumed to exist, since such assumptions allow for a great many applications in all contexts where infinite pops up (differential and integral calculus being one of them).

[Anonymous No.9786354]

>>9786314
and even if we assume reals, operations on the are not defined we just approximate everything and add ... at the end. so much for MUH pure mathematics

[Anonymous No.9786396]

>>9785881
What's the length of the diagonal of a 1 by 1 square ?

[Anonymous No.9786402]

>>9786396
you tell me

[Anonymous No.9786404]

>>9786402
I say square root of 2. You'll tell me there is no square root of 2. What is it then ?

[Anonymous No.9786409]

>>9786314
We assume all numbers exist, the naturals are no more "real" than the reals. Otherwise there are shitloads of proofs of irrationals that show us that there are 'real' numbers.

[Anonymous No.9786415]

>>9786404
what is √2 equal?

[Anonymous No.9786418]

>>9786409
naturals are well defined numbers whose operations are also defined and can be written down (ie exist)

[Anonymous No.9786420]

>>9786415
To itself. What else ?

[Anonymous No.9786429]

>>9784384
That was great, really interesting. THanks

[Anonymous No.9786440]

>>9786420
square root is an operator not a number. what is that operation equal to? come on it can't be that hard to write a number down can it?

[Anonymous No.9786442]

>>9784395
leafs are just savage Americans

[Anonymous No.9786444]

>>9786440
>it can't be that hard to write a number down can it?
therein lies the problem: it is in fact quite difficult to write down an infinite number of decimal digits

[Anonymous No.9786451]

>>9784384
He may have put nails in a coffin, but he's failed to actually put modern mathematics in that coffin. He's burying an empty coffin and making a huge spectacle out of it.

[Anonymous No.9786524]

>>9786440
You can't write down infinite digits. Do you think you need to "write down" a number for it to exist ?

[Anonymous No.9786552]

>>9785998
> If I restrict the existence of things as tightly as an ulta-finitists but believe in existence of the very finite, I'd say I'm a good kind of Platonist
No, you're a terrible platonist who denies the existence or reality of mathematical constructions involving the infinite, or even the very large.
>>9786219
>even computers can't and never will because our universe is ultimately finite
That's why ultrafinitism is the opposite of platonism: a platonist believes that mathematical truths are real, even if our particular section of physical universe doesn't allow for them.

[Anonymous No.9786560]

>>9786552
wildberger calls himself an "aristotelian realist" (whatever that shit even is lol)

[Anonymous No.9786570]

I really don't get you guys. We divide the rationals into subsets with no lower bound, no greatest element and bounded from above, then we see some of those sets don't have a rational least upper bound so we say "hey, whatever should be here is a new number" and identify it with the set for which it would be an upper bound. Then we define operations between this subsets and it turns out they're not only well-defined but also coincide with the operations on the rationals. So we keep the system. How can you even think something we literally made up doesn't exist? If I glued a stone with a poker card on a stick and called it "shmerfle", would you argue shmerfles don't exist? I just fucking invented them, of course they do.
>inb4 you can't just make up new numbers
Yes, I can. That's how literally all of the numbers came to be. Why do you accept some made up concepts and not others when they're both shown to be equally valid? (ie the reals are consistent iff the rationals are).

[Anonymous No.9786587]

>>9784544
He doesn't "disagree"; clearly he just doesn't understand.

[Anonymous No.9786668]

>>9786524
you can't even do basic arithmetic with infinite digits you can't literally do anything with it.
"real" numbers is just code for rational approximation. you never ever deal with real numbers, EVER!
>real numbers are mystical creatures that we will never be blessed to feast our eyes upon
>there's supposedly infinitely many of them and yet nobody can even give 1 example

[Anonymous No.9786682]

>>9786668
self-serving false flag.

[Anonymous No.9786707]

>>9786682
what does that even mean?

[Anonymous No.9786822]

>>9786440
Write down 1/3 for me. Go on, do it. 0.333333... what a mess! It's infinite! Keep writing. I can buy a plane ticket to Australia, have a beer with Wildberger, get into a bar fight over the existence of reals, and be home while you are still trying to approximate 1/3.

Division is an operator. 1/3 is a rational number with the property that the inverse operator (multiplication) yields a whole number.

Square root is an operator. √2 is an irrational number whose inverse operator (squaring) yields a whole number.

Logarithms are an operator. log(2) is a transcendental number whose inverse operator (exponentiation) yields a whole number.

All of these are reals and are essential to build a theory of functions (analysis). But analysis is probably the first time in math that we actually need reals and their associated operators.

