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Can laws like the Commutative and Associative laws of Addition and Multiplication be proved or only demonstrated?

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    What is the difference between ''proved'' and ''demonstrated'' ? – 2017-02-14
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    sorry. i meant to ask that are they defined or taken as axioms or is there a way to prove them from more basic principles? – 2017-02-14
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    Depends on the exact question. Given a $K$-algebra over a field $K$, we have a bilinear product $(x,y)\mapsto x\cdot y$. Nothing can prove that $x\cdot y=y\cdot x$ for all $x,y$, because it need not be true. – 2017-02-14
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    I am referring in specific to elementary algebra, as in with positive integers. A book on elementary algebra states them as laws, and then attempts to prove them, but the proof doesn't seem to be quite formal. – 2017-02-14

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There is an axiomatic theory of arithmetic (that is, of algebra on the natural numbers $\mathbb N = \{0,1,2,\ldots\}$) known as Peano arithmetic. That axiomatization is enough to prove most "ordinary" facts about the natural numbers. One axiomatization can be found here on Wikipedia. You will note that none of the axioms explicitly state that addition or multiplication are associative or commutative, but these facts can be derived from the axioms.

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    thanks, but could you tell me if when elementary books like Hall and Knight's elementary algebra are expounding the algebraic principles, do the follow the method set forth by Peano is it an outdated way ? because then i find myself in the diemma of which approach is the right one. – 2017-02-14
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    @SaitamaSensei, I'm not familiar with the book, but leafing through the first few pages [here](https://archive.org/details/elementaryalgebr00hall) I get the impression that they are not using an axiomatic approach at all. Since they are not, they presumably find commutativity and associativity of addition and multiplication "obvious" (and in some sense it is). – 2017-02-14
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In the definition of an abstract algebraic structure (as a field), the properties of the operations in such structure are axioms, that are essential part of the definition.

If we want to prove that a set, with operations defined on its elements, is some algebraic structure (e.g. a field) we have to prove that the definition of the operation is such that the axioms are verified, and this means that we have to prove the required properties of the operations.