In strictest sense, an expression such as $1+2+3$ isn't even defined, only $(1+2)+3$ and $1+(2+3)$ are. It is the law of associativity that allows us to interchange the latter two expressions and motivates the usual convention of dropping parentheses altogether in such sums. In view of this, the "right" to evaluate a sum in arbitrary order is already implicit in the legitimiacy of leaving out parentheses. Similar for products.
The fact that we write $1+2\cdot 3$ without parentheses, although $(1+2)\cdot 3$ and $1+(2\cdot 3)$ differ, is not based on an arithmetic law, but rather on the convention that multiplication and division precede addition and subtraction. That is, $1+2\cdot 3$ is really a shorthand for $1+(2\cdot 3)$ whereas there is no shorthand for $(1+2)\cdot 3$.
Finally, a similar convention, namely that the non-associative operation of subtraction (as well as division) are to be done left to right. That is, of the two expressions $(1-2)-3$ and $1-(2-3)$, only the first has a shorthand notation of $1-2-3$.
Note however that exponentiation is not associative, e.g. $(2^3)^4\ne 2^{(3^4)}$. Since the former can be written simply as $2^{3\cdot4}$, we have the convention that the expression $2^{3^4}$ is a shorthand for $2^{(3^4)}$.
I think the best way to view this is in for of a tree where each node is either a leaf labelled with a number or (if the node has a left and a right subtree) an operator $+,-,\cdot,/$ or epxonetiation. You may additionally introduce negative signs and functions as unary operators (nodes with one subtree). For associative operations such as addition and multiplication, you may loosen these rules and allow more than two subtrees. This tree determines the order of operation (note that there are no parentheses needed to build the tree): In order to perform an addition, subtraction etc. you need to first compute the two subtrees and then combine these two results accordingly. Note that the overall sequence of operation is only very loosely defined/restricted by this: You can first evaluate the left tree, then the right tree, or vice versa, or intertwined. Only the "top" operation must be last. This is all there is behind the rules of precedence and parentheses: They clarify which of several possible trees is intended.
Given two sets $A$ and $B$ of cardinality $a$ and $b$, respectively, the cardinality of the cartesian product $A\times B$ is called the product of $a$ and $b$, and is denoted by $a\cdot b$.
Update
When I wrote this answer I didn't have infinite sets in mind. I just wanted to convey a mental picture of multiplication that does not involve repeated addition.
Best Answer
Multiplication has higher priority than addition because first of all, multiplication is indeed repeated addition. Take the expression $4 + 3 \times 9$. This is 4 plus 3 groups of 9. So the mutliplication can be dismantled as $4 + 9 + 9 + 9 = 31$. It isn't $7 \times 9 = 63$. When you simplify an expression, you start with the parentheses, which are the highest and the most complex parts and evaluate what's in there and then combine the result to what else is there. Then, the middle parts come, which are exponents, which is repeated multiplication, then multiplication, which is repeated addition; and finally, we add everything up.
Also, this draws to the distributive property. The distributive property states: $$a(b+c) = ab+ac$$ If addition had higher priority, the property would look awkward: $$a \cdot b + c = (ab) + (ac)$$ It is worse with implied multiplication: $$ab+c=(ab)+(ac)$$ With multiplication having higher priority, you don't have to add parentheses. That's why PEMDAS is so useful. Without PEMDAS, $P(x) = ax^4 + bx^3 + cx^2 + dx + f$ would have to be written as $P(x) = (a(x^4)) + (b(x^3)) + (c(x^2)) + (dx) + f$ for us to understand.
In short, multiplication has higher priority because of the distributive law of multiplication over addition as well as it being repeated addition.
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