But still we can’t claim to be “absolute” about anything for the reasons I set out. We can have a qualified “absolute” at best: “X is absolutely true but only inasmuch as human minds are capable of understanding such matters”, but that’s all.
Since the system is entirely human-made, the idea that we don't understand it is somewhat bizarre. The logic that goes from the axioms to the conclusion is absolutely true because it is pure logic. Unless you reject logic itself, we have an absolute proof.
Yes, but assumptions can be wrong – and so therefore can the conclusions they lead to be wrong.
But mathematical statements exist within a context that may or may not be true. That context is that 2+2 actually = 4, which assumes that you’re not a SIMS-style algorithm programmed to think that 2+2=4 when a reality of which we’re oblivious says otherwise.
This is potentially more interesting. In fact, 2 + 2 = 4 is true
by definition. It's truth comes simply from the way we've defined '2', '4', '+', and '='. We've just labelled quantities and the process of addition. Take the quantity of rocks we call 'two', put two more next to them, a process we call 'addition', and you've now got the quantity we call 'four'.
In fact, mathematics goes back and defines the numbers in terms of set theory, so as to make it more formal. My signature (the axiom of infinity from set theory) can be considered as a construction of the natural numbers from nothing but the empty set, but I digress.
What's interesting is that, unlike geometry, we
can't fully axiomatise all the true theorems of arithmetic (Gödel's incompleteness theorems).
The statement “if the axioms of plane geometry are true then the angles of a triangle add up to 180 degrees” still doesn’t get you to the internal angles of a triangle adding up to 180 degrees being an absolute truth therefore. It’s “absolutely” true only within the confines of its own conditions and of our ability to understand such things, but it's not necessarily universally true too.
We actually
know that Euclidean geometry is a special case, that
doesn't, in general, apply to the real world. However, the proof is still absolutely correct.