/sci/ - Science & Math

>>11577324
For gods sake, you come up with negative numbers by pretending that for any natural number a there is a natural number "-a" such that a + (-a) = 0.
That is the exact same principle and somehow that doesn't upset people...
Negative numbers are as "real" as imaginary numbers.

>Like, say if I were to tell let's pretend sqrt(1)^0.3 = -2 because why not
That clearly is a contradiction, while defining sqrt(-1) such that sqrt(-1)·sqrt(-1) = -1 is obviously not.

no

>>11577339
any number multiplied by itself is positive how the fuck does sqrt(-1)·sqrt(-1) = -1 make any sense

>>11577324
Complex numbers are the set of pairs of real numbers (a,b) with multiplication and addition defined as:
(a,b)+(c,d) = (a+c, b+d)
(a,b)*(c,d)=(ac-bd, ad+bc)
The number (a,b) is denoted as a+bi.
You can easily verify that
i^2 = (0+1*i)^2 = (0,1)*(0,1)=(-1,0)=-1
What's so hard to understand about this?

>>11577345
>any number multiplied by itself is positive
Any REAL number which isn't zero. Yes, so obviously sqrt(-1) isn't a real number.

But your argument is retarded. Do negative numbers not exist because "all natural numbers are positive or equal to zero"?

>how the fuck does sqrt(-1)·sqrt(-1) = -1 make any sense
It is the definition of sqrt(-1).
Just like a + (-a) = 0 is the definition of -a.

Seriously, do you think negative numbers make sense?
Because imaginary numbers make exactly the same amount of sense.

>>11577351
so you could basically create a field as big as complex numbers by defining some random set of rules you come up with?

>>11577324
It is IMAGINARY.

>>11577357
>by defining some random set of rules you come up with?
That is EXACTLY how you define negative numbers and fractions.

>by defining some random set of rules you come up with?
Exactly. Many important mathematical objects are defined that way.
BUT you have to make sure that your definition EXTENDS what you have had previously and doesn't contradict it.

>>11577355
I don't understand how making up some bullshit rule like a +(-a) = 0 or sqrt(-1)·sqrt(-1) = -1 can make sense with the rest of math

>>11577357
congratulations, you figured out what pure mathematics is

>>11577324
>If the whole idea behind complex numbers is based on pretending that sqrt(-1)·sqrt(-1) = -1
It's not, roots of negative numbers aren't defined. It's based on i^2=-1.
What's the third root of -8?

>>11577364
>BUT you have to make sure that your definition EXTENDS what you have had previously and doesn't contradict it.
Guess I can live with this explanation. Thanks

>>11577365
>I don't understand how making up some bullshit rule like a +(-a) = 0 or sqrt(-1)·sqrt(-1) = -1 can make sense with the rest of math
Then read a book on how you define the negative numbers.
This is really basic stuff.


If you believe that negative numbers are consistent with the rest of mathematics, then CERTAINLY imaginary numbers are too.

>>11577365
thats why you should fuck off back to kfc, nigger

simsalabim you are now 10% nigger
>nooooo my IQ I need to make a retarded threat right now!!!

>>11577365
all of math is bullshit rules, they're called axioms

>>11577324
The whole idea is flexibility of square roots over the domain of numerical representation.

>>11577357
A big part of algebra in a math course id understanding that there's a limit to the rulesets that make sense and that many rulesets belonging to seemingly different fields are abstractly the same. For example, homotopies are a subject of algebraic topology where you can learn how a looping around circumference is in some sense equivalent to addition of integer numbers.

>>11577324
>that mean you could create any other set of numbers by pretending some other stupid rule is true?
Sure, if you can develop a useful framework around it
>Like, say if I were to tell let's pretend sqrt(1)^0.3 = -2 because why not
no, because none of those numbers are new

>>11577324
The complex numbers ARE the "real" numbers, by that I mean "the numbers that are objectively real and exist", because they are algebraically closed and they allow us to perform all the typical operations in ways that we expect.
When you have 1 apple, you ACTUALLY have 1+0i apples. EVERYTHING is actually complex in this world. "natural numbers" "rational numbers" etc. are the subsets of this but the set of numbers that ACTUALLY exists are the complex numbers. It took humanity a long time to figure this out but it also took us a long time to understand QM and that's how reality actually works as well.

>>11577390
is there some other bs rule that somehow makes sense with the rest of math that makes it so that there's even something above complex numbers?

