Comment by tsimionescu

> Physical world is grainy and probabilistic at quantum scale and smooth amd deterministic at larger scale.

This is almost entirely backwards. Quantum Mechanics is not only fully deterministic, but even linear (in the sense of linear differential equations) - so there isn't even the problem of chaos in QM systems. QFT maintains this fundamental property. It's only the measurement, the interaction of particles with large scale objects, that is probabilistic.

And there is no dilemma - mathematics is a framework in which any of the things you mentioned can be modeled. We have mathematics that can model both deterministic and nondeterministic worlds. But the mathematical reasoning itself is always deterministic.