554 19. LOGIC AND SET THEORYif we apply Cantor's argument to this mapping, we are led to consider S — [E :
Å £ E}. By definition of the mapping / we should have f(S) = S, and so, just
as in the case of Cantor's argument, we ask if S € S. Either way, we are led to a
contradiction. This result is known as Russell's paradox.
After Russell had straightened out the paradox with a theory of types, he collab-
orated with Alfred North Whitehead on a monumental derivation of mathematics
from logic, published in 1910 as Principia mathematica.3.2. Formalism. A different view of the foundations of mathematics was ad-
vanced by Hilbert, who was interested in the problem of axiomatization (the ax-
iomatization of probability theory was the sixth of his famous 23 problems) and
particularly interested in preserving as much as possible of the freedom to reason
that Cantor had provided while avoiding the uncomfortable paradoxes of logicism.
The essence of this position, now known as formalism, is the idea stated by de Mor-
gan and Boole that the legal manipulation of the symbols of mathematics and their
interpretation are separate issues. Hilbert is famously quoted as having claimed
that the words point, line, and plane should be replaceable by table, chair, and beer
mug when a theorem is stated. Grattan-Guinness (2000, p. 208) notes that Hilbert
may not have intended this statement in quite the way it is generally perceived
and may not have thought the matter through at the time. He also notes (p. 471)
that Hilbert never used the name formalism. Characteristic of the formalist view is
the assumption that any mathematical object whatever may be defined, provided
only that the definition does not lead to a contradiction. Cantor was a formalist in
this sense (Grattan-Guinness, 2000, p. 119). In the formalist view mathematics is
the study of formal systems, but the rules governing those systems must be stated
with some care. In that respect, formalism shares some of the caution of the earlier
constructivist approach. It involves a strict separation between the symbols and
formulas of mathematics and the meaning attached to them, that is, a distinction
between syntax and semantics. Hilbert had been interested in logical questions in
the 1890s and early 1900s, but his work on formal languages such as propositional
calculus dates from 1917. In 1922, when the intuitionists (discussed below) were
publishing their criticism of mathematical methodologies, he formulated his own
version of mathematical logic. In it he introduced the concept of metamathemat-
ics, the study whose subject matter is the structure of a mathematical system.^17 A
formal language consists of certain rules for recognizing legitimate formulas, certain
formulas called axioms, and certain rules of inference (such as syllogism, general-
ization over unspecified variables, and the rules for manipulating equations). These
elements make up the syntax of the language. One can therefore always tell by fol-
lowing clearly prescribed rules whether a formula is meaningful (well formed) and
whether a sequence of formulas constitutes a valid deduction. To avoid infinity in
this system while preserving sufficient generality, Hilbert resorted to a "finitistic"
device called a schema. Certain basic formulas are declared to be legitimate by
fiat. Then a few rules are adopted, such as the rule that if A and  are legitimate
formulas, so is [A B. This way of defining legitimate (well-formed) formulas
makes it possible to determine in a finite number of steps whether or not a formula
is well formed. It replaces the synthetic constructivist approach with an analytic
(^17) This distinction had been introduced by L.E.J. Brouwer in his 1907 thesis, but not given a
name and never developed (see Grattan-Guinness, 2000, p. 481).