MathFoundations190: Numbers, polynumbers and arithmetic with vexels I
https://www.youtube.com/watch?v=OFzl37nX58A
When will undergrad learn Mset theory?
>>8194906
>another wildburger post
>>8194906
>the mset
Mind = blown
>>8194906
Would every vector space have a basis in Wildburgers world?
what's the motivation here?
Wildberger seems like a pretty cool guy. Eh questions set theory and doesn't afraid of anything.
Excuse me, but are polynomials not vector spaces?
How is this new?
>>8194963
Polynomials are considered vectors in certain linear algebra settings, but they have a kind of confusing / undefined quality (where the variables of the polynomials are used to define the objects but are not part of the vector space).
It seems like he's building this up to avoid this.
Definitely pretty interesting ideas here, and one can more easily see why the combinatoric n choose r gives us the coefficients of a polynomial multiplication.
>>8194906
>replace sets with ordered lists
>call them mset
>ASKING THE REAL QUESTIONS
>MUH CONSTRUCTIVISM
I hope nobody lets this guy near undergrads who would actually believe his stupid shit
>>8195004
multisets aren't ordered lists and they show up everywhere in math
>>8194968
Algebra books ever since seem to make it work by just informally introducing more and more notation hovering around the core theory.
It's what you get form choosing foundations without lambda calculus, i.e. [math] \mapsto [/math] as integral part of your language.
"""
[math] p ( X ) := 3 + 5 X - 2 X^2 + 7 X^3[/math]
"""
[math] p:=X \mapsto 3 + 5 X - 2 X^2 + 7 X^3 [/math]
[math] p := \lambda X.\, 3 + 5 X - 2 X^2 + 7 X^3 [/math]
What you can still do is secretly work with tuples like [math]v_p=(3,5,-2,7)[/math] use polynomial notation as charade. Then you may define an evaluation at
[math] eval (v_p,a) := \sum_{k=1}^4v_p^ka^k[/math]
="""p(a)"""
Tuples is what every implementation will work with anyway, at one level or the other.
Given that we arrived in the century where people must regularly implement math on computers, which are stricter with them than their school teachers, it may be that it's better to learn math in an implementable way, and so that kids (like all of us) have to understand how to translate their 19'th century like cheating notation into expressions that will actually compile, and in an efficient way.
>>8194968
>see why the combinatoric n choose r gives us the coefficients of a polynomial multiplication.
explain
Why is /sci/ obsessed with this guy?
>>8194911
WildHotDog isn't the inventor of multisets.
>>8194906
why the fuck does he write vectors and polynomials backwards?
[math]
\left.\begin{matrix}
x&+&y &+ &z & \\
2x&+&2y&+&2z &
\end{matrix}\right\}
[/math]
becomes
[math]
\begin{pmatrix}
1 & 1 & 1\\
2 & 2 & 2
\end{pmatrix}
[/math]
>>8194968
There is nothing confusing about polynomials as vector spaces. It's super easy to show that they satisfy the vectorspace axioms. Not just "in some settings", in every setting they have this structure, even with classes of objects that aren't the real numbers (for example matrices).
Wildberger is an intelligent guy. There is very little that he does which is mathematically incorrect. A lot of what he says is bullshit though.
What I mean is that the guy uses sound mathematics, but rejects certain mathematics because of suspicions about infinities, etc. I think that the concept of infinity is well defined in many different contexts, and Wildberger is putting his head in the sand to ignore highly successful and consistent bodies of mathematical knowledge that use these ideas.
He has an interesting and practical philosophy that math has to be grounded in physics, but I think that's a major flaw in his reasoning, because math is independent of physics and environment. We could live in universes with different physics and environments and still derive the same mathematics truths.
>>8195956
>It's super easy to show that they satisfy the vectorspace axioms.
Lol, prove multiplication of polynomials is closed for n degree polynomials.
Wildberger's system literally makes the vector space axioms almost immediate for polynomials.
There's some very clever simplifying going on here. I've never had a textbook point out that polynomials can be represented as msets (which are a far more basic / abstract kind of object). That's actually completely fascinating and (to me) unexpected.
>>8196436
>Lol, prove multiplication of polynomials is closed for n degree polynomials.
I don't understand how this is supposed to be hard.
>>8196506
yeah it's not
>>8194906
>memematician
>>8195956
but what is the variable X in a polynomial then/
>>8196585
Parameter
Newton was called a heretic and forced to recant his theory of planetary rotation.
Just saying, until he proved them wrong, the esteemed scientists of the day thought he was a moran and lunatic.
Maybe give the guy a chance?
