The so-called Maverick Turn in the Philosophy of Mathematics has been marked by an increasing focus on the mathematical practice. One pressuposition, found in many maverick authors as Lakatos (1976), Kitcher (1984), Hersh (1997), Ernest (1998) and Cole (2013), is that taking the practice at face value incline us to reject any realist explanation for mathematical objects, or even for mathematical truth. As they mostly argued, any reasonable account of the practice is incompatible with theories that explains mathematics in aprioristic terms. In this talk, I want to evaluate this pressuposition in the following twofold manner: first, by showing that some of the realist features are actually pressuposed in the practice, and second, that authors that seek to explain mathematics aprioristically were not completely blind to the practice. The paradigmatic case will be Frege: the realist spokesman and common target of the mavericks.
SEMINAR - Finished
We will read and discuss the second chapter of Luca Incurvati's book Conceptions of Sets and the Foundations of Mathematics, Cambridge University Press, 2020.
Nesta fala apresentarei uma teoria da validade para a lógica LP, que chamaremos de Val-LP. Noutras palavras, estenderei a linguagem de LP com um predicado primitivo de validade Val(x) tal que Val(‘φ’) é verdadeiro se φ é uma fórmula válida de LP. Veremos que em Val-LP duas noções de consistência são formalizáveis e quais são as propriedades delas. Posteriormente, apresentaremos a lógica LPS0.5, que estende LP com uma modalidade não-normal []. Para LPS0.5 apresentaremos um sistema de tablôs completos e corretos. Por fim, mostramos que LPS0.5 é a teoria de validade de LP mediante uma tradução r.
As Fine would say, There is the following view. In addition to individual objects, there are arbitrary objects. But for how long has this view been held? If we look close enough, we find talk about arbitrary objects, disguised in different terminologies, when Locke talks about general ideas, or even when Plato talks about Ideas. Yet, many have cast aside the possibility of such sort of objects, claiming them to be dispensable, or even outright contradictory. This is due to their characteristic feature: a certain arbitrary P must have the properties common to all individual P's, and only those properties. This feature, however, is remarkably close to the one presented by another important concept, essence. In fact, for Locke, at least in one understanding of essence (nominal essence), an arbitrary P and the essence of P are the same thing. Zalta's theory of objects also implies a similar reduction. This suggests a close connection between arbitrary objects and essence.
In this talk, we shall analyze the general form of arguments for and against the existence of arbitrary objects, and see how these arguments have been presented by different authors throughout the history of Philosophy. We, then, present different theories of arbitrary objects and of essence put forward by some of these authors, and see how the two concepts are connected in each of them. At last, we analyze the formal aspects of these theories, and how a same logical framework, with different interpretations, may be used to explain how essence and arbitrariness work.
We will read and discuss the first chapter of Luca Incurvati's book Conceptions of Sets and the Foundations of Mathematics, Cambridge University Press, 2020.