The special case of quantum mechanics is represented. It mathematised reality in the tradition of physics. However it was forced to involve infinity in its model in a way radically different from that of classical physics. No finite model can unify quantum and smooth motions, which is the main problem of quantum mechanics. Thus it was forced to make the step of mathematizing an external reality independent of any observer to mathematical reality. The boundary between mathematics and physics, between a model and reality was crossed: This can be designated as mathematizing ontology rather than a science.
The thesis is:
Quantum mechanics can be described both as mathematizing an external reality, which is not mathematical by itself, and as studding mathematical reality. Thus quantum mechanics occupies a unique place among all sciences including the physical and mathematical ones in relation to the mathematizing of sciences.
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