Bernard Bolzano

How to use this course

Students of analysis meet Bolzano’s name twice: in the intermediate value theorem, often called Bolzano’s theorem, and in the Bolzano–Weierstrass theorem. Behind those two labels is a much larger body of work. Bolzano defined continuity in modern terms in 1817, stated the convergence criterion usually credited to Cauchy, built a continuous function with no derivative, constructed the real numbers from rationals, studied one-to-one correspondences between infinite sets, and wrote one of the major works of nineteenth-century logic. Most of it lay unread for decades.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements in modern notation, proof sketches and exercises with folded solutions. Where Bolzano’s own argument differs from the modern one, or has a gap, the lesson says so. The order follows his work: the printed papers of 1816–1817, the manuscripts of the 1830s, the Wissenschaftslehre of 1837, and the posthumous Paradoxien des Unendlichen of 1851.

Timeline

A Prague childhood, 1781–1796

Bernard Bolzano was born in Prague on 5 October 1781, the fourth of twelve children. Only he and his brother Johann survived to adulthood. His father, Bernard Pompeius Bolzano, had come to Prague from northern Italy and dealt in art; he was a devout Catholic and helped found an orphanage. His mother, Maria Cecilia Maurer, came from a German-speaking merchant family in the city and was deeply religious. Bolzano grew up speaking German in a bilingual Bohemian city, and he had weak lungs all his life.

From 1791 to 1796 he attended the Gymnasium of the Piarist order in Prague. The Piarists were a teaching order that ran schools across Catholic Central Europe. Two concerns that marked his adult work, the moral duties of the educated and the search for clear foundations, both took shape in these years of Catholic schooling in a city of two languages.

In 1796 he entered the Philosophy Faculty of the University of Prague, which at that time was a preparatory faculty taken before law, medicine or theology. He studied philosophy, physics and mathematics.

Mathematics or the priesthood, 1796–1805

The mathematics Bolzano found at Prague was the textbook tradition of Abraham Gotthelf Kästner of Göttingen, whose Mathematische Anfangsgründe he read closely. Kästner tried to prove things that others took as obvious, and Bolzano admired this. In 1799–1800 he worked on mathematical problems with the engineer and mathematician František Josef Gerstner, the professor of mathematics at Prague.

In the autumn of 1800 he began three years of theology, against his father’s wishes, and kept working on mathematics alongside. He was drawn to the priesthood less by a pastoral calling than by a conviction that religion should serve the moral improvement of society. He later wrote that his real vocation was teaching.

His first publication, Betrachtungen über einige Gegenstände der Elementargeometrie (1804), reworked the foundations of elementary geometry, treating the straight line, the triangle and parallels with a demand for proofs from clear first principles. In 1804 he competed for two vacant posts: the chair of mathematics, left open by the death of Stanislav Vydra, and a new chair of religious doctrine that the Habsburg government had created in every university to strengthen the loyalty of students. He placed first in both competitions. The university gave the mathematics chair to Ladislav Jandera and the religion chair to Bolzano. He was ordained priest on 7 April 1805, received his doctorate in philosophy the same month, and gave his first lectures on 19 April 1805.

The chair of religious doctrine, 1805–1819

The post was meant as an instrument of the conservative state. Bolzano used it to preach a rational, ethical Christianity. His Sunday exhortations to students (Erbauungsreden) attracted large audiences from the city as well as the university. He argued that the highest moral law is to promote the general good, criticized militarism and war, and spoke against discrimination between the Czech- and German-speaking peoples of Bohemia and against the treatment of Jews. In 1816 he published an essay on the relations between the two nationalities of Bohemia.

He became a full professor in 1806 and in 1818 was elected dean of the Philosophical Faculty. All the while he wrote mathematics. The Beyträge zu einer begründeteren Darstellung der Mathematik (1810) set out his view that mathematics needed a better order of definitions and proofs, in which each truth is derived from those that are its objective grounds. Then came a remarkable run of short papers: Der binomische Lehrsatz (1816), which tried to free the calculus from infinitely small quantities; and in 1817 Rein analytischer Beweis, the subject of Lessons 04 and 05, and Die drey Probleme der Rectification, der Complanation und der Cubirung, on lengths, areas and volumes, which contains early topological definitions of line, surface and solid.

These were printed in Prague and read by very few. Meanwhile his preaching had made enemies in Vienna and within the Church.

Continuity and the intermediate value theorem

For the general reader. If a continuous curve starts below a line and ends above it, it must cross the line. Before Bolzano, mathematicians proved this with pictures and motion. In 1817 he argued that a picture proves nothing about numbers, gave a definition of continuity that uses only inequalities, and proved the crossing exists from properties of the real numbers alone.

