I am appropriately corrected. This actually doesn't change my view at all, because when I said "moderately complex," I assumed that the natural numbers where included in that. If you go back to read Descartes and Kant you'll see much of the same treatment - the addition operation as defined by our algebra of the set of the natural numbers is used many times as an example of a priori knowledge.
You are completely correct though - it was very late when I first commented on this and was tired and simply wrong. I'll try and be much more specific in my treatment of mathematics in the future, although I am not a mathematician. I just want to note that I never claimed that Gödel proved that Peano arithmetic was incomplete or inconsistent (although if I remember correctly, he could not prove that the whole of PA was consistent), but simply that the nature of Peano arithmetic does not imbue any a priori knowledge of the universe or our existence. This is supported by Gödel in the broad sense that we cannot generate a "universal theory" of mathematics. However, my main point is that mathematics is not truth, it is only a model of our definitions and observations -- a tool, if you will -- and an incomplete model at that.
I didn't think that your view would be changed. Quite honestly I'm not sure I understand the philosophy behind this. But the fact that there is only one model to the second order axioms of arithmetic (Peano's axioms with induction included) is a bit surprising to me.
We can't come up with a computable system for finding all mathematical truth but there appears to be a hardwired number system in the universe. It's not computable but it is unique. The natural numbers lead naturally (no pun intended) to the integers in a unique way. The integers uniquely lead to the rationals and the completion of the rationals is a unique object called the real numbers. The unique algebraic closure of the reals is the complex numbers. There is uniqueness at each step. This coupled with the utility of using mathematics to describe natural processes is...strange to me and some others.
I don't know what this has to do with your points because I didn't understand them. Not because you didn't write clearly but because I don't know enough philosophy. I'm a mathematician and know very little about philosophy.
You are completely correct though - it was very late when I first commented on this and was tired and simply wrong. I'll try and be much more specific in my treatment of mathematics in the future, although I am not a mathematician. I just want to note that I never claimed that Gödel proved that Peano arithmetic was incomplete or inconsistent (although if I remember correctly, he could not prove that the whole of PA was consistent), but simply that the nature of Peano arithmetic does not imbue any a priori knowledge of the universe or our existence. This is supported by Gödel in the broad sense that we cannot generate a "universal theory" of mathematics. However, my main point is that mathematics is not truth, it is only a model of our definitions and observations -- a tool, if you will -- and an incomplete model at that.