Foundations and Applications depend ultimately for their existence
on each other.
The main links between them are education and the axiomatic method.
Those links can
be strengthened with the help of a categorical method which was concentrated
forty years ago by Cartier, Grothendieck, Isbell, Kan, and Yoneda.
I extended that method
to extract some essential features of the category of categories in 1965,
and I apply it here in section 3 to sketch a similar foundation within the
smooth categories which provide the setting for the mathematics of change.
The possibility that other methods may be needed to clarify a
contradiction introduced by Cantor,
now embedded in mathematical practice, is discussed in section 5.