Artículos con la etiqueta ‘Fundamentos de matemática’

Formalizing set theory in weak logics, searching for the weakest logic with Gödel’s incompleteness property

Por • 16 nov, 2011 • Category: Opinion

We show that first-order logic can be translated into a very simple and weak logic, and thus set theory can be formalized in this weak logic. This weak logical system is equivalent to the equational theory of Boolean algebras with three commuting complemented closure operators, i.e., that of diagonal-free 3-dimensional cylindric algebras (Df_3’s). Equivalently, set theory can be formulated in propositional logic with 3 commuting S5 modalities (i.e., in the multi-modal logic [S5,S5,S5]). There are many consequences, e.g., free finitely generated Df_3’s are not atomic and [S5,S5,S5] has G\»odel’s incompleteness property. The results reported here are strong improvements of the main result of the book: Tarski, A. and Givant, S. R., Formalizing Set Theory without variables, AMS, 1987.

Meaning in Classical Mathematics: Is it at Odds with Intuitionism?

Por • 6 nov, 2011 • Category: Ciencia y tecnología

We examine the classical/intuitionist divide, and how it reflects on modern theories of infinitesimals. When leading intuitionist Heyting announced that «the creation of non-standard analysis is a standard model of important mathematical research», he was fully aware that he was breaking ranks with Brouwer. Was Errett Bishop faithful to either Kronecker or Brouwer? Through a comparative textual analysis of three of Bishop’s texts, we analyze the ideological and/or pedagogical nature of his objections to infinitesimals a la Robinson. Bishop’s famous «debasement» comment at the 1974 Boston workshop, published as part of his Crisis lecture, in reality was never uttered in front of an audience. We compare the realist and the anti-realist intuitionist narratives, and analyze the views of Dummett, Pourciau, Richman, Shapiro, and Tennant. Variational principles are important physical applications, currently lacking a constructive framework. We examine the case of the Hawking-Penrose singularity theorem, already analyzed by Hellman in the context of the Quine-Putnam indispensability thesis.

Alan Turing and the Origins of Complexity

Por • 9 oct, 2011 • Category: Filosofía

The 75th anniversary of Turing’s seminal paper and his centennial year anniversary occur in 2011 and 2012, respectively. It is natural to review and assess Turing’s contributions in diverse fields in the light of new developments that his thoughts has triggered in many scientific communities. Here, the main idea is to discuss how the work of Turing allows us to change our views on the foundations of Mathematics, much like quantum mechanics changed our conception of the world of Physics. Basic notions like computability and universality are discussed in a broad context, making special emphasis on how the notion of complexity can be given a precise meaning after Turing, i.e., not just qualitative but also quantitative. Turing’s work is given some historical perspective with respect to some of his precursors, contemporaries and mathematicians who took up his ideas farther.