>>11577393
quaternions

>>11577393
>is there some other bs rule that somehow makes sense with the rest of math that makes it so that there's even something above complex numbers?
Yes. There are ALOT of these "bs" rules.
See, e.g https://en.wikipedia.org/wiki/Quaternion

>>11577397
So any real number is actually a quaternion? Mind giving an example like with
>3=3+i·0

>>11577403
3 + 0i + 0j + 0k

>>11577403
>So any real number is actually a quaternion?
1 = 1+ 0*i + 0*j +0*k
See https://en.wikipedia.org/wiki/Quaternion for the rules of i, j and k.

>>11577409
Could you extend this for n dimensions? Like instead of just i,j,k make it i,j,k,l,m,n..

>>11577412
I guess:
https://en.wikipedia.org/wiki/Hypercomplex_number
but it is just getting more boring the more non-real shit you have

>>11577371
It's not bullshit because it allows you to expand the set of numbers whilst not breaking the previous rules. Addition and multiplication stay the same in positive integers whether or not you see them as a subset of all integers.

And by the way, the rigorous way of introducing negative integers isn't even by assuming that for every integer n there exists m such that n+m=0: in number theory, you take the set NxN of the couples (a,b) of positive integers and create an equivalence relation ~ with the following rule: given two couples (a,b) and (c,d)
(a,b)~(c,d) iff b+c = a+d

Then you take the set (NxN)/~ of all equivalence classes and you give it an operation * so that (a,b)*(c,d) = (a+b,c+d)
and you notice that if you do
(a,b)*(b,a) you get (a+b,b+a)=(a+b,a+b)
but by the rule defined by ~ a+b+0=a+b+0
and therefore (a+b,a+b) is equivalent to (0,0)
We obtained the desired rule of for every element (a,b) of this set, we can find another one (b,a) such that they add to (0,0). So we built a set that is structurally identical to integers by using couples of positive integers and without the need to introduce the - symbol.

Analogous processes are used to build thr set Q of ratios (from positive integers) and the set C of compled numbers (from real numbers). A more difficult problem is how the set R of real numbers is built from Q.

>>11577412
>Could you extend this for n dimensions?
Yes. But to get something remotely useful, it turns out that n needs to be a power of 2. So you get reals, complex numbers (2D), quaternions (4D), octonions (8D).

Each one has fewer properties than its predecessors. Complex numbers aren't totally-ordered, multiplication of quaternions isn't commutative, multiplication of octonions isn't associative.

>>11577412
you can generalize real and complex numbers in many ways (for example look up clifford algebras) to arbitrary dimension, but the point is that in general the extended operations will not satisfy the usual rules of arithmetics.

the real numbers, complex numbers, quaternions and octonions are special, because there's division available: any non-zero element has a multiplicative inverse. n=1,2,4,8 are the only dimensions where this is possible.

>>11577364
This goes against the idea of cuaternions and octonions because when you get to those instances you actually lose properties

>>11577324

complex numbers were made so you can have a solution to the equation

x^2 + 1 = 0

it's not "made up"

>>11577351
wrong
>>11577364
wrong
>>11577429
wrong and cancer

>>11577409
>>11577408
>>11577412
>>11577431
correct

Remember OP, math is about solving problems- not making shit up because it's complicated. Complex numbers are the *solution* to a problem. Likewise, Quaternions are the *solution* to a problem, as are octonions and any 2^n dimensional coordinate system.

set theory on the other hand was *not* invented to solve ANY problems and has only created more problems than it solves so you should be skeptical of that. If you are going to be call bullshit on a mathematical "theory" do it to set theory. it is utterly redundant and doesn't solve any long outstanding problems that previous mathematical methods could not.

But not complex numbers. Complex numbers are good guys. have you ever seen plots of complex functions? they're cool as shit and electrical engineers use them.

>>11577324
yes
the whole system is made up
natural numbers does even exists in the physical world

we've only kept are the useful ones

>>11577324
uhh why not just like uhhh
like [math]\frac{x}{0} = xn[/math] where [math]n[/math] is the null number that equals [math]1[/math] once multiplied by 0?

like this
[math]\frac{3}{0} = 3n[/math]
[math]3n * 0 = 3[/math]

>>11577324

Look how fucking cool this function is. Are you telling me you wouldn't want to be able to compute this function in your head? How much of a boss would you be if you could visualize the output to a function like this?

>>11577355
>it is the definition of sqrt(-1)
>just like a + (-a) = 0 is the definition of -a

WRONG. It is not just a definition that someone made up arbitrarily. Don't teach people that.

