Circular reasoning is when two statements depend on each other in order to be true (consistent with the model).
- The inside angles of a triangles are supplementary
- The co-interior angles of a line intersecting parallel lines are supplementary
Both statements can be used to prove the other but neither of the above statements can be proven from the existing set of axioms in Geometry unless one of them is assumed as true without proof (an axiom). This "straightens" the argument so as to begin at this assumption. This is not fallacious at all unless it produces a contradiction with existing assumptions. It just limits or rather refines the set of statements that can follow as a logical consequences of the additional assumption.
Alice's Argument that Mathematical Proofs are Circular can be understood as:
To prove any one of these two statements will require the other one. This is circular. To "straighten" it we must assume one of the two to be true and the other can follow by the rules of inference/logic.
Is Mathematics then a tautology? If you define a tautology as a system of statements that is true by necessity or by virtue of its logical form, then I think it unashamedly is, and has to be based on the rules of inference/logic and the assumption just made: Each and every proof must be based on axioms or restated: All theses must follow by the rules of inference (proof) from the set of axioms.
Is Alice's argument valid that, in form, it is equivalent to that of axiomatic system of Mathematics? Is it consistent with the model produced by the assumption: all theses must follow by the rules of inference (proof) from the set of axioms.
Proof by contradiction:
Assumed it is true that a proof contains its thesis within its assumptions.
1) The thesis is an existing axiom (assumption without proof).
If the thesis is an existing axiom, then of course her model is equivalent. A leads to A.
2) The thesis is not an existing axiom (assumption without proof).
If the thesis is not an existing axiom but it is an axiom because it is assumed without a proof, then is leads to a logical contradiction.
Which means her proof is only valid if her thesis is indeed an existing axiom.
So if her argument is indeed circular, then her solution is to admit that one of the statements that she is making must be is an necessary assumption to develop the model/argument/worldview etc. Then it could be equivalent to the axiomatic system of Mathematics if it does not contradict any existing assumptions/axioms.