Why is a+b=b+a ?

We define it that way.

>>7709248
So truth is arbitrary?

Why isn't it?

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

>>7709215

Because the binary operation of addition takes number (of some one type of thing) as an abstract starting point, and then takes two heaps, throws them both them together into another heap, and provides an answer as to how many (things) are in the resulting heap. The issue of comparable quantity on both sides is typically resolved before an addition (throwing two quantity-heaps together into one big quantity-heap, having its own third quantity, or sum) is performed.

I work in a produce department at a supermarket. Every bag of apples contains three apples. Every bag of oranges contains twenty oranges. Apples and oranges are kept in the same staging zone, we get one small pallet of each per day, one has one hundred bags of apples, and one has fifty bags of oranges. Inventory is kept in terms of grand-total bags, not pieces of fruit, so I log 100 + 50 = 150 new bags of inventory each day at receiving. In order for the operation (of addition) to make sense in my accounting context, a unit is agreed ahead of time-bags. In the abstract, one need not have a unit - because when you heap two like groups of things together, no information is lost-at least where the performance of the sum is concerned. They can mix, co-mingle, and as long as they are countable, the sum remains the same.

>>7709249
No, because litteraly thats the way humans created the operation. You could give shittier rules and "proove" commutativity, but it is less formal.

>>7709260

Addition to this :^) : Say instead of apples, I get a straight shipment of two pallets of identically-packaged oranges, every single day: 50 bags, 100 bags. I still receive 150 bags per day. Even though I would be hard-pressed to take the original 50 orange-bags of the small pallet and parse (subtract) them back out once summed together (and physically co-mingled), I would still be able to easily parse (subtract) out /some fifty/ bags, and I could even do the same, treating apples and oranges as the same earlier.

The fungible unit here is not a particular fruit, but /bags-of-fruit/. Obviously a supervisor with more granular inventory concerns would take issue with me if I were to have a laff and mix items where they don't belong. I would then point out that it might be a good idea to create different codes for apples and oranges, thereby separating the two for accounting purposes.

>>7709215
One of the axioms of [math]\mathbb{R}[/math]

>>7709215
+ usually denotes an associative operation, so...

>>7709215
https://en.wikipedia.org/wiki/Proofs_involving_the_addition_of_natural_numbers#Proof_of_commutativity

>>7710555
It has nothing to do with associativity :^)

>>7712057
:^)

Though I like how A*B is not B*A in case of matrices.

>>7709249
Not necessarily. We defined symmetry to be a necessary requirement for what we call "equivalence relations" and then go on to prove that equality under certain constructions is an equivalence relation.
It amounts to a lot of diagram chasing, but what it all really comes down to is you have to have faith on the existence of 0, and the existence of a number after 0.

If you accept those two things, then all of arithmetic has a very rigorous foundation. Read more about number systems and abstract algebra.

>>7709267
>>7709279
>>7709260
These are not helpful answers, as they don't emphasize the fact that there are HUNDREDS of number systems, metric spaces, and even groups where addition is not commutative and equality is not symmetric.

Saying "it's real because apples" is like an economist trying to speak math. Let the big boys talk.

>>7709249

Philosophy fags plz go

>>7712118

Incorrect post; an appeal to the existence of noncommutative algebras, and to the variety of algebra in general does not directly answer OP's question (although it can and does throw light on old assumptions). I did.

I correctly and helpfully (if word-ily) articulated (and this is the key word, OP's word) WHY "regular" addition is commutative. Because implicit in the operation is the fungibility of its operands. Not even the induction proof which has been cited in this thread really gets at the WHY; note how the Peano axioms involved are simply a subtler expression of how arithmetic does what it does, from which the proof is then developed. And no, it is not satisfactory to spit 'n' sputter at this point, and say something like "the reason why addition is commutative is because it was proven to be so using Peano, asshole."

>>7712118
Please define the general concept of "addition" in the big boy way.
Hint: "a binary law denoted by the symbol +" is not an acceptable definition.

>>7709255
thank you for posting this tune

>>7709215
You can prove it using Dedekind-Peano axioms

>>7709249
Mathmatics doesn't deal with truth. Yes your axioms are fairly arbitrary.

