Integrals themselves are, except in very special cases (piecewise functions with rational values), only definable as limits - specifically the limit of the Riemann sum. (You can also use measure theory, but measures themselves are only definable on sigma algebras, which in the non-finite case are also not explicitly constructable.)

In what comparable way does truth (e.g. the true process underlying gene expression) have a role in statistics? We neither measure the truth nor model it; it is absent.

I don't quite understand. You are arguing that statistics doesn't care about truth simply because some biologists are using a model they know to be wrong? That doesn't even make sense.

In applied math in general (which includes but is not necessarily limited to statistics), the following equation holds:

    error = |true model - actual approximated model|

We can use the triangle inequality to show:

    error <= |true model - best model in class X| + |best model in class X - actual approximated model|

Presumably you all have decided that |true - best| is adequately small via scientific investigation. Or maybe not, maybe your workplace just doesn't care, I don't really know.

Various mathematical techniques, or increasing sample size in a statistical scenario, can be used to reduce |best - actual|. Due to the triangle inequality, this brings you closer to truth.

Statistics is also concerned with expanding class X in such a way as to more easily reduce the model error.

I really feel like I'm missing something, because I truly can't comprehend what you are trying to argue.