Patrick Suppes (Stanford):

A Neuroscience Perspective on the Foundations of Mathematics

Abstract: I mainly ask and partially answer three questions. First, what is a number? Second, how does the brain process numbers? Third, what are the brain processes by which mathematicians discover new theorems about numbers? Of course, these three questions generalize immediately to mathematical objects and processes of a more general nature. Typical examples are abstract groups, high dimensional spaces or probability structures. But my emphasis is not on these mathematical structures as such, but how we think about them. For the grounding of mathematics, I argue that understanding how we think about mathematics and discover new results is as important as foundations of mathematics in the traditional sense.

Luca Incurvati (Cambridge):

Abstract: I will begin by reviewing and further defending the minimalist approach to sets which I advanced in earlier work. Then, I will consider the prospects for extending the minimalist approach to the case of semantics. I will conclude by examining whether a minimalist approach prevents us from giving a common account of grounding assumptions in set theory and semantics.

Toby Meadows (Bristol):

Local Dependence: Complexity & Groundedness

Abstract: The purpose of this paper is to lay out some groundwork for a general understanding of relationships between: notions of dependence and groundedness; and notions of complexity and cardinality. We commence by sketching a pleasing framework linking dependence with the theory of inductive definitions. We pose a kind of problem for this framework and attempt to explain why this is interesting using a case study from the semantic theory of truth literature. We then attempt to formulate a definition of local dependence which addresses this issue. Finally, we consider limitations of this definition and consider possibilities for its further generalisation.

Casper Storm Hansen (Aberdeen):

Alternative Supervaluation for Kripke's Theory of Truth.

Abstract: I will present a method of supervaluation for Kripke's theory of truth. It is different from Kripke's own method in that it employs trees, results in a compositional semantics, assigns the intuitively correct truth values to the sentences of a particularly tricky example of Gupta's, and is acceptable as an explication of the correspondence theory of truth.

Dependence and Groundedness

Abstract: In this talk, I will discuss whether the notion of groundedness can be analyzed directly in terms of dependence upon non-semantic states of affairs or whether a detour via truth and falsity is always necessary.

Ground and Partial Content

Abstract: I provide a ground-theoretic account of partial content.

Pure Logic of Iterated Ground

Abstract: The presently existing logics of ground have not had anything to say about iterated grounding claims, that is, claims of the form: “A grounds that (B grounds C)”. I develop a pure logic of iterated ground providing a systematic account of such iterated grounding claims. The logic is developed as a Prawitz style natural deduction system; the grounding operators are provided with both introduction and elimination rules, and normalization can be proved. The resulting logic is a conservative extension of Kit Fine's Pure Logic of Ground.

Grounding Class Theory

In this talk, I propose a conception of proper class. It brings together two separate strands of research: metaphysical and semantical groundedness.

On the one hand, I use the *metaphysical* concept of grounding. Whether or not something is a member of a class, I will argue, must be grounded in other facts.

On the other hand, the method of *semantical* groundedness will provide a formal theory that captures this conception of class.

Francesca Boccuni (Milan):

In philosophy of logic and mathematics, the notion of groundedness is hotly debated, in particular as applied to truth and individuation. In this article, I am going to investigate the notion of groundedness as applied to reference.

The informal counterpart of groundedness is dependence, so first of all I will argue for referential dependence: the possibility of referring to a depends upon the possibility of referring to b.

Secondly, the very notion of the possibility of reference has to be analysed. In this respect, the so-called thesis of ideal reference (TIR: Martino 2001, 2004) will provide the logico-philosophical background for clarifying such a possibility.

I will show how TIR implies a reformulation of Russell's Vicious Circle Principle: (VCP*) no entity can be referred to only in terms of a totality it belongs to. VCP* nevertheless poses some tight restrictions on the fixing of the reference of terms: namely, such a fixing has to be strictly predicative.

These too tight restrictions can be liberalised through grounding of reference: given a language and a domain of entities for that language, we can fix the reference of the terms in stages, each one bearing upon the previous ones.

Finally, I will sketch a semantics accounting for grounded reference.

References

Martino, E. (2001), “Arbitrary Reference in Mathematical Reasoning”, Topoi, 20:65-77.

Martino, E. (2004), “Lupi, pecore e logica”, in M. Carrara and P. Giaretta (eds.), Filosofia e logica, Catanzaro, Rubettino: 103-33.

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Øystein Linnebo (Oslo):

I outline some desiderata for a theory of properties and then examine

some attempts to develop an account of ‘grounded’ properties..