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Seminario di Logica e Filosofia della Scienza

Organizzazione Unità di ricerca LOG-LAB

 

 

Venerdì 1 marzo 2024 ore 11:00
Dipartimento di Lettere e Filosofia
via della Pergola 60 - Firenze
Sala Altana

 

Paul Gorbow (University of Stockholm and University of Oslo)

Dynamic Structures -- a Modal Explanation of Certain Refinement Processes in Mathematics

 

Abstract: The iterative conception of the natural numbers has been given a modal potentialist explanation by Linnebo and Shapiro. The universe of natural numbers is given as a potentialist structure, i.e. a possible worlds model, P, where each world is an initial segment of the standard model, N, of the natural numbers, and the accessibility relation is given by the substructure relation. The so-called Mirroring theorem establishes a correspondence to the effect that a formula holds in N iff its full modalization holds in P. Any fully modalized formula A is stable in P, meaning that A→□A. However, the richer modal language of P has many unstable formulas, e.g. "x is the largest number". The modal potentialist is committed to that at any stage of the process there is a maximal actualized number. I introduce the notion of dynamic numbers to track such entities. E.g. letting max by the dynamic maximal number, we obtain:
• □"max is maximal"
• □(◊"max is even" ∧ ◊"max is odd")
• For each standard n: ◊□(n < max).

 

Although not in the range of the Mirroring theorem, these truths in P bring forth characteristics of modal potentialism. Generalizing the above, I introduce the notion of dynamic structures. These include:
• Potentialist structures corresponding to standard models.
• Structures corresponding to models constructed as ultrapowers. 

 

Dynamic structures have the following theoretical virtues:
• They explain the properties of the corresponding mathematical structures.
• They are ontologically advantageous compared to the corresponding mathematical structures.
• They are free from the arbitrary choices of the corresponding mathematical structures.
• They are often recursive, even when the corresponding mathematical structure is not.

 

As for applications, philosophical explanations emerge for the following entities:
• "Benign" non-standard numbers.
• The class of truths in the revision semantics of the truth predicate.

 

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Martedì 27 febbraio 2024 ore 11:00
Plesso didattico Capponi, via Capponi 9 - Firenze
Aula 6

 

Josè Ferreiros (Universidad de Sevilla)

Conceptual Structuralism

 

Abstract: In this talk, I defend a conceptualistic version of structuralism as the most convincing way of elaborating a philosophical understanding of structuralism in line with the classical tradition. The argument begins with a revision of the tradition of “conceptual mathematics”, incarnated in key figures of the period 1850 to 1940 like Riemann, Dedekind, Hilbert or Noether, showing how it led to a structuralist methodology. Then the tension between the ‘presuppositionless’ approach of those authors, and the platonism of some recent versions of philosophical structuralism, is presented. In order to resolve this tension, we argue for the idea of ‘logical objects’ as a form of minimalist realism, again in the tradition of classical authors including Peirce and Cassirer, and we introduce the basic tenets of conceptual structuralism. The remainder of the talk is devoted to an open discussion of the assumptions behind conceptual structuralism, and—most importantly—an argument to show how the objectivity of mathematics  can be explained from the adopted standpoint. This includes the idea that advanced mathematics builds on hypothetical assumptions (Riemann, Peirce, and others), which is presented and discussed in some detail. Finally, the ensuing notion of objectivity is interpreted as a form of particularly robust intersubjectivity, and it is distinguished from fictional or social ontology.

 

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Venerdì 23 febbraio 2024 ore 11:00
Dipartimento di Lettere e Filosofia
via della Pergola 60 - Firenze
Sala Altana

 

Casey McCoy (Yonsei University)

Pursuitworthiness and Epistemic Justification of Scientific Theories

 

Abstract: Much of modern philosophy of science with an epistemological orientation has focused on questions of justification, following on from the classic distinction between the context of discovery and the context of justification. Lately, Laudan's idea of a context of pursuit has been revived as a means of deflating certain justificatory accounts centered on justification, such as inference to the best explanation and meta-empirical confirmation. In this paper, we criticize such efforts to decouple assessments of pursuitworthiness from epistemic assessments, and argue for two main points: that assessing the viability of a theory is essential to assessing its pursuitworthiness, and that pursuitworthiness assessments are worthwhile only to the extent that pursuitworthy theories tend to be epistemically successful.

 

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Ultimo aggiornamento

23.02.2024

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