Fairy Tail(s) or Tale(s) ?

In the current debate about Evolution versus Intelligent Design…let’s not forget to do the math. Whether pixie-like, or playfully mischievous, or pixy-led, and bewildered...let's not forget that like human beings, fairys live in a universe of contradictions.

Farey's occur all over the place in fractal & chaos theory; they're kind of like an eye in the center of the storm. Not to be one to generalize, but the generalization for fractions are "Farey Numbers."

Farey Numbers are ordinary fractions, but are endowed with a funny addition:
a/b + c/d == (a+c) / (b+d)

Note that the following farey sequence has a fibonacci number, which is always an integer, in the numerator & denominator.
e.g. 0/1 + 1/1 = 1/2
0/1 + 1/2 = 1/3
1/2 + 1/3 = 2/5 etc.
(although most farey's do not have fibonacci's in them -- e.g.
0/1 + 1/3 = 1/4, etc.)

The farey sequences of orders 1 to 8 are :
F1 = {0⁄1, 1⁄1}
F2 = {0⁄1, 1⁄2, 1⁄1}
F3 = {0⁄1, 1⁄3, 1⁄2, 2⁄3, 1⁄1}
F4 = {0⁄1, 1⁄4, 1⁄3, 1⁄2, 2⁄3, 3⁄4, 1⁄1}
F5 = {0⁄1, 1⁄5, 1⁄4, 1⁄3, 2⁄5, 1⁄2, 3⁄5, 2⁄3, 3⁄4, 4⁄5, 1⁄1}
F6 = {0⁄1, 1⁄6, 1⁄5, 1⁄4, 1⁄3, 2⁄5, 1⁄2, 3⁄5, 2⁄3, 3⁄4, 4⁄5, 5⁄6, 1⁄1}
F7 = {0⁄1, 1⁄7, 1⁄6, 1⁄5, 1⁄4, 2⁄7, 1⁄3, 2⁄5, 3⁄7, 1⁄2, 4⁄7, 3⁄5, 2⁄3, 5⁄7, 3⁄4, 4⁄5, 5⁄6, 6⁄7, 1⁄1}
F8 = {0⁄1, 1⁄8, 1⁄7, 1⁄6, 1⁄5, 1⁄4, 2⁄7, 1⁄3, 3⁄8, 2⁄5, 3⁄7, 1⁄2, 4⁄7, 3⁄5, 5⁄8, 2⁄3, 5⁄7, 3⁄4, 4⁄5, 5⁄6, 6⁄7, 7⁄8, 1⁄1}

For every rational p/q, there is a corresponding farey; you can use the above to fill out the mapping from rational's to farey's.

The mapping for farey number to reals is bizzarre: its infinitely differentiable, its derivatives are all zero at all rational numbers -- i.e. its infinitely flat at all rational numbers. But its not a straight line; its a bumpy curve that is increasing ...

Thus, brought back to this broggers topical question tangentially concerning the current theological versus scientific brouhaha...is mathematics an accident, or was math cleverly designed by an omnipotent God?"

We will never know, since knowing would probably violate Kurt Gödel's (1906-1978) theorem, which states that some things can be true without being provably true!

But, let's not forget that "... we have learned from much experience that all philosophical intuitions about what nature is going to do fail." -- Richard Feynman