I apologize if a similar question has been asked here, but I haven't found it.
Are mathematical statements necessary truths?
By 'mathematical statements', I mean both mathematical axioms as well as statements such as 2+2=4, but I'm not referring to statements such as 2+2 alone.
Truth is a property of propositions. In the simple case of propositonal logic this means that any proposition is either true or false.
Either it rains or it does not rain, at a fixed date and time at a fixed point of Manhattan.
The most widespread definition of a proposition being true is due to Tarski: A proposition is true if and only if the actual situation, which the proposition is about, is a matter of fact. The proposition
It rains in Manhattan at a fixed date and time and at a fixed point
is true if and only if it actually rains. Otherwise the proposition is false. Hence Tarski's truth definition compares a proposition with a real situation.
Because mathematics does not deal with reality - it is not a natural science like physics - one cannot apply Tarski's definition to mathematical statements. Instead, one has two related, but different concepts:
A mathematical statement of an axiomatized theory is provable iff and only it derives by a logical conclusion, i.e. a syllogism, from the axioms of the theory.
A mathematical statement is true if and only if it holds in any model of the axioms.
Any provable statement is true. The converse holds in simple theories like propositional calculus. But it does not hold in more general mathematical theories.
Taking your question literally, the answer is NO: There exist false mathematical statements by trivial reasons, e.g. There is a maximal prime number is a false mathematical statement.
There exist mathematical statements, the truth of which is undecidable, e.g. the continuum hypothesis is undecidable within ZF set theory.
There exist true mathematical statements. They are true in all possible worlds where our logic is valid, which means necessarily true.
This depends upon your starting point.
Generally speaking, they are as close to truth by neccessity as we generally find them.
Kant, however, advised caution. He stated in his Critique of Pure Reason that there was no necessary reason why the angles in Euclidean geometry must add up to 180 degrees. And lo and behold within the next few decades Gauss, Lobachevsky and Bolyai had invented non-Euclidean geometry. Gauss had read the critique, so it's quite possible that the possibility that Kant opened up prompted him to his discovery.
If you are a Spinozist, then everything other than Absolute Being is contingent and this includes logic and mathematics.
I'll keep it short as it is rather difficult to answer the question in all his aspects. Take your example: 2+2=4. Why and when should you say that 2+2=4 is true? Using axioms of the natural numbers you has to end with the number four, hence being 'necessarily true' means here that you have applied the axioma's in a correct way to get the a priori known answer. And if 2+2=4 stands for this whole process of counting then one can say that 2+2=4 is true. I don't. I have given you 2 apples and shortly after that again 2 appels, hence I have given you four apples. So it is necessarily true that I have given you four apples? What is meant by saying this? Suppose you only know the axioms and only the numbers 0 and 1. How would you get 2+2=4?
The only even number that is prime is the number two, and all other prime numbers are odd.
It is true because an evil demon, as Descartes proposed, could not make me go wrong in believing this incorrect.
It's, dare I say, categorically true.
