The Practice Of Theory
“Pure” mathematics is a collection of philosophical constructs, and as such its most important feature is consistency with itself. Where it fails
to achieve this, it can be modified to bring it into self-harmony.
Resolving such anomalies can be accomplished entirely within the domain of mathematics, and this feature reveals mathematics as a product of our minds
constructed entirely of theory.
The fact that mathematics was originally derived from practical needs does not change the fact that it is something we invented for our use, rather
than something “found in nature” (or as I like to call it, “perceptual reality”). It is a system for interpreting and (ideally) predicting
natural phenomena, but we are its creators.
It is, in other words, a work consisting purely of intellectual art.
Applied or “practical”mathematics, however, must specifically contend with the differences between the domain of pure mathematics and the domain
of perceptual reality.
The inherent problem with applied mathematics lies in the fact that it is not necessarily founded on the same rules that perceptual reality is founded
on -- whatever those may be.
The rules for mathematics are known by definition, while the rules for perceptual reality are largely unknown. And while mathematics can be regulated
by the constraint of consistency, it is not clear that perceptual reality is governed by the same constraint.
Consequently, to the extent they disagree, applied mathematics will fail to accurately model perceptual reality.
Moreover, the extent of this disagreement is subject to unexpected and unexplained changes over time.
The Grand Irony
I think
Protector is grappling with the inconsistencies inherent to mathematics both as pure theory and as an applied science.
The Grand Irony is that despite the (admittedly arbitrary) differences I expressed above, the two are inextricably linked.
If a “pure” mathematical theory cannot be shown to correspond to some sort of natural phenomenon, it will tend to be discarded or modified.
What's more intriguing to me, however, is the urge -- so often indulged -- to redefine natural phenomena that do not agree with theory.
In other words, if the “real world” doesn't match the elegance of theory, it is not uncommon for scientists to want to “fudge” observations
to agree with theory. Of course, “good” scientists don't do this, but scientists are human.
This tendency to “fudge” observation can be summarized by the expression “That's impossible!” which signals the scientist's decision to
reject observations which don't reinforce presumption.
Scientists aren't the only ones who do this, however. It's a very human trait. Everyone habitually and frequently discards perceptions they don't
want to believe -- even those which take place “right before their very eyes”. The catch phrase: “I don't believe it!”
What results is
modification of perception to agree with theory. Sometimes the degree of modification is subtle, sometimes gross, but it is
always present to some extent.
We believe what we want to believe.
In theoretical physics, models are constantly being created, modified and discarded in attempts to explain observed phenomena.
The irony is that the same thing occurs in mathematics as well, and that's what I think you're pointing out.
Mathematics, like nature, is not a closed set.
Or I could be dead wrong.
... 
the harder it is to describe to the outside world. 
