I learned math with the Peano axioms and we considered the symbol 2 to refer to the 1+1, 3 to (1+1)+1 and so on. However even if you consider it to be more complicated it still stays an analytic statement and isn’t a synthetic one.
If you define 2 differently what’s the definition of 2?
When you write “1+1” you may mean two things: “the result of doing the adding operation to 1 and 1“, and “the successor of 1”. It just happens that we use “+1” to denote both of those. The fact that successor(1) = add(1,1) isn’t completely trivial.
Principia Mathematica, though, takes a different line. IIRC, in PM “2” means something like “the property a set has when it has exactly two elements” (i.e., when it has an element a and an element b, and a=b is false, and for any element x we have either x=a or x=b) and similarly for “1” (with all sorts of complications because of the hierarchy of kinda-sorta-types PM uses to try to avoid Russell-style paradoxes). And “m+n” means something like “the property a set has when it it is the union of two disjoint subsets, one of which has m and the other of which has n”. Proving 1+1=2 is more cumbersome then. And PM begins from a very early point, devoting quite a lot of space to introducing propositional calculus and predicate calculus (in an early, somewhat clunky form).
I learned math with the Peano axioms and we considered the symbol
2to refer to the1+1, 3 to(1+1)+1and so on. However even if you consider it to be more complicated it still stays an analytic statement and isn’t a synthetic one.If you define 2 differently what’s the definition of 2?
When you write “1+1” you may mean two things: “the result of doing the adding operation to 1 and 1“, and “the successor of 1”. It just happens that we use “+1” to denote both of those. The fact that successor(1) = add(1,1) isn’t completely trivial.
Principia Mathematica, though, takes a different line. IIRC, in PM “2” means something like “the property a set has when it has exactly two elements” (i.e., when it has an element a and an element b, and a=b is false, and for any element x we have either x=a or x=b) and similarly for “1” (with all sorts of complications because of the hierarchy of kinda-sorta-types PM uses to try to avoid Russell-style paradoxes). And “m+n” means something like “the property a set has when it it is the union of two disjoint subsets, one of which has m and the other of which has n”. Proving 1+1=2 is more cumbersome then. And PM begins from a very early point, devoting quite a lot of space to introducing propositional calculus and predicate calculus (in an early, somewhat clunky form).
One popular definition (at least, among that small class of people who need to define 2) is { { }, { { } } }.
Another, less used nowadays, is { z : ∃x,y. x∈z ∧ y∈z ∧ x ≠ y ∧ ∀w∈z.(w=x ∨ w=y) }.
In surreal numbers, 2 is { { { | } | } | }.
In type theory and some fields of logic, 2 is usually defined as (λf.λx.f (f x)); essentially, the concept of doing something twice.