We know from Rand that the base of epistemology and number is equally shared through reliance on the concept "unit". We measure qualities through units of relations. We measure quantities through units of cardinal numbers. It is easy to see where ordinal numbers come from because that merely requires we remember our experiences. "When we (first) conceptualize, we focus on an attribute perceptually, not conceptually." (Peikoff in OPAR pg 86) We just have to be able to relate things in order to compare and contrast them. This describes how our minds function most of the time. It should be the mode of consciousness you are in as you read this because ordinality is the mode of change and approximation. Cardinality is the mode of conceptual exactness and it is important to know how the two modes of consciousness are related.
Introspect and think about some habit of yours that you do without thinking. Driving a car with a clutch comes to mind. If you have become a skillful driver you need not waste you attention on the clutch, it is all mentally automatized. Recall how hard it was to learn the skill. This describes a crucial relation between ordinal and cardinal as well as perception/ conception.
The Greeks were the first to see the relation between the ordinal and the cardinal but it was only for a short time because of the ignorance of Pythagoras. Boyd, again says that the Ionians rearranged knowledge into a deductive scheme based largely on verified experience. That's describing the integration of the ordinal to the cardinal. Verified experience uses a calculus to arrive as close to certainty as we please which is Aristotle's way of describing the potential infinite.
George Cantor is creditied with our mathematical terms: ordinal and cardinal. But though he knew there were problems with using Pythagorean concepts of number he based his theory of the actual infinite on them. Thales gave us mind/body integrity through the integration of ordinal/cardinal thinking. Boyd: "The oriental mysticism of Pythagoras, however, reversed this state of affairs and gave to mathematics a supru-sensuous reality of which the world of appearances was a counterpart." This is what led to the opposites of Heraclitus and Parmenides and the paradoxes of Zeno. Cantor, to give him his due is, it seems to me, trying to accomplish Thales' intgration but fails because of Pythagoras.
The actual infinite is a term that requires one to know how to integrate incommensurables. Thales did it, Pythagoras couldn't. When Cantor attempted to define the actual infinite he was in a sense repeating Thales but with Pythagoras in the way. Cantor took Bolzano's paradox which states that the part is equal to the whole and used a contradiction to make sense of incommensurables. Of course it doesn't work and Canter went through several mental breakdowns, I assume there was a connection.
The major complaint to my last few posts has been my insistence that cardinal number be used united to practical application. My way, this way leads to integrity, Cantors and Pythagoras way leads to contradiction and perhaps madness.