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Ultimately truth is determined by physical consequences. We want
to know what will happen if we make particular choices. A system
that helps us to know this is to at least some degree true. One
that misleads us is false. Abstract mathematics was developed
because it helped us make predictions. But mathematical truth seems
to be of a special absolute nature. 
is not so
much a statement about the world as it is a statement about
language. It follows from the definition of the terms. As long as
mathematics is talking about finite objects it can be regarded as
as stating what is true by definition.
Mathematical assertions that refer to all integers are useful. Each of the individual events such statements refer to is finite but the collection of all of them is infinite. Such assertions still seem to be true by definition. However we cannot be so certain about these statements. Their truth is dependent on laws of induction. There is little doubt that first order induction is valid. But we know from Gödel's Incompleteness Theorem that there will always be valid laws of induction that are outside of any formal mathematical system.
Logical truths about finite objects refer to something that can exist physically. Logical truths about all integers do not refer to any single object that could exist physically at least as far as we know. They are a creation of the human mind. However each of the events they refer to is a finite thing that can exist physically there is an objective basis for the truth of these created statements.
Existing mathematics in the form of Zermelo Frankel Set Theory provides powerful principles of induction by treating infinite sets in the same way it treats finite sets. Philosophically mathematicians have a variety of beliefs about the nature of these completed infinite totalities. Practically few would like to give up the enormous mathematical power that is implicit in this formalization of mathematics.
A problem with this approach to mathematics is the disconnect that exists between the higher axioms of mathematics and physical reality. It is our intuition about physical reality that is the starting point for mathematics. When the axioms become disconnected from reality how are we to develop our intuition about them?
The philosophical approach we are proposing keeps much of the power of contemporary mathematics intact and retains a connection with physical reality. Our definition of meaningful mathematical truth is imprecise. The problem is the phrase ``determined by a recursively enumerable set of events.'' Deciding which statements meet that criteria is a creative act that ultimately will run up against the limitations of Gödel's Incompleteness Theorem. We discuss that in Section 7.
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