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In mathematics if one is to prove a property of arithmetic, such as the associativity of addition, without going into greater detail about the numbers themselves, I feel like I'm missing something important.

Should one first prove what a number is, and how does one accomplish such?

For example the property of associativity of addition states that

$$a+b+c = (a+b)+c = a+(b+c)$$

But if those numbers are found in a number line or in a ring, then what proves that at some point those three sentences are equal to $$a+b+c = d$$

Maybe I'm thinking too hard about it, and I don't really know anything about rings, which might be part of the answer to my question. How do you prove what a number is?

nathandelane
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  • For the usually agreed basics on "what a number is" you could start with an internet search for "Peano axioms". – David Feb 10 '14 at 23:00
  • After posting this I found a Wikipedia page about this, and I see that they use zero as a base vase in$$(a+b)+0=a+b=a+(b+0)$$ and I don't know that this helps - it also seems like used induction is a form of assumption about natural numbers - maybe I'm just overthinking this. – nathandelane Feb 10 '14 at 23:01
  • Thanks David, I think that helps answer my question - I was curious about whether Peano axioms were the thing I was looking for, and upon reading more I think that it is. – nathandelane Feb 10 '14 at 23:05
  • Induction is one of Peano's assumptions, it's number 9 in the Wikipedia version. This might help too. – David Feb 10 '14 at 23:06

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Your question is completely clear.

In general you don't prove what a number is, you construct them. The construction will often be such that the various laws are satisfied.

I could, for example, first define the set of $2\times 2$ real matrices, and two rows and two columns of numbers that I write like this $$ \pmatrix{a & b \\ c& d} $$ The I can defined addition by $$ \pmatrix{a_1 & b_1 \\ c_1& d_1} + \pmatrix{a_2 & b_2 \\ c_2 & d_2} = \pmatrix{a_1 + a_2 & b_1 + b_2 \\ c_1 + c_2 & d_1 + d_2}. $$ Here the associativity (and commutativity) of the addition is clear since the real numbers have this property. Likewise, I can define multiplication the usual way and it is not hard to verify that the axioms for a ring are satisfied.

Now you might then ask: how do you we prove/know that the real numbers are a ring with this associative property?

For this you could consider the construction of the real numbers as Cauchy sequences of rational numbers. If you take a look at teh Wikipedia article, it again becomes clear that the associativity for the real numbers follows from the associativity of the rational numbers. And if you look at the construction/definition of the rational numbers, the associativity is clear from the associativity of the integers. So in this way we have reduced the question to one about the integers. If you look at this Wikipedia article this boils down to considering the natural numbers.

So how do we construct the natural numbers you ask. One way is to essentially define a natural number as a set. $0$ is the empty set. $1$ is the set with one element being the empty set and so on. You might want to take a look at this question with answers: How is addition defined? where the associativity is proved!

Community
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Thomas
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  • This idea of construction seems clear to me, but since construction depends on properties of operations, isn't a constructed integer dependent on a property which is dependent on an integer? That seems like a recursive definition, like for $$1$$ to be constructed you could say $$5 - 4$$ but that depends on the idea that four and five are equally distant values as one. I think your saying that associativity and commutativity of addition have a property of resulting in integers, while integers in addition have the properties of commutativity and associativity? But it must start somewhere. – nathandelane Feb 10 '14 at 23:56
  • @nathandelane: So the idea is that part of the construction is the definition of addition. Does that make sense? – Thomas Feb 10 '14 at 23:59
  • I think that makes sense. – nathandelane Feb 11 '14 at 00:03
  • Thomas' point-and it's really one of the hardest to get students just learning how to carefully prove things understand-is that sometimes things are true in mathematics because that's how we defined them. That was one of the critical insights the early pioneers of axiomatic thinking had-they realized the reason so many mathematicians had failed for centuries to prove the parallel postulate was because it was simply a derived property of the axions of Euclidean geometry. In other words,it couldn't be false without the entire system being false. Hence non-Euclidean geometry. – Mathemagician1234 Mar 21 '14 at 22:51