It is the brilliant and creative solution to the very simple but misleading polynomial equation:

x^2 + 1 = 0

Similarly, negative numbers are the SOLUTION to the equation

x + 1 = 0

which requires slightly less ingenuity although similar by way of "extending" a number system to allow for an inverse operation.

Math is about solving problems. It is not just some arbitrary collection of definitions that someone pulled out of their ass. The answer to these questions is never "It's just how it's defined." that is WRONG.

>>11577484
>sOmEoNe JuSt MaDe It Up UhHhH dUrRrR

Low IQ take

>>11577476
>>11577491
Wrong.

>>11577511
Wrong.

>>11577489
You can easily break that
n*0 = 1
(n*0)*0 = 1*0 = 0
but n*(0*0) = n*0 = 1

>>11577637
1*0 is just [math]\frac{1}{1n}[/math] though

>>11577476
Sometimes solving problems requires making shit up that's complicated. And set theory as per ZFC axioms was born specifically to solve the paradoxes that naïve set theory presented, such as Russell's. So first you say that math is valid when it stems from a real life problem (false: oftentimes we avoid real life problems for the very reason that we simulate them before they present) and then you go on to attack a theory that literally stems from real life problems.

>>11577648
Ok then 1/n = 1 which is just as absurd

>>11577637
>NOOOOOOOOOOOOOOOOOOOOOOOOOO ITS NOT COMMUTATIVE

>>11577654
Then logically, 1 and n cannot be separated.
A better notation would be some f(1), where f(1)*0=1.

>>11577324
>look at all this made up math!
>*complex numbers used fucking constantly in engineering*
>NOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO ITS MADE UP YOU CANT DO THAT

>>11577668
I don't get it
If the codomain of f is real numbers, f(1)=n for some n either way

>>11577651

I enjoy doing math for the sake of it. Nowhere in my post do I say it needs a "real life" application. In fact, I resent people who need a "real life" application. People who ask "when are we going to use this?" are fucking cancer.

The point I am making is that complex numbers were invented as a solution to an unsolved equation. All of the most interesting inventions of math are a result of unsolved equations. Set theory is different because it was not invented to solve unsolved equations. Set theory is not a "technique" to solving an equation. Set theory not only is not a number system, it also does not extend a number system to allow for solutions to an unsolvable equation. Set theory is literally just a rebranding of predicate logic. The two languages are logically equivalent. This is a well established, basic result in logic and you can look it up. The only difference between set theory and predicate logic is the "membership operator" which literally does fucking nothing other than give you a symbol for the word "in" or "from" or the expressions "member of" or "element of." It is linguistics, not math. Math is about solving equations, not renaming fucking words.

Yes, cantor's first paper on "set theory" was about trigonometric series, but since that paper, who has actually used the techniques he supposedly developed in that paper to go on to solve the problem he was discussing in his paper in regards to trigonometric series?

For an even better perspective, think of it like this. When you're working on a problem involving a trigonometric series, whose techniques are you most likely to use: Fourier's or Cantor's?

Now ask yourself, if set theory doesn't provide any techniques in math, why are we forced to accept that it is the fucking FOUNDATION of ALL mathematics? The answer is simple. Look at who Cantor really is. Look at who the leading set theorists are. Look at who promotes set theory. That might answer your question.

>>11577351
[math] \displaystyle
a+ib \leftrightarrow
\begin{bmatrix}
a & -b \\
b & a
\end{bmatrix}
\\
\begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}
\begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}
=
\begin{bmatrix}
-1 & 0 \\
0 & -1
\end{bmatrix}
[/math]

>>11577843
>I enjoy doing math for the sake of it. ... In fact, I resent people who need a "real life" application.
>Math is about solving equations

>>11577868
>equations are "real life"

should we tell him?

>>11577868

Why are dumb people so fucking confident. Jesus christ equations are not "real life." Equations are metaphysical objects. They are abstract. You cannot observe a fucking equation under a microscope. Solving equations is not a "real life" application. Fuck I hate this board so fucking much.

>>11577910
haha loser

>>11577843
Wrong.

>>11577324
You being too stupid to understand why we use something is not an argument against it.

>>11577324
Complex numbers aren't fundamentally about the square root of -1. Fundamentally, they are about the geometric fact that scaling and rotation have the same properties.

