Toward a General Theory of Process

An imaginary unit vector in time requires a mathematical foundation that is an extension of Hamilton’s quaterions. This applies also to the imaginary unit vectors associated with the second- and third-order spacetime del operators given later in the text. The hypernumber arithmetics required were worked out by Charles Musès during the 1960s and 70s. See “Applied Hypernumbers: Computational Concepts”, Applied Mathematics and Computation, 3, 1977 and 4, 1978. See also Consciousness and Reality, edited by Charles Musès and Arthur M. Young (Avon, 1972), which contains material on hypernumbers and an interesting popular essay by Young on the roles of velocity, acceleration, and time rate of change of acceleration in engineering of the rotor for the Bell helicopter. Cayley’s 8-tuples and Grassmann algebras are also required for complete handling of unit vectors relative to 3-fold imaginary time. Unfortunately, it appears that the formulations of Musès regarding -- which is not equal to +1 or -1 and involves a twist more hypercomplex then that from the real number plane over onto the plane of imaginary numbers -- have not been embraced by the mathematical establishment for extra-mathematical reasons.

The fundamental concepts involved here are far more elaborate than the passive notion of imaginary time Stephen Hawking presented a decade later in his book A Brief History of Time (and in the associated technical papers). Spacetime del operators and temporal curl involve time actively operating on space so as to effect changes of its topology; such temporal operators are no part of Hawking’s conception of imaginary time. Charles Musès coined the term “chronotopology” in the late 1960s and wrote extensively about his concept of the “shapes” of time in a long unpublished manuscript entitled Synchrony. This is, again, a passive-time concept, useful as it may be. In cascade theory, we are not describing “chronotopology”, but active temporal operations on the topology of space. Time, as an experiential “somewhat”, is knowable only through change of space. Time must be held responsible for those spatial changes by which it is known. This epistemological injunction is borne by operator-time.

In 1934, J. W. Dunne, in his book An Experiment in Time, argued that linear-time requires another kind of time to measure its rate of flow, and that this other time requires another, and so on to infinity. Dunne’s argument does not apply to orders of temporal operators acting at limiting velocities, limiting accelerations, and limiting time rates of change of acceleration. That three orders of operator-time are necessary and sufficient to decompose and recompose a transfinite set of orders of logical-value on a multivalued reference space can be demonstrated in the context of G. Spencer Brown’s calculus of distinctions or indications (see: THE MOON OF HOA BINH, Vol. II, pp. 343-4).