The genius of Wildberger is that he shows we can develop a complete treatment of many fields of mathematics (trigometry, hyperbolic geometry, etc.) using only the division operator on whole numbers (rationals). That's fucking amazing!

That is, rational numbers are *foundational* to all branches of mathematics.

Real numbers and function analysis are interesting, and elucidate connections between many fields of mathematics. But they are *not* foundational (though textbooks and math curriculum incorrectly treat them as such).

[Anonymous No.9786848]

>>9786822
repeating decimals, unlike """"reals"""", can be written down and most importantly all operations on them are well define.
also the ... after a repeating decimal is again well defined where the ... following 4.28649964.... (a "real" number i just made up) is not

[Anonymous No.9787399]

>>9786848
(((reals))) are a Zionist conspiracy.

[Anonymous No.9787418]

>>9786848
Please give an example of operations on the reals not being well defined. I'm curious

[Anonymous No.9787419]

>>9786396
Show me this so called diagonal.
I've never seen a "1 by 1 square and don't believe such a thing exists.

[Anonymous No.9787421]

>>9787418
literally any, from addition to multiplication and all in between

[Anonymous No.9787422]

>>9787418
What is the algorithm for adding two reals?
We certaintly can't just "add the digits forever".

[Anonymous No.9787424]

>>9787419
Not that guy
please get it through your head that an object doesn't have to exist in the physical world to be worthy of inquiry. We can achieve the most accurate results if we save the rounding for the end. You'd like us to truncate all our numbers before we ever reach a conclusion, that's dumb. Literally nobody is suggesting you can make a perfect unit square in real life, that doesn't mean it isn't worth considering as an abstract mathematical concept

[Anonymous No.9787427]

>>9787421
>>9787422
If we have two real numbers that we can compute to an arbitrary level of precision (the limit of the cauchy sequences exists), then we can compute their sum or product to arbitrary precision. Therefore the resultant number satisfies our definition for a real as well, which is to be expected since it's a consistent system.

[Anonymous No.9787435]

>>9787427
so the solution is ignoring the real numbers and using a rational approximation instead

[Anonymous No.9787438]

>>9787424
So [math] \sqrt{2} [/math] doesn't exist then?

>>9787427
You're just passing the problem of infinity onto the Cauchy sequences then. Infinite sequences are just as non-existent as the reals.

[Anonymous No.9787441]

>>9787427
Can you tell me what 2.537878...+3.462121... equals?

[Anonymous No.9787445]

>>9787435
for practical purposes yes, if you try to apply an irrational number to something physical then you will eventually have to approximate it.
So the point is this:
If you define "existence" to mean perfectly representable in the physical world, then yes, irrational numbers don't exist.
If you take a looser definition of existence (the one mathematicians use), then they do exist because they're logically consistent (please google this term before arguing, it only means that they do not result in a contradiction).
You have a different idea of the word "existence" than mathematicians, that's fine and all, but when two people argue without even agreeing about the definitions they'll just run around in circles.

[Anonymous No.9787454]

>>9787438
No, they don't exist in that you can't represent them perfectly in the physical world. Any mathematician will agree with you that irrational numbers don't exist in that sense. That doesn't mean they're not useful.
>>9787441
since you haven't given an algorithm to obtain the nth digit of those numbers, no I cannot. Without specifying all 2.537878... can really be is the set of numbers between 2.537878 and 2.537879, and you can't really add sets that way.

[Anonymous No.9787456]

>>9787445
>if you try to apply an irrational number to something physical then you will eventually have to approximate it.
no examples of non approximated "reals" in either the physical or the metaphysical world

[Anonymous No.9787458]

>>9787456
>the metaphysical world
what

[Anonymous No.9787463]

>>9787458
as opposed to something physical, call it whatever you want, still no reals to be found

[Anonymous No.9787480]

I'm undecided on whether his autism is of any importance. He makes wonderful videos, though.

[Anonymous No.9787484]

>>9786524
>Do you think you need to "write down" a number for it to exist ?
yes, that's wildberger's whole point

[Anonymous No.9787745]

>>9786293
rationals are reals you big dummy

[Anonymous No.9787889]

>>9787745
Sorry, the reals are not well-defined hence statements like Q is a subset of R are also not well-defined.

[Anonymous No.9787917]

>>9787889
That is like saying that the statement "a dog has four legs" is not well-defined, because a dog could be any dog, and so is not well-defined.

[Anonymous No.9787932]

>>9786587
The problem of finitism long predates Wildberger and has basically existed since the introduction of an infinite object, arguably longer if about the exact nature of mathematical objects more broadly. I don't think there is a right answer, or certainly no one on 4chan is qualified to say so. You can build a consistent mathematics from either viewpoint (and many have done so).

