How does platonic realism affect th

How does platonic realism affect the status of axiomatic set theory? From the point of view of
platonic realism, mathematical objects are given to us ready-made, with all their features and all their
properties. It follows that to say a mathematical theorem is true means it expresses a correct
statement about the relevant mathematical objects. (For example, the proposition 2 + 2 = 4 is not
merely a formal statement provable in arithmetic; it states an actual fact about numbers.) Now—if
we admit that mathematical objects are given to us with all their properties, it follows, in particular,
that the notion of set is a fixed, well-defined concept which we are not free to alter for our own
convenience. Thus the “sets” created by Zermelo and von Neumann do not exist, and theorems which
purport to describe these nonexistent objects are false! In conclusion, if we were to accept a strict
interpretation of platonic realism, we would be forced to reject the systems of Zermelo and von
Neumann as mathematically invalid.
Fortunately, the trend, for some time now, has been away from platonism and toward a more
flexible, more “agnostic” attitude toward mathematical “truth.” For one thing, developments in
mathematics have been conforming less and less to the pattern dictated by platonic philosophy. For
another, the cardinal requirement of platonism—that every mathematical object correspond to a
definite, distinct object of our intuition (just as “point” and “line” refer to well-defined objects of our
spatial intuition)—came to be an almost unbearable burden on the work of creative mathematicians
by the nineteenth century. They were dealing with a host of new concepts (such as complex numbers,
abstract laws of composition, and the general notion of function) which did not lend themselves to a
simple interpretation in concrete terms. The case of the complex numbers is a good illustration of
what was happening. Classical mathematics never felt at ease with the complex numbers, for it lacked
a suitable “interpretation” of them, and as a result there were nagging doubts as to whether such things
really “existed.” Real numbers may be interpreted as lengths or quantities, but the square root of a
negative real number—this did not seem to correspond to anything in the real world or in our intuition
of number. Yet the system of the complex numbers arises in a most natural way—as the smallest
number system which contains the real numbers and includes the roots of every algebraic equation
with real coefficients; whether or not the complex numbers have a physical or psychological
counterpart seems irrelevant.
The case of the complex numbers strikes a parallel with the problem of axiomatic set theory. For
the “sets” created by Zermelo and von Neumann arise quite naturally in a mathematical context. They
give us the simplest notion of set which is adequate for mathematics and yields a consistent axiomatic
theory. Whether or not we can interpret them intuitively may be relatively unimportant.
Be that as it may, many mathematicians in the early 1900’s were reluctant to make so sharp a break
with tradition as axiomatic set theory seemed to demand. Furthermore, they felt, on esthetic grounds,
that a mathematical theory of sets should describe all the things—and only those things—which our
intuition recognizes to be sets. Among them was Bertrand Russell; in his efforts to reinstate intuitive
set theory, Russell was led to the idea that we may consider sets to be ordered in a hierarchy of
“levels,” where, if A and are sets and A is an element of , then is “one level higher” than A.
For example, in plane geometry, a circle (regarded as a set of points) is one level below a family of
circles, which, in turn, is one level below a set of families of circles. This basic idea was built by
Russell into a theory called the theory of types, which can be described, in essence, as follows.
Every set has a natural number assigned to it, called its level. The simplest sets, those of level 0,
are called individuals—they do not have elements. A collection of individuals is a set of level 1; a
collection of sets of level 1 is a set of level 2; and so on. In the theory of types the expression a ∈ B
is only meaningful if, for some number n, a is a set of level n and B is a set of level n + 1. It follows
that the statement x ∈ x has no meaning in the theory of types, and as a result, Russell’s paradox
vanishes for the simple reason that it cannot even be formulated.

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