Paul Rusnock

Bernard Bolzano (1781-1848) was a philosophical mathematician, especially interested in foundations and the analysis of important mathematical concepts. The notion of continuity was a subject of sustained reflection throughout his life. He deals with the notion in many settings: the theory of space (geometry), the theory of time (chronometry), the theory of functions (analysis), physics (continuous processes, matter), and numerical continuity (the theory of measurable numbers). One can also distinguish earlier and later versions of most of his work in these areas. This chapter provides an overview of his thoughts on these matters, along with some indications of the historical and philosophical context of his work.

Bolzano’s best-known and arguably best mathematical work, the _Rein analytischer Beweis_ of 1817, promises to deliver a _ground-revealing_ proof of an important theorem from the theory of equations, which Bolzano shows to follow from a generalization of the Intermediate Value Theorem. This paper explains and assesses this promise against the background of Bolzano’s early account of mathematical method, in which the idea of grounding plays a central role. In addition, it shows how Bolzano’s early views on proof, though still in an early stage of development, provided the basis for decisive criticisms of previous attempts to prove the theorem. Finally, Bolzano’s early views on proof and grounding are briefly considered in light of the later development of his thought.

This chapter contains translations of texts in which Bolzano develops and applies his mature theory of grounding. The heart of the chapter are excerpts from Bolzano’s main work _Theory of Science_, where he gives the most comprehensive account of his conception of grounding and discusses the role of grounding in several philosophical areas ranging from the methodology of science, to the philosophy of causation, to metaphysics. In addition to that, the chapter includes a brief excerpt from a survey of Bolzano’s _Theory of Science_ and his _Theory of the Science of Religion_, in which Bolzano gives a succinct summary of central ideas of his mature theory. Finally, the chapter includes the complete translation of Bolzano’s essay _What Is Philosophy?_ which defends a ground-theoretical conception of philosophy.

This chapter shows how epistemic coherence can be understood in terms of maximization of constraint satisfaction, in keeping with computational models that have had a substantial impact in cognitive science. It is shown how explanatory coherence subsumes Haack's recent “foundherentist” theory of knowledge. An account of deductive coherence is provided, showing how the selection of mathematical axioms can be understood as a constraint satisfaction problem. Visual interpretation can also be understood in terms of satisfaction of multiple constraints. After a brief account of how conceptual coherence can also be understood in terms of constraint satisfaction, the chapter concludes with a discussion of how the “multicoherence” theory of knowledge avoids many criticisms traditionally made of coherentism.

Patrick Grim, The incomplete universe:totality, knowledge, and truth. Cambridge, Mass, and London: The MIT Press, 1991. xiv + 165pp. £22.50 Jan SebestikLogique et mathématique chez Bernard Bolzano. Paris:Vrin, 1992. 522 pp. 198Fr J. De Lorenzo, Kant y la matemâtica. El uso constructivo de la razön pura Madrid:Editorial Tecnos, 1992. 180 pp. No price stated F. Coniglione, R. Poli And J. Woleintski, Polish scientific philosophy:The Lvov-Warsaw school. Amsterdam and Atlanta, Georgia: Rodopi, 1993. xi + 358pp. No price stated I. Kant Lectures on logic. Translated and edited by J. Michael Young. Cambridge:Cambridge University Press, 1992. xxxii + 695 pp. £55.00 Michael Dummett, The seas of language. Oxford:Clarendon Press, 1993. xvii + 482 pp. £40.00 Stephen Read,, Sophisms in medieval logic and grammar. Acts of the Ninth European Symposium for Medieval Logic and Semantics held at St Andrews, June 1990. Dordrecht, Boston, London:Kluwer Academic Publishers, 1993. xvii + 421pp. $157.50 James D. McCawley, Everything that linguists have always wanted to know about logic∗ ∗but were ashamed to ask. Second edition. Chicago and London:The University of Chicago Press, 1993. xxi + 633pp. £25.95/$37.50 ; £79.25/$113.75 Theodor Ebert, Dialektiker und frühe Stoiker bei Sextus Empiricus. Untersuchungen zur Entstehung der Aussagenlogik. Göttingen:Vandenhoeck & Ruprecht, 1991. 347 pp. DM 85.00 Entre la logique et la foi. Entretiens avec Joseph-M. Bochenski recueillis par Jan Parys. Translated from the Polish by Eric Morin-Aguilar. Montricher :Editions Noir-sur-Blanc, 1990. 353 pp. SFr 40.00 Noel Balzer, The human being as a logical thinker. Amsterdam and Atlanta:Rodopi 1993, 148 pp. No price stated R. I. G. Hughes, A philosophical companion to first-order logic. Indianapolis:Hackett, 1993, viii + 309 pp. No price stated Shahid Rahman, Über Dialoge, Protologische Kategorien und andere Seltenheiten, Frankfurt, Peter Lang Verlag, 1993. 318 pp. No price stated Claire Ortiz Hill, Word and object in Husserl, Frege, and Russell. The roots of twentieth century philosophy. Athens, Ohio:Ohio University Press, 1991. xiii + 215pp., $34.95

