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October 24

Commutative, Associative, and Distributive Properties

Do you have to prove the Commutative, Associative, and Distributive Properties for addition and multiplication of natural numbers? Do you have to prove that a + 0 = a, that a - a = 0, that a*1 = a or a*0 = 0?

Or is it the other way round, we define the natural numbers as the numbers which have those properties?

Couldn't we just do the same (=define) for -(a + a) = -(a) + -(b) or that -(a*b) = -(a)*b, and not bother about proving them?

That is, at the basis of everything algebra, is it all a question of definitions of axioms vs proofs, but with the intent of keeping the number of definitions to the lower number possible? Bumptump (talk) 21:19, 24 October 2022 (UTC)[reply]

It depends on what you take to be axioms. You can take these properties as axioms, or you can start with simpler assumptions, define things like addition and multiplication from more basic ideas, and prove these properties. A good place to start is Edmund Landau's Foundations of Analysis. You can go even deeper though and define natural numbers in terms of sets. But it turns out the going gets more difficult when you try to define everything from scratch. --RDBury (talk) 03:07, 25 October 2022 (UTC)[reply]

A well-known approach is to start with the Peano axioms and use these to define the operations; see the section § Defining arithmetic operations and relations. Then

a + 0 = a

and

a \cdot 0 = 0

by definition. But

a \cdot 1 = a

needs a proof, which is given in the section linked to. Subtraction is trickier, because integer subtraction is a partial operation when restricted to the natural numbers. Instead one can use "truncated subtraction", commonly denoted by the symbol ∸ . Then

a ∸ a = 0

is a theorem that is easily proved by natural induction. --Lambiam 06:51, 25 October 2022 (UTC)[reply]

One of the issues with getting down to the foundational aspects of Mathematics is the turtles all the way down nature of mathematical proof. At some point one needs to start with an unproven statement, and then to expand from that statement to build the rest of the system. These unproven statements are called axioms, and choosing which axioms one accepts is not a trivial thing. The most famous (or perhaps infamous) attempt to make the minimum possible assumptions and still build a consistent and cohesive system of mathematics from the ground up was the Principia Mathematica of Whitehead and Russell, where in one part they spend 379 pages of dense symbolic logic not quite proving that 1+1=2. This is not trivial stuff. --Jayron 32 12:49, 28 October 2022 (UTC)[reply]