Everything, he says -- our universe, and the many other universes with which it coexists -- are not only explainable mathematically but are in fact mathematical structures.
There are two versions of mathematical structuralism: a realist view according to which mathematical structures exist independently of their concrete instantiations; and an eliminativist position according to which statements about mathematical structures are disguised generalisations about their instantiations that exemplify them (see Shapiro 1997, 149 50).
I believe that this distinction between the beauty of mathematical structures and the usefulness of mathematics to model physical and social phenomena is a false dichotomy.
When interpreted physically, these mathematical structures or constraints correspond to physical structural fields (Strukturfelder).
The Boussinesq equation also is of considerable mathematic interests because of its rich mathematical structures.
In general, according to Platonists, mathematics is the study of the nature of various mathematical structures, which are abstract in nature.
But rather than supporting the existence of mathematical objects, the argument would only support commitment to modal-structural translations of mathematical theories and the possibility of mathematical structures.
It may be argued that the modal structural account does not require an isomorphism between (applied) mathematical structures and those describing the material situation.
On the modal structural conception, mathematics is applied by establishing an appropriate isomorphism between (parts of) mathematical structures and those structures that represent the material situation.
Take fractals, mathematical structures or sets with intermediate dimensionality.
His grasp of biological theory was extremely firm, and all his major works draw on mathematical structures.