[Anonymous No.9788052]

>>9787917

[Anonymous No.9788055]

>>9784533
Set /believers/ on suicide watch

[Anonymous No.9788058]

>>9786219
Dab status: dabbed on

[Anonymous No.9788397]

>>9784384
>Wilderberger
>Saying anything of value about math
Pick only one.

[Anonymous No.9789279]

how can it not exist if its real??

[Anonymous No.9789299]

Not a math major so excuse me if dumb question. If mathematics is self-consistent, isn't the set of "axioms" arbitrary so long as they can be used to show the rest of the stuff? Can't his approach be just as right as the set theory approach if both can be used to prove the same stuff?

[Anonymous No.9789325]

>>9789279
i dunno lol

>>9789299
Not dumb at all
If your axioms would lead to the same results it wouldn't matter what you take as axioms. But, different axioms can lead to different theorems. like the axioms of euclidean versus non euclidean geometry. leaving out one of the axioms of euclidean geometry, leads to a whole new theory, where some things are new, and some things aren't true anymore. What wildberger is suggesting, would leave out a LOT of stuff. The fundamental theorem of algebra - wildberger calls it the fundamental dream of algebra - no longer holds for example.

The validity of the system is kind of a philosophical question. If you say a self consistent system is valid - then they are both equally valid, but they won't have the same results. Philosophy of mathematics is very interesting, so you should look into it if you're really interested.

[Anonymous No.9790136]

>>9786314
>There exists no complete proof of the existence of the real numbers; they are simply assumed to exist, since such assumptions allow for a great many applications in all contexts where infinite pops up (differential and integral calculus being one of them).
What about the standard 1st semester proof everybody does by showing that sqrt(2) is not a rational and thus a bigger framework has to exist.

[Anonymous No.9790141]

>>9790136
>because sqrt(2) doesn't exist it should exist...

[Anonymous No.9790149]

>>9790136
>What about the standard 1st semester proof everybody does by showing that sqrt(2) is not a rational and thus a bigger framework has to exist.

This just shows that you do not know logic. What that proof actually says is that if you assume the square root of 2 exists (in a number system with only rational) you get a contradiction. What that proof actually says in its original context was that the square root of 2 does not.

In fact, to prove that the square root of 2 is irrational you must prove two things
1) That it exists
2) That it is not rational

If you have a system of real numbers you can appropiate the classical proof that root 2 does not exist, which in the new context would say "if root 2 exists, it is not rational". Then you would have to prove that it exists, which is really only possible using limits.

[Anonymous No.9790161]

>>9790149
It's the length of the diagonal of a 1x1 square

[Anonymous No.9790176]

>>9784384
>Has Wilderberger finally put the nail in the coffin of modern mathematics?
no

[Anonymous No.9790391]

>>9790161
but how do you know the diagonal has a length in the first place?

[Anonymous No.9790398]

>>9784384
Lads, I don't know enough to meaningfully contribute to the discussion. But, with finitism vs. infinitism, which can produce a superior mathematics? Do they have strengths and weaknesses? If so, what's the problem with having separate systems or maybe even mixing them? Or will finitism always result in mathematics that is inferior as a tool?

[Anonymous No.9790402]

>>9790398
You have a very rich system already developed based in part on the existence of infinite sets. Actually you'd have to dispense with a large chunk of modern mathematics.
Now that doesn't mean you'd be unable to develop a comparably rich and elaborate system of mathematics based on finitism, just that it has not been done, or even attempted yet (as far as I know).

[Anonymous No.9790404]

>>9787889
>the reals are not well defined

[Anonymous No.9790407]

>>9790398
the problem is that only the believers in ZFC have the monopole of education.
Also, it is hard to judge a framework, because besides those philosophical appeals, there is no quantization of en effectivity of a framework.
for instance, in a framework,
-does a statement has a proof?
-if a statement has a proof, how many proofs does it have?
-if there are statements which have the truth-value of TRUE, but which have no proof, how many statements do not have proof? ie how many theorems are there?

there is no way to answer those, like in ZFC. We are in the total dark, we do not know how many theorems there are, we do not know if a statement has automatically a proof or not and if it has at least 1 proof how to find the proof.

Most math guys do not care at all about this and just use ZFC because they have been taught this. At the end of their career, they reflect a bit on why they have been doing during 40 years, rather doing something else and they have no reply ''it just work so far, so let's continue this way''

[Anonymous No.9790412]

>>9790407
>At the end of their career, they reflect a bit on why they have been doing during 40 years, rather doing something else and they have no reply ''it just work so far, so let's continue this way'
What are you basing this statement on?