>>8197050
(although I'm not the guy you reply to, but the guy with the logic screenshot)
see
http://math.stackexchange.com/questions/157047/axiomatic-approach-to-polynomials
>>8197098
so purely logically, X is a constant in the language?
>>8194906
what is best wilburgercore ?
>>8195011
This. I'm shocked by the number of retards in this thread who have never heard of multisets.
This confirms that all wildburger deniers are only acquainted with babby math.
His definition of multisets as tuples only works for some fields. For example, no such mapping between a real vectorspace and multiset exists. Not that Wildebeest would care, cause reals are 2pooky4him.
>>8197766
The normal definition of multiset is as a function from a set to the set of naturals.
eg. the multiset {3,3,3,3,5,5} is actually formalized by letting A={3,5} and [math]f:A\to \mathbb{N}[/math] where f(3)=4 and f(5)=2.
Reals are kind of garbage math when it comes to lots of problems dealing with formalization.
>>8197947
To add to this, you can think of a multiset as a generalization of the indicator function.
https://en.wikipedia.org/wiki/Indicator_function#Definition
In some fields, indicator functions are used to deal with set membership (like computability where one says that a set exists if there exists an indicator function for the set).
What does the sqrt(2) equal to?
>>8197992
You don't need the reals to construct sqrt(2)
See: Second semester algebra
>>8198018
What if I want to measure it?
>>8198025
You didn't say you wanted to
boinmp
>>8194906
steve martin???
Its instructive to watch the greatest mathematical mind of his generation work out some small part of 6th grade math.
bupn
>>8197012
Underfuckingrated post.
>>8194906
Now we just wait for someone to formulate some paradoxes to prove just how lame foundation theories are again.
>>8197012
And so where thousands of other people and yet they didn't turn out to be right
He """"""""""rigorously"""""""""" defined naturals as strokes on board. How can anyone even try to take him seriously
>>8203077
Yeah and brainlets """"""""""rigorously"""""""""" define natural numbers as curly brackets.
bup
>>8198018
>>You don't need the reals to construct sqrt(2)
lel yes you need
>>8203756
draw a right triangle with legs each of length 1 :)
>>8203756
no, you don't. algebraic numbers are actually countable, and Q(sqrt(2)) is a 2-dimensional Q vector field
>>8203760
yeah and you do not know that you can ascribe a number to the diagonal, in the same fashion that you can ascribe the number 1 to the sides.
but IF you can ascribe a number, there is a formal system which makes it so, and in this, the usual number is sqrt[2].
>>8203766
compass-straightedge constructions define a number system, a subset of C.
>>8203767
New dude here....
Logic comes from the space you are in. The reason deduction "works" the reason you feel the inference of a syllogism is because big things don't fit in small things....
When you start messing with the space like you are doing, you need to be careful that you are not attributing the training in logic learned in Euclidean space to the new space.
This is why we use groups and rings and fields...
so you don't use the "ah ha!" feeling of "That makes sense!" you learned in one space and put it in another...
SO! What space are you in? Can you justify making up your arithmetic using deduction and induction if those properties don't hold in your space?
>>8203761
you really think that x^2 = 2 has a solution??? how old are you ??
>>8203779
4chan is for 13-25 year olds.
Reddit is for 22-35 year olds.
Very rarely are there any top level graduates online in forums anymore.
They jump to conclusions, straw man and contradict well known consensus-based concepts, and in some areas, they even reject axioms.
They don't seem to understand the importance of coherency or source.
In this case, anon straw mans and then refuses to point to a source, just dictating anons memory justifies anons emotional retort and denial.
Debate etiquette calls for references, which I posted, and logical arguments without presumptions, which I posted, but I doubt anyone will take LOGIC for what it's worth when people can try to rely on self-serving biases and interpretations.
>>8203779
>>8203780
That has never been proven.
Intellectual (me):1
Pseudo-intellectual:0
I am an atheist as well.
I'm just an educated atheist.
Here are my beliefs:
Empiricism, falsifiability, fallacy checking, the scientific method, the socratic method, humility, scientific consensus, etc.
I don't believe in jumping to conclusions or siding with an unproven concept and calling it proven with emotional fervor.
That's irrational.
The only rational thing is to remain neutral until something is proven true with experimentation or some form of evidence.
Presumption is never evidence.
>>8203783
>Reddit is for 22-35 year olds.
>>8203795
Look, if you're still going to troll or act retarded, that's fine.
- Swear
- Ad hominem; Call people names
- Don't provide counter-arguments
- Reject realism and the scientific consensus
That's ok.
Just don't loop.
Looping is cancer.
Personal incredulity and the argument from ignorance are fallacies. You're ignorant.
You imply you have no knowledge of the other kinds, therefore they don't exist.
