Two hundred years ago Bernard Bolzano published a booklet on the philosophy of mathematics that is the first major step forward in this area since Pascal’s De l’esprit géométrique. Following Aristotelian lines Bolzano distinguishes in his opusculum two kinds of proofs, those that simply show that something is the case, and those that explain why something is the case. In his Wissenschaftslehre this contrast reappears as that between derivability and consecutivity . Husserl takes up some of Bolzano’s key concepts in (...) his Prolegomena to Pure Logic. In this paper I discuss, among other things, the question whether consecutivity can be regarded as a special case of derivability, as Husserl seems to think, and I contrast Bolzano’s rejection of any appeal to self-evidence with Husserl’s reliance on this notion.Genau vor zweihundert Jahren erschienen Bernard Bolzano Beyträge zu einer begründeteren Darstellung der Mathematik. Dieses Büchlein ist der erste Meilenstein in der Geschichte der Philosophie der Mathematik seit Pascals De l’esprit géométrique. Im Anschluss an Aristoteles unterscheidet Bolzano in seinem Opusculum zwei Arten von Beweisen: solche, die nur dartun, dass etwas der Fall ist, und solche, die erklären warum etwas der Fall ist. Im seiner Wissenschaftslehre erscheint dieser Kontrast als der zwischen Ableitbarkeit und Abfolge. Husserl übernimmt einige von Bolzanos Grundbegriffen in seinen Prolegomena zur reinen Logik. In diesem Aufsatz diskutiere ich u.a. die Frage, ob Abfolge ein Spezialfall von Ableitbarkeit ist, wie Husserl anzunehmen scheint, und ich kontrastiere Bolzanos Ablehnung jeder Berufung auf Evidenz mit Husserls Einstellung. (shrink)

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 We also discuss the second, unfinished and unpublished issue, where Bolzano develops his views on universal mathematics.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. (shrink)

In his booklet "Contributions to a better founded presentation of mathematics" of 1810 Bernard Bolzano made his first serious attempt to explain the notion of a rigorous proof. Although the system of logic he employed at that stage is in various respects far below the level of the achievements in his later Wissenschaftslehre, there is a striking continuity between his earlier and later work as regards the methodological constraints on rigorous proofs. This paper tries to give a perspicuous and critical (...) account of the fragmentary logic of Beyträge, and it shows that there is a tension between that logic and Bolzano's methodological ban on ?kind crossing? (shrink)

This paper analyzes and evaluates Bolzano's remarks on the apagogic method of proof with reference to his juvenile booklet "Contributions to a better founded presentation of mathematics" of 1810 and to his ?Theory of science? (1837). I shall try to defend the following contentions: (1) Bolzanos vain attempt to transform all indirect proofs into direct proofs becomes comprehensible as soon as one recognizes the following facts: (1.1) his attitude towards indirect proofs with an affirmative conclusion differs from his stance to (...) indirect proofs with a negative conclusion; (1.2) by Bolzano's lights arguments via consequentia mirabilis only seem to be indirect. (2) Bolzano does not deny that indirect proofs can be perfect certifications (Gewissmachungen) of their conclusion; what he denies is rather that they can provide grounds for their conclusions. (2.1) They cannot do the latter, since they start from false premises and (2.2) since they make an unnecessary detour. (3) The far-reaching agreement between his early and late assessment of apagogical proofs (in the Beyträge of 1810 and the Wissenschaftslehre of 1837) is partly due to the fact that he develops his own position always against the background of Wolff's and Lambert's views. (shrink)

Two centuries ago Bernard Bolzano published his Contributions to a more well-founded presentation of mathematics which Goethe praised as “an opusculum of very high value”. Bolzano still seems to accept the traditional principle that that intension and extension of a concept stand in an inverse relation . In particular he claims that the concept of a genus proximum is always a component of the concept of the species which are subordinated to it. However, this does not harmonize with his simultaneous (...) assumption that there are simple species-concepts. In section 1 of this paper I shall try to bring to light this tension in Bolzanos Contributions. In section 2 I shall try to reconstruct Bolzano ’s arguments to the effect that the properties an object must have if it is to fall under a certain concept are not always components of that concept. In section 3 I shall try to reconstructs his arguments against the Canon in his Theory of Science .In seinen vor zwei Jahrhunderten erschienenen Beyträgen zu einer begründeteren Darstellung der Mathematik, die Goethe als „ein Werkchen von besonderem Werte“ pries, scheint Bernard Bolzano den traditionellen Lehrsatz der Reziprozität noch zu akzeptieren, demzufolge Inhalt und Umfang eines Begriffs in inversem Verhältnis stehen. Insbesondere akzeptiert er die traditionelle These, dass der Begriff eines genus proximum immer ein Bestandteil der Begriffe der Spezies dieses Genus ist. Diese Annahme schient aber im Gegensatz zu seiner gleichzeitigen Akzeptanz der These zu stehen, dass es einfache Spezies-Begriffe gibt. Im Paragraphen 1 bespreche ich diese Frage. Im Paragraphen 2 versuche ich, einige der Argumente zu rekonstruieren, die Bolzano in seiner Wissenschaftslehre zur Widerlegung der These verwendet, dass die Begriffe der Beschaffenheiten, die ein Gegenstand haben muss, um unter einen bestimmten Begriff zu fallen, immer Teile dieses Begriffs sind. Im Paragraphen 3 präsentiere ich die wichtigsten Argumente Bolzanos, die in der Wissenschaftslehre dafür verwendet werden, um den Kanon Allgemeingültigkeit abzusprechen. (shrink)