[Anonymous No.9790418]

>>9790412
experience

[Anonymous No.9790431]

>>9787454
>since you haven't given an algorithm to obtain the nth digit of those numbers
There are uncountable many real numbers but only countable many algorithms. Not having such algorithm is the common case.

[Anonymous No.9790435]

>>9790398
>as a tool
No, mathematics IS NOT and WILL NEVER BE about being "useful" insofar as engineering etc. goes.
It is about discovering the underlying truths of the platonic realm and the theorems of whatever axiomatic system you're working with.

[Anonymous No.9790436]

>>9786560
So he's anti-Platonist

[Anonymous No.9790534]

>>9786214
The most tiresome part of people who fight new paradigms is when they flip to saying everyone already knew
Has happened with every god damn conspiracy theory proven true
People ridicule said theory and then say that it was obvious all along

[Anonymous No.9790822]

>>9790391
If your theory can't accomodate simple objects like triangles and circles then there is no reason to use it.

[Anonymous No.9790829]

>>9786214
Literally everything other than mathematics is a "spook" (I hate that word so much)

[Anonymous No.9790851]

>>9786304
They have the mental capacity to learn it because their brain is flexible enough to absorb the information rapidly if given a solid foundation. Most high school classes are just busywork and there is no inherent difference in intelligence between a hs student and a uni student so if we could just gradually shift the material into high school by making more efficient use of time, then in 50 years we will be able to teach CT to the brightest hs students

[Anonymous No.9791005]

>>9786848

... is an operator not a number. what is that operation equal to? come on it can't be that hard to write a number down can it?

[Anonymous No.9791013]

>>9787441

5.999999...

[Anonymous No.9791016]

>>9791013
>

[Anonymous No.9791024]

>>9791005
[eqn]0.\overline{3}[/eqn]

[Anonymous No.9791120]

>>9790534
He's saying "real numbers don't correspond to physical measurements" -- which most people agree on -- "and therefore we shouldn't use them as a basis for so much of math" -- which is the crazy part. If he developed a working version of topology that was powered by natural number arithmetic and didn't mention infinite sets, I would be impressed.

[Anonymous No.9791425]

>>9790435
I know this is the perspective of pure mathematics, that's why I specified 'as a tool', you retard. Because philosophical meandering is irrelevant to utility.

[Anonymous No.9791944]

>>9791425
yes people do not like to wonder why they do things and whether doing is a good thing to do.

[Anonymous No.9792107]

>>9788058
uh, no, but nice try

[Anonymous No.9793949]

>>9791944
Pointless without exploring 'why' and 'good'.
And all irrelevant to mathematics.

[Anonymous No.9794033]

>>9784429
>real mathematician

[Anonymous No.9794407]

>>9790407
Mathematicians are not "believers" in ZFC.
A colleague of mine works in number theory. Computational stuff. His results, and the results of his colleagues, largely require the assumption that the Riemann hypothesis is true. Whether or not they "believe" that it is true is immaterial - they see what they can do when they accept that it's true, and it turns out, they can do a lot.
This guy knows that most of modern mathematics is based on the acceptance that the ZFC axioms are true. You can set up alternative sets of axioms and go nuts, but those that do quickly find most of anything is impossible.
"Two sets are equal if they have the same elements"
If you don't say "OK, that sounds good" to this statement, what are you going to say instead? Are two sets never the same? Are they the same under some other set of conditions? Do you want to create different axioms that mean this is a conclusion derived from those axioms? Why? What makes those better? And what are they?
He's right that the foundations are important. But ZFC is fine. You have to assume something, and complaining about irrational numbers as some sort of "fiction" sounds just like people who complained about negative numbers, or zero. They're useful fictions and we tell powerful stories with them and they go on to help society build amazing stuff.
He's a troll. A great troll, which is why he comes up so much. He knows just enough to sound smart to people who aren't mathematicians, which should also explain why he's so popular here.

[Anonymous No.9796104]

>>9784384
He's just tapping at the coffin that Brouwer constructed.

[Anonymous No.9797564]

>>9784384
Still don't understand why he finds modern mathematics "problematic"

[Anonymous No.9797579]

>>9786354
>doesn't get it
engicuck spotted

[Anonymous No.9797588]

>>9785695
Why do you put "they do" in brackets there as if "existence of numbers" would be a sharp and sweat well captured notion. If you say that real numbers are a concept pushed around by humans and people work with it, then that's obviously true. If you say "they do" exist in any Platonic sense, then that's extremely disputable. So why do you even add this ontological statement as if it would help you point?

