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This trivial question is all about reasoning (intuition) and obviously not proving. I know $a\cdot b = b\cdot a$ from very early school years and it's considered intuitive. A simple proof is by taking a rectangle that is $2 \cdot 7$ and calculate the area which is $14$. The same is true if we rotate the rectangle. That however is a just the proof, it just does not explain the intuition behind this trivial theorem.

But, how can you say that when we add up seven twos, the result is equal to adding up two sevens?

Edit: It's about the why and not how and in fact I think it needs a bit of philosophical answer.

Jack Crawford
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doc_id
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    I really think counting the number of tiles in the rectangle with 2 columns and 7 rows in two different ways is the intuition. – Julian Mejia May 31 '19 at 21:55
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    Usually when we say “reversible” we mean something like invertibility, but the “reversibility” you’re talking about here is actually a property we call “commutativity”, or in the case of groups, “Abelian”. As for why multiplication is commutative: this isn’t always the case. The sets of objects that do have commutative multiplication, however, are exactly those which can be represented as objects in a preadditive category. – Jack Crawford May 31 '19 at 23:13
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    When you say it’s about the why and not the how, I think you should be careful about expecting mathematics to have a “why” for being the way that it is. It certainly doesn’t have any motivations like a conscious agent might. Certain structures just intrinsically have certain properties. The “why” is the “how” in some sense; any proof that a given group is abelian is the “why” at a very deep level. If you just want to find proof that a given algebraic structure is commutative, you need only venture to learn a bit of abstract algebra, this does not require a “philosophical” answer. – Jack Crawford May 31 '19 at 23:22
  • @JulianMejia that is just an observation. Obviously I'm not a mathematician but I know that no matter how many times we see evidences, curiosity will always help understanding things we don't question now or even unaware of its existence. – doc_id Jun 01 '19 at 03:29

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Suppose that you have $7$ red balls, numbered from $1$ to $7$, and $7$ green balls, also numbered from $1$ to $7$. If you pick the red balls and then you pick the green ones, in the end you will have $2\times7$ balls. And if you pick first the balls with the number $1$, then the balls with the number $2$, and so on, in the end you wil have $7\times2$ balls. But its the same set of balls. Therefore, $2\times7=7\times2$.

José Carlos Santos
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It seems that you are referring to multiplication in $\mathbb R$.

The geometric answer to this question is

Multiplication in a division ring $D$ is commutative precisely when the plane $D\times D$ satisfies the theorem of Pappus.

So in principle, you can take $\mathbb R\times \mathbb R$ and "forget" that commutativity works, and then use the fact that Pappus' theorem holds to deduce that multiplication is commutative.

I think these books at least contain proofs

Artin, Emil. Geometric algebra. Courier Dover Publications, 2016.

Kaplansky, Irving. Linear algebra and geometry: a second course. Courier Corporation, 2003.

Hartshorne, Robin. Geometry: Euclid and beyond. Springer Science & Business Media, 2013.

In Hilbert, David. The foundations of geometry. Open court publishing Company, 1902. Hilbert proves that real multiplication is commutative because Pascal's theorem holds in the real plane, but I think the idea is that you also need the division ring to be Archimedian to imply that it also satisfies Pappus' theorem.

This information is sort of hard to find because so often we axiomatize $\mathbb R$ by assuming commutativity. In contrast, the above books discuss constructing the real line and real plane with geometric axioms, and then one can build an "algebra of segments" which turns out to be nothing more or less than $\mathbb R$.

rschwieb
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This is not a rigorous proof but I can give sort of an intuition.

It's because addition is commutative and Every Natural number comparatively larger can be represented as sum of two or more smaller numbers. for example:

$2 \cdot 4$ basically means adding two four times, i.e $2+2+2+2$ which turns out to be $8$: $$2 \cdot 4 = 2+2+2+2 = 8,$$ whereas $4 \cdot 2$ means adding four two times, i.e $4+4$ BUT $4 = 2+2$. Therefore $4+4$ becomes $(2+2) + (2+2)$. Since addition is commutative I can remove the brackets and it becomes $2+2+2+2$ which is equal to $8$: $$4\cdot 2 = 4+4 = (2+2)+(2+2) = 2+2+2+2 = 8.$$ Hence multiplication is commutative. And if you ask a bit more like why addition is commutative it's what seems so at least from our human reasoning and observation. Another reason I have stated this is precisely why division ain't commutative – because subtraction is not commutative.

