conceptual

  Toda epistemología realista de las matemáticas, es decir, toda epistemología que acepte que hay objetos matemáticos reales , es decir, independientes de nuestras prácticas históricas concretas, nuestras convenciones y/o aparato cognitivo, y que la disciplina que llamamos así se dedica a su estudio, enfrenta tres retos. Por un lado, (1) necesita explicar nuestro acceso epistémico a los objetos matemáticos, es decir, nuestra capacidad de tener pensamientos acerca de los objetos propios de las matemáticas. Este reto es igualmente difícil para racionalistas como para empiristas. En la tradición racionalista, este acceso epistémico se suele lograr apelando a algún tipo de intuición racional (Brown, Parsons, Gödel, etc.). Mientras que en la tradición empirista, lo mismo se logra a través de la postulación de un mecanismo, que podríamos llamar de abstracción en un sentido amplio, ya sea lógico (como el propuesto por logicistas y neo-logicistas), psicológico (como el propuesto por los ...

Back when I was trying to find a modal (also sometimes called "intensional" or "combinatorial", in the literature) definition of intrinsicality, everyone was banging their head against a point discovered by Lewis: that there are two different ways of understanding what it means for some one aspect of reality (a parameter)  E to be sensitive to another one  F  (commonly, a contextual feature, but not necessarily): 1. Dependance : The value of E  depends on F . 2. Shiftiness : The value of E  changes with F . One might think that both things are just different aspects of the same phenomenon and that they are, at least, extensionally equivalent. However, a little bit of reflection reveals that this is not so. It is perfectly possible for a parameter to get the same value in different ways depending on different features of its context. This phenomenon is most easily seen in natural language, as Kaplan clearly showed in his logic of indexicals . Thus, it is perfec...

  One might think that this is obviously true: Two expressions are synonyms if they have the same content, and having the same content is a transitive relation. This is a direct corollary from a more general principle that if we define a relation by an identity (or, in general, we have a relation that is equivalent to some identity), the relation will inherit from the relevant identity its structural features, including transitivity. In other words, for every R if there is an F that R(x,y) =def F(x)=F(y) then R is transitive. This applies not only to the definition of synonym as sameness of meaning, but also to the definition of synonym as substitution salva veritate : φ is cognitively synonym to ψ if and only if, for all allowed sentential  contexts χ[. . .], we have χ[φ] iff χ[ψ]. Here, the double conditional in the final equation hides also an identity (of truth value). This can be made explicit thus: φ is a cognitive synonym of ψ if and only if, for all allowed sente...

There are some problematic cases for defenders of the hypothesis that logical form is topic-transparent, i.e. the claim that two statements that differ only in the logical composition off its atomic components cannot differ in topic:  On the one hand, many people, myself included, claim that tautologies are not about anything in the content of the atomic components that occur in them, but about the logical operators themselves. Presumably, sentences of the form ( P → ( Q → P )) are not about whatever P and Q are about, but about material implication: in particular, they tell us that if the consequent of an implication is true, the whole implication is true as well. The basic argument for this later claim is that whatever P and Q are about makes no difference to the content of the tautology.  Based on Wittgenstein, Lazerowitz and Ambrose, and myself , we have argued that even though sentences like “Triangles are my favorite geometrical figures” are about triangles, i...

¿En qué sentido se habla de "relevancia" en las lógicas relevantes? ¿Qué relación tienen con el lenguaje natural, si alguno? Trato de responder estas y otras preguntas en esta entrevista.

Notes on Franz Berto’s first talk of the 2023 Cátedra Gaos Just as to be an intentionalist (like Kripke, but unlike Quine) is to take extentionality to be too coarse as tool for philosophically analyzing some phenomena, to be a hyperintentionalist (like Dunn and Restall, Berto, Chalmers, Yablo, Fine, etc. but unlike Lewis (sometimes), Stalker   (sometimes) , Montague, etc.) is to take intensionality (traditional modality) to be too coarse as tool for philosophically analyzing some phenomena. Intentionality is to material equivalence as hyper-intentionality is to necessary equivalence, but this can be extended ad infinitum, at least in principle, thus: Let R be an equivalence relation on a representational domain D , then H is a hyper-R operator iff not- R(H(p), H(q)) in spite of R(p, q) . In this schema, intentionality is hyper-extensionality, and hyper-intentionality is, well, hyper-intentionality. Thus, there can be hyper-hyper-intentionality and hyper-hyper-hyp...

What I saw at Marco Panza and collaborators’ Euclid session at Chapman University this Friday October 13th: Looking, right now at mathematicians workshop looking for a non-standard model to a formal system of axioms, it is very interesting to see how the whiteboard is used. For example, by explicitly writing out the formulas, it raises the salience of the operations involved. Thus, once it was shown, by explicitly writing the relevant formula, that a certain parameter was being calculated by a quadratic equation, this immediately suggested the use of irrationals to find the desired non-standard model.  Similarly, using ad-hoc formulas i.e. displays that belong to no actual standardized formal language but share a superficial grammar with them, for example, at some time Marco Panza wrote “SAS : Ang → Sea” to represent the fact that, in Euclid, the so-called Segment-Angle-Segment axiom (which they never called that way, always using the acronym “SAS”) states that, given c...