Bear with me for a second. Consider the vector space R^2. Scaling of vectors is commutative (doesn't matter which order you apply scalings) and it distributes over vector addition ([math]a(u+v)=au+av[/math] where u,v are vectors and a is a scalar). Now consider rotation. Rotation is also obviously commutative and distributes over vector addition. Considering the similarities between these two operations, it's natural to try and come up with a combined scaling and rotation operation which encodes both of them. This is exactly what complex numbers do.

So where does sqrt(-1) come in? Consider the real number line. Multiplying a by b effectively scales a by a factor of b to reach the product ab on the line. When we multiply by negative numbers, it's equivalent to reflecting and then scaling. Consider the equation x^2 = -1. No matter what real number x is, this equation is not satisfied. However, we only need to reframe the question geometrically. Reflecting when multiplying by negative numbers is the same as rotating by 180 degrees. Therefore, to get from 1 to -1, we have to rotate 90 degrees twice (in either direction). The complex number "i" represents the concept of rotating 90 degrees counterclockwise, and its conjugate "-i" represents the concept of rotating 90 degrees clockwise. Remember since scaling and rotation have the same exact properties, it is entirely self consistent to introduce new numbers which rotate instead of scale when they multiply. In this way, we find two solutions to x^2=-1 which are geometrically and algebraically self-consistent.

>>11577521
Wrong.

>>11577656
associative* you fucking mongoloid

>>11578025
I like this.

>>11577843
Ok bro other posters gave you shit but honestly I think your bitterness towards set theory isn't completely unjustified. But you still pretty much need it to make proofs work without inducing paradoxes.
Plus I think it's neat that you can progressively expand number sets by taking the cartesian product of a number set and then quotient it over some equivalence relation.

>>11577958
he's so right

>>11577489
triangle in the pic is real but badly drawn
in the complex plane, numbers "point up" vs the reals "pointing sideways" so i and 1 should really be drawn in the same direction, overlapping. so the length between the ends of those line segments is zero

>>11578025
So if you rotate 90 degrees clockwise and then 90 degrees anticlockwise the vector is back where it started, and not "negative"

>>11577910
lol I wasn't trying to say that equations are real life. on one hand you act all patrician, on the other hand you reduce all of math to solving equations and you shit on set theory because it doesn't help you solve an equation. it's a bit ironic.

>>11578355
i*-i = 1

>>11578300
but the angle is not 90 degrees if they're on top of one another as you're implying

>>11577324
I do think that negative numbers are a hack, and amth would make more sense if it wasn't allowed to mix the plus/minus operator yto the actual fucking number,
But I do agree that it gives some measure of convenience in several cases, despite it being a hack.

>>11578359
his point is that solving general algebraic equations is an important, well defined, mathematical problem in itself, regardless of its applicability on other fields (physics in particular).

actually, it is so important that the actual construction of the complex numbers is irrelevant. set theory now provides a useful set of tools to create other sets and objects (if sometimes a bit artificially). but just because you can is not a reason. in that sense 'set theory' is not 'foundational'. not everything that can be built out of it (as the OP opinion could be interpreted) is interesting or even math just for that. perhaps not everything even should be allowed.

>>11578384
yes, abstract thinking is a hack and provides a measure of convenience

>>11577831
>If the codomain of f is real numbers, f(1)=n for some n either way
That doesn't make any sense
f(n), where f(n)*0=n, is clearly not real. So how would f(1) be n, where n is real?

>>11577324
read a book on analysis
you seem confused on how math is done

>>11577324
>because why not
Because everyone will think I meant 1/3 and put 0.3 by mistake and assume I'm a brainlet.

>>11577489
You are in your complete right to try to allow this number to exist. But then it is not difficut to show that it will force the equality 0 = 1 (because of the fundamental properties of both these integers). In particular, the whole theory "collapses" (meaning you do nothing with it, so it is uninteresting). This is why "division by 0" is not defined for numbers (and their extensions, up to complex).

>>11577455
and with complex numbers you lose
[math]\sqrt{xy}=\sqrt{x}\sqrt{y}[/math]
and some logarithim rules

Thing is, that's an emergent property of the square root in the reals, not a defined property. Similarly quaternions (if i remember correctly) do not break defined properties, they break emergent properties

>>11578025
based geometric interpret

>>11579063
you keep it outside a selected branch tho

>>11579090
sure, but the fact of the matter is that that there's no need for such care to be taken when working exclusively in the reals
the complex numbers have changed the conditions that allow that emergent behavior

>>11577345
>tfw he doesn't realize math is just made up

>>11578025
tell us more

>>11578070
>>11578355
>>11579068
>>11579713

He's referencing an explanation from this video, which is part of a larger video series, which you should all watch.

https://www.youtube.com/watch?v=T-c8hvMXENo

>>11577324

The way that mathematicians came up with [math]i[/math] was not by saying "let's accept this equation and see what happens lol."