The problem

The full title of the 1817 paper announces its aim: a purely analytic proof that between any two values giving results of opposite sign, an equation has at least one real root. Earlier proofs, including those of Carl Friedrich Gauss for the fundamental theorem of algebra, used the fact that a continuous curve crossing a line meets it. Bolzano objected that this imports geometry into analysis, and that arguments from motion import time and mechanics. Both, he said, put a truth on grounds that are not its own.

The definition

Bolzano called \(f\) continuous at \(x\) if the difference \(f(x+\omega)-f(x)\) can be made smaller than any given quantity by taking \(\omega\) sufficiently small. In modern form:

Augustin-Louis Cauchy gave an equivalent definition in the language of infinitely small increments in his Cours d’analyse of 1821. Bolzano’s version uses no infinitesimals at all.

Proof

Put \(h=f-g\), continuous with \(h(\alpha)<0<h(\beta)\). Let \(S=\{t\in[\alpha,\beta]: h(s)<0\ \text{for all } s\in[\alpha,t]\}\). It is non-empty (\(\alpha\in S\)) and bounded by \(\beta\), so it has a least upper bound \(c\). Bolzano’s key lemma (Lesson 05) is precisely the existence of \(c\).

• If \(h(c)>0\), continuity gives \(\delta\) with \(h>0\) on \((c-\delta,c]\), so no point of \((c-\delta,c]\) lies in \(S\) and \(c-\delta\) is a smaller upper bound. Contradiction.
• If \(h(c)<0\), then \(c<\beta\), and continuity gives \(h<0\) on \([c,c+\delta)\). Since \(h<0\) on \([\alpha,c)\) (every \(s<c\) lies below some point of \(S\)), points slightly beyond \(c\) are in \(S\). Contradiction.

So \(h(c)=0\). Bolzano’s proof follows the same plan, with the least upper bound applied to the property “\(h\) is negative on all smaller values”.

Convergence and the least upper bound

For the general reader. How can you tell that a sequence of numbers settles down to a limit if you do not already know the limit? Bolzano’s answer, in 1817, was to check that the terms eventually stay as close to one another as you like. The same paper contains the principle that any bounded set of numbers has a smallest upper bound. Both are forms of the completeness of the real line.

Bolzano stated this criterion for sequences of sums, in exactly this “for every \(r\)” form. He showed that the limit, if it exists, is unique, and argued that the hypothesis assumes nothing impossible. He could not yet prove that the limit exists, because there was no definition of the real numbers from which to prove it. Cauchy, four years later, took sufficiency for granted. Bolzano’s own construction of the reals in the 1830s (Lesson 08) finally supplied the missing step.

Proof from the criterion (Bolzano’s bisection)

Let \(A\) be non-empty and bounded above. Choose \(a_0\in A\) and an upper bound \(b_0\). Given \([a_n,b_n]\) with \(b_n\) an upper bound and \([a_n,b_n]\cap A\neq\varnothing\), let \(m=(a_n+b_n)/2\). If \(m\) is an upper bound of \(A\), set \([a_{n+1},b_{n+1}]=[a_n,m]\); otherwise some point of \(A\) exceeds \(m\), and set \([a_{n+1},b_{n+1}]=[m,b_n]\). Then \(|b_{n+r}-b_n|\le(b_0-a_0)2^{-n}\), so \((b_n)\) is a Cauchy sequence with a limit \(U\). Each \(b_n\) is an upper bound, so \(U\) is one. Each \([a_n,b_n]\) meets \(A\), so for every \(n\) some element of \(A\) exceeds \(U-(b_0-a_0)2^{-n}\). Hence \(U\) is the least upper bound. Bolzano’s 1817 argument divides by powers of two in the same way.

The binomial theorem (1816)

The year before, Bolzano had used the same ideas to prove the binomial series \((1+x)^\mu=\sum_{k\ge0}\binom{\mu}{k}x^k\) for real exponents \(\mu\) and \(|x|<1\). Earlier proofs, from Leonhard Euler onward, manipulated infinite series without asking whether they converged. Bolzano insisted that a series should be equated with a quantity only when it converges, and only in its domain of convergence.

Bisection and Bolzano–Weierstrass

For the general reader. Infinitely many points packed into a bounded interval must crowd together somewhere. Repeatedly halving the interval and keeping a half that still holds infinitely many points traps the crowding place. This halving method runs through Bolzano’s 1817 paper. Karl Weierstrass made the resulting theorem a cornerstone of his Berlin lectures decades later, independently of Bolzano.

Proof

Let \((x_n)\subset[a,b]\). Halve \([a,b]\); at least one half contains \(x_n\) for infinitely many \(n\). Call it \(I_1\) and repeat, producing nested intervals \(I_1\supset I_2\supset\cdots\) of length \((b-a)2^{-k}\), each containing infinitely many terms. Choose \(n_1<n_2<\cdots\) with \(x_{n_k}\in I_k\). Then \(|x_{n_{k+r}}-x_{n_k}|\le(b-a)2^{-k}\), so the subsequence is Cauchy and converges by Lesson 05.