[Anonymous No.9797610]

>>9794407
Regarding the bulk of your statement, the point is that of course you can write down any theory in first order logic, say, and study it. And you should. But I can well argue that the 9 axioms that together make for that theory that is called ZFC is not a theory that speaks of "sets" in any good sense. You have this first order theory with a domain of discourse and some variables x,y,z,.. that you quantify over, and you start with some axioms in that theory and derive more statements. And while you started out with some notions that sets should have, the theory as a whole is too strong to be speaking about "sets" as your mom would understand it. Instead, mathematicans since Cantor say "set" and really they mean the objects that fulfill the rules of said theory.

If somebody like Wildberger says he doesn't believe in infinite sets, then it's a valid requirement. (But he's imho nutty when he handwavingly says "the reals are not well defined" simply due to his fundamental rejection of the philosophical position of mathematical formalism.)

Basically, the good set axioms are

https://en.wikipedia.org/wiki/Kripke%E2%80%93Platek_set_theory

and beyond that you're in the terrain where you force "set" to mean this handy object which you mold to model other stuff.

[Anonymous No.9797633]

>>9797610
I don't see how advocating KP solves anything. Very few mathematicians actually care about what a set is, set theory is just a language to work with. Barwise in 'Admissible Sets and Structures' says on the importance of KP over ZFC that "important distinctions on the nature of sets asserted to exist are completely lost." Meaning that from a set theoretic lens mathematical objects can be indistinguishable, whereas in their respective theories that have important differences. He goes on to give some algebra examples, where the predicative definition of a collection of sets is increasing in complexity but still remain the same. I just don't see any algebraist caring about the foundations of their field to that level of scrutiny, they simply won't get anything done. Anyone who does is going to be working in set theory proper or with somewhere in effective descriptive set theory.

[Anonymous No.9797641]

>>9797610
>https://en.wikipedia.org/wiki/Kripke%E2%80%93Platek_set_theory
>Axiom of extensionality: Two sets are the same if and only if they have the same elements.
Then how are they "two sets" if they're the same? Bizarre logic.
>Axiom of empty set: There exists a set with no members, called the empty set and denoted {}.
How can a "set" exist without any elements? This implies that "sets" are more fundamental in nature than the objects within these "sets". No object can exist without a set, but a set can exist without any objects in "it". Are mathematicians ruled by arbitrary brackets?
>Axiom of pairing: If x, y are sets, then so is {x, y}, a set containing x and y as its only elements.
x, y can never be separate sets in the first place, again making the same mistake as the extensionality axiom by separating a set into two sets that cannot be separated
>Axiom of union: For any set x, there is a set y such that the elements of y are precisely the elements of the elements of x.
Same fucking mistake again, x and y cannot be separate sets if they are the same, motherfuckers got double vision or something?
>Axiom of Σ0-separation: Given any set and any Σ0-formula φ(x), there is a subset of the original set containing precisely those elements x for which φ(x) holds. (This is an axiom schema.)
>Axiom of Σ0-collection: Given any Σ0-formula φ(x, y), if for every set x there exists a set y such that φ(x, y) holds, then for all sets u there exists a set v such that for every x in u there is a y in v such that φ(x, y) holds.
Motherfucker I don't believe it. How many ways do they want use to say the same thing? Creating unnecessary separation, why?

[Anonymous No.9797652]

>>9797633
I was doing not much more than pointing to a collection of axioms that is hard to reject for "sets".
Yes of course, set theory provide a language. I was making the point that you can argue that sets theory isn't about "sets". Not as sets in the sense that your mom would understand them, i.e. a certain notion of collection. E.g. I can fully understand that one would want to have a set theory which doesn't involve collections of uncountable things. It might be the best and nice theory for mathematicans, but not a theory of sets.

[Anonymous No.9797700]

>>9797641
>Then how are they "two sets" if they're the same? Bizarre logic.
Two objects represent the same object insofar as all the features of the one are contained in the other, and all of the features of the other are contained in the one. So what's the problem? It's not like sets are material objects like chairs; granted, it'd be nonsensical to say two distinct chairs are the same, since each occupies different positions in space. But sets are abstractions, not material objects, so the notion of a set occupying different points in space compared to another is nonsense. Thus there is no reason to make a distinction between two sets if they have the same elements.

[Anonymous No.9797709]

>>9797641

A = A is bizarre logic. Interesting.

