Some theoretical mathematicians go so far as to swear allegiance to G.H. Hardy's toast to pure mathematics: "May it never find an application!" And, some, like Plato, believe their a priori deductions reveal the metaphysical organization of reality.
Often, the prediction gap between so-called pure theory and pragmatic result is filled by empirically derived constants. For example, the ideal gas law, a synthesis of Boyle's Law, Charles's Law, and Avogadro's Law, is
where P is pressure, V the volume[/B], n the quantity of gas, R the gas constant, and T the absolute temperature. The ideal gas equation works perfectly for an ideal gas--unfortunately, there are not too many ideal gasses available for testing. In the lab, the technician makes "adjustments" for the particular gas under examination.
PV = nRT
When science is looked at in this manner, most of the entities talked about in scientific theory do not exist: e.g., frictionless planes, freely falling bodies, black bodies that absorb all radiation, rigid bars, black holes, and so on.
Mathematics in epistemology and metaphysics leads a kind of double life: (1) organization of phenomena in nature is selected by our perceptual processes and (2) mathematics is created a priori which coincidentally describes the organization of phenomena.
The lessons suggested here seem to me to be exemplified by the recent disappointment of string theory in fundamental physics. Peter Woit's book, Not Even Wrong describes how string theory has become a physics disaster. Sharon Begeley writes in the Wall Street Journal;
Sharon Begley, "Has String Theory Tied Up Better Ideas in Physics," Wall St. Journal (June 23, 2006).)
String theory, which took off in 1984, posits that elementary particles such as electrons are not points, as standard physics had it. They are, instead, vibrations of one-dimensional strings 1/100 billion billionth the size of an atomic nucleus. Different vibrations supposedly produce all the subatomic particles from quarks to gluons. Oh, and strings exist in a space of 10, or maybe 11, dimensions. No one knows exactly what or where the extra dimensions are, but assuming their existence makes the math work.
Problems include the fact that with so many dimensions involved, the equations have such multifarious solutions, verifiability or testability is applicable to many different possible worlds. In a word, string theory, at this point, has become a fantastic excursion into abstract numerology.