Bernard Bolzano is increasingly recognized as one of the greatest nineteenth-century philosophers. A philosopher and mathematician of rare talent, he made ground-breaking contributions to logic, the foundations and philosophy of mathematics, metaphysics, and the philosophy of religion. Many of the larger features of later analytic philosophy first appear in his work: for example, the separation of logic from psychology, his sophisticated understanding of mathematical proof, his definition of logical consequence, his work on the semantics of natural kind terms, or his anticipations of Cantor's set theory, to name but a few. To his contemporaries, however, he was best known as an intelligent and determined advocate for reform of Church and State. Based in large part on a carefully argued utilitarian practical philosophy, he developed a program for the non-violent reform of the authoritarian institutions of the Hapsburg Empire, a program which he himself helped to set in motion through his teaching and other activities. Rarely has a philosopher had such a great impact on the political culture of his homeland. Persecuted in his lifetime by secular and ecclesiastical authorities, long ignored or misunderstood by philosophers, Bolzano's reputation has nevertheless steadily increased over the past century and a half. Much discussed and respected in Central Europe for over a century, he is finally beginning to receive the recognition he deserves in the English-speaking world. This book provides a comprehensive and detailed critical introduction to Bolzano, covering both his life and works.

The impressive volume before us started out as an attempt to write “the history of epistemology since Kant, the way Carnap would have written it had he been Hegel.” Coffa began his project in 1981 while a fellow at the Center for Philosophy of Science in Pittsburgh and had finished a “good penultimate draft” when he suddenly died, after a brief illness, on 30 Dec., 1984. The title alludes to Edmund Wilson’s classic study of revolutionary ideology, To the Finland Station. This is no accident and would be presumptuous in a lesser work. The two studies cover roughly the same period, from the late eighteenth century to the first part of the present one, and they are both sparkling narratives that boldly project a large picture and then fill it in with canny detail. Coffa begins with Kant, the villain of the piece, and ends with a group of thinkers “in the neighborhood of Vienna” who came to terms with the restrictive framework they had inherited from Kant, finally overcoming it in favor of a new philosophy in which meaning, rather than mind, played the central role.

…all men begin… by wondering that things are as they are…as they do about…the incommensurability of the diagonal of the square with the side; for it seems wonderful to all who have not yet seen the reason, that there is a thing which cannot be measured even by the smallest unit. But we must end in the contrary and, according to the proverb, the better state, as is the case in these instances too when men learn the cause; for there is nothing which would surprise a geometer so much as if the diagonal turned out to be commensurable.

Received accounts of nineteenth-century thought make little or no room for the substantial contributions of the Bohemian philosopher Bernard Bolzano . This thesis constitutes a sustained attempt to show that this omission is a serious mistake, that Bolzano is more rightly treated as one of the major figures of western philosophy. To his contemporaries, Bolzano was known primarily as a teacher, religious leader, and social reformer. Outside of central Europe, though, he became known primarily through his logical and mathematical writings. The thesis focuses primarily on the latter, showing in detail how Bolzano's methodological work advanced revolutionary changes in both mathematics and logic, with effects which would be felt throughout philosophy.