It was more like "let i be the symbol that generates the splitting field of the polynomial [math]x^2+1=0[/math] over [math]\mathbb{R}[/math]."

What's the difference? There are theorems that say this construction is justified. see https://ncatlab.org/nlab/show/splitting+field, in particular:

> Theorem 2.1. Splitting fields of a polynomial are unique up to (non-unique) isomorphism.

This means that there is exactly one field that contains the root of our polynomial, which we refer to as [math]i[/math].

Except, with one caveat:

> (non-unique) isomorphism

In the case of [math]x^2+1[math] over [math]\mathbb{R}[/math], this means that you can define a symbol[math]i_2 = -i[/math] which behaves the same way as the normal [math]i[/math]. When you take the conjugate of a complex number, you are using this fact.

>>11581577
>let me use these higher-order abstractions I derived from the lower-order abstraction to explain how I derived the lower-order abstraction

>>11581601

it's actually the opposite. Taking "sqrt(-1) = i" as a definition is a higher order abstraction to using the actual construction.

the answer to OPs question is "no" because sqrt(1)^0.3 = -2 does not involve any symbols. If you want to define new symbols using the same construction as i, you need to be able to define your new symbol as the solution to a polynomial.

>>11581577
>>11581623

Splitting fields are a generalization that is derived from the progression in algebra seen in complex numbers, quaternions, octonions, and progressively greater n's for 2^n coordinate systems.

You can't derive complex numbers from splitting fields. It's like trying to say that real numbers are derived from the concept of a field. No, the concept of a field is generalization of the real numbers, which are themselves derived from solutions to certain but not all polynomial equations, which are themselves derived from equations that again produce the generalizations we see in things like rings and groups.

You can't derive things from their generalizations. That's circular reasoning.

>>11581577
I'm fairly confident it was "let's accept this equation and see what happens lol"

>>11581687

> It's like trying to say that real numbers are derived from the concept of a field.

The real numbers are constructed by stating "Let the real numbers be the field which has at least these properties and no other assumed properties. " So, they are constructed from the concept of a field.

> No, the concept of a field is generalization of the real numbers

wrong, the finite fields and the rationals are not generalizations of the real numbers

>>11581701

Sure, such a definition works for some purposes, but it is not what you need to answer OP's question

>>11577324
How about, because it isn't?

>>11581482
actually discovered this myself, but i guess that video works too

>>11577491
>Math is about solving problems. It is not just some arbitrary collection of definitions that someone pulled out of their ass. The answer to these questions is never "It's just how it's defined." that is WRONG.

Math is about solving problems BUT it's also about finding new ones and creating new weird shit.

>>11581740
>the reals are constructed

the reals are constructed IN HINDSIGHT motherfucker. you don't begin with the construction of the fucking reals. you BEGIN with solutions to polynomial equations that force you to use fractional exponents as solutions thereby "creating" irrational numbers and "completing" the rationals.

Sets are created by polynomials. You do not begin with sets. You begin with polynomials. You can, in hindsight, begin with a set, but again, that is always in hindsight, after you have produced the set from the polynomial.

n + 1 > n produces the naturals

n + 1 = n produces the naturals with zero

n + 1 = 0 produces the integers

n*m = 1 produces the rationals

x^2 = 2 produces the irrationals

x^2 = -1 produces the complex numbers

and so on

THEN, once you have produced the sets, you can in HINDSIGHT observe that when you restrict yourself to only using certain solutions to certain polynomials (certain "sets"), you realize under this restriction some polynomials have no solutions for certain operations and that is when you begin creating the generalization of groups, fields, rings, and so on.

You do not begin with a fucking field. That is such a stupidly literal interpretation of mathematics, like you started with abstract algebra and have no idea where the generalizations came from.

>finite fields and the rationals are not generalizations of the real numbers

I didn't say this. I said fields are a generalization of the reals. One of the cool things about generalizations is that you can find instances of the generalization that are not the same thing as the original thing you generalized from. You generalize the properties of a field from the reals, and then re-instantiate. Finite fields end up being an instance of a field when considering modular systems.

Do you not understand how abstraction works? You don't begin with a field. It's an abstract concept, it has to be derived from something.

>>11581806

I fully agree with this and recognize I might have been a little too harsh.

