Title

Platonism or constructivism in mathematics?

Abstract

The ontology and epistemology of mathematical objects is a
controversial topic. The core problem for an empiricist is that
conceiving mathematical objects as existing independently of human
thinking, i.e., platonism, makes it impossible to understand how we
can have mathematical knowledge. The alternative, a constructivist
conception according to which mathematical objects are the results of
actual performed constructions, resolves the epistemological
problem, but is associated with the identification of truth with
provability, the basic principle of intuitionism. That entails that the
law of excluded middle must be dismissed as a generally valid logical
principle, hence indirect proofs are not allowed. Another difficulty is
that proofs must be based on axioms, but how do we know that these
axioms are true?

Suppose axiomhood were a decidable property. That would mean that
one could formalize constructivist provability. Then we could apply
Gödel’s first incompleteness theorem and show that there are true but
unprovable sentences in this system. This contradicts the very basic
principle of intuitionism in particular and constructivism in general
that truth is provability.

In this talk I will propose a modified constructivism, which keeps the
distinction between truth and provability. The core idea is that
mathematical objects are not individually constructed, as in Bishop &
Bridges book, but en masse, so to say. For example, introducing into
discourse the predicate ’natural number’ by adopting the axioms
guiding this predicate, entails that all objects satisfying this predicate
are thereby accepted in one’s ontology; they are in a sense constructed
independently of any existence proofs. This view on the relation
between a concept and the objects satisfying that concept is similar to
Kant’s; objects are objects of judgement and the judgement is
primary.

Litterature:
Bishop & Bridges: Constructive Analysis, Springer, 1985.
Kant, I.: Critique of Pure Reason, St. Martins Press, 1965.
Johansson, L-G. Empiricism and Philosophy of Physics, Springer,
2021 (ch. 4)