In a series of publications beginning in the 1980s, John Etchemendy has argued that the standard semantical account of logical consequence, due in its essentials to Alfred Tarski, is fundamentally mistaken. He argues that, while Tarski's definition requires us to classify the terms of a language as logical or non-logical, no such division is guaranteed to deliver the correct extension of our pre-theoretical or intuitive consequence relation. In addition, and perhaps more importantly, Tarski's account is claimed to be incapable of explaining an essential modal/epistemological feature of consequence, namely, its necessity and apriority. Bernard Bolzano (1781-1848) is widely recognized as having anticipated Tarski's definition in his Wissenschaftslehre (or Theory of Science ) of 1837. Because of the similarities between his account and Tarski's, Etchemendy's arguments have also been extended to cover Bolzano. The purpose of this article is to consider Bolzano's theory in the light of these criticisms. We argue that, due to important differences between Bolzano's and Tarski's theories, Etchemendy's objections do not apply immediately to Bolzano's account of consequence. Moreover, Bolzano's writings contain the elements of a detailed philosophical response to Etchemendy

This edition provides the first complete English translation of Bernard Bolzano's four-volume Wissenschaftslehre or Theory of Science, a masterwork of theoretical philosophy. First published in 1837, the Wissenschaftslehre is a monumental, wholly original study in logic, epistemology, heuristics, and scientific methodology. Unlike most logical studies of the period, it is not concerned with the "psychological self-consciousness of the thinking mind." Instead, it develops logic as the science of "propositions in themselves" and their parts, especially the relations between these entities. It offers, for the first time in the history of logic, a viable definition of consequence (or deducibility), and a novel view of probability. Giving constant attention to Bolzano's predecessors and contemporaries, with particular emphasis on Kant, this richly documented work is also a valuable source for the history of logic and philosophy.

This paper surveys Bolzano's Beyträge zu einer begründeteren Darstellung der Mathematik (Contributions to a better-grounded presentation of mathematics) on the 200th anniversary of its publication. The first and only published issue presents a definition of mathematics, a classification of its subdisciplines, and an essay on mathematical method, or logic. Though underdeveloped in some areas (including,somewhat surprisingly, in logic), it is nonetheless a radically innovative work, where Bolzano presents a remarkably modern account of axiomatics and the epistemology of the formal sciences. We also discuss the second, unfinished and unpublished issue, where Bolzano develops his views on universal mathematics. Here we find the beginnings of his theory of collections, for him the most fundamental of the mathematical disciplines. Though not exactly the same as the later Cantorian set theory, Bolzano's theory of collections was used in very similar ways in mathematics, notably in analysis. In retrospect, Bolzano's debut in philosophy was a remarkably successful one, though its fruits would only become generally known much later

In the works of Kant and his followers, the notion of form plays an important role in explaining the apriority, necessity and certainty of logic. Bernard Bolzano (1781–1848), an important early critic of Kant, found the Kantians' definitions of form imprecise and their explanations of the special status of logic deeply unsatisfying. Proposing his own conception of form, Bolzano developed radically different views on logic, truth in virtue of form, and other matters. This essay presents Bolzano's views in the light of his criticisms of the Kantian logicians

Largely ignored in Germany during the nineteenth century, Bolzano was certainly better known in Austria, in particular among Brentano’s students, who enthusiastically studied his Theory of science. In this respect it makes sense to speak of Bolzano as belonging to a tradition of Austrian philosophy. Yet an examination of the reception of Bolzano’s ideas among Brentano’s students indicates that he was not always well understood. This article discusses a particular case, Twardowski’s reaction to Bolzano’s theory of representation.

This paper is devoted to an examination of Bolzano’s notion of necessary existence, which has so far received relatively little attention in the literature. We situate Bolzano’s ideas in their historical context and show how he proposed to correct various flaws of his predecessors’ definitions. Further, we relate Bolzano’s conception to his metaphysical and theological assumptions, arguing that some consequences of his definition which have been deemed counterintuitive by some of his interpreters turn out to be more reasonable given the broadly Leibnizian background of his metaphysics. Finally, we consider some difficulties that arise from Bolzano’s evolving views on freedom, which, at least in his early thought, was intimately linked with contingency. In an appendix, we discuss a recent debate on Bolzano’s notion of necessary truth between Textor and Rusnock that has some bearing on our overall line of interpretation of Bolzano’s notion of necessary existence.