It's just because I keep seeing this trend in newer math students and posters like >>11581577 who have these very literal and anachronistic notion about math and appear to be incapable of understanding how and where newer and more abstract mathematical concepts come from, either because they are skipping ahead too far, or they genuinely can't think abstractly and as a result develop a flawed circular reasoning to justify their arguments.

>>11581740
>the reals are constructed

the reals are constructed in hindsight. you don't begin with the construction of the reals. you begin with solutions to polynomial equations that force you to use fractional exponents as solutions thereby "creating" irrational numbers and "completing" the rationals.

Sets are created by polynomials. You do not begin with sets. You begin with polynomials. You can, in hindsight, begin with a set, but again, that is always in hindsight, after you have produced the set from the polynomial.

n + 1 produces the naturals

n + 1 = 1 produces the naturals with zero

n + 1 = 0 produces the integers

n*m = 1 produces the rationals

x^2 = 2 produces the irrationals

x^2 = -1 produces the complex numbers

and so on

then, once you have produced the sets, you can in hindsight observe that when you restrict yourself to only using certain solutions to certain polynomials (certain "sets"), you realize under this restriction some polynomials have no solutions for certain operations and that is when you begin creating the generalization of groups, fields, rings, and so on.

You do not begin with a field. That is such a literal and anachronistic interpretation of mathematics, like you started with abstract algebra and have no idea what the source of the abstraction is

>finite fields and the rationals are not generalizations of the real numbers

I didn't say this. I said fields are a generalization of the reals. One of the cool things about generalizations is that you can find instances of the generalization that are not the same thing as the original thing you generalized from. You can generalize the properties of a field from the reals, and then re-instantiate into something that isn't the reals. Finite fields end up being an instance of a field that are not "the reals" when considering modular systems.

I re-edited the post and toned it down. The other poster is right I should be more civil.

>>11581817

> you BEGIN with solutions to polynomial equations that force you to use fractional exponents as solutions thereby "creating" irrational numbers and "completing" the rationals.

Do you understand that:

1. there are real numbers which aren't the solutions to polynomials
2. there are solutions to polynomials which can't be written using fractional exponents?

> Sets are created by polynomials. You do not begin with sets. You begin with polynomials. You can, in hindsight, begin with a set, but again, that is always in hindsight, after you have produced the set from the polynomial.

I don't understand what you're on about. I think you are trying to argue that one does not need to understand foundational stuff such as ZF set theory to come up with a working definition of a polynomial. I'm going to assume that you're saying "HISTORICALLY, polynomials came before sets."

Everything I said reflects the modern understanding of things, which math scholars all agree upon. I don't care about historical developments or "what came first."

>I said fields are a generalization of the reals

which is bullshit because the rationals and the finite fields are not generalizations of the reals

> You do not begin with a fucking field. That is such a stupidly literal interpretation of mathematics, like you started with abstract algebra and have no idea where the generalizations came from.

You sound like a theologist, "the bible is not meant to be taken literally!!!"

I'm done replying to you, schizo. Have a nice day.

>>11581854
>which is bullshit because the rationals and the finite fields are not generalizations of the reals

Again, you seem to not understand how generalization works. You begin with an example, and abstract out the properties of that example that are needed for your generalization. Once you have the generalization, you can re-instantiate an example that is different from the one you started with. Finite fields are a re-instantiation of a field, that was originally generalized from the reals.

Why is this difficult to understand?

Look I apologize for calling you stupid. That doesn't make for a good discussion. You'll notice I deleted and re-edited the post.

>you sound like a theologist

Quite frankly, that is very ironic of you to say.

If we do mathematics your way, we are beginning with abstract notions without knowing where they came from. This is more of a religious model of understanding, where you simply accept things as true and then find examples of them. If we do mathematics my way, it's the opposite. You begin with concrete examples, notice a pattern or rule, create a generalization, and then find more examples that satisfy the generalization. It's more akin to an empirical process of observation -> hypothesis -> experiment -> theory, whereas you are doing the inverse: where you begin with theory, and then find examples that confirm it.

>you're saying "the bible is not meant to be taken literally!!"
>you're a schizo

Again, ironic because schizophrenics are notorious for being both highly religious and citing the bible out of context as well as their inability to think abstractly which results in them often interpreting the bible very literally.

I hope you have a nice day as well.

>>11577365
https://www.youtube.com/watch?v=_h49ilnTmW4

>>11577324
Yes OP, you can declare any inane rules you want, but if they have zero useful applications/cannot further math in any sense, why bother?

