/sci/ - Science & Math

whose says anything is truly conintuous?

>>9527638
√2

>>9527738
Well meme'd.

Usually, the proper approach to prove something in the reals is to prove it for something more general.
There are plently issues with representation about which men like Wildberger worry about, but the thinking mathematician doesn't really give a damn.

>>9527638
It is reasonable to reject the existence of the real numbers and infinity. It is not reasonable to pretend that this rejection is based on "logical difficulties", "problematic foundations" and other things that Wildbergers repeats over and over. He cannot find deductive flaws with the standards theorems of any analysis book. This is why his MathFoundations videos are actually garbage. "Why is this proof wrong?" "Duh, because the real numbers don 't exist!" gets old pretty fast.

TL;DR Wildberger is a hack pretending that his philosophical disagreements are logico-mathematical.

>>9527816
Godel
You stupid cunt.

>>9527823
>namedropping Godel
Detected the pseud who has only read about Godel from pop-sci articles

>>9527652
You mean like a line? Gee, I wonder what use a line would have in mathematics.

>>9527836
kek
>pretending that his philosophical disagreements are logico-mathematical.
Just neck yourself.

>>9527839
A line is complete with rational numbers.
Do what this guy should do >>9527836

>>9527839
Not him, but a line is an abstraction that is doubtful to exist in real life.

>>9527823
So Wildburger is also wrong because you can't do standard arithmetic without triggering first Gödel.
Fuck off to /lit/ you retarded poser.

>>9527846
"standard arithmatice"
>assuming what you are arguing against is wrong from the outset
are you are jew? Serious question. Everyone else in western academia does not entertain the idea that we should start with a conclusion and argue outwards. Only jews do that (pilpul).

>>9527843
Name drop Godel? K, Godel.
>>9527845
Yeah, fuck geometry.

>>9527850
So this is the power of wildberger mathematics...

>>9527857
>Yeah, fuck geometry.
PERFECT CIRCLES EXIST
EVEN THOUGH I ASSUME THEY DON'T
LISTEN TO ME

>>9527850
How do you want to construct rational numbers without triggering Gödel?
Or are you telling me that Presburger arithmetic is all that we need to do mathematics?

>>9527858
Yep. Don't assume your conclusions. We are radical.

>>9527868
Read the line of thought. Godel was brought up against:
1.. This is why his MathFoundations videos are actually garbage. "Why is this proof wrong?" "Duh, because the real numbers don 't exist!" gets old pretty fast.
Which clearly violates the incompleteness theorom. So either you are an idiot or you didn;t read the thread.

>>9527867
You're yelling too loud.

>>9527872
I don't understand what his problems, does he think that we should consider the algebraic non-complex numbers an extension of the number line somehow?

I DEMAND THE RIGHT TO ASSUME MY CONCLUSIONS AND ANYONE TELLING ME OTHERWISE IS AN ANTI-SEMITE
THE ADL WILL KNOW ABOUT THIS THREAD
DISGUSTING
ASSUMING YOUR CONLCUSIONS IS A CORE COMPONENT OF JEWISH IDENTITY.

>>9527872
Retard Wildberger only needs ONE video then. A video where he says that real numbers don't exist and then it'd be trivial for anyone with half a brain do deduce that PI according to his philosophy doesn't exist too.
All the standard proofs of real analysis are not flawed, and Godel is irrelevant. Now kill yourself, highschooler

>>9527878
He doesn't think you can define a numberline, any numberline or field, as a set.

>>9527878
He's a finitist. He doesn't believe in any infinite sets.

>>9527880
You literally just listed three conclusions with zero backing. Your agument is:
1. THESE STATEMENTS ARE TRUE
...
...
...
...
Nothing.
I was correct in labeling you a jew. Only jews think this is a valid form of argument.

>>9527882
So he thinks there's a largest integer?

>>9527888
Yes, he actually does, L0L

>>9527888
Yes.

>>9527888
I'm not him, but the largest integer is the largest number we can compute before the universe collapses. This is trivial.

I'm laughing because anti-semitism is basically accepted because jews are so horrible to deal with in real life.

>>9527891
Serious question: what do you mean with compute? How do you compute something like 12 for example?

>>9527941
In N we count from 1 to 12. Jesus.

>>9527948
Alright mother fucker you just sealed your fate, and the fate of this universe. I have now dedicated myself to becoming the head of a universe destroying cult. We will count up day and night, pass down our counting from generation to generation. Soon we will hit that final number, and the death of the entire universe will be on your hands. Remember that.

>>9527948
A list of numbers doesn't take any time to count, dumbass.

>>9528088
Tell that to the computer you are typing on or your own brain.

gauss proved that every degree n polynomial has at least n roots.

>>9528088
> Genuinely thinks a list is a thing that has to be counted to qualify as a list

You are now aware that a list as an entity is conceptually different from counting that list.

