Why is the distributive law an axiom? I've always heard the distributive law of multiplication described as an axiom, an assumed property of a given set of numbers of any kind. In the same category as what I would call more fundamental axioms such as $a\cdot 1=a$.
The distributive law of multiplication in a simple form states the rule used in algebra to multiply "into brackets" (or vice versa to "take outside the bracket"):
$$(a+b)\cdot c=a\cdot c+b\cdot c.$$
I feel this should be provable or at least justifiable rather than simply being assumed as an axiom. Can anyone shed some light on the inclusion of this law among the fundamental axioms of our number sets? Am I totally wrong in feeling that this law seems categorically different from other typical fundamental axioms?
 A: The axioms are the starting point for whatever system you are talking about. If you are talking about natural numbers (counting numbers) probably the most famous set of axioms are the Peano axioms, which don't actually talk about addition or multiplication at all. You can define addition and multiplication in terms of these axioms and prove that they have all the properties we usually want, including the property that multiplication is distributive over addition. You can also extend the system in steps to complex numbers, and again the various properties we know can be proved.
On the other hand if you are doing abstract algebra you start in a different place. You will define a system using some set of axioms and find out what can be proved from those axioms. The normal number system may fit those axioms or it may not, and even if the normal number system does fit there might be other systems that also fit.
There are different sets of axioms which can be used to define real numbers. While it is possible to start with the Peano axioms or even start from the axioms of set theory (see Whitehead and Russell's Principia Mathematica) this is a long process and does not teach anything new to most people. Many Maths textbook writers instead select a different set of axioms to define real numbers, and normally the distributive law is one of them. These sets of axioms are just as valid, and an interested reader could prove the Peano statements from them.
A: 
I've always heard the distributive law of multiplication described as an axiom, an assumed property of a given set of numbers of any kind.

This is actually simply not the case.
You're touching on the notion of algebraic structures, and how we define them. The sets of the integers, rationals, reals, complex numbers, and other such numbers, are all what we call "commutative rings." (In fact, the latter three are special kinds which we call "fields.")
To be a ring, the set - under a defined addition and multiplication operation - must satisfy certain axioms, and distribution is one of them.
Bear in mind the distinction -- we do not assume that these properties hold to name them this. Rather, it indeed must be verified that it holds. How you prove it for the prescribed sets earlier ultimately depends on how you define them, which is a very nontrivial question to address first.

For instance, we can consider our set of "numbers" ("number" is a very vague term, so I'll focus on just sets) to be a near-ring. Let $S$ be our set with $+,\cdot$ our operations. Then $S$ is a near-ring if

*

*$(S,+)$ is a group, i.e.

*

*Addition associates

*Addition has an identity element

*Each element has an additive inverse



*Multiplication via $\cdot$ distributes on the right side, but not the left. (Or on the left, but not the right, for a "left near-ring.")

*Multiplication associates

Various subsets of the functions $f : G \to G$ for a group $G$ form near-rings when properly given the necessary operations.
You can also look at ring-like structures that have no distribution at all (rather than just one kind); see here for a possible example. It requires, ultimately, just a carefully-defined addition and multiplication operation on the proper set; after all, addition and multiplication are interacting in the definition of distributivity, so both must play a role.


I feel this should be provable or at least justifiable rather than simply being assumed as an axiom.

Of course, now addressing this, in the axioms which define a ring -- sadly, no, this would not be provable from the others. Loosely speaking, to echo jjagmath's answer, it's because it is the only ring axiom which specifies relations between both addition and multiplication. (You may find the axioms here.)
A: There are algebraic structures with two operations "sum" and "product" that satisfies all the other axioms about the "sum" and the "product" but not distributivity. That means that distributivity can't be proved from the rest of the axioms.
A: 
I've always heard the distributive law of multiplication described as an axiom, an assumed property of a given set of numbers of any kind.

If we're talking about a general abstract system, rather than one intended to model the natural numbers, "multiplication" is just a label we put on one of the binary operators. It doesn't imply any of the properties of "normal" multiplication, beyond what is given in the axioms. We can call anything we want "multiplication". For instance, we could define a number system in which exponentiation is referred to as "multiplication". Then this "multiplication" would not distribute over addition.
Also, an axiom can be thought of as more of as part of the definition of the number system, rather than an "assumed property" of it.
A: 
I've always heard the distributive law of multiplication described as an axiom, an assumed property of a given set of numbers of any kind.

That is not true. In our current number system up to complex number, we have a bottom up approach to define everything, including rules of addtion and multiplication. All the properties such as commutative law can be deduced and proven using their definition. The axioms of number system only contain the Peano's Axioms, with some usage of set theory so you can consider axioms of set theory is used.
This bottom up approach is like this:
Building natural numbers using Peano's Axioms. There are 5 of them. Addition(natural number limited) can be constructed using the successor notation(which is the 2nd Peano's Axiom). Within this range, we can prove some property of addtion(natural number limited), like communtation and association, using mathematical induction, which is the 5th Peano's Axiom. On this base we can define multiplication as repeated addtion, an prove some of its properties.
Then we use natural number and addtion to construct and define negative numbers. While doing this, we must prove that in this newly defined system, all the previously defined operations still hold. We must check and prove each of additive commutation and so on still hold for negative numbers, and we can say that expansion from natural number to integer is successful. This process will be somehow tedious and trivial, but if we want it to be must rigorous it will be necessary.
Then we use integers and multiplication to define rational numbers. Still, check if each existing rules still hold.
Finally, we use the concept of limit to define real number. This step is less trivial because it involves some new concept, but all these concept do not come from axioms but can be defined using rational numbers. Note that if we want to do this rigorously bottom up, we can only use sequences of rational numbers, and the concept of distance must be limited to rational. Once this step is done, we can easily see that all the operations, properties, concepts can be immediately apply to real numbers, and finally we have what we are very familiar with.
In a top down approach, we will say like this: We want to prove, say distribution law for real numbers, then we must know how real number is defined using rational numbers, and how to prove distribution law in rational numbers. Again for rational numbers, we will trace back to integers, and proof of integer distribution law is done by mathematical induction which is 5th Peano's Axiom. Here is the axiom really on the bottom.
Terry Tao in his Analysis book did what we just mentioned in huge and tireless details, which cost 3 or 4 chapters, so that we can completely see through the entire approach of how our number system is built. Actually I learned all this from his book.