>>11577843
None of this is true. This dude always posts here with his retarded takes. There is no result that set theory is equivalent to predicate logic, whatever that means, set theory is first order theory that uses the syntax and semantics of predicate logic. People have solved problems in algebra, topology, analysis, graph theory, combinatorics, and more with techniques of set theory. This dude knows no set theory, he just bitches about it.
>Math is about solving equations
lol

>>11581845
What polynomial construction over [math]\mathbb{C}[/math] needs the quaternions to solve? [math]\mathbb{C}[/math] is closed.

[math]
i^{2}\eq-1
But
i\neq\sqrt{-1}
[/math]
What's up with that?

>>11583865
[eq]i^{2}=-1[/eq] but [eq]i\neq\sqrt{-1}[/eq]
What's up with that?

[math]i^{2}=-1[/math] but [math]i\neq\sqrt{-1}[/math]
What's up with that?

>>11583877
I could be viewed as sqrt(-1). You can define sqrt on the whole of C such that it's continuous everywhere except at the positive real numbers. and coincides with sqrt of the positive reals.

>>11583727
>What polynomial construction over C
How about over Z?
x^2 = y^2 = z^2 =xyz = -1
is a system of polynomial equations. Try to solve them over C.

>>11577345
To put it in the terms of the "negative numbers don't upset people"
'Any number being added to another number results in a greater one. '

>>11583519
>there is no result that set theory is equivalent to predicate logic

You prove this in any Logic I course.

There are no statements you can make in set theory that you can't make in FOL.

>techniques of set theory

There are NO "techniques" in set theory dumbass. Set theory is a descriptive language. You don't "do" math with set theory.

>snickering about the importance of solving equations

cancer

>>11584418
Cite the theorem them. Who proved it? What year? I can prove it's not true, take ZF+V=L and ZF+there is a measurable cardinal, these are both first order theories of set theory. They are inequivalent, since neither proves the existence of a model of the other. So which one is FOL equivalent to? An analogy to what you are saying is that French is equivalent to language. It just makes no sense.
>There are NO "techniques" in set theory dumbass
What makes people like you speak so confidently about things you don't know. This is honestly really pathetic. Forcing, ultrafilters, definable equivalence relations to name three. Every single one of these set theoretic methods has applications in all major areas of math.
>Set theory is a descriptive language
This is horrendously wrong, set theory is a first order theory. It has a language, but so does every other theory.
>snickering about the importance of solving equations
I do think it's important, I've actually done research in it. There is much more to math than solving equations though.
You might want to read a book about mathematical logic one day.

>tfw just imagine complex numbers as vectors of length 2

It's the same shit

>>11584827
>vectors of length 2
cs major detected

>>11581845
>>11581687
>The real numbers are derived from solutions to polynomial equations
You've outed yourself as an absolute retard.

>>11584776

This is the problem with mathematical logic. You are completely out-of-touch with these nth-order abstractions that you can't even see your origins. Specifically, yes, set theory is equivalent to FOL if you add in equality relations and membership. They both use quantifiers and all the usual connectives. You can make the same formulas. Sets are Predicates and Predicates are sets. That's not controversial and if you're as familiar with the subject as you claim you should know this basic fact about the languages. You can say the same things without equality relations but you end up becoming long-winded, in the same way that you can use propositions with indexed connectives to make predicate-like statements without using predicates. Again, it's equivalent but it takes more work. That's why we create higher-order concepts, you can say more with less work.

>forcing and ultrafilters have applications in ALL major areas of math

I could only find one paper on google scholar that mentions forcing and something "tangible" like primes. It's mentioned once and has barely any relevance to the rest of the paper most of which is him talking about roots (he's *solving an equation*, go fucking figure). Funny enough written by Noam Elkies. Like I keep saying, these concepts are higher-order and descriptive. You don't actually use them to "do" math, you use them to define the boundaries of where you can and can't do math. Yes, obviously that's part of being a mathematician is defining boundaries, but at a certain point you have to ask yourself, have I gone too far above to say anything meaningful? Is this actually abstract or am I just blindly following a chain of ad-hoc tangents that lead nowhere?

>>11585374

Read history. The "Reals" were not born modern and constructed. It's been a slow progression of solving polynomials with increasingly relaxed parameters, i.e. becoming more general, abstract- which is fine, but just needs to be acknowledged.

>>11585508
I never claimed they were born modern or constructed. You cannot obtain the reals from solutions to polynomial equations. Everything else you say is discredited because we know it comes from a braindead moron.