[Anonymous No.9797811]

>>9794407
Except the irrational numbers aren't a "fiction" they are objectively real in the platonic realm, they are more real than anything in this universe.

[Anonymous No.9797843]

>>9784384

Its not the work that Wilderberger does, or even the nature of his objections, its the fact that he willing to question the fundamentals of mathematics. Right down to the most basic conceptions.

For far too long maths and its most elementary fundamental principles have enjoyed a free ride, remained unquestioned. It has taken on the attributes of a dogma, more like a religion than a science. That is not progressive, its stagnant.

Discussion on maths today reek more of Catholic priests debating how many angels can dance on a pin head, rather than creative, innovative and logically sound individuals looking into, and QUESTIONING the fundamental principles of our current models.

The very fact you have many people attacking Wilderberger as a "crank", deriding his efforts, reinforces my view that the Mathematical fraternity is inhabited by far too many narrow and blinkered little minds who are the modern day equivalent of Jesuit priests, attempting to stamp out Heresy. They are like little anarchisms who are a hindrance to innovative thought and progress.

I do not believe Wildberger's efforts will by themselves lead to ground breaking results, but may build a foundation for further inquiry. Certainly his willingness to reexamine and call into question the fundamentals of mathematics may encourage others to do the same. That can only be healthy for the future of how Mankind perceives of the nature of the Universe and applies rational methods to make sense of it.

If so then I have no doubt that within the nest few generations we shall see revolutionary thoughts and re-conceptualizations that will overthrow most, if not all, of current math and render it as obsolete as Newtonian physics as compared to General and Special Relativity.

I am sure that those who were so ready to condemn the likes of people like Wilderberger will remain very quiet then, and hope their disparaging comments are forgotten.

[Anonymous No.9797862]

>>9797843

Its not the work that Wilderberger does, or even the nature of his objections, its the fact that he willing to question the fundamentals of mathematics. Right down to the most basic conceptions.

For far too long maths and its most elementary fundamental principles have enjoyed a free ride, remained unquestioned. It has taken on the attributes of a dogma, more like a religion than a science. That is not progressive, its stagnant.

Discussion on maths today reek more of Catholic priests debating how many angels can dance on a pin head, rather than creative, innovative and logically sound individuals looking into, and QUESTIONING the fundamental principles of our current models.

The very fact you have many people attacking Wilderberger as a "crank", deriding his efforts, reinforces my view that the Mathematical fraternity is inhabited by far too many narrow and blinkered little minds who are the modern day equivalent of Jesuit priests, attempting to stamp out Heresy. They are like little anarchisms who are a hindrance to innovative thought and progress.

I do not believe Wildberger's efforts will by themselves lead to ground breaking results, but may build a foundation for further inquiry. Certainly his willingness to reexamine and call into question the fundamentals of mathematics may encourage others to do the same. That can only be healthy for the future of how Mankind perceives of the nature of the Universe and applies rational methods to make sense of it.

If so then I have no doubt that within the nest few generations we shall see revolutionary thoughts and re-conceptualizations that will overthrow most, if not all, of current math and render it as obsolete as Newtonian physics as compared to General and Special Relativity.

I am sure that those who were so ready to condemn the likes of people like Wilderberger will remain very quiet then, and hope their disparaging comments are forgotten.

[Anonymous No.9797958]

>>9797843
>>9797862
Philosophical schools of mathematics have existed since the greeks you tars. Finitism is nothing new.

[Anonymous No.9798881]

>>9784384
>this guy still makes videos
Jesus fucking christ I was watching his shit 6 years ago...when I was a junior in high school.
I remember trying to explain his 'rational trigonometry' or something to my math teacher and my teacher was confused as all fuck.

[Anonymous No.9798907]

>>9797843
>For far too long maths and its most elementary fundamental principles have enjoyed a free ride, remained unquestioned
Stopped reading there

[Anonymous No.9798913]

>>9797641
>How are "two sets" the same.
They aren't. What you're describing is a rule of discourse established to give meaning to the symbols we use to make mathematical arguments. If two sets have been described separately from each other and the words used to describe their elements refer to the same elements, then both descriptions do indeed describe the same set.

A collection is a primitive notion, as is an object which may be an element of a collection. A class is something which has some defining property and a set is a collection of unique elements.

