The Fundamental Act of Counting and Recounting
Counting is something that we learn to do at such an early age, and it is an act we perform so routinely, that we mostly remain unaware of how the manner in which we count ultimately reinforces a particular understanding of mathematical number, and how, through the act of physical measurement that depends on the counting operation, it ultimately determines our modern scientific understanding of the world as well. So we must ask the fundamental question whether there are other equally legitimate ways of counting multitudes. An examination of these alternatives may help make us aware of the basic - and possibly limiting - cognitive habit we are cultivating when we moderns count in the way we do.
Our point of departure in investigating the problem of counting has been the profound epistemological researches of Jacob Klein in his Greek Mathematical Thought and the Origin of Algebra. In that work, Klein diagnosed a far-reaching shift in the intentionality of the number concept - or how we aim our minds when doing mathematics - that took place in the late Renaissance. However, Klein did not pay sufficient attention to exactly how numbers were understood in Greek times, and the relation of the number concept to the counting process, which has been the focus of our own researches.
To date, our hindsight research into the number concept has yielded three central results:
1) a way of conceptualizing number in terms of the partitioning rather than the aggregation of a multitude;
2) a clearer distinction between number and magnitude; and
3) the discovery of dyadic counting in the Greek conceptualization of time as number.
At the same time, the recognition of certain asymmetries in classical Greek analysis and synthesis, reflection on the known result that many higher order algebraic equations cannot be solved using radicals, and considerations of the broader ramifications of an ancient astrological procedure that links pairs of planets as "time-lords ( chronocrators ), have led us to postulate that certain manifolds cannot be recounted in the same manner that they are counted out. We will be examining the direct mathematical consequences of this postulate, as well as its possible implications for the spatial/temporal manifolds of modern physics.