Preproceedings 2006 - Austrian Ludwig Wittgenstein Society

Our thesis is that the dichotomy between the Realist
and the Constructivist views which dominates the modern
philosophy of mathematics is misleading. With time not
only the doctrines but also what is considered as the
opposites change.
Wittgenstein’s approach is in a way transversal to
both. What is essential to Wittgenstein’s philosophy of
mathematics is that philosophy is a descriptive activity. So
the aim of the philosopher of mathematics is to understand
mathematics as it is. It is no part of the philosopher’s task
to rewrite all or part of mathematics in new and better
ways. The philosopher wants to describe mathematics, but
needs to be careful to separate mathematics itself from the
philosophy of mathematics.
For Wittgenstein philosophy and mathematics have
nothing to say to one another. It would seem that no
philosophical opinion could affect the procedures of the
mathematicians.
Wittgenstein saw the distinction between
mathematics itself, on one hand and what is said about
mathematics, on the other, as being of fundamental
philosophical importance.
Wittgenstein asserts in “Philosophical
Investigations” (Wittgenstein 1956) that there are no
mathematical propositions or truths.
Mathematics has its foundation in the activity of
calculating.
Thus for Wittgenstein an empirical regularity lies
behind a mathematical law.
The mathematical law does not assert what the
regularity obtains.
Wittgenstein claims that what we are doing in
mathematics is calculation. If we do not prove anything in
logical sense, so no possibility of producing a logical
contradiction does arise. It means that a calculus is
consistent if we calculate in it. Indeed, no proof could tell
us that “this calculus is inconsistent or consistent”,
because there is no calculus which itself can prove
anything.
Wittgenstein emphasizes that the calculi which
make up mathematics are extremely diverse and
heterogeneous – what he calls the “motley” mathematics.
Wittgenstein insists that to accept a theorem is to
adopt a new rule of language. It follows from the statement
that our concepts cannot remain unchanged at the end of
the proof. For an intuitionist, every natural number is either
prime or composite. So we have a method for deciding
whether this number is prime or not. But Wittgenstein
argues that we do not have reliable method even to
construct the numbers.
Wittgenstein’s main reason for denying the
objectivity of mathematical truth is his denial of the
objectivity of proof in mathematics, his idea is that a proof
does not compel acceptance.
This idea is closely connected with Wittgenstein’s
doctrine that the meaning is the use.
Wittgenstein’s constructivism is seen as a much
more restricted kind than Brouwer’s intuitionism.
It seems that Wittgenstein is a strict finitist, because
he held that the only comprehensible and valid kind of
proof in mathematics takes the form of intuitively clear
manipulations with concrete objects.
18
About Wittgenstein’s mathematical epistemology - Olga Antonova / Sergei Soloviev
Developing further Brouwer’s views Heyting asserts
(cf. Heyting 1974) that proofs are mental constructions with
fulfilments of intensions.
Heyting explained mathematical propositions as
expressions of intentions, «intensions» none only refer to
the states of affairs thought to exist independently of us
but also to the experiences thought to be possible.
Martin-Löf’s views in this are similar to Heyting’s.
Martin-Löf asserts that a proof is not an object, but
an act, (cf. Martin-Löf, 1996).
So a proof is a cognitive act or a process before it is
an object, an act or a process in which we come to see or
grasp something intuitively.
Also Martin-Löf’s system of intuitionistic type theory
uses four basic forms of judgment, among which are the
two that «S is a proposition» and «a is a proof
(construction) of the proposition S». Martin-Löf notes that
we can read these, equivalently, as «S is an intention» and
«a is a method of fulfilling the intention S». Thus, we can
understand his system as a formalization of the informal
concepts of intentionality, intuition and evidence.
Let us now turn briefly to Dummett’s views, because
Dummett has a very original and interesting view on
problems of the philosophical basis of proof.
Dummett, who agrees with Wittgenstein on the role
of language in mathematics, believes there is no way to
approach the question about proof independently of
investigation in the philosophy of language.
Dummett contrasts his view with the position that
the meaning of proposition is determined by its truth
conditions.
For Dummett only the use of language determines
the meaning of an individual statement. One of the things
that disappears with the idea of mental acts and processes
in Dummett’s approach is the philosophical objection to the
form of intuitionism based on solipsism. But we shall argue
that Dummett goes too far here. Dummett’s account of
intuitionism contains no theory of intentionality, fulfilled
intentions and evidence. He suggests no theory of
mathematicians as cognitive information “processors”, the
cognitive structure of mental acts and meaning of mental
representations and the like. We think that if the proof is
really to be understood as either a cognitive act or as an
objectification of an act then these notions must figure in
our understanding of the philosophical basis of intuition.
Thus it is clear that the philosophical basis of
intuitionistic mathematics is best understood along the
lines suggested by Heyting’s investigations. So according
to the views of intuitionists, the main source of intuitionistic
mathematics and criterion of the validity of its constructions
is intuition. Conclusions of intuitionistic mathematics are
not obtained by precisely established rules which could be
united in a logical system. On the contrary evidence of any
separate conclusion is considered directly. The essence of
the mathematical proof consists not in logical conclusions
and in design of mathematical systems. It is obvious that
mathematical activity is defined not with language and
logic but as constructive activity of pure thinking.
Proceeding from the position that the intuitionistic
mathematics is a mental process, obviously, any language
including the language of formalization cannot be an
equivalent model of the given system as the thinking
cannot be reduced to a finite number of formal rules. Logic
is only a true imitation of mathematical language. Logic