Nobody takes ZFC as truth because mathematics is about arguing within logical systems. We use axioms to develop theories, as that is their purpose. Within theories we have models, which we desire to be consistent with our axioms. There are axioms for all theories. Axioms for arithmetic, axioms for group theory, etc. They are all "axioms." That being said, the number "2" refers to "2" within the theory, or context, which it's being described, NOT necessarily to the description {{}} in set theory one might use to represent it. While you can represent all these mathematical concepts with sets, and it is important that you can, the notion of "2" is not referring to those sets or any part of ZFC, it is referring to "2" as defined by whatever axioms for the theory we are working in, usually the Peano axioms for arithmetic. ZFC is important because it purports to be a set of axioms, of which there are many, for which an extremely broad set of theories can be built upon with consistency. Consistency is the key word there.

Nobody in mathematics can reasonably debate whether ZFC is "true" or "not true," only whether it is axiomatically consistent and the limitations of arguments one can make out of ZFC set theory.

[Anonymous No.9798924]

>>9797843
>>9797862
How are there actually people like this?
Mathematics over N or Q aren't special. Why do you think they will "replace" the math system we have? (they won't).

[Anonymous No.9798928]

>>9798907
>>9798924
The jump from "mathematics is a tool to describe reality" to "mathematics is a collection of theories which provide insight into the nature of rigorous logical arguments from any chosen axiomatic foundation or primitive notions" is one most of this board still needs to make.

[Anonymous No.9798952]

>>9784384
>All this is built up with set theory at the base
This is not true though.
I try to explain that here: >>9798913

Theories can, for the most part, be thought of independently from each other, each with their own axioms which must be consistent for all models within the theory. ZFC as a set theoretical foundation incidentally allows one to represent a huge array of mathematical theories, but when discussing those theories independently, the relation to ZFC is only incidental. Other theories aren't "relying" on ZFC for anything. They have their own axioms and primitive notions (things that are not defined within the theory but which can only be defined within a larger theory for consistency. This goes back to Godel's theorems and so even ZFC can be used to prove its own consistency) to rely on.

[Anonymous No.9799039]

>>9798928
>a collection of theories which provide insight into the nature of rigorous logical arguments from any chosen axiomatic foundation or primitive notions
That is how most (it not all) mathematicians see maths. Anyway, what does that have to do with your idiot claim that maths and its foundations have remained unquestioned for too long?

[Anonymous No.9799051]

>>9799039
oh I'm not the guy who claimed that lol. I didn't even read his post. I don't think math foundations have remained unquestioned at all. I was just trying to throw in a comment.

I'm here:>>9798952
>>9798928

[Anonymous No.9799056]

>>9798952
>>9799051
Also that should say "ZFC cannot be used to prove its own consistency" not "ZFC can be used to prove its own consistency." That was a typo.

[Anonymous No.9800020]

>>9786314
t. Wilderberger

[Anonymous No.9800716]

>>9790822
basically this is the level I'm at with wild burger

[Anonymous No.9800759]

>>9790404

[Anonymous No.9800868]

>>9797958
>The very fact you have many people attacking Wilderberger as a "crank", deriding his efforts, reinforces my view that the Mathematical fraternity is inhabited by far too many narrow and blinkered little minds who are the modern day equivalent of Jesuit priests, attempting to stamp out Heresy. They are like little anarchisms who are a hindrance to innovative thought and progress.

[Anonymous No.9800870]

>>9798907
The very fact you have many people attacking Wilderberger as a "crank", deriding his efforts, reinforces my view that the Mathematical fraternity is inhabited by far too many narrow and blinkered little minds who are the modern day equivalent of Jesuit priests, attempting to stamp out Heresy. They are like little anarchisms who are a hindrance to innovative thought and progress.

[Anonymous No.9800873]

>>9800870
Spam violates global rules, enjoy your ban

[Anonymous No.9800949]

>>9787422
The algorithm is you add the sequences of which the reals are defined. You do know that reals can be defined as limits of sequences, right?

[Anonymous No.9801008]

>>9800949
>sequences
There is no such thing as an infinite sequence. Stop using circular logic to justify things.

If I reject the existence of the reals then I obviously reject the existence of Cauchy sequences. Same goes for any other bullshit method of "constructing" them (Dedekind cuts). I'm not an idiot.

[Anonymous No.9801018]

>>9801008
> I'm not an idiot.
A troll pretending to be an idiot?

[Anonymous No.9801026]

>>9800870
That is ridiculous. This scammer posts videos on you tube where he presents whatever he manages to pull directly out of his ass. He then spams 4chan with announcements of his videos. To make idiots give him enough views that he can make advertising money on.
His crackpot ideas are totally useless.

