| | "With the grasp of the (implicit) concept "unit" man reaches the conceptual level of cognition, which consists of two interrelated fields: the conceptual and the mathematical. The process of concept-formation is, in large part, a mathematical process."
My 1979 paperback copy of ITOE has this on page 8.
If you look carefully at these statements, you will see that it divides "the conceptual level of cognition" into two parts, the conceptual, and the mathematical. Formally, that is an error of categorization. The conceptual is not just one part of the conceptual. Further, mathematical knowledge should not be implied to be non-conceptual, as this statement does.
It is generally given that numbers are the result of matching sets of things. Like the blinded cyclops who picked up one pebble for every sheep that passed through his door, and used that set of pebbles again to make sure they all returned at night, a one-to-one correspondence between sets proved that the group was the same size as before, that its "number" hadn't changed.
In Mesopotamia, the number of containers of grain that a given farmer put into storage was recorded by a correspondence set of tokens, which were sealed in an envelope. If he added or subtracted measures of grain, a corresponding number of tokens was added or taken away from his envelope. After a while, the accountants made a corresponding number of marks on the outside of the envelope, so they didn't have to open it to tell how many it contained. Again, after a while, the groups of marks on the envelopes were transformed into a set of single marks that each represented a different number of groups of marks, so, for example, lll became 3, and llll became 4, etc.
The point of reviewing this history is to see that mathematics is conceptual in its most primitive forms. The items used in matching by one-to-one correspondence have to belong to a concept before the correspondence can be made. If Cyclops sometimes added a pebble to his pile when a sheep went by, and sometimes added some sand, his efforts would be futile. If the Mesopotamian accountants sometimes added a token for a measure of grain added to the stores, and sometimes added a token for each day on which the farmer came to add grain, even though he might add two or three measures on some of these days, their accounts wouldn't be accurate. Each set for which a correspondence is made has to be defined as to type of entity. And that is a conceptual classification.
As I have said before, I think Rand's theory of concept-formation was heavily influenced by her desire to thwart the Analytic/Synthetic Dichotomy. I think this was a distorting influence, as polemics often are when they become the guiding light of theory. That speculation is neither here nor there regarding the issues, though.
The whole project of explaining concept-formation as being mathematical begs the question of how mathematical concepts themselves are formed. It might be worth noting, also, that even if mathematics were non-conceptual, its genesis would need explaining, and ought to be explained before we rely on it to explain how we form "concepts."
-more later
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