This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society

The collected works

A non-monotonic logic, the Logic of Plausible Reasoning (LPR), capable of coping with the demands of what we call complex reasoning, is introduced. It is argued that creative complex reasoning is the way of reasoning required in many instances of scientific thought, professional practice and common life decision taking. For managing the simultaneous consideration of multiple scenarios inherent in these activities, two new modalities, weak and strong plausibility, are introduced as part of the Logic of Plausible Deduction (LPD), a deductive logic specially designed to serve as the monotonic support for LPR. Axiomatics and semantics for LPD, together with a completeness proof, are provided. Once LPD has been given, LPR may be defined via a concept of extension over LPD. Although the construction of LPR extensions is first presented in standard style, for the sake of comparison with existing non-monotonic formalisms, alternative more elegant and intuitive ways for constructing non-monotonic LPR extensions are also given and proofs of their equivalence are presented.

https://www.jstor.org/stable/info/27653468

A logical expression of reasoning

Novice students have difficulty with the topic of limits in calculus. We believe this is in part because of the multiple perspectives and shifting metaphors available to solve items correctly. We investigated college calculus instructors' personal concepts of limits. Based upon previous research investigating introductory calculus student metaphorical reasoning, we examined 11 college instructors' metaphorical reasoning on limit concepts. This paper focused on previous research of metaphor clusters observed among students to answer the following: (a) Do college instructors use metaphorical reasoning to conceptualize the meaning of a limit? (b) Can we characterize instructor metaphorical reasoning similar to those observed among students? (c) Will an instructor's self-identification of metaphor clusters be consistent with our metaphor coding? We found that college instructors' perspectives vary, either graphical or algebraic, in their explanations of limit items. All the instructors used metaphors, and instructor metaphorical reasoning aligned with student metaphor clusters. Instructors tended to change their metaphors with respect to the limit item. Instructors were not aware of their use of metaphors, nor were they aware of their inconsistency in their choice of metaphor. We believe that instructor awareness of their own distinct perspectives and metaphors would assist students' understanding of limit concepts.

Metaphor Clusters: Characterizing Instructor Metaphorical Reasoning on Limit Concepts in College Calculus

The general purpose of this paper is to show a practical instance of how philosophy can benefit from some ideas, methods and techniques developed in the field of Artificial Intelligence (AI). It has to do with some recent claims [4] that some of the most traditional philosophical problems have been raised and, in some sense, solved by AI researchers. The philosophical problem we will deal with here is the representation of non-deductive intra-theoretic scientific inferences. We start by showing the flaws with the most traditional solution for this problem found in philosophy: Hempel’s Inductive-Statistical (I-S) model [5]. After we present a new formal model based on previous works motivated by reasoning needs in Artificial Intelligence [11] and show that since it does not suffer from the problems identified in the I-S model, it has great chances to be successful in the task of satisfactorily representing the non-deductive intra- theoretic scientific inferences.

Is Plausible Reasoning a Sensible Alternative for Inductive-Statistical Reasoning?

O objetivo desse artigo e realizar o que podemos chamar de uma analise conceitual da nocao de inducao, tomando como ponto de partida o problema das ambiguidades indutivas. Tentaremos mostrar que existe nao apenas uma, mas duas maneiras igualmente autenticas de lidar com o problema das ambiguidades indutivas, e que quando certos aspectos logicos dos dois conceitos de plausibilidade oriundos dessas duas abordagens da inducao sao considerados, muito da controversia a respeito das ambiguidades indutivas e dissolvido. Dentre tais aspectos esta uma suposta relacao entre essas duas nocoes indutivas de plausibilidade e os conceitos de paraconsistencia e paracompletude. Nossa analise e materializada, por assim dizer, em uma logica modal paraconsistente e paracompleta capaz de dar conta dos aspectos mais basicos dessas duas abordagens da inducao bem como dos dois conceitos de plausibilidade delas oriundos. Na exemplificacao do tipo de controversia o qual estamos nos referindo, bem como a solucao que estamos propondo, nos detalharemos sobre uma das mais ilustres instâncias do problema das inconsistencias indutivas: o paradoxo da loteria. The purpose of this paper is to undertake a conceptual analysis of the notion of induction, taking as starting-point the problem of inductive ambiguities. We shall try to show that there is not only one, but two equally authentic ways of dealing with the problem of inductive ambiguities, and that when certain logical aspects of the two plausibility concepts that arise from these two approaches to induction are considered, much of the controversy about the inductive ambiguities disappears. Among such aspects is a relation that supposedly exists between these two notions of inductive plausibility and the concepts of paraconsistency and paracompleteness. Our analysis is materialized, so to speak, in a paraconsistent and paracomplete modal logic able to deal with the most basic aspects of these two approaches to induction. In exemplifying the kind of controversy we are talking about, as well as the solution we are proposing, we shall focus on one of the noblest instances of the problem of inductive inconsistencies: the lottery paradox.

/pdf/ambiguidades-indutivas-paraconsistencia-paracompletude-e-as-38sfomzadz.pdf

Ambiguidades indutivas, paraconsistência, paracompletude e as duas abordagens da indução

The field of nonmonotonic logic, sixteen years old now, is devoted to solve the problem of reasoning under incomplete knowledge, whose good understanding is essential to the construction of AI as a science and whose relevance reaches far beyond AI applications. During these years, many insights have been accumulated in the form of desirable properties the proposed formalisms should exhibit and of criticisms on the available solutions. This paper takes advantage on this experience to derive from them a sort of canon to be imposed to nonmonotonic formalisms. This canon is translated as a set of etiquette rules guiding knowledge representation into theories framed within the Inconsistent Default Logic, IDL. It is then established the important result that IDL produces a unique extension for a theory constructed according to these rules. This result calls forth IDL as an interesting alternative to credulous common sense reasoning formalization fulfilling many desired properties.

Marcelino Pequeno

Papers

A logical expression of reasoning

Well-Behaved IDL Theories

A Multiple Worlds Semantics for a Paraconsistent Nonmonotonic Logic