# Bolzano, Bernard (1781–1848)

# BOLZANO, BERNARD

*(1781–1848)*

Bernard Bolzano, a philosopher, theologian, logician, and mathematician, was born in Prague, where his father, an Italian art dealer, had settled; his mother was a German merchant's daughter. Bolzano studied mathematics, philosophy, and theology in Prague and defended his doctor's thesis in mathematics in 1804; he was ordained a Roman Catholic priest the following year. Shortly thereafter he was appointed to a temporary professorship in the science of religion at Karlova University in Prague and two years later was given a newly established chair in this field. Some time later he was accused of religious and political heresy and was removed from his teaching position in December 1819. Bolzano spent much of his time thereafter with the family of his friend and benefactor, A. Hoffmann, at their estate in southern Bohemia. He had difficulty getting his later publications through the Metternich censorship. Some of his books were put on the Index, and many appeared only posthumously. Some manuscripts are yet to be published; the most important of these are in the National Museum and the University Library in Prague, others are in the Österreichische Nationalbibliothek in Vienna. In December 1848, Bolzano died of a respiratory disease from which he had suffered for most of his life.

## Mathematics

Bolzano's mathematical teachings were not quite understood by his contemporaries, and most of his deep insights into the foundations of mathematical analysis long remained unrecognized. A famous theorem in the early stages of a modern presentation of the calculus is known as the Bolzano-Weierstrass theorem, but another masterful anticipation (by more than forty years) of Karl Theodor Wilhelm Weierstrass's discovery that there exist functions that are everywhere continuous but nowhere differentiable remained buried in manuscripts until the 1920s. But perhaps more important than Bolzano's actual discoveries of new theorems was the meticulousness with which he endeavored to lay new foundations for the *Grössenlehre*, the science of quantity—which was how Bolzano, using a very broad interpretation of "quantity," designated mathematics. In particular, his insistence that no appeal to any intuition of space and time should be acknowledged for this purpose and that only "purely analytical" methods were to be recognized put him in opposition to the then current Kantian ways of thinking and back into the Leibnizian tradition.

Bolzano's most famous posthumously published work is *Paradoxien des Unendlichen* (F. Prihonsky, ed., Leipzig, 1851; translated by D. A. Steele as *The Paradoxes of the Infinite*, London, 1950), in which he anticipated certain basic ideas of set theory, developed only a generation later by Georg Cantor, who fully acknowledged his indebtedness to Bolzano in this respect. This anticipation should, however, not be overrated. Bolzano was not quite able to rid himself of all the prejudices of his time and was, therefore, unable to reach a clear and fruitful conception of equivalence between infinite sets.

## Ethics and Philosophy of Religion

Bolzano was, in his time, much more influential as a theologian and social moralist than as a mathematician. An advocate of the Bohemian Catholic enlightenment, he lectured on religion and moral philosophy with strong pacifistic and socialistic overtones. He used the pulpit to proclaim before hundreds of impressed students a kind of utopian socialism. In his sermons he tried to prove the essential equality of all human beings, attacked private property obtained without work, and exhorted his listeners to sacrifice everything in their struggle for human rights. These sermons served him as a preparation for what he regarded as his most important book, *Von dem besten Staate*, which he finished in 1837 but was unable to publish. It first appeared in Prague in 1932.

Bolzano's philosophy of religion is presented in the books *Athanasia oder Gründe für die Unsterblichkeit der Seele* (Sulzbach, 1827) and *Lehrbuch der Religionswissenschaft* (4 vols., Sulzbach, 1834), the latter being a revised version of his lectures at the Prague university. He tried to prove that Catholicism is in full harmony with common sense. To this end he either disregarded or interpreted allegorically all mystical elements of Catholicism.

Bolzano derived his utilitarian ethics from a "highest ethical principle": "Of all actions possible to you, choose always the one which, weighing all consequences, will most further the good of the totality, in all its parts" (*Lehrbuch der Religionswissenschaft*, Vol. I, Sec. 87). This reminds one, of course, of Jeremy Bentham. "The most important idea of mankind" Bolzano took to be the "essential" equality of all human beings, which he tried to prove from historical, rational, and ethical considerations.

## Logic and Epistemology

It is as logician, methodologist, and epistemologist that Bolzano, after a long period of neglect, regained philosophical attention in the twentieth century. Mainly in order to combat radical skepticism, he found it necessary to base his teachings in these fields on certain ontological conceptions. He was convinced that there exist truths-in-themselves (*Wahrheiten an sich* ) prior to and independent of language and man. These truths he carefully distinguished from truths expressed in words and conceived truths. The set of truths-in-themselves is a subset of the set of propositions (in-themselves) (*Sätze an sich* ), again to be distinguished from propositions expressed in words and conceived propositions. Propositions consist of terms (ideas-in-themselves, *Vorstellungen an sich* ). These are likewise to be distinguished, on the one hand, from the words or word sequences by which they are denoted and, on the other, from subjective ideas that occur in our mind. Although linguistic entities and conceived entities exist concretely, terms, propositions, and truths do not. Terms were equally carefully distinguished from their objects, whether or not these objects themselves existed concretely. Though Bolzano was a Platonist (in the modern sense), his ontology was rather remote from that of Plato or, for that matter, from that of Immanuel Kant, in spite of the common *an sich* terminology.

