Thesis: All knowledge is synthetic. There is no such thing as an "obvious" truth, and both mathematics and logic are empirical sciences. Every "axiom" is open to question and the axiomatization of arithmetic was nothing more than an attempt to find a "spanning set" of mathematical statements which are logically independent. Which is to say the logicist programme was flawed in ways much more fundamental than just being limited by incompleteness; it was doomed as soon as Frege mocked Mill.

Problem: Logic is a means of specifying the object under consideration (assuming everyone knows what I'm talking about since this is discussed in one of the sequence articles). Throwing out an axiom, say by defining a non-commutative form of addition, is changing the subject. We cannot make any assertion about arithmetic (and hence cannot make any discoveries about arithmetic) without first logically circumscribing the subject. Perceiving a given quantity of object in the visual field relies on a contingent definition of the unit. Synthetic knowledge of arithmetic seems impossible.

Proposed resolution: Weaken definition of knowledge from "justified true belief" simply to "true belief". Some might want to haggle over what "justification" is and object that we shouldn't throw out the whole concept because both logicism and empiricism seem to fail to provide it infallibly. Mathematical knowledge is then attained only insofar as we guess correctly with respect to the independence of our axioms.

Interested in hearing what would be regarded as "standard" objections to the above (which is not to say I'm disinterested in original objections; just believe in respecting others who have worked on a problem by learning what they've done).