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