Sabe-se que o wittgensteiniano ver aspectos está intrinsecamente relacionado com a atividade matemática -- ao menos isso fica bastante claro no caso da geometria. No caso da álgebra, contudo, as coisas não são tão evidentes. Foi em um artigo de 1996 que Curtis e Lowe aprofundaram essa ideia ao proporem traduzir proposições matemáticas em termos de grafos. Nesse contexto, motivados pela expressividade dessa nova linguagem, os autores formularam um cálculo relacional orientado a fluxo para predicados binários e mostraram como as famosas sentenças de Lyndon podem ser facilmente derivadas. Contudo, questões fundamentais como consistência e completude foram deixadas para trabalhos futuros. Mais recentemente, no artigo 'On graph reasoning' (2009), as mesmas ideias de 1996 foram retomadas. Nesse artigo, os autores introduziram a linguagem +RG e demonstraram a sua consistência e completude (fraca) com relação ao cálculo relacional positivo. Desde então, no esforço de desenvolver as ideias seminais de Curtis e Lowe, novos trabalhos surgiram nesse mesmo sentido. Neste seminário, pretendo ilustrar como a relação de consequência pode ser compreendida em termos de homomorfismo entre grafos, avaliando a possibilidade de incorporar tal mecanismo em assistentes de prova. Para isso, irei sugerir a construção de algoritmos voltados para a derivação e decisão e esboçarei uma análise a complexidade computacional envolvida na tarefa.
SEMINAR - Finished
In this seminar it will be presented and discussed the recently published book "Conceptions of sets and the foundations of mathematics" (Cambridge University Press, 2020)".
Although both Cantor and Frege broadly used the same abstractionist strategy to define
numbers (in particular, cardinal numbers), Cantor’s conception has generally been viewed
as less justifiable and, overall, more problematic than Frege’s. Frege himself sharply
criticised Cantor’s conception in his review of Cantor (1890) (Frege (1892)). In this paper, we
aim to reassess Cantor’s conception on new grounds, show its fundamental
indistinguishability from Frege’s conception, and, above all, suggest that Cantor’s
abstractionism might help establish the inevitability of Hume’s Principle as the logical basis
of the definition of number, in particular, of the extension of the notion of ‘cardinality’ to
the infinite. In order to do this, we take into account several alternative notions of cardinal
number, including that provided by the recently developed theory of numerosities (in,
among other works, Benci-Di Nasso (2003)), and show that ‘good company’ arguments, put
forward by Heck (1997) and further discussed by Mancosu (2016), based on all such
alternative notions of cardinal number, fail to challenge the inevitability of Hume’s principle,
if the latter is construed in a way which does justice to a refined version of Cantor’s
abstractionism.
We explore many linguistic facets that permeate mathematical proofs. To do so, we analyze proofs through the lens of speech act theory, focusing on those illocutionary aspects which are peculiar to the mathematical discourse. Our analysis includes both the single speech acts that occur in proofs and the delicate interplay of such acts.
There are many different things in the world. There are trees, cars, buildings and pianos. But there are also thoughts, songs, universities and fictional characters. These objects seem to be organized in two sorts: concrete and abstract objects. Or, maybe we can divide them, more classically, between individuals, or particulars, and universals. There is also the view according to which there are arbitrary objects, as opposed to individual objects, and arbitrary objects are those which have a specific range, and present only properties common to all objects in their range. Or maybe not, arbitrary objects present those properties common to most objects in their range, like types. But are types arbitrary objects?
In this talk, we shall analyze these concepts, and others, finding out in what ways they are different - and why we seem to have come up with them -, and in what ways they are similar and intertwined.