"Deflationism and objectivity"

Site: Seminario Anscombe, 3rd floor, Faculty of Philosophy

University of Santiago de Compostela (Spain)

April 29-30, 2024

 

Workshop Organizers

Concha Martínez-Vidal (USC)

Ismael Miguéns (USC)

PID2020-115482GB-I00 funded by MCIN/AEI/10.13039/501100011033

 

 

Monday April 29

15:30 – 16:30    Matteo Plebani (University of Torino) ― Counterpossibles in relative computability theory: A closer Look

16:30 – 17:30    Ismael Miguéns (University of Santiago de Compostela) ― Aristotle meets Tarski. Minimal reasons for domain expansionism

17:30 – 17:45 Break

17:45 – 18:45    José Ferreirós (University of Sevilla) ― On arbitrary collections in logic and set theory

 

Tuesday April 30

16:30 – 17:30    Mary Leng (University of York) ― Mathematically natural kinds. Or, must be the fictionalist be a (Groucho) Marxist? (Online)

17:30 – 18:30    Øystein Linnebo (University of Oslo) ― What is an extensionally determinate domain?

 

 

BRIEF DESCRIPTION OF THE TALKS

Matteo Plebani  (University of Torino) 

Title: Counterpossibles in relative computability theory: a closer look

 

A counterpossible is a counterfactual with an impossible antecedent, like “if zero were equal to one, two would be equal to five”. Matthias Jenny has argued that the following is an example of a false counterpossible: HT If the validity problem were algorithmically solvable, then arithmetical truth would be also algorithmically decidable. According to the standard analysis of counterfactuals all counterpossibles are vacuously true. If HT is false, then, the standard analysis of counterfactuals is wrong.

I will argue that HT admits two readings, which are expressed by two different ways of formalizing HT. Under the first reading, HT is clearly a counterpossible. Under the second reading, HT is clearly false. Hence, it is possible to read HT as a counterpossible and it is possible to read HT as a false claim. However, it is unclear that it is possible to do both things at once, i.e. interpret HT as a false counterpossible. It can be proven that the two readings are not equivalent. The formalization expressing the first reading is a mathematical theorem, which means that under the first reading, HT is a true counterpossible. On the other hand, I will argue that under the second reading HT, while false, is best interpreted as a counterpossible with a contingent antecedent.It can be proven that the two readings are not equivalent. The formalization expressing the first reading is a mathematical theorem, which means that under the first reading, HT is a true counterpossible. On the other hand, I will argue that under the second reading HT, while false, is best interpreted as a counterpossible with a contingent antecedent.

      

 Ismael Miguéns (University of Santiago de Compostela) (In person)

Title: “Aristotle meets Tarski. Minimal reasons for Domain Expansionism”

 

Russell’s paradox arises from the instability generated by the combination of an unrestricted second-order comprehension principle, a one-to-one mapping from second-order entities into first order entities, and absolute generality. Therefore, a way out of the paradox demands reasons to deny at least one of these three items. A ―nowadays standard― solution urges us to restrict the first one when dealing with pluralities: not every condition characterises a plurality. As a result, we can reify pluralities into objects and accept domain expansionism, the thesis that the domain of first-order quantifiers ―under some specific conditions― can be expanded. In turn, the Liar paradox arises from the instability generated by the introduction of truth predicates, enough self-referential resources, and absolute generality. As in the previous case, a standard solution to this paradox urges us to restrict the former item: open languages cannot express their own truth. However, this solution opens the door to language expansionism, formulating truth conditions in a richer language.

 

In this talk, I will elaborate an analogy between Russell’s paradox and the Liar paradox. Then, I will argue that language expansionism gives us new, minimal reasons for domain expansionism: by expanding the language we can expand the domain.

 

 

José Ferreirós (University of Sevilla) (In person)

Title: On Arbitrary Collections in Logic and Set Theory'

My purpose is to underscore the fact that some developments in logic involve a strong commitment to arbitrary collections, and to recommend a more restrained understanding of “logic”. We shall consider some key aspects of the semantics of second-order logic, of plural logic, and of set theory, in the light of basic considerations about logical principles. I will argue that so-called quasi-combinatorial conceptions have their origins purely and exclusively in mathematics (real numbers, functions) completely independent from logic. This has consequences for logical theory.

 

 Mary Leng (University of York)(on-line)

Title: 'Mathematically Natural Kinds' (Or, Must the Fictionalist be a (Groucho) Marxist?)

 

Pluralism about mathematical theories (according to which any consistent axiomatic theories are as good ―metaphysically speaking― as any other) is currently en vogue. Rather than defend any collection of mathematical principles as capturing the 'true' nature of sets, for example, the mathematical pluralist is happy to follow Groucho Marx in declaring "these are my principles, and if you don't like them...well, I have others". Insisting, as Kurt Gödel did, on a single objective answer to an open question about the sets (such as whether the continuum hypothesis holds), as opposed to saying 'it's true if you adopt these principles, but if you don't like that, then I can offer you some other principles according to which it is false', is thought to embroil the objectivist in inescapable epistemic difficulties. Yet reading Gödel's paper on the continuum problem, I find my sympathies are increasingly with Gödel and objectivism. Is the combination of a fictionalist view about the nature of mathematical objects with an objectivist view about the truth of particular set theoretic principles coherent? I will argue that it is.

According to mathematical fictionalists, there are no mathematical objects. Mathematicians are engaged in uncovering what is 'true in the story (or, perhaps better, stories)' of mathematics, where 'the story' is either given by a background set theory as an ultimate foundational theory within which all of the rest of mathematics can live, or alternatively any coherent axioms are viewed as setting the scene for their own mathematical 'story', with the story of elliptic geometry, for example, as being given by the theory's axioms and their consequences. In order to fit intuitions about truth in mathematics outrunning provability, many fictionalists will insist that mathematical stories can be told using second-order axioms, and insist on a semantic, rather than deductive, consequence (making use of primitive modality rather than set theoretic models to elaborate on this notion of consequence). But with all this in place, it would appear that fictionalists ought to be liberal pluralists about mathematical stories: metaphysically-speaking, any coherent system of axioms is as good as any other in telling a story about how mathematical objects could be (though perhaps some stories will be of more interest than others). Indeed, fictionalism here seems to be in step with a general trend in the philosophy of mathematics towards embracing some form of mathematical pluralism (as seen e.g. in Balaguer's full blooded Platonism and Shapiro's ante rem structuralism). This paper will consider whether fictionalists have to be pluralists, or whether it is possible - and even desirable - within a fictionalist approach to recognise sources of mathematical objectivity that might vindicate some mathematical theories over others. In particular, I will consider whether fictionalists could (and indeed should) embrace Putnam's indispensability argument as an argument for taking some mathematical concepts to be objectively vindicated by their use in empirical science, or Maddy's considerations of mathematical depth as an argument for taking some mathematical concepts to be objectively vindicated by their use in mathematics.

 

Oystein Linnebo (University of Oslo) (In person)

Title: What is an extensionally determinate domain?

Hermann Weyl articulates an interesting notion of a domain being “extensionally determinate”. He analyzes the notion in terms of the availability of quantification over the domain subject to classical logic. I explain Weyl’s notion and his elegant logical analysis of it. I also explore the conditions under which a domain enjoys this form of determinateness.