An intuitive perception of science is the concept of the Arrow of Time. Biological entities do not seem to get younger, and are evolving in an irreversible manner. On the other hand, most classical theories of physical evolution, such as Hamiltonian mechanics, are exposed as having a built-in time reversibility. Much effort (mostly unsuccessful) has been used in attempts to resolve this paradox, but the issues usually have been buried by the analytic machinery and assumptions used in the development. Most of this work has been based upon the credo that a physical theory is of merit only if, given initial data, a unique prediction can be made. Indeed such possibilities, which are at the foundations of current technology, are possible with respect to evolutionary process which are diffeomorphically reversible.

On the other hand if the evolutionary process is not reversible, not representable by a homeomorphism -much less a diffeomorphism-but is C1 differentiable, it is possible to prove that with respect to Tensor Fields the concept of unique prediction is impossible Kiehn 1977c!

When topology changes in a continuous manner, the evolutionary process is irreversible, and the classic scientific credo fails.

However, functional substitution in the Cartan calculus permits a deterministic retrodiction process to be formulated with respect to continuous evolutionary processes that involve changing topology. The mathematical logic of the Cartan Calculus
generates a Topological Arrow of Time.

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