The mathematical works of Bernard Bolzano published between 1804 and 1817

The purpose of the thesis is to assess the mathematical achievements of Bernard Bolzano on the basis of the five early published works. The material is divided into the areas of the foundations of mathematics, geometry and analysis. In making this assessment there have been two principal considerations. Firstly, any judgement of the significance of Bolzano's work should be made in the light of the historical context, so considerable space is devoted to the relevant 18th century sources. Secondly, as a general framework to the thesis there is the question of how Bolzano's general views about mathematical proofs and concepts are related to his achievements. The main claim and conclusion of the thesis is that this relationship was unusually clear and significant in the case of Bolzano's work. There is an Appendix containing the first English translation of all five of Bolzano's works as well as the German texts of their first editions

Stefan Roski, Bolzano’s Conception of Grounding

I review Stefan Roski's "Bolzano's Conception of Grounding"

When, and Why, did Frege read Bolzano?

Constructivism, Judgement and Meanin

Structure and Gestalt: Philosophy and Literature in Austria-Hungary and Her Successor States

The majority of the papers in the present volume were presented at, or prepared in conjunction with, meetings of the Seminar for Austro-German Philosophy, a group of philosophers interested in the work of Brentano and Husserl and of the ..

Naturalized Rationality. A Glance At Bolzano\u27s Philosophy Of Mind

Bernard Bolzano\u27s philosophy of mind is closely related to his metaphysical conceptions of substance, adherence and force. Questions as to how the mind is working are treated in terms of efficient (causal) faculties producing simple and complex representations, conclusive and non-conclusive judgments, and meta-representational attitudes such as believing and knowing. My paper outlines the proximity of Bolzano\u27s account of mental forces to contemporary accounts of faculty psychology such as Modularity Theory and Simple Heuristics. While the modularist notions of domain specificity and encapsulated mental faculties align with Bolzano\u27s allotment of domain specific tasks to correspondingly specified psychological forces (e.g. judging to judgmental force , inferring to inferential force etc.), the emphasis of Simple Heuristics on accurate fast and frugal processes aligns with Bolzano\u27s views regarding cognitive resources and the importance of epistemic economy. The paper attempts to show how Bolzano\u27s metaphysics of mind supposes a conception of bound rationality that determines his epistemology. Combining the rationalist concern for epistemic agent responsibility in the pursuit of knowledge with a strong confidence in the reliability of causal processes to generate the right beliefs, his epistemology shows close affinities with contemporary Virtue Epistemology. According to Virtue Epistemology, knowledge requires that true beliefs be generated by reliable processes typical of a virtuous character. The thesis that Bolzano anticipates virtue epistemological considerations is corroborated by his discussion of heuristic principles that set the norms for the acquisition of knowledge. The paper explores possible relations between such principles and the presumed low-level heuristics of cognitive processes

Acts of Time: Cohen and Benjamin on Mathematics and History

This paper argues that the principle of continuity that underlies Benjamin’s understanding of what makes the reality of a thing thinkable, which in the Kantian context implies a process of “filling time” with an anticipatory structure oriented to the subject, is of a different order than that of infinitesimal calculus—and that a “discontinuity” constitutive of the continuity of experience and (merely) counterposed to the image of actuality as an infinite gradation of ultimately thetic acts cannot be the principle on which Benjamin bases the structure of becoming. Tracking the transformation of the process of “filling time” from its logical to its historical iteration, or from what Cohen called the “fundamental acts of time” in Logik der reinen Erkenntnis to Benjamin’s image of a language of language (qua language touching itself), the paper will suggest that for Benjamin, moving from 0 to 1 is anything but paradoxical, and instead relies on the possibility for a mathematical function to capture the nature of historical occurrence beyond paradoxes of language or phenomenality

Investigations in Belnap's Logic of Inconsistent and Unknown Information

Nuel Belnap schlug 1977 eine vierwertige Logik vor, die -- im Gegensatz zur klassischen Logik -- die Faehigkeit haben sollte, sowohl mit widerspruechlicher als auch mit fehlender Information umzugehen. Diese Logik hat jedoch den Nachteil, dass sie Saetze der Form 'wenn ..., dann ...' nicht ausdruecken kann. Ausgehend von dieser Beobachtung analysieren wir die beiden nichtklassischen Aspekte, Widerspruechlichkeit und fehlende Information, indem wir eine dreiwertige Logik entwickeln, die mit widerspruechlicher Information umgehen kann und eine Modallogik, die mit fehlender Information umgehen kann. Beide Logiken sind nicht monoton. Wir untersuchen Eigenschaften, wie z.B. Kompaktheit, Entscheidbarkeit, Deduktionstheoreme und Berechnungkomplexitaet dieser Logiken. Es stellt sich heraus, dass die dreiwertige Logik, nicht kompakt und ihre Folgerungsmenge im Allgemeinen nicht rekursiv aufzaehlbar ist. Beschraenkt man sich hingegen auf endliche Formelmengen, so ist die Folgerungsmenge rekursiv entscheidbar, liegt in der Klasse $\Sigma_2^P$ der polynomiellen Zeithierarchie und ist DIFFP-schwer. Wir geben ein auf semantischen Tableaux basierendes, korrektes und vollstaendiges Berechnungsverfahren fuer endliche Praemissenmengen an. Darueberhinaus untersuchen wir Abschwaechungen der Kompaktheitseigenschaft. Die nichtmonotone auf S5-Modellen basierende Modallogik stellt sich als nicht minder komplex heraus. Auch hier untersuchen wir eine sinnvolle Abschwaechung der Kompaktheitseigenschaft. Desweiteren studieren wir den Zusammenhang zu anderen nichtmonotonen Modallogiken wie Moores autoepistemischer Logik (AEL) und McDermotts NML-2. Wir zeigen, dass unsere Logik zwischen AEL und NML-2 liegt. Schliesslich koppeln wir die entworfene Modallogik mit der dreiwertigen Logik. Die dabei enstehende Logik MKT ist eine Erweiterung des nichtmonotonen Fragments von Belnaps Logik. Wir schliessen unsere Betrachtungen mit einem Vergleich von MKT und verschiedenen informationstheoretischen Logiken, wie z.B. Nelsons N und Heytings intuitionistischer Logik ab