The Metaphysical Status of Concepts
[L]et us identify a problem in regard to concepts which has bedeviled philosophers from Greece to the present: what is the relationship of concepts to existents? To what precisely do concepts refer in reality?[1]
The formation of a concept involves a process of abstraction---but these abstractions do not actually exist, so the problem at hand is: what, if anything, is a concept in reality?
Aristotle's answer was that:
a concept refers to what all the concretes in a given class possess in common. In this view, "manness" or "humanity," for instance, refers to the attribute(s) that is the same in every instance of the species.[2]
The issue with such a view is that on the purely perceptual level there is no identity between different concretes that could serve as the basis for a concept. This is because A is A, not non-A---that dog is that dog, not this dog. There may be many similar aspects in common to dogs, but it could very easily be the case that every aspect changes in some minute way---each dog varies in its exact weight, its height, its breed, the length of its hair, the speed at which it can run, etc. In attempting to treat concepts qua metaphysics as referring to shared attributes between every item in the set, one is attempting to equate the different concretes in some way---you are not treating individual dogs "as more or less similar, but in some way as identical: as equally, interchangeably, members of the group."[3]
Ayn Rand's solution to this problem is found in recognising the essential connection between the processes of conceptualisation and measurement:
In both cases, man identifies relationships among concretes. In both cases, he takes perceived concretes as the base, to which he relates everything else, including innumerable existents outside his ability to perceive. In both cases, the result is to bring the whole universe within the range of human knowledge.[4] And now a further, crucial observation: in both cases, man relates concretes by the same method---by quantitative means. Both concept-formation and measurement involve the mind's discovery of a mathematical relationship among concretes.[5]
On the Objectivist theory of concepts, man equates the different concretes not by finding some specific aspect that they all share in common, but rather by dispensing with the measurements of the different concretes, and focusing only on what remains. Thus, one abstracts the concept "dog," not by identifying the weight of "dogness" or the speed of "dogness" or anything like that---but rather by ignoring all of the quantitative features of any individual dogs. "When we form a concept, [...] our mental process consists in retaining the characteristics, but omitting their measurements."[6]
This is the relationship between concepts and concretes: the concept is a stand-in for a concrete whose attributes exist with any quantity---"Length must exist in some quantity, but may exist in any quantity. I shall identify as 'length' that attribute of any existent possessing it which can be quantitatively related to a unit of length, without specifying the quantity."[7] With respect to concepts, the puzzle laid out above has it that the differences between specific concretes is apparent, but it is unclear what is the same between them. Rand has it that what is similar is the characteristics sans the specific measurments or quantities of those characteristics---the formation of a concept involves retaining all of the characteristics and omitting all of the measurements (within certain bounds).
"Manness," [...] is men, the real men who exist, past, present, and future; it is men viewed from a certain perspective.[8]
This is not to say that the specific quantities of these entities are unreal, or subjective, or irrelevant---"It means that measurements exist, but are not specified. That measurements must exist is an essential part of the process. The principle is: the relevant measurements must exist in some quantity, but may exist in any quantity."[9] Ayn Rand perfectly highlights the connection between concept-formation and algebra in connection to this: namely, in the equation