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Philosphy

Invariants are algebra's approximations

In the way that approximations in analysis can be improved to be closer to the object they are approximating (and limiting to), some algebraic invariants can also be improved. Invariants like Betti numbers, Bass numbers, and multiranks are (at least in the graded setting), graded in a way such that increasing the homological degrees (take all homological degrees below or at m for Betti and Bass numbers) or the level (take all levels at or below m for multiranks) of the invariants increases their specificity. There may not be infinitely many homological degrees or levels, and they may not limit to the desired value, but they still follow the format of an approximation. 

With perfect data, sensitivity isn't an issue for invariants; only specificity (essentially, the number of false positives) is. Without perfect data, stability becomes an important topic in determining sensitivity and specificity.