>>11585536
>you cannot obtain the reals from solutions to polynomial equations

The reals are literally the union of algebraic, irrational, and rational numbers. Where the fuck do you think irrational and algebraic numbers come from?

>>11577324
>If the whole idea behind complex numbers is based on pretending that sqrt(-1)·sqrt(-1) = -1
Everything that comes after this idiotic assumption is just mute. Try to learn something OP, it would make you look less like a moron.

>>11585536
>You cannot obtain the reals from solutions to polynomial equations
*with integer coefficients

>>11585559
>>11585536

Was just about to respond to him because I realized I have a bad habit of referring to polynomials in a more general sense than people are used to. In primary school you learn that a polynomial ONLY has certain types of numbers as it's coefficients and exponents. I literally from the first time I heard it have rejected that definition as obviously it's more useful to consider the more general form as a "polynomial" and then specify what KIND of polynomial it is by restricting what kind coefficients and exponents you can have.

I forget not everyone has the more general view and that most people are stuck with the primary school definition of a polynomial. My apologies bro.

>>11585574
whole number exponents is a taylor polynomial, it's hardly "primary school", it's the most typical type of polynomial discussed
even Laurent polynomials only allow integer exponents
anything more broad than that becomes a definition of polynomial broad enough it rapidly becomes difficult to discuss any emergent properties since you can essentially describe any analytic function whatsoever

>>11577476
I heard someone say that we should encourage doing shit even if it doesn't solve anything because it may in the future, what are your thoughts on that?

>>11585595

Well I've always been an experimental type. I was skipped ahead a grade and was always in advanced math classes as a kid. I remember the first time they showed us polynomials I asked the teacher what would happen if we put numbers with radicals in the exponents. It's just how my mind works. I don't follow the rules people give me if they're too limiting. A polynomial to me is just a sum of terms with coefficients and exponents. The coefficients and exponents can be anything in my mind. Like I said I often forget this is not the general, accepted definition and that people don't view them so broadly but look that's just how I roll.

>>11585598

I'm more open minded than I often come off when I post. I do a lot of experimenting myself but I don't force everyone else to be as experimental as I am, and I also just don't think it's a good idea for most people. Have you ever met the average person? They don't even like math. They can barely grasp fractions. Now you're going to throw a structure at them? Fuck that shit. People need to start with something concrete and work their way up to the abstract. You don't start with abstract and then instantiate. I mean, you CAN do that but why the fuck would you? It's much more fun to generalize yourself then to be given a generalization that you have to instantiate like a little servant for his master.

>>11585559
Irrelevant, due to the word "obtain". You don't obtain something if you already started with it.
>>11585574
Are you trying to tell me that you think a polynomial is literally any differentiable function? And you think that's more useful than the conventional definition?

>>11585636
>Irrelevant, due to the word "obtain"
fair

>>11585598
>>11585635

I just want people to see where the abstraction comes from. Everyone knows Mondrian's later work, the primary colors on grid lines. Less people understand that these grid lines and primary colors were progressively abstracted over the course of his life. He began with very recognizable illustrations, and then slowly began to generalize the objects by removing their particular properties, until eventually the only thing left was primary colors and straight lines. It's a beautiful thing to see and I wish more people could see it. When you start with abstraction you're treating it like a creation myth instead of an evolutionary process, which is why I'm so opposed to it- not to mention the absolute degenerate snobbery that comes along with it. I'm just fighting for the little guy.

>>11585636

Man, c'mon just read what I fucking said and don't be so fucking literal. Fuck why do you have to be such a fucking square.

>>11585636
dont be petty

>>11577476
Fucking imbecile.
This type of retard is why you should have to show proof of formal education before posting in this board

>>11585681

This person is probably a french, wiglicking monarchist and we should disregard their opinion. If you like doing math, then do it. Don't let old order european values limit your education. who cares how many fields medals they've won.

>>11585727
Based French-hater.

>>11585595
>whole number exponents is a taylor polynomial
???

>>11585756
A Taylor polynomial is of the form a+bx+cx^2+..., i.e. natural number exponents.

>>11577476
Hey guys, just wanted to let you know that I've solved finally and decisively the deepest question in the universe that philosophers have debated for millennia, fundamental question of metaphysics: why is there something rather than nothing?

My solution is thus: Define $b$ such that $b$ answers the question of why there is something rather than nothing. Then $b$ is the reason why there is something than nothing.

>>11585772
Based automatic use of normal LaTeX.

>>11585756
a polynomial that can be described by a finite taylor series

>>11577324

Agreed