[Math] What exactly is circular reasoning

Question

The way I used to be getting it was that circular reasoning occurs when a proof contains its thesis within its assumptions. Then, everything such a proof "proves" is that this particular statement entails itself; which is trivial since any statement entails itself.

But I witnessed a conversation that made me think I'm not getting this at all.

In short, Bob accused Alice of circular reasoning. But Alice responded in a way that perplexed me:

Of course my proof contains its thesis within its assumptions. Each and every proof must be based on axioms, which are assumptions that are not to be proved. Thus each set of axioms implicitly contains all theses that can be proven from this set of axioms. As we know, each theorem in mathematics and logic is little more than a tautology: so is mine.

Not sure what should I think? On the one hand, Alice's reasoning seems correct. I, at least, can't find any error there. On the other hand, this entails that… Every valid proof must be circular! Which is absurd.

What is a circular proof? And what is wrong with the reasoning above?

Answer 1

Of course my proof contains its thesis within its assumptions. Each and every proof must be based on axioms, which are assumptions that are not to be proved.

Hold it right there, Alice. These specific axioms are to be accepted without proof but nothing else is. For anything that is true that is not one of these axioms, the role of proof must be to demonstrate that such a truth can be derived from these axioms and how it would be so derived.

Thus each set of axioms implicite contains all thesis that can be proven from this set of axioms.

Implicit. But the role of a proof is to make the implicit explicit. I can claim that Fermat's last theorem is true. That is a true statement. But merely claiming it is not the same as a proof. I can claim the axioms of mathematics imply Fermat's last theorem and that would be true. But that's still not a proof. To prove it, I must demonstrate how the axioms imply it. And in doing so I can not base any of my demonstration implications upon the knowledge that I know it to be true.

As we know, each theorem in mathematics and logics is little more than a tautology:

That's not actually what a tautology is. But I'll assume you mean a true statement.

so is mine.

No one cares if your statement is true. We care if you can demonstrate how it is true. You did not do that.