You are now aware that you are retarded.

>>9528100
>>9528093

Meant to reply to this retard.

>>9528100
tell that to any computation system that defines that list

>>9528104
> Thinks a physical system is the same as a conceptual thing.

Why are you so fucking dumb, anon?

>>9528115
What does this even mean? And if you are as smart as you think you are, a sudden quip won't do. You should be able to define "physical system" and "concept" to show how the first both applies to my post and negates the second.

You can't.

>>9527895
Now your getting it.

>>9528115
Why would I try and reason with a retard who isn't being reasonable?

A list can be counted. That is not the same as saying a list has to be counted.

A list of length 21 does not need to be counted for it to be of length 21.

If you can't see why this makes your "hurr there's a largest number because computation before the universe ends" argument completely ridiculous, why are you even posting here? Shouldn't you go back to grade school?

>>9527845
So is... everything in math. Numbers don't "exist" but they are used to describe things that do. Or not. This argument has nothing to do with math.

>>9527738
10/10 image

>>9527638
This guy is not up to the task of defending irrational numbers. I watched five minutes and already see he's not understanding the problem.

There is no problem with the square root of 2 as an algebraic entity. Wildberger has no issues resolving square roots by field extensions of rationals. It's almost certainly the case that he is fine considering arbitrary but finite extensions of the rational numbers to handle any square root we like though I don't think I've ever heard him say this explicitly. His problem lies in the collapse of such extensions into the "real numbers" and he has made his case against Cauchy sequences ("almost all cauchy sequences are the same up to our ability to check them") and Dedekind cuts ("that's a cute definition of the square root of two now show me e to the pi in your system"). These arguments are quite solid criticisms.

Personally I like to think of the real numbers as the completion of the rationals not in the sense of them having some prior existence but in the sense that explicit epsilon-delta requirements are always satisfiable. Like, "if you need this number, it can appear." Which is roughly the same kind of promise algebraic extensions make. This is why I like the continued fraction definition of reals so much.

Non-constructive existence "proofs" are always garbage, though.

>>9528095
also known as the "fundamental dream of algebra"

>>9527846
His system isn't wrong, at least as far as I can tell. But the idea that a physical object can only represent the number 1, but not represent any other number is ridiculous.

>>9528229
Fta doesn't require anything besides natural numbers. It's nice to use reals, but it's not necessary.

He can't even prove theorems within his own fucked up system. Why would anyone take him seriously?

>>9528310
true integer polynomials are weird man
how does that even work? Just ignore non-integer shit?

>>9528310
>It's nice to use reals, but it's not necessary.
I would love to see the proof you're implying exists. Do you have a link?

>>9528325
the fundamental theorem of algebra is like famous
I think euler proved it in his PhD thesis or some shit
I don't remember any actual math from textbook, just the little side notes

>>9527888
How does that follow? If you make a finite set of natural numbers, I can make a finite set whose largest member is larger that your set's largest member.

>>9528316
>>9528325
You can do it with congruence arithmetic. It's taught in undergrad number theory.

As for the prime factorization part, that can also be proven with the naturals, or integers. That proof is simple enough, just Google a proof and show me where it invokes reals anywhere.

>>9528362
>You can do it with congruence arithmetic. It's taught in undergrad number theory.
Then it should be no trouble for you to produce such a proof. Also you're wrong.

>>9528346
His argument is that the most effecient way to represent a number is through atoms. Then you assign a value of 1 to each atom in the universe. Then the largest natural number is the number you get if you counted all the atoms in the universe. He assumes the total number of aTom's in the universe is both finite and fixed.

>>9528382
crypto.stanford.edu/pbc/notes/numbertheory/poly.html

>>9528403
That is not the fundamental theorem of algebra.

>>9528408
>it's not true because I say so

It's literally fta for a ring.

>>9528546
Do you have to try to be that retarded or does it just come naturally?

>>9527891
>but the largest integer is the largest number we can compute before the universe collapses.
w0t?

>>9528656
Every instant you don't spend applying the successor function to an element, the largest POSSIBLE integer shrinks.

>>9527850
>Y-You're probably an ethnic minority!
Hmm, I wonder why this movement isn't gaining more traction.

>>9528992
I thought jews were white?

>>9527960
you are such ass :(

>>9528391
what if i want to count all of the protons?

that wilderberger guy is a retarded hack and i strongly disapprove making any mentions of him ever again on this board

>>9529051
>what if i want to count all of the protons?
what are you going to mark them with?

>>9529099
the same argument holds for counting all of the atoms, doesn't it.
We can even go further and count with quarks or some other made-up shit.
Also, how many possible permutations in open space of atoms are there? well, even if we admit space is discrete, still huge-ass fucking number. and we can always devise some higher.

again, this wildergerb guy is fraud

>>9529117
>the same argument holds for counting all of the atoms, doesn't it.
I can mark neutral atoms by removing an electron. But this is really beside the point.