Timur Bakiev
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Skeptic
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The proof that the natural numbers is commutative is the “why”. Constructive proof of something is the closest thing to an objective “why” that we can get. Mathematics does not admit motivations and aspirations like a human agent might — there may very well be no deeper “why” as to why this is the case. Certain structures just have intrinsic properties.

Here is the proof that integer multiplication is commutative. It’s the best answer you’re going to get as to “why” this is the case. There may very well not be some deeper philosophical answer, and I suspect that any attempt to find one would be itself a bit of a philosophically unsound endeavour.

Jack Crawford
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  • However, it is much more clear "why" addition is commutative (thanks for correcting the term). Think of something that is broken down into pieces, it's now valid to say that pieces A and B are needed to complete the thing, and it's equally valid to say B and A are needed, because they both belong to one thing, or quantity. This is not true with multiplication; let alone it's not exclusive to "natural numbers". – doc_id Jun 01 '19 at 03:38
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    Is it? I'm not sure I find the commutativity of addition any less intuitive than the commutativity of multiplication (over fields). I like to think of the operation in a monoid as a sort of addition (such as concatenating strings to form longer strings) and that usually isn't commutative either. It's moreorless the case that we tend to use words like "addition" and "multiplication" to describe things that behave nicely, and not the other way around. If matrix addition didn't have any of these nice properties and was just a boring magma, we wouldn't have called it addition in the first place. – Jack Crawford Jun 01 '19 at 04:02
  • If I break a bowl of cereal down into pieces; the bowl, the milk, and the cereal - it's clear that these things are needed to put the bowl of cereal back together. But if I put the milk down, and then the bowl, and then the cereal, I get something very different to if I put the bowl down and then the cereal and then the milk. I'm not sure your "breaking things down" analogy really does explain commutativity so much, either. Maybe the milk and the cereal commute with one another but they do not commute with the bowl. The cereal and milk form the "centre" of the joke bowl-cereal-milk group. – Jack Crawford Jun 01 '19 at 04:06
  • One way you could formalise arithmetic is by seeing it as just the cardinality of different sets under set operations. If I have sets $A$ and $B$, then define addition of $|A|, |B|$ to be the cardinality of their disjoint union. Similarly, define multiplication to be the cardinality of their Cartesian product. The question of commutativity then reduces to whether you can make the same number of ordered pairs with one element from $A$ and one element from $B$ as you can with one element from $B$ and one element with $A$. It follows from commutativity of "and" in natural language, if you like. – Jack Crawford Jun 01 '19 at 04:16
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In mathematics, one does not discover rules that already exist. One makes up rules, and if something interesting comes out of those rules, you keep them. The made-up rules are called axioms. What comes out of them is a theorem.

In most cases, the commutativity of multiplication is an axiom. We keep it around because all of arithmetic, algebra, trigonometry, calculus, differential equations, etc., is based around this rule, so it is very interesting.

weux082690
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  • Describing some rule as "very interesting" makes it persuasive to question why such rule exist, and while this does not answer the question, proving why a question cannot be answered is in fact the answer. Do you think it is possible to prove that this "why" cannot be answered? – doc_id Jun 01 '19 at 03:23
  • Communitivity isn't always taken "as the rule." There are some examples that you might have experienced where multiplication isn't communitive: matrix multiplication, the cross product for vectors, composition of functions, moves on a Rubik's Cube, etc. When you don't include this axiom, there is some interesting math that happens. Fields are structures that include multiplicative communitivity while rings and groups are things that do not necessary include multiplicative communitivity. So, it doesn't always have to be true, but in the field of real numbers you are used to, it is. – weux082690 Jun 01 '19 at 03:55
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    You cannot "prove" your question is unanswerable because your question is not based in a formal mathematical system. – weux082690 Jun 01 '19 at 04:00
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    The reason why multiplication over the real numbers is communitive is explained by the example you gave in your question. Let me word it differently to see if it helps. Consider multiplication as the area of a rectangle. It doesn't matter if you count the number of columns and the number of pieces in each column or if you count the number of rows and the number of pieces in each row. Thus the area can be counted in two different ways which means that the two different ways of counting are equivalent. – weux082690 Jun 01 '19 at 04:06