More on not being able to find π, as I'm piecing it together: given only the field structure, you can't construct an equation identifying π or even narrowing it down, because if π is the only free variable then it will work out to finding roots of a polynomial (you only have field operations!) and π is transcendental so that polynomial can only be 0 (if you're allowed to use not-equals instead of equals, of course you can specify that π isn't in various sets of algebraic numbers). With other free variables, because the field's algebraically closed, you can fix π to whatever transcendental you like and still solve for the remaining variables. So it's something like, the rationals plus a continuum's worth of arbitrary field extensions? Not terribly surprising that all instances of this are isomorphic as fields but it's starting to feel about as useful as claiming the real numbers are "up to set isomorphism, the unique set whose cardinality matches the power set of the natural numbers", like, of course it's got automorphisms, you didn't finish defining it.

You need some notion of order or of metric structure if you want to talk about numbers being "close" enough to π. This is related to the property of completeness for the real numbers, which is rather important. Ultimately, the real numbers are also a rigorously defined abstraction for the common notion of approximating some extant but perhaps not fully known quantity.