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A brief excerpt from Leonard Peikoff's Objectivism: The Philosophy of Ayn
Rand
The answer to the "problem of universals" lies in Ayn Rand's discovery
of the relationship between universals and mathematics. Specifically, the answer
lies in the brilliant comparison she draws between conceptformation and
algebra.
This is more than a mere comparison, as she shows, since the underlying method
in both fields is the same.
The basic principle of conceptformation (which
states that the omitted measurements must exist in some quantity, but may exist
in any quantity) is the equivalent of the basic principle of algebra, which
states that algebraic symbols must be given some numerical value, but may be
given any value. In this sense and respect, perceptual awareness is the
arithmetic, but conceptual awareness is the algebra of cognition.
The relationship of concepts to their constituent particulars is the same as the
relationship of algebraic symbols to numbers. In the equation 2a = a + a, any
number may be substituted for the symbol "a" without affecting the
truth of the equation. For instance: 2 x 5 = 5 + 5, or: 2 x 5,000,000 —
5,000,000 + 5,000,000. In the same manner, by the same psychoepistemological
method, a concept is used as an algebraic symbol that stands for any of the
arithmetical sequence of units it subsumes.
Let those who attempt to invalidate concepts by declaring that they cannot find
"manness" in men, try to invalidate algebra by declaring that they
cannot find "aness" in 5 or in 5,000,000.
For centuries, rationalist philosophers have venerated mathematics as the
model of cognition. What they have admired about the discipline is its deductive
method. Objectivism, too, regards mathematics as an epistemological model, but
for a different reason.
The mathematician is the exemplar of conceptual integration. He does
professionally and in numerical terms what the rest of us do implicitly and have
done since childhood, to the extent that we exercise our distinctive human
capacity.
Mathematics is the substance of thought writ large, as the West has been told
from Pythagoras to Bertrand Russell; it does provide a unique window into human
nature. What the window reveals, however, is not the barren constructs of
rationalistic tradition, but man's method of extrapolating from observed data to
the total of the universe.
What the window of mathematics reveals is not the mechanics of deduction, but of
induction. Such is Ayn Rand's unprecedented and pregnant identification in the
field of epistemology.
"Ayn Rand's philosophy has changed thousands of lives, including my own, and has the power to
change the course of history. Her views, however, are spread across more than a dozen books and hundreds of articles and speeches. The present book is the first comprehensive
statement of her philosophy." — Leonard Peikoff 
