I'm reading David Steward's "Foundations of Mathematics" and in chapter 8 he is building an axiomatic system for the natural numbers with addition defined using Peano's Axioms. I don't really fully understand the need for this. Why can't I just define addition in the same way I learnt it in primary school - using an addition table - but define it for all natural numbers?
Thanks,
drd26
Elaborating on a comment. It's impossible to use that table as the definition. Because since the table is infinite you can't write it all down, so you have nowhere to look up the result of 3+17.
Really. definitions have to be finite. You can use a table to define the addition in $\Bbb Z_2$; that amounts to simply defining $$0+0=1+1=0, 0+1=1+0=1.$$Now if you have to add to elements of $\Bbb Z_2$ you can simply look up the result in that display.
But that doesn't work for the infinite table you'd need to add integers. No matter how large a finite part of the table you're given, there is some addition problem that it doesn't cover.
Hmm, look at it this way: If we haven't defined addition it's impossible for you to tell me exactly what "table" you're talking about! You can't write it all down for me, the best you can do is say "the table with $x+y$ in row $x$ and column $y$", and if you say that I won't have any idea what you're talking about, because addition is undefined.