Reflections on the Foundations of Mathematics
Univalent Foundations, Set Theory and General Thoughts
This volume 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 of mathematical practice and computer science.
The volume shares the work of some of the most important scholars in the fields of set theory (S. Friedman), non-classical logic (G. Priest) and philosophy of mathematics (P. Maddy).
Mathesis Universales, Computability and Proof
In a fragment entitled Elementa Nova Matheseos Universalis (1683?) 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” (“objets quelconques”). 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” (“Gründe”) of others, and the latter are “consequences” (“Folgen”) of the former. The reason-consequence relation seems to be the counterpart of causality at the level of a relation between true propositions. A rigorous 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.
Essays on Husserl’s Logic and Philosophy of Mathematics
Essays on Husserl’s Logic and Philosophy of Mathematics sets out to fill up a lacuna in the present research on Husserl by presenting a precise account of Husserl’s work in the field of logic, of the philosophy of logic and of the philosophy of mathematics. The aim is to provide an in-depth reconstruction and analysis of the discussion between Husserl and his most important interlocutors, and to clarify pivotal ideas of Husserl’s by considering their reception and elaboration by some of his disciples and followers, such as Oskar Becker and Jacob Klein, as well as their influence on some of the most significant logicians and mathematicians of the past century, such as Luitzen E. J. Brouwer, Rudolf Carnap, Kurt Gödel and Hermann Weyl. Most of the papers consider Husserl and another scholar – e.g. Leibniz, Kant, Bolzano, Brentano, Cantor, Frege – and trace out and contextualize lines of influence, points of contact, and points of disagreement. Each essay is written by an expert of the field, and the volume includes contributions both from the analytical tradition and from the phenomenological one.
Versuche über Husserl
Husserls Werk ist an begrifflichen Herausforderungen gewiss nicht arm, was durchaus seiner messianischen Überzeugung entspricht, »dass aus meiner Lebensarbeit eine völlige Umwälzung des ganzen Stils, der notwendigen Problemstellung der gesamten Philosophie der Jahrtausende hervorgeht«.
Dieser Band versammelt acht leicht verständliche Interpretationen namhafter Husserl-Forscher zu den zentralen Begriffen der Intentionalität, Reflexion, Eidetik und Evidenz, zur Rolle des Handelns, zum Personenverstehen und zur Rechtfertigung sowie zu Husserls Position im Psychologismus-Streit. Zugleich versuchen die Autoren, die Diskussionsgrundlagen zwischen Husserl und einigen seiner wichtigsten und einflussreichsten Gesprächspartner wiederherzustellen: Bernard Bolzano, Franz Brentano, Gottlob Frege, Martin Heidegger und Ludwig Wittgenstein. Ergänzt wird der Band durch einen Abriss zu Husserls Leben, Werk und Wirkung.
Mit Beiträgen von: Christian Beyer, Stefania Centrone, Dagfinn Føllesdal, George Heffernan, Wolfgang Künne, Eduard Marbach und Markus Stepanians