- PHILOSOPHIES OF MATHEMATICS 555
approach (which can be reversed, once the analysis is finished, to synthesize a given
well-formed formula from primitive elements).
The formalist approach makes a distinction between statements of arithmetic
and statements about arithmetic. For example, the assertion that there are no
positive integers x, y, æ such that x^3 + y^3 = z^3 is a statement of arithmetic.
The assertion that this statement can be proved from the axioms of arithmetic is
a statement about arithmetic. The metalanguage, in which statements are made
about arithmetic, contains all the meaning to be assigned to the propositions of
arithmetic. In particular, it becomes possible to distinguish between what is true
(that is, what can be known to be true from the metalanguage) and what is prov-
able (what can be deduced within the object language). Two questions thus arise
in the metalanguage: (1) Is every deducible proposition true? (the problem of con-
sistency); (2) Is every true proposition deducible? (the problem of completeness).
As we saw in Section 2, Godel showed that the answer, for first-order recursive
arithmetic and more generally for systems of that type, is very pessimistic. This
language is not complete and is incapable of proving its own consistency.
3.3. Intuitionism. The most cautious approach to the foundations of mathemat-
ics, known as intuitionism, was championed by the Dutch mathematician Luitzen
Egbertus Jan Brouwer (1881-1966). Brouwer was one of the most mystical of
mathematicians, and his mysticism crept into his early work. He even published
a pamphlet in 1905, claiming that true happiness came from the inner world, and
that contact with the outer world brought pain (Franchella, 1995, p. 305). In his
dissertation at the University of Amsterdam in 1907, he criticized the logicism of
Russell and Zermelo's axiom of choice. Although he was willing to grant the va-
lidity of constructing each particular denumerable ordinal number, he questioned
whether one could meaningfully form a set of all denumerable ordinals.^18 In a
series of articles published form 1918 to 1928, Brouwer laid down the principles
of intuitionism. These principles include the rejection not only of the axiom of
choice beyond the countable case, but also of proof by contradiction. That is, the
implication "A implies not-(not-A)" is accepted, but not its converse, "Not-(not-A)
implies A." Intuitionists reject any proof whose implementation leaves choices to
be made by the reader. Thus it is not enough in an intuitionist proof to say that
objects of a certain kind exist. One must, choose such an object and use it for the
remainder of the proof. This extreme caution has rather drastic consequences. For
example, the function f(x) defined in ordinary language as
is not considered to be defined by the intuitionists, since there are ways of defining
numbers ÷ that do not make it possible to determine whether the number is negative
or positive. For example, is the number (-1)", where ç is the trillionth decimal
digit of ð, positive or negative? This restrictedness has certain advantages, however.
The objects that are acceptable to the intuitionists tend to have pleasant properties.
For example, every rational-valued function of a rational variable is continuous.
The intuitionist rejection of proof by contradiction needs to be looked at in
more detail. Proof by contradiction was always used somewhat reluctantly by(^18) This objection seems strange at first, but the question of whether an effectively defined set
must have effectively defined members is not at all trivial.