Without a clear answer to the infinite regress problem, can we justify anything without resorting to 'a priori' truths? And if not, how is there a reliable standard for testing the validity of a priori reasoning?
Agrippa's Trilemma demonstrates that all systems must ultimately be grounded in one of three way: infinite regress, circular reference, or unsupported axioms. So, ultimately, you are going to have to resort to one of those options.
Reliability, then, is generally redefined appropriately.
To give a more detailed answer would require reference to a specific philosopher or philosophical tradition.
One attitude we might adopt, assuming a negative attitude to the first question, is that while justification for a posteriori truths depends on a priori truths, the epistemology for a priori propositions has more of a coherentist flair to it - we are justified in our use of mathematics and logic because a given mathematical theory or logic is a self-enclosed system, and these systems have different epistemic virtues depending on how their propositions and objects "connect together" rather then how they depend on things like observable phenomena. Reasoning about a posteriori propositions could then safely partially depend on this coherent base.
David Hilbert thought that this might be a good way to account for mathematical truths. Hilbert wanted to give, among other things, a stronger and more logically rigorous axiomatic account for Euclidian geometry, given the emergence of new kinds of geometric space and questions over what geometric systems were, and whether they could be considered correct or usefully applied. Hilbert's observation was that a certain amount of normativity, even in supposedly a priori mathematical systems of geometry, could be found in requiring a proof of the consistency of the set of axioms which give rise to the theories in question. This consistency would be enough, he thought, to identify content for mathematicians to study without skeptically worrying about whether some result was going to come along and render the whole enterprise a waste of time.
The key positive contribution in Hilbert's work, motivated by this view of the normativity of mathematical claims, was how he proved the logical independence of the various axioms in his geometric system from one another. Once we accept that consistency is all we require, we can appeal to two forms of mathematical reasoning that might otherwise be in question:
- We can safely define models that satisfy our consistent collections of axioms. This means that we can show that there are collections of objects in some proven consistent field of maths, in which all of a set of initial axioms are true, but where extending ones are not, and others where all of the axioms can be consistently true. In this way, we show that the extending axioms are independent of the initial axioms. This was how the "new" geometries could be interpreted without undermining the value of the "old" geometries.
- We can be non-constructive and use logical deductions like proof by contradiction to determine the truth of existence claims without requiring that we actually be able to produce the mathematical object that makes the claim true.
Eventually, Godel's Incompleteness proofs showed that no complete axiomatic system can prove its own consistency. This is something of a problem given the role that consistency proofs play in Hilbert's formal understanding, because of the mathematical requirements of proof theory itself. In order to prove the consistency of, say, Peano-Dedekind arithmetic, either we need to prove it in a stronger theory than arithmetic or the theory itself must be understood to be either incomplete or inconsistent (even if it proves its own consistency, it could still do so inconsistently).
Hilbert's project is generally thought to have been stopped in its tracks, but the questions it has generated about how we might make logic and maths secure continue to be relevant. Current study in the philosophy of logic tends to focus around how theories about models or proofs can be motivated and develop, whether by appeal to an a priori Kantian intuition, through complex language or symbol games, through human psychology or through some relation between mathematical practice and its role in scientific theorising.
No, we cannot do away with a priori postulates. Please take note that we call them postulates, not truths.
Empirical science relies on the principle of induction by simple enumeration, but the principle itself is simply assumed. It cannot be proved empirically.
Bertrand Russell suggested that "the postulates required to validate scientific method may be reduced to five." They are:
a. The postulate of quasi-permanence.
b. The postulate of separable causal lines.
c. The postulate of spatio-temporal continuity in causal lines.
d. The postulate of the common causal origin of similar structures ranged about a center, or, more simply, the structural postulate.
e. The postulate of analogy.
Source: Russell, Bertrand. Human Knowledge, Its Scope and Limits. New York: Simon and Schuster, 1948