Stefania Centrone

The name “Becker’s rule” was coined by C. West Churchman in 1938 to denote the modal inference rule according to which the theoremhood of a strict implication of the form may be inferred from the theoremhood of a strict implication of the form. According to Churchman, such a rule was introduced explicitly by Oskar Becker in his On the Logic of Modalities (1930)—hence the name he gave to it, which is still current in the literature. The aim of this note is to point out that “Becker’s rule” is not at all Becker’s rule (a fact which has remained unnoticed until now, as far as we know) by explaining where and how Churchman misunderstood what Becker intended.

Dieser Artikel untersucht Gottfried Wilhelm Leibniz als Erfinder einiger Konzepte, die der künstlichen Intelligenz zugrunde liegen. Leibniz Ideen einer lingua characteristica und eines calculus ratiocinator werden an ihrem Entstehungsort, der Dissertatio de Arte Combinatoria (1666), untersucht. Zudem wird auf ihr Vorbild, die ars magna von Raymund Lull, hingewiesen und einige wichtige Ausarbeitungen der Leibnizschen Konzepte betrachtet, wie die logische Grammatik von Edmund Husserl, die Algebra der Logik von George Boole und die Begriffsschrift von Gottlob Frege. Darüber hinaus wird eine Interpretation der Leibnischen Konzepte im Lichte der Gödelschen Sätze versucht und die wichtigste Praezisierung des Begriffs eines Algorithmus, die Turingmaschine, analysiert. Schließlich werden Leibniz’ Dualsystem und Leibniz’ Rechenmaschine dargestellt.

We aim at clarifying to what extent the work of the English mathematician George Boole on the algebra of logic is taken into consideration and discussed in the work of early Husserl, focusing in particular on Husserl’s lecture “Über die neueren Forschungen zur deduktiven Logik” of 1895, in which an entire section is devoted to Boole. We confront Husserl’s representation of the problem-solving processes with the analysis of “symbolic reasoning” proposed by George Boole in the Laws of Thought and try to show how and why Husserl, while praising Boole’s calculus, strongly criticizes his attempt at a philosophical clarification and justification of it.

In the last few years research on Husserl has more and more brought attention to his contributions to logic and to philosophy of mathematics. Phenomenology and Mathematics participates in this trend; ‘[i]t gathers the contributions of the main scholars of the field into one publication for the first time’ (p. xxi) and is remarkably successful in giving ‘an overview of the current debates and themes in the phenomenology of mathematics’ (loc. cit.). As the editor, Mirja Hartimo, declares in her Introduction, the volume can be read as an answer to the question, ‘What kind of philosophy of mathematics is phenomenology?’ (loc. cit.). In the following I will select some questions that mathematics poses to philosophical reflection and see to what extent they are dealt with and answered in the volume and to what extent the volume is successful in justifying the underlying claim that we have to use a phenomenological reading-glass to look at what is going on in mathematical reasoning.

In a fragment entitled Elementa Nova Matheseos Universalis Leibniz writes “the mathesis [...] shall deliver the method through which things that are conceivable can be exactly determined”; in another fragment he takes the mathesis to be “the science of all things that are conceivable.” Leibniz considers all mathematical disciplines as branches of the mathesis and conceives the mathesis as a general science of forms applicable not only to magnitudes but to every object that exists in our imagination, i.e. that is possible at least in principle. As a general science of forms the mathesis investigates possible relations between “arbitrary objects”. It is an abstract theory of combinations and relations among objects whatsoever. In 1810 the mathematician and philosopher Bernard Bolzano published a booklet entitled Contributions to a Better-Grounded Presentation of Mathematics. There is, according to him, a certain objective connection among the truths that are germane to a certain homogeneous field of objects: some truths are the “reasons” of others, and the latter are “consequences” of the former. The reason-consequence relation seems to be the counterpart of causality at the level of a relation between true propositions. Arigorous proof is characterized in this context as a proof that shows the reason of the proposition that is to be proven. Requirements imposed on rigorous proofs seem to anticipate normalization results in current proof theory. The contributors of Mathesis Universalis, Computability and Proof, leading experts in the fields of computer science, mathematics, logic and philosophy, show the evolution of these and related ideas exploring topics in proof theory, computability theory, intuitionistic logic, constructivism and reverse mathematics, delving deeply into a contextual examination of the relationship between mathematical rigor and demands for simplification.

In this chapter, the objective of this work, which is to introduce many-valuedness to meta-logical notions like consequence, consistency/inconsistency, tautologihood, etc. involved in a logical discourse, is stated. To arrive at this end the issues that have been sailed through are (i) three levels inherent in a logic discourse, (ii) from many-valued logics, fuzzy logics to graded consequence: a brief overview, (iii) a general discussion on uncertainty and vagueness, (iv) notion of consequence in classical logic and (v) finally some motivations for lifting many-valuedness to the meta-level.

This edited work presents contemporary mathematical practice in the foundational mathematical theories, in particular set theory and the univalent foundations. It shares the work of significant scholars across the disciplines of mathematics, philosophy and computer science. Readers will discover systematic thought on criteria for a suitable foundation in mathematics and philosophical reflections around the mathematical perspectives. The first two sections focus on the two most prominent candidate theories for a foundation of mathematics. Readers may trace current research in set theory, which has widely been assumed to serve as a framework for foundational issues, as well as new material elaborating on the univalent foundations, considering an approach based on homotopy type theory (HoTT). The further sections then build on this and are centred on philosophical questions connected to the foundations of mathematics. Here, the authors contribute to discussions on foundational criteria with more general thoughts on the foundations of mathematics which are not connected to particular theories. This book shares the work of some of the most important scholars in the fields of set theory (S. Friedman), non-classical logic (G. Priest) and the philosophy of mathematics (P. Maddy). The reader will become aware of the advantages of each theory and objections to it as a foundation, following the latest and best work across the disciplines and it is therefore a valuable read for anyone working on the foundations of mathematics or in the philosophy of mathematics.

The notion of mathesis universalis appears in many of Edmund Husserl’s works, where it corresponds essentially to “a universal a priori ontology”. This paper has two purposes; one, largely exegetical, of clarifying how Husserl elaborates on Leibniz’ concept of mathesis universalis and associated notions like symbolic thinking and symbolic knowledge filtering them through the lesson of the so called “bohemian Leibniz”, Bernard Bolzano; another, more properly philosophical, of examining the role that the universal mathesis is allowed to play, and the space it occupies in Husserl’s intuition-based epistemology.

In the present paper, we discuss Husserl's deep account of the notions of ?calculation? and of arithmetical ?operation? which is found in the final chapter of the Philosophy of Arithmetic, arguing that Husserl is as far as we know the first scholar to reflect seriously on and to investigate the problem of circumscribing the totality of computable numerical operations. We pursue two complementary goals, namely: (i) to provide a formal reconstruction of Husserl's intuitions, and (ii) to demonstrate on the basis of our reconstruction that the class of operations that Husserl has in mind turns out to be extensionally equivalent to the one that, in contemporary logic, is known as the class of partial recursive functions