We recall some of the early occurrences of the notions of interactivity and symmetry in the operational and denotational semantics of programming languages. We suggest some connections with ludics.

In this talk I will introduce two spaces: the first space is the usual n-dimensional vector space with the unusual feature that n is a non-integer; the second space is composed of the linear matrices acting on the previous space (physicists are particularly interested in studying the limit as n goes to zero). These two spaces are not known to most mathematicians, but they are widely used by physicists. It is possible that, by extending the notion of space, they can become well defined mathematical objects.

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.