>>9529117
>Also, how many possible permutations in open space of atoms are there? well, even if we admit space is discrete, still huge-ass fucking number. and we can always devise some higher.
The fact that you don't understand various people's positions is not actually evidence that they're wrong, stupid, or engaging in some kind of fraud. I cannot speak for Wildberger, he has his own opinions and it's not my place to defend him from anonymous shitflinging, but I can tell you I take exception to the thought that "we can always device some higher [number]." We quite certainly hit practical limits very quickly. If you wish to do arithmetic with very large numbers and get an answer in your lifetime you'll need carry bits to propagate beyond the speed of light.

I appreciate angels on pins arguments and take pleasure in them myself. But that's all your promise of "some higher" amounts to, because you'll never exhibit this number, nor add 3 to it, nor evaluate some cubic polynomial at that value... This is a fact which is easily disproven by you exhibiting such a massive number and then performing arithmetic with it. Such a demonstration would shut up all finitists for all time. Just take Graham's number, write it down, then apply the successor operation just like set theory says it should be applied. By the way you have to write down Graham's number as it "actually is" since set theory is the foundation of all of mathematics, right? No cheating.

I'll wait.

>>9529218
You don't have to. Math is platonic.
If I take 1 atom and 1 atom, I won't necessarily have 2 atoms. Forcing math to have a physicality is flawed. You are also making an assumption which cannot be asserted. By your logic, the larest number should be 1000 or so, since you probably can't keep that many atoms together long enough to count.

Stop defending this retarded thought.

>>9529233
>You don't have to. Math is platonic.
I see. So when Wildberger says, "There's no such thing as an infinite set of counting numbers," and then someone says, "Yes there is, there's no largest number because you can always add one," what they mean is.... "math is platonic so I can do whatever I say I can do"?

>>9529218
thanks for your answer.

my anger comes from understandable and generally held position that mathematics is abstract tool, not meant to deal with physical limitations -- there is physics for that. I understand that "we cannot express some numbers because they are too big", but from my understanding, it does not follow that general properties of arithmetic stop hold after some point, because we abstract from such phenomena.

I don't have to explain my position, as you understand it already, and it's a generally held view and interpretation of function of mathematics.

>The fact that you don't understand various people's positions is not actually evidence that they're wrong
Sure, however, what we can disagree with is that i am not willing to entertain such notions of something that is so loony and contrarian and look like something that is in fact a product of a misunderstanding.

this is a waste of time and resources

>>9529258
>but from my understanding, it does not follow that general properties of arithmetic stop hold after some point, because we abstract from such phenomena.
Actually the general properties of arithmetic do stop after some point since eventually you have written down a godel sentence and will be unable to compute it.

>>9528665
Underrated
>muh largest integer is changing according to new physics discoveries

>>9529218
>you'll never exhibit this number, nor add 3 to it
t. brainlet
N + 3

>>9527638
>>9527888

>>9527850
>Everyone else in western academia does not entertain the idea that we should start with a conclusion and argue outwards. Only jews do that (pilpul).
It's called "begging the question." And it's probably one of the most common informal fallacies there is. So your jew spotting false positive rate is going to be pretty high if you assume every time you see that fallacy the poster was a jew.

>>9529249
Yes, as long as what you say is consistent with everything else you say. Point is that numbers don't exist physically. Addition doesn't exist physically. So basing math construction as something physical is at best, arbitrary.

>>9529284
I count all the atoms in the universe except 1. Then I assign the counted number to the last aTom and count again.

I don't have to write the number to know exactly what it is. I know that the assigned value and another value will be larger than either term. In this way, I can keep tabs on any arbitrary large number and know that there is a number bigger than it, because I never run out of atoms with which to represent it.

Wildburger is an egoist completely unaware of the last 50 years of development in logic. He should stay in education because he'ssomewhat good at it and avoid forcing his shit to actual working mathematicians.

>>9529249
>math is platonic so I can do whatever I say I can do
Platonic means it exists as an idea as opposed to a real world structure.
e.g. The abstract content of a floor plan isn't necessarily anything that corresponds to a real world building. You might reuse a general floor plan as a guide for constructing real world buildings, but the floor plan doesn't stop existing just because you don't have real world buildings modeled off it.
And you can definitely do *more* with an abstract structure than with a real world structure since you don't have to worry about implementation details just to express a high level idea in some way, but that isn't the same as being able to do whatever you want since your idea still needs to be coherent and consistent.