[Anonymous No.9801062]

apparently pythagoreans killed a guy who showed the diagonal of a 1x1 square was irrational
wildsperger should be on some kind of watchlist

[Anonymous No.9801065]

>>9801018
>Hurr Durr I don't understand your argument so I'll call you a troll.
Real numbers don't exist. Infinity does not exist. There is no such thing as an infinite sequence. To define real numbers as equivalence classes of limits of Cauchy sequence does not in any way adress this issue since infinite sequences are just as make-believe as the reals.

If you believe otherwise then it should be easy for you to provide a counterexample. Until then I'm going to stick to doing maths with axioms that are reasonable.

[Anonymous No.9801078]

>>9801065
>infinite sequences are just as make-believe as the reals
wtf integer dont exist
pls help i suddenlycan not count

[Anonymous No.9801081]

>>9801078
Go on then. Post an infinite sequence of integers, I dare you. Might take you a while.

[Anonymous No.9801088]

>>9801081
N

[Anonymous No.9801091]

Surely you jest. Thats just one term.
Unless you're planning to post each term of the sequence in a different post?

[Anonymous No.9801093]

>>9801088
thats a letter lmao wilburg wins again

[Anonymous No.9801094]

>>9800949
>limits
>sequences
back to brainlets general with you

[Anonymous No.9801099]

>>9801093
>responding to yourself

[Anonymous No.9801238]

>>9798913
>mathematics is about arguing within logical systems
Non-mathematician spotted.

[Anonymous No.9801250]

>>9790431
Of course, but the real point is I can't add them with any more precision than they have already provided.

[Anonymous No.9801775]

this man explicitly said during a talk that there are NOT infinitely many prime numbers
this isn't taken out of context either
https://youtu.be/Lme-uNPrry8?t=51m40s
that is the 51 minute and 40 second mark

>What about Euclid's argument about primes past Z
>So we all know that there are supposedly an infinite number of primes
>Well let me tell you my friends there are not an infinite number of primes
>THERE AREN'T

[Anonymous No.9801827]

>>9801775
HAHAHAHAHAHAHA HOLY SHIT
"number out of range"
HAHAHAHAHA

[Anonymous No.9801854]

>>9801775
>write down 2z
>therefore it is impossible to write down a prime bigger than z

[Anonymous No.9801991]

>>9801094
>>9801008
Please do tell me how limits of sequences are not rigorously defined or stop trolling

[Anonymous No.9802160]

>>9784384
You can make up whatever axiom system you want with whatever objects you wish. But if stuff like Weierstrass' approximation theorem, the fundamental theorem of algebra, pretty much every foundational result in analysis with Cauchy's name on it or hell even basic shit like the mean value theorem or Pythagoras' theorem isn't valid on your theory it is just a neat thing and nothing more.
You could think of mathematics very abstractly as just "pick a bunch of axoms -> beat them like pinatas until theorems come out" but that goes against how we actually do mathematics. There's stuff we expect or even know to be true for a multitude of reasons and the theory of real numbers includes a lot of such things.

[Anonymous No.9802171]

>>9785993
>MathOverflow posts of the top set theorists, they indeed consider ZFC in some sense the >muh only real set theory.
>E.g. this guy
>https://mathoverflow.net/users/1946/joel-david-hamkins
>They don't take a pluralistic approach at all, as you may expect.

Hamkins is a pluralist, you absolute nincompoop.

[Anonymous No.9802185]

>>9785993
>Joel David Hamkins
>not pluralistic

[Anonymous No.9802188]

>>9802160
>But if stuff like Weierstrass' approximation theorem, the fundamental theorem of algebra, pretty much every foundational result in analysis with Cauchy's name on it or hell even basic shit like the mean value theorem or Pythagoras' theorem isn't valid on your theory it is just a neat thing and nothing more.

If you drop the law of excluded middle (LEM), i.e. if you work only constructively, then you can prove just as many theorems as when you assume LEM.
More concretely (via Gödels double negation), you have that for every classical theorem CL, there is a constructive theorem CO, which is classically equivalent to CL. The LEM just collapses a bunch of theorems, making semantics of the sentences less rich.

For example, LEM states
A ∨ ¬A
>Either A is true or A is absurd
and this is not constructively provable.
However, the sentence
¬(¬A ∧ ¬(¬A))
>If it's the case that 'A leads to an absurdity' and also that the statement 'A leads to an absurdity' does lead to an absurdity, then this together leads to an absurdity.
Is constructively provable.
And in classical logic, the later sentence reduces to the former.

That is to say, if you work constructively, you don't lose any theorems. However, all proves have computational content (and are harder to obtain, because you didn't axiomize stuff you'd like to be true).

Pic also related, regarding your complains.

[Anonymous No.9802208]

>>9784411
Or if all the people who watch his videos did something useful...