"Truth": reflections on mathematics and G-d

In previous posts I've noted that

1. I'm an agnostic

... for me the truth-value of the proposition "God exists" is unknown and unknowable

2. I watch a LOT of TV evangelicals

... recently I've started watching Todd Friel's Wretched.
In my usual fashion, after watching a few episodes, I Googled "Todd Friel", and found a recording of Friel going one-on-one with "public intellectual", Iraq war apologist, and atheist Christopher Hitchens.
The basis of the "interview" is Friel's "What if...?" game:

Friel: “What if God exists, and what if he has provided everything for you… life, health, food, trees, royalties… would he not have been good to you?”

Anyway - Hitchens pretty much demolishes Friel.
[Note: I would likely take a different tack than Hitchens, thanking G-d for the gifts, but walking away. When someone gives me a gift, I thank 'em. I don't owe 'em anything in return - it was a gift!... and if the giver EXPECTS something in return, it wasn't a GIFT, it was a sales transaction to which I did not agree.
... Oh, and by the way: if G-d created me so that I could worship him, well - that makes him a rather vainglorious, petty deity!]... and I have a hard time not turning the tables on Friel:

“What if Allah - the deity who dictated the Qur'an to Mohammed - exists, and what if he has provided everything for you… life, health, food, trees, royalties… would he - Allah - not have been good to you?”

... anyway, that's not really the point of this note.
This note was prompted by one of Friel's other rhetorical tricks:

On this desk is Hinduism, Buddhism, Mormonism, Roman Catholicism... They all have one thing in common: they assert that your WORKS gain you salvation.

On this other desk is Biblical Christianity, which asserts that salvation is from the Grace of God alone.

They can't both be true.

Well - sure enough, Todd - they can't both be true.
BUT: they can all be FALSE!

From this simple observation doth this treatise flow.
[note: file this post under "mental masturbation"... I'm just having fun playing with myself.]

Mathematics is the most highly developed logical system devised by man.
From a very few primitives (axioms & rules of inference) a magnificent, self-consistent logical system follows.
I'll do my best to avoid the various takes on the 'meaning' of mathematics, and leave it at, "Mathematics is the most highly developed logical system devised by man."

One of the most famous propositions of elementary geometry is the Pythagorean Theorem:

The sum of the squares of the legs of right-triangle equals the square of the hypotenuse.

There are hundreds (thousands?) of proofs of this theorem.
It underlies tons of mathematics & statistics (statistical sums-of-squares decompositions in the Analysis of Variance are essentially Pythagorean).

So, is the Pythagorean Theorem true?
(dear readers: please pause for a moment to reflect on this question... then read on!)

Every proof of the Pythagorean Theorem depends on Euclid's infamous Parallel Postulate:

At most one line can be drawn through any point not on a given line parallel to the given line in a plane.

Note: this is the high school geometry version of the postulate. There are LOTS of equivalent formulations. Euclid's original is a bit opaque:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

Does anyone remember that the interior angles of a triangle sum to 180 degrees?
This, too, is equivalent to the Parallel Postulate.

Fun exercise: Get a piece of paper, a straight-edge, a pencil, and scissors.
Draw a triangle.
Now, cut out the triangle you just drew.
Tear off the three corners of the triangle.
Arrange these three torn-off corners side-by-side.
Whaddaya get? A straight line! - 180 degrees!!!

Euclid's formulation seemed so bizarre that for close to 2000 years mathematicians were convinced that this infamous Fifth Postulate could be derived from the first four.
Some convinced themselves they'd succeeded.
None did.

In the early-mid 19th century several daring souls took another tack: what happens if we DENY the Parallel Postulate?
Turns out, you get perfectly consistent geometries!
Two denials present themselves:

1. NO line can be drawn through any point not on a given line parallel to the given line.

2. MORE THAN one line can be drawn through any point not on a given line parallel to the given line.

Careful readers will note the omission of the qualifying condition, "... in the plane".

The first alternative (NO parallel exists) leads to spherical geometry - the geometry of the surface on which we in fact live!
The second alternative (more than one parallel exists) leads to hyperbolic geometry - which is amazingly difficult to visualize!

In spherical geometry (the geometry in which we in fact live - the surface of the earth), the Pythagorean Theorem is NOT true.
In hyperbolic geometry, the Pythagorean Theorem is NOT true.
The Pythagorean Theorem is true only in Euclidean, PLANE geometry - the geometry of flat surfaces (planes).

So, is The Pythagorean Theorem true?
If one accepts Euclid's Parallel Postulate, yes - it is.
If one denies Euclid's Parallel Postulate, no - it isn't.

I note that it is perfectly possible for the conscientious mathematician to prove theorems in Euclidean geometry today and in non-Euclidean geometry tomorrow, without damage to his conscience and without being simply absurd.

What then is one to make of the truth-value of the proposition, "God exists."???

Accepting it as an axiom doubtless entails some theology.
Denying it implies a very different understanding of the world.

Can it be proven from a priori principles (axioms/postulates)?
Well - just what might these be?
... and what are the rules of inference allowed?

I suspect that Todd Friel will be quite happy to answer me by waving his Bible and asserting,

"God said it, I believe it, that settles it!"

... without ever reflecting that many others wave their Bibles at me, make the same assertion, and come to completely different understandings of G-d's Word!