>>7712116
Interesting. I was wondering why literally the second axiom introduced in a book I'm reading was the existence of objects 0, and 1, with 0=/1. I've seen equivalence relations before. Perhaps I should revisit and study them further

>>7709249
what's a truth senpai?

math major reporting in

>>7709215
>>7709248
Yes we defined it that way, but I think this says a lot more about what we want out of [math]=[/math] than what we want out of [math]+[/math].

>>7712455
You need [math]1[/math] because otherwise everything can be done in the field [math]\mathbb{F} = \{0\}[/math]. No big deal.

>>7712174
If you're restricting yourself to the reals, it's the operation [math]\mathbb{R}^2\to\mathbb{R}[/math] given by [math](a, b)\mapsto a+ b[/math].

More generally, you can think of all fields as "extending" groups. The addition operation is the one that works in both the group and the field.

>>7712269
Is that the whole "equality is reflective, symmetric and transitive" axiom?

>>7712148
You did not answer OP's question. You literally said "because it exists in the real world" when that is complete nonsense.
OP did not ask about whether two apples is the same as two apples. He asked about addition.

You are dense. You like to wave your dick around, but in reality you seem to lack knowledge.

>>7712174
Repeated iteration/composition of the successor function.

>>7713521
But anon, the reason we went with the Peano axiom in the first place is because it ends up giving the natural arithmetic that we expect to find, not the other way around.

>>7709215
Take two sticks and lay them end to end. Do you get any further by laying one in front of the other?

No? Then you shouldn't with numbers.

>>7712556
Something that is correct in it's correlation the settlements of tautologies in regarding to a ontological framework of knowledge, manifested through definitions and assumptions.

South could be North, but why create mutiny aboard me ship, matey?.

It's a commutative property of algebra. You can a+b to be anything you want in some obscure space, but over the real numbers, this is how addition is defined.

It makes sense empirically. If you have some arbitrary number of apples a and add an arbitrary number of oranges b, you'll have the same amount as if you added the other way around.

>>7709215
It isn't necessarily you idiot. Not all + operations are commutative.

>>7713529
Engineer detected.

>>7709260
>pleb trying to demonstrate an abstract mathematical concept using oranges an apples

>>7713663
Math major, if you think the most basic concepts of addition humans created weren't designed to model counting or distance then you're dumb m8

>>7712120

/thread

>>7713521
>>7713928
>>7713657

You have all completely and repeatedly missed the point, and yes, I /did so/ >>7709260 answer >>7709279 OP's question. The "things" are immaterial - quite the opposite, the fact that you're hung up on my dopey produce department (immaterial details) strongly suggests that you're still not getting this.

Babby addition is commutative, /exactly because/ it treats it operands, or units, as fungible, having been agreed upon as interchangeable /a priori/. /Ahead of time/. /Of course/ the operation will be commutative if the entities to be added are agreed upon as interchangeable! To point out "b-but did U know dat there are noncommutative algebras!!1" like a sophomore is obfuscatory, and not (really) germane to OP's thing.

>>7713526 This guy (who is not me) both correctly characterizes Peano and pushes back on the poster who is wrong. This guy gets it. This >>7713949 is also a good post.

Speaking of details... had any of you actually /read the posts/ instead of instantly kekking over Muh Produce, you would know that the things being added in my example are /not apples and oranges/.

>>7712732
https://en.wikiversity.org/wiki/Natural_number
See the addition part how it can be proved. Right above are Peanos axioms.

cant you just rearrange the right side so it looks like a+b?? then u have a+b=a+b

>>7709215
1+2=3
2+1=3
1+2=2+1
its not complicated

>>7714367
You can rearrange it that way IF a+b=b+a. So the fact that you can says nothing for why you can.

>>7714384
>proof by example
ISHYGDDT

>>7714395
My point is that the most common algebra we use comes directly from our experiences in physical reality. We can count two groups of things out in either order. However, in other algebras this might not be the case. As,

Nah. I was going to give a more logical answer but I remembered I am on /sci/. I'm gonna meme it up.

Here's my real reply:

>>7714395
>>7714395
prove me wrong, guy. you literally cant prove me wrong. i hate braniacs like this. look at me i can do proofs. is this your first semester? i take it your not stem, are you? humanities need to stop coming on my board!!

File: abbacus.gif (240KB, 250x170px) Image search: [Google]

240KB, 250x170px

Because