Many contemporary philosophers favor coherence theories of knowledge (Bender 1989, BonJour 1985, Davidson 1986, Harman 1986, Lehrer 1990). But the nature of coherence is usually left vague, with no method provided for determining whether a belief should be accepted or rejected on the basis of its coherence or incoherence with other beliefs. Haack's (1993) explication of coherence relies largely on an analogy between epistemic justification and crossword puzzles. We show in this paper how epistemic coherence can be understood in terms of maximization of constraint satisfaction, in keeping with computational models that have had a substantial impact in cognitive science. A coherence problem can be defined in terms of a set of elements and sets of positive and negative constraints between pairs of those elements. Algorithms are available for computing coherence by determining how to accept and reject elements in a way that satisfies the most constraints. Knowledge involves at least five different kinds of coherence - explanatory, analogical, deductive, perceptual, and conceptual - each requiring different sorts of elements and constraints.

The history of speculation on a notion or notions called analyticity, now usually characterized as truth in virtue of meanings and independently of fact, is often viewed from the perspective of the Quine-Carnap dispute. Previous characterizations, due to Kant, Frege and others, are then seen as being of a piece with Carnap’s various definitions of analyticity, and thus open to Quine’s objections. Seen from this point of view, Bolzano’s claims about analyticity appear downright bizarre: for on his conception, analyticity is not only non-linguistic, but also independent of both apriority and necessity. In this paper, it is argued that the problem lies not with Bolzano, but rather with the received historical account, especially its interpretation of Kant.

Russell, in his History of Western Philosophy, wrote that modern analytical philosophy had its origins in the construction of modern functional analysis by Weierstrass and others. As it turns out, Bolzano, in the first four decades of the nineteenth century, had already made important contributions'to the creation of "Weierstrassian" analysis, some of which were well known to Weierstrass and his circle. In addition, his mathematical research was guided by a methodology which articulated many of the central principles of modern philosophical analysis. That Russell was able to discover philosophical content within mathematical analysis was thus not surprising, for it had been carefully put there in the first place. Bolzano can and should, accordingly, be viewed as a founder of modern analytical philosophy, and not necessarily as an uninfluential one. This paper considers his work in mathematical and philosophical analysis against some of the relevant historical background.

This essay presents a new interpretation of Bolzano's account of necessary truth as set out in ?182 of the Theory of Science. According to this interpretation, Bolzano's conception is closely related to that of Leibniz, with some important differences. In the first place, Bolzano's conception of necessary truth embraces not only what Leibniz called metaphysical or brute necessities but also moral necessities (truths grounded in God's choice of the best among all metaphysical possibilities). Second, in marked contrast to Leibniz, Bolzano maintains that there is still plenty of room for contingency even on this broader conception of necessity

Bernard Bolzano of Prague was one of the few thinkers of his time who combined real talent in mathematics and philosophy. He was especially drawn to the common ground between these fields, interested in questions of method and what would today be called foundations . Interestingly, he was neither a professional mathematician nor a professional philosopher. As a young man, he had decided that his first priority must be to work for the reform and improvement of society. This led him, after much reflection, to become a priest and to occupy a newly-created chair in religious science at the Charles University in 1805. In this post, he was widely known as what would later be called a dissident, denouncing abuses of power and the ills that were the consequence of the still largely feudal institutions and attitudes of the Austrian Empire, and inspiring a generation of students to transform these institutions from within. While his activities brought him a large and committed following, they also gained him powerful enemies in Vienna and Rome, who eventually convinced the Emperor to remove him from his post in December of 1819. Amazingly, during these years of energetic public life, in which he suffered horribly from lung disease and a variety of other ailments, Bolzano remained quite active as a mathematician. As his notebooks attest, he read the latest mathematical literature voraciously and systematically.1 He also made a number of important discoveries, especially in the foundations of real analysis, and published four mathematical works between 1810 and 1817 [Bolzano, 1810; 1816; 1817a; 1817b].2 After his dismissal, he continued his work, pursuing his earlier inquiries with greater depth and precision, …