C-ALPHA Workshop on the Philosophy of Logic and Mathematics

The Center for the Advancement of Logic, its History, Philosophy, and Applications (C-ALPHA) organizes a Philosophy of Logic and Mathematics Workshop on December 9 2017, from 2-5:30pm.

Program:

2:00-3:00  Benedict Eastaugh "Does nonclassical truth impair mathematical reasoning?"

3:15-4:15  Marianna Antonutti Marfori "Indirect Naturalism"

4:30-5:30 Toby Meadows "Two puzzles and a programme in mathematical hermeneutics"

Venue: Social & Behavioral Sciences Gateway Building, room 1321.

Abstracts:

-Benedict Eastaugh: 

Does nonclassical truth impair mathematical reasoning? 

Joint work with Carlo Nicolai.  

Volker Halbach has argued that when we give up classical logic, we thereby lose important non-semantic patterns of reasoning, in particular in mathematics. This loss is made particularly clear when we compare two axiomatic theories of truth, the classical theory KF and the theory PKF formulated in the logic of first-degree entailment. These two theories axiomatize the same Kripkean concept of truth, but differ significantly in the mathematical reasoning they support: KF validates transfinite induction up to epsilon_0, while PKF does not.   

In this talk I will argue for two claims. Firstly, while Halbach’s argument shows that by giving up classical logic we are unable to carry out some patterns of mathematical reasoning, it fails to show that this loss is significant. Secondly, such significance should be assessed in terms of the mathematical consequences of this loss, i.e. in terms of significant mathematical theorems that are validated by KF but not by PKF. I close the gap in Halbach’s argument and show that such mathematical consequences do in fact exist, by proving that an ordinary mathematical theorem concerned with indecomposable linear orderings is proof-theoretically reducible to KF, but not to PKF.

-Marianna Antonutti Marfori: 

Indirect Naturalism

Scientific realistic naturalism entails some degree of revisionism about mathematical practice: those parts of mathematics that are too far removed from empirical justification (such as the higher reaches of set theory and uncountable infinitary mathematics) are not legitimately parts of science. The naturalist who wants to resist revisionism and grant mathematics the same autonomy that the scientific naturalist grants to science is liable to being unable to discern mathematics from other pseudo-scientific enterprises. 

In this paper, I present a new framework for the mathematical naturalist that accounts for those parts of mathematics that are far removed from applications as legitimate parts of scientific practice, not on a par with pseudo-scientific enterprises. This framework is non-revisionist about mathematical practice and mathematical language, and allows the mathematical naturalist to be neutral about mathematical ontology, modulo a naturalistic commitment to the weakest theory that can prove the arithmetised completeness theorem.

-Toby Meadows: 

Two puzzles and a programme in mathematical hermeneutics

I am interested in the problem of selecting the best - or just better - mathematical theories. However, this kind of investigation faces a difficult initial hurdle. How do we compare different theories? How do we know even know that we're talking about the same stuff? Mathematical logic provides a means of addressing these questions via the relative interpretability. I will discuss two dilemmas for the application of this approach and conclude with a programmatic suggestion about what a good interpretation should do, which borrows from recent results in set theory.