DUANE, you wrote,

"Jacob, surely you're willing to consider that there may be truths which exist outside of your particular philosophy...."

D-SP considers THE TRUE (as referred to by Plato, and from which Truths derive) to be of three kinds:

The Theological, called Dogmas
The Scientifical, called Axioms
The Philosophical, which D-SP calls Natural Truths, or The Three Laws of Being.

My "particular" philosophy, as embodied in D-SP essays and dialogues, has not touched yet on ALL particular truths, although ALL are within its frame of reference. Being there phenomena that I can not understand, yet I accept as true, they are not considered within the province of D-SP. Therefore, I limit myself to describe them. For that purpose, I use my site http://www.delphi.com/ghitis in the Folder called Madam-X Files. For the time being there are two files.

All that I post as D-SP I understand as being facts, or very educated and learned postulates that deserve to be made known.

"...Different branches of Mathematics exist because there's not one single system of axioms, rules, and theorems which can express all mathematical truths..."

Different branches of every discipline exist because those so-called 'branches' are disciplines by themselves. They are placed, for reasons of taxonomic convenience, under a Super-discipline. Each branch of the Super-discipline called Mathematics has its own axioms, being a discipline on its own merit.

As for "mathematical truths," they might be defined as being the formulas that consider all the possible aspects of the Laws of Physics. Not all of the discoverable formulas have been discovered by H. sapiens yet.

"...In fact, I contend that Godel's Theorem applies not merely to set theory, but to all philosophical systems as well--viz., all systems of thought are either consistent and incomplete, or complete and inconsistent."

Duane, of Godel's theorem I scarcely know what I have read from commentators. I have written that my knowledge of Philosophy has followed a similar pattern. I simply consider that I enjoy a capability of grasping essentialities. I have written about intuition, in educated biochemically- oriented language. The public has no interest in learning about Godel and his Theorem, which caused a shock to the esoteric world of mathematicians when first postulated. Still, it is my impression that I can offer graspable ideas, which can explain the conundrum of H. sapiens, us, not being in a position to explain certain transcendental realities, as follows.

D-SP states that:

1. Everything is the result of the Laws of Physics acting on a physical Universe.
2. Homo sapiens is one of the results of those Laws.
3. Mathematics is the core of Physics.

I have written that we can not know the origin of the Universe because we are inside it. We would have to study it from the outside. Those ideas of mine developed gradually, and are published in three mini-essays on The Umbilicus, the third of them also posted in the Files.

Now, Mathematics --in the guise of the Laws of Physics-- surrounds us, we are inside it, because we are one of its products. We would have to position ourselves in an external vantage point allowing us to see, understand and PROVE all the mathematical formulas possible. Only then would we be able to see them "complete and consistent."