That is wrong irrational.
:D
>>8203798
Argumentum ad hominem is not namecalling.
>>8203798
kek. does anyone recall the origin of this pasta?
>>8203783
Well, you'd probably make a good Pragmatist. Leaving ethos and pathos out as emotional and instinctual "convincing" of the consistency of any narrative through rhetoric, you are left with logos, but that gets to be tricky as well if you are not careful.
But, regardless of any narratives consistency, it has to reflect the world at least for the intent of the narrative.
Maths are fun, and surely can be "useful to be believed" but one of its strongest points is this:
All we ever have is a story, and the story is never the world, but the way we approach it is by making the pathways (narratives) in our brain first (connecting diverse narratives as consistently as possible if you are a mathematician, and by your feels if you are everyone else), then we check them against the other pathways (narratives) for consistency. But ultimately for them to be useful we have to see if they reflect the world, which assumes the narrative comes first. In reality, it is a recursion, and our givens are just recursions in equilibrium, not some magical platonic forms. The fault is that we only know our narratives and we always mistake them for the world.
The world is not the picture in our heads. It is out of conveneience that we think so.
So, I feel your frustration. Cutting out the informal first is noble, but necessary. Finding consistency in narratives not readily accessible to your senses, however, is where the effort pays off.
That takes sobriety, and it is no fun to be around a bunch of drunks.
>>8203798
Nigga I am not even the guy you were arguing with.
I just saw that shit and started laughing like a maniac.
Are you retarded by any chance?
>>8203827
Well, you'd probably make a good Pragmatist. Leaving ethos and pathos out as emotional and instinctual "convincing" of the consistency of any narrative through rhetoric, you are left with logos, but that gets to be tricky as well if you are not careful.
But, regardless of any narratives consistency, it has to reflect the world at least for the intent of the narrative.
Maths are fun, and surely can be "useful to be believed" but one of its strongest points is this:
All we ever have is a story, and the story is never the world, but the way we approach it is by making the pathways (narratives) in our brain first (connecting diverse narratives as consistently as possible if you are a mathematician, and by your feels if you are everyone else), then we check them against the other pathways (narratives) for consistency. But ultimately for them to be useful we have to see if they reflect the world, which assumes the narrative comes first. In reality, it is a recursion, and our givens are just recursions in equilibrium, not some magical platonic forms. The fault is that we only know our narratives and we always mistake them for the world.
The world is not the picture in our heads. It is out of conveneience that we think so.
So, I feel your frustration. Cutting out the informal first is noble, but necessary. Finding consistency in narratives not readily accessible to your senses, however, is where the effort pays off.
That takes sobriety, and it is no fun to be around a bunch of drunks.
thanks /sci/, now im at vid 4 in the series and have been proving along with him. Im on the verge of learning real maths so now im forecer colored by these meme understandings
Pic very related
Also im going to do the whole series then start on linear algebra, unless theres a better recommendation
>>8203903
*forever
Also my pic, hes australian right
>>8203903
Despite what a lot of autists say, most maths doesn't really depend on the details of the formal system you work in, outside of a few fringe cases, so this won't really "colour" your understanding.
>>8203919
Could you name a couple examples for me to research?
>>8203903
What the fuck is wrong with your notes. That's utterly ridiculous.
>>8204029
probably a girl
>>8203903
>actually taking notes from wildburger
>>8204063
It doesn't matter what gender or sexuality one is. Those notes are ridiculous. Compiled with that it's notes on Wildburger, it's the most absurd thing I've encountered in a while.
>>8203903
>Also im going to do the whole series then start on linear algebra, unless theres a better recommendation
better recommendation: start with linear algebra directly.
no need for quack maths
>>8195538
he's like the official professor version of the 0.999999... != 1 retards
some subtraction """"""proofs""""""
For you
>>8197992
It is the element of [math]{\bar \mathbb{Q}}[/math] that is a root to [math]{x^2} - 2 \in \mathbb{Q}\left[ x \right][/math].
>>8204471
how do you know that you can solve x^2- 2 in q[x]
>>8204894
Because I can construct [math]\mathbb Q[x]/(x^2−2)[/math] and pass to the quotient. People can bitch about "transcendental" numbers (even though these also have an incredibly simple and valid construction), but very simple algebra provides you with algebraic numbers over any base field.
>>8195442
Since no one is helping you out, I'll give you two keywords as sufficient hints:
Pascal's Triangle and Binomial Expansion.
Combinatorics teaches us that we can find the coefficients of x, y for (x+y)^n, n > 0, for a given x^p*y^n-p. Apparently, according to the person you quoted, the video gives a solid background for why that is possible from another perspective.
It's pretty neat.
