Abductive Reasoning in Mathematics and the Multiverse of Sets

(Logic & Philosophy of Mathematics, English)

hat guides theory choice in mathematics? The fact that certain mathematical statements, like the statement that there is no cardinality strictly between the cardinality of the natural numbers and the cardinality of the real numbers, known as Cantor's Continuum Hypothesis, have been shown independent of the standard axiomatization of set theory, ZFC, has led set theorists to explore further axiom candidates that might be added to ZFC. How should one justify the acceptance of new axioms for set theory? One suggestion is that set theoretical principles can be justified extrinsically by some kind of abductive reasoning, where the axiom candidate is accepted not for its immediate intuitiveness but its explanatory power, simplicity or elementary consequences.

One might think that the use of extrinsic evidence is a staple of any scientific practice and therefore also legitimate in mathematics. I want to resist such a quick legitimatization of the method, and argue that the notion of extrinsic evidence might be problematic in the case of set theory. I do this by appealing to what I call the criterion of match, which is a claim to the effect that proper methodology within a scientific field must match up with the ontology of the subject matter. I then investigate in particular how the view that there is a multiverse of set theoretical universes might block the appropriateness of certain uses of extrinsic evidence in set theory, especially in the case of establishing new set theoretical principles. The thought is that, if the multiverse exists, then some uses of extrinsic evidence will not be legitimate due to a mismatch between methodology and ontology.

Chair: Stefan Forster

Time: 15:10-15:40, 15 September 2017 (Friday)

Location: HS E.002

Hans Robin Solberg

(University of Oslo, Norway)

I am from Oslo, Norway. Currently, I am taking a master's degree in philosophy at the university of Oslo, supervised by Øystein Linnebo and Peter Fritz, on the topic of set theoretic pluralism. My BA is also in philosophy with a minor in linguistics. Before I started my studies I worked in a kindergarten here in Oslo.

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