Lost innocence
Once there was a time in which Mathematics was an innocent subject. In order to produce a mathematical system, all you had to do was develop some concepts, add some operations on these concepts (a calculus) and top it off with a few self-evident axioms. Euclid was the first to do this systematically. He started off with points, straight lines, distances and angles, added a calculus in which lines intersect to create points and angles and points join to create lines, added some axioms which any farmer on a plain would agree with, and ended up with what is now known as Euclidean geometry. Mathematicians believed that this approach was sound. If you did everything correctly, you couldn't lie, and you could prove anything that was true.
This innocence began to crumble when Bertrand Russell demonstrated that some mathematical statements that appeared to be perfectly sound nevertheless contained contradictions. His paradox can be illustrated with the example of Jacques, the barber of a French village. Jacques shaves all men of the village who don't shave themselves, and none of those that do. The question is: does Jacques shave himself? If he does, he defies the rule by shaving somebody who shaves himself, namely Jacques. If he does not, he defies the rule by not shaving a man who does not shave himself, namely Jacques. In either case, there is a contradiction. Despite the very best efforts of generations of mathematicians, no generally satisfactory way has been found to avoid such contradictions.
The innocence was further undermined in 1934 by the Austrian mathematician, Kurt Gödel. In what is now known as Gödels proof, he demonstrated that any system of mathematics must contain statements that can be expressed in terms of the system but not proved (nor disproved) from within that system. It got even worse when it was shown that there are mathematical statements for which you can choose whether to regard them as true or false, and either way you get a consistent system. A case in point is the continuum hypothesis: the hypothesis that there is no set of numbers larger than the set of rational numbers but smaller than the set of real numbers.
To conclude, we now have a situation in which you can produce nonsense, choose for yourself what to believe, and will always fall short. In other words, mathematics is in a mess. Why have faith in mathematics if this is the case? How can mathematics be useful in understanding the universe?
QO accounts for how this mess has arisen and offers a way out of it. See Mathematics - innocence regained. This has an Occam score of 0000, relative to QO. Mainstream mathematics only observes that mathematics is in a mess, but does not explain why that is so. Implicitly, it speculates that an explanation that fills in the gap exists. Such an explanation requires multiple, interacting explanatory elements, and therefore has an Occam score of at least 0300.