# Högre seminarum i vetenskapsfilosofi: Lars-Göran Johansson (Uppsala)

Seminarium

Datum:

torsdag 21 oktober 2021

Tid:

13.15 – 15.00

Plats:

D700

Platonism or constructivism in mathematics?

**Title**

Platonism or constructivism in mathematics?

**Abstract**

TBA

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)

Senast uppdaterad: 6 oktober 2021

Sidansvarig: Department of Philosophy