Beyond these negative determinations, Bolzano had little positive to say on the ontological status of terms and propositions except that they are the matter (*Stoff* ) or sense (*Sinn* ) of their correlates in language and thought.

Terms can be either simple or complex and either empty (*gegenstandslos* ) or nonempty (*gegenständlich* ); if nonempty, they are either singular or general. Examples of empty terms are −1, 0, Nothing, Round Square, Green Virtue, and Golden Mountain; absolutely simple terms are Not, Some, Have, Be, and Ought, but Bolzano was uncertain about others. Simple, singular terms he called intuitions (*Anschauungen* ).

Propositions are composed of terms and are perhaps best regarded as ordered sequences of terms, while the content (*Inhalt* ) of a proposition is the (unordered) set of the simple terms out of which the terms constituting the proposition are composed. The content of a complex term is similarly defined. The terms 3^{5} and 5^{3} are different, though they have the same content. The terms 2^{4} and 4^{2} are different, though they have not only the same content but even the same object. With this conception of content, the traditional doctrine of the reciprocity between the extension of a term (the set of objects falling under it) and the content of a term can easily be seen to be invalid.

Among Bolzano's many idiosyncratic convictions, perhaps the most interesting, but also the most strange to the modern mind, was his belief that each branch of science has a unique, strictly scientific presentation, which for him meant not only a unique finite axiom system (a belief he shared with many) but also an essentially unique entailment (*Abfolge* ) of each theorem of this science by the axioms, a belief which might well be unique to Bolzano.

This relationship of entailment, as presented by Bolzano, is very peculiar and obscure. Bolzano was never quite sure that he understood it himself, though he was convinced that there objectively must exist some such relationship, that each science must have its basic truths (*Grundwahrheiten* ) to which all other truths of that science stand in the peculiar relation of consequence (*Folge* ) to ground (*Grund* ). Bolzano was constantly struggling to differentiate this relation of entailment from the relation of derivability (*Ableitbarkeit* ), which was the basic relation of his logic. Though he did not succeed in putting his theory of entailment into consistent and fruitful shape—and could not possibly have done so, in view of the chimerical character of his goal—his acumen, mastery of the contemporary logical and methodological literature, intellectual honesty, and lifelong self-criticism more than made up for his numerous shortcomings. Bolzano remains a towering figure in the epistemology, logic, and methodology of the first half of the nineteenth century.

** See also ** Bentham, Jeremy; Cantor, Georg; Kant, Immanuel; Logic, History of; Propositions, Judgments, Sentences, and Statements.

## Bibliography

### additional works by bolzano

Bolzano's masterwork is his *Wissenschaftslehre*, 4 vols. (Sulzbach, 1837; edited by Wolfgang Schultz, Leipzig: F. Meiner, 1929–1931). *Grundlegung der Logik* (Hamburg, 1964) is a very useful selection by Friedrich Kambartel from the first two volumes of the *Wissenschaftslehre*, with summaries of omitted portions, an excellent introduction, and a good index.

### works on bolzano

Bolzano's philosophical work was virtually disregarded until Edmund Husserl called attention to it at the start of the twentieth century. Hugo Bergmann's monograph, *Das philosophische Werk Bernard Bolzanos* (Halle: M. Niemeyer, 1909), increased the revived interest in Bolzano's ideas. Heinrich Scholz's articles, especially "Die Wissenschaftslehre Bolzanos," in *Abhandlungen des Fries'schen Schule*, n.s, 6 (1937): 399–472, reprinted in *Mathesis Universalis*, pp. 219–267 (Basel: B. Schwabe, 1961), presented Bolzano's contributions to logic, semantics, and the methodology of the deductive sciences in a modernized form. The best recent study in English of Bolzano as a logician is J. Berg's *Bolzano's Logic* (Stockholm: Almqvist and Wiksell, 1962). D. A. Steele's historical introduction to his translation of Bolzano's *Paradoxien des Unendlichen* is useful. Among other secondary works the most important are Eduard Winter's *Bernard Bolzano und sein Kreis* (Leipzig: J. Hegner, 1933), Günter Buhl's *Ableitbarkeit und Abfolge in der Wissenschaftstheorie Bolzanos* (Cologne: Cologne University Press, 1961), and (from a Marxist viewpoint) A. Kolman's *Bernard Bolzano* (in Russian, Moscow, 1955; in Czech, Prague, 1957; and in German, Berlin, 1963).

*Yehoshua Bar-Hillel (1967)*

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