Argument against atheism

[Perhaps]

Mathematics is a form of logic with it's own rules, it need not conform to the rules of outside systems (or other systems in the same overall family) except in cases where it is to be applied to those systems.

No, I'm not. False positives would be evidence. Reason and evidence have their own words and definitions for a reason..hehehe, zing.

We trashed many of those lines of reasoning by applying them to situations in which evidence could be leveraged. Whittling down a system on the basis of certain parts of the system not conforming to its own rules is fine. To say, "this is illogical because it does not conform to the rules of logic" is one thing. To say that "this is illogical because evidence we have found in the material world annihilates this or that assertion" is an entirely different thing. It's a difference that should be appreciated.

I think what we're missing in this discussion is the understanding of how evidence is gathered or obtained. What is considered evidence? Are there criteria? If so, how did that criteria originate?

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A fun example of this is the
"All swans are white"
"Dave is a swan"
"Therefore Dave is white"

That works, there's no tearing that down with logic alone. What you'd need to do is provide evidence that dave is not a swan, that all swans are not white. Even so, all swans may not be white, and Dave might still be a white swan. Or perhaps All swans are white, but dave is not a swan, and still dave is white. Evidence, evidence, evidence.

Once again, what is evidence? How is it gathered? Even the idea that the application of evidence is a method through which things can be confirmed or denied is an assumption. A useful assumption as it relates to the world as we know, but an assumption none the less.

All systems have limitations.

Correct. Which are based on the original assumption.