Even though it is obvious that mathematics involves social activities, this rather trivial fact is rarely considered as important for its subject matter, mostly due to its undesired ontological consequences. An attempted solution for this tension was developed by Julian Cole's institutional account of mathematics, named Practice-Dependent Realism. In the present paper, Cole's account is evaluated, and its lights and shadows assessed concerning the * I am indebted to Giorgio Venturi, Luca San Mauro, Edson Bezerra and two anonymous referees for helpful comments and ideas on early drafts of the paper. The paper was written under a CNPq grant (170390/2017-9).
We analyze axioms and postulates as speech acts. After a brief historical appraisal of the concept of axiom in Euclid, Frege, and Hilbert, we evaluate contemporary axiomatics from a linguistic perspective. Our reading is inspired by Hilbert and is meant to account for the assertive, directive, and declarative components of modern axiomatics. We will do this by describing the constitutive and regulative roles that axioms possess with respect to the linguistic practice of mathematics.
Entre as principais contribuições de Frege à filosofia da matemática encontra-se a sua famosa definição de Ancestral de uma Relação. Apresentada primeiro na Be-griffsschrift de 1879, tal definição possui um papel crucial no logicismo fregeano ao estabelecer as condições para transformar a relação predecessora em uma série line-armente ordenada. Uma das objeções à definição de Frege provém de sua impredi-catividade e da circularidade resultante. Tal objeção foi primeiramente apresentada por Benno Kerry em 1887, e mais recentemente por Ignacio Angelelli em 2012. Neste artigo, argumento, a partir da perspectiva fregeana, que o Ancestral não é cir-cular, não obstante sua inevitável impredicatividade.
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