>>9527850
Back to /pol/ you insufferable cunt

>>9527638
Holy fuck how embarrassing. The absolute state of high school math teachers

>>9529642
>The abstract content of a floor plan isn't necessarily anything that corresponds to a real world building. You might reuse a general floor plan as a guide for constructing real world buildings, but the floor plan doesn't stop existing just because you don't have real world buildings modeled off it.
I understand. If this was the only thing the real number people said I would have no problem with them. However they make actual claims, which are then refuted (e.g. you cannot write down graham's number in ZFC or some other foundational mathematical system), which they then retreat to things being done "in principle," and then they cannot exhibit that principle either as Wildberger shows in his videos that wreck (equivalence classes of) Cauchy sequence definitions and Dedekind cut definitions. So they are left without facts and without principles.

Then they retreat to philosophy.

>And you can definitely do *more* with an abstract structure than with a real world structure
He does not have a problem with abstract structures, neither do I. The rational numbers and integers are already such a structure, and even if you took them as given (somehow) then algebraic extensions of these are just such a structure (in which case the square root of two is no more and no less interesting than the square root of negative one).

>that isn't the same as being able to do whatever you want since your idea still needs to be coherent and consistent
Sure. But the arithmetic of real numbers is totally incoherent. It is either reduced to "in principle" (but there's no principles), symbolic (algebraic), or "approximate as close as we like" (rational).

>>9527841
His arguments hinge on the fact he rejects the axiom of infinity (and presumably the axiom of choice). It is perfectly valid to do so, you can assume as many axioms as you want to do mathematics, whatever. He crosses the line when he says the axiom of infinity isn't real because it leads to conclusions he doesn't like. That is a philosophical disagreement.

>>9529815
You don't need the axiom of choice with finite sets.

>>9529789
>However they make actual claims, which are then refuted (e.g. you cannot write down graham's number in ZFC or some other foundational mathematical system), which they then retreat to things being done "in principle," and then they cannot exhibit that principle either as Wildberger shows in his videos that wreck (equivalence classes of) Cauchy sequence definitions and Dedekind cut definitions.
Not the person you're arguing with, but can you explain this part? I don't see that at all.

>But the arithmetic of real numbers is totally incoherent.
How so?

>"approximate as close as we like" (rational).
Right, you can define a notion of approximation schemes over rational numbers. You can do arithmetic and algebra on those. We call the result "real numbers", and Cauchy sequences (or equivalence classes thereof) are a particular scheme like this (though probably not the most enlightening one). What problem do you see with this, exactly?

>>9529945
I see no problem with arithmetic with rationals. It's what I prefer. It's what I use.

>>9528204
>Non-constructive existence "proofs" are always garbage, though.
Algebra and its applications are also to blame for those
>hurr durr if you say there's no fixed point you can construct a retraction which would give an injective homomorphism from an infinite cyclic homology group to a trivial one, contradiction

>>9528095
at MOST n roots retard. Consider x^3

>>9530549
kek

>>9529218
What do you mean by "as it actually is" ?
What notations are allowed and why?

>>9529284
you are just memeing by now

>>9531054
War. War never changes.

bunmp

>mfw my Calc 3 professor says a sequence "converges to infinity"

>>9527638
Retarded. Dr. Normal Wilderger does not say we don't need irrational numbers. He says that the way we deal with those numbers today is inadequate.

In his most recent series (Algebraic Calculus) he gave a very nice example of why, as well as an introduction to how he would reform Calculus. He showed two Greek problems, which were both solved, but with a subtle difference.

The first problem was that of finding the area of a parabolic segment. This problem was proposed and solved by the Greeks. It was solved perfectly, actually. This means that an exact solution can be written with finitely many digits. Furthermore, a very "simple" proportionality constant was found, [math] \frac{4}{3} [/math].

Then he presented the problem of finding the area of a circle. This problem was also solved but in a different way. A formula was given, but this formula needed the much more complex proportionality constant " [math] \pi [/math]".

Wildberger then comments: Isn't that weird? Why is one problem so easy, while the other is so hard? Why can one problem be solved exactly, while the other basically needs an algorithm (pi approximating algorithms).

Then Wildberger lays out his philosophy: There should be a distinction between "area" and "approximate area". This means that today's irrational numbers do not belong in a number system, but in a more abstract structure below the numbers. No mathematician (other than himself) probably knows how complex this structure is. That is why we ignore it and say "Isn't this infinite decimal just a number haha please don't ask questions!"

>>9529769
>High school teacher

Oh honey...

>>9532314
>Then Wildberger lays out his philosophy: There should be a distinction between "area" and "approximate area".
But Pi isn't an approximation of the area of a circle.

>This means that today's irrational numbers do not belong in a number system, but in a more abstract structure below the numbers.
Why? Wildberger is free to dislike & not use irrational numbers, but he seems to think they're funder mentally broken in some way and I don't understand why.

>That is why we ignore it and say "Isn't this infinite decimal just a number haha please don't ask questions!"
Nobody does that.