Bolzano’s 1817 paper establishes the least upper bound principle, from which this follows in a few lines, and the Functionenlehre of the 1830s treats results that depend on it, with proofs not all complete by modern standards, including that a function continuous on a closed interval is bounded and attains its maximum. Weierstrass taught the theorem in Berlin from the 1860s and used it systematically. The joint name records two independent routes to the same idea.

Proof. If \(f\) were unbounded, pick \(x_n\) with \(|f(x_n)|>n\); a subsequence converges to some \(c\in[a,b]\), and continuity at \(c\) contradicts \(|f(x_{n_k})|\to\infty\). Let \(M=\sup f\) and choose \(f(x_n)\to M\); a subsequence converges to \(c\), and \(f(c)=M\).

A continuous nowhere differentiable function

For the general reader. For most of the history of calculus, mathematicians assumed that a continuous curve has a tangent at most of its points. Around 1830 Bolzano built a curve, by endlessly replacing straight pieces with zigzags, that is continuous but has a tangent nowhere. The manuscript was not published until 1930, so the example usually taught is Weierstrass’s of 1872.

The construction

Start with a segment from \((a,c)\) to \((b,d)\), with width \(w=b-a\) and rise \(\Delta=d-c\). Replace it by the polygon through

The four pieces have widths \(\tfrac38,\tfrac18,\tfrac38,\tfrac18\) of \(w\) and rises \(\tfrac58,-\tfrac18,\tfrac58,-\tfrac18\) of \(\Delta\). Their slopes are therefore \(\tfrac53\sigma,\ -\sigma,\ \tfrac53\sigma,\ -\sigma\), where \(\sigma=\Delta/w\) is the original slope. Apply the rule to every segment, again and again. Starting from the diagonal of the unit square gives piecewise linear functions \(B_1,B_2,\dots\), and Bolzano’s function is \(B=\lim_k B_k\).

Continuity

Bolzano assumed that a pointwise limit of continuous functions is continuous, which is false in general. The repair is uniform convergence. On a segment of rise \(\Delta\), the new polygon departs from the old line by at most \(\tfrac14|\Delta|\) (at \(\tfrac38 w\) the line is at \(c+\tfrac38\Delta\) and the polygon at \(c+\tfrac58\Delta\); at \(\tfrac78w\), at \(c+\tfrac78\Delta\) versus \(c+\tfrac98\Delta\)). Every stage-\(k\) segment has rise at most \((\tfrac58)^k\) in absolute value. Hence

a summable bound, so \(B_k\to B\) uniformly by the Weierstrass M-test, and \(B\) is continuous.

No finite derivative

The vertices of \(B_k\) are never moved later, so \(B\) agrees with \(B_k\) at them. Fix \(x\) and choose stage-\(k\) segments \([a_k,b_k]\ni x\), nested, with \(b_k-a_k\to0\). The chord slope \(\sigma_k=(B(b_k)-B(a_k))/(b_k-a_k)\) equals the slope of \(B_k\) there, and \(\sigma_{k+1}/\sigma_k\in\{\tfrac53,-1\}\).

If \(B'(x)=L\) existed and were finite, then \(\sigma_k\to L\), because each chord slope over an interval containing \(x\) is a weighted average of the two difference quotients from \(x\). Now \(|\sigma_k|\) never decreases and \(|\sigma_0|=1\), so \(L\neq0\). But then \(\sigma_{k+1}/\sigma_k\to1\), impossible when every ratio is \(\tfrac53\) or \(-1\). So \(B\) is differentiable nowhere.

Measurable numbers

For the general reader. To prove that limits exist, you must first say what a real number is. In manuscripts of the early 1830s, Bolzano defined real numbers as expressions that can be pinned between neighbouring fractions with any denominator you choose. This came decades before the constructions of Richard Dedekind and Georg Cantor in 1872. It was published only in 1962 and is still debated.

In modern terms, \(S\) determines for each \(q\) the integer \(p=\lfloor qS\rfloor\), and the sequence \(p_q/q\) converges to \(S\) with error less than \(1/q\). Bolzano first called two measurable numbers equal if they have the same measuring fractions for every \(q\). He then realized that an expression such as \(1/(1+1+1+\cdots)\) has measuring fraction \(0/q\) for every \(q\) without being literally zero; he called it infinitely small. In a later inserted note he revised the definition so that \(A\) and \(B\) count as equal when their difference is infinitely small, judged by the measuring fractions of \(A-B\). The new relation is an equivalence relation, so Bolzano was among the first to need one, and the classes form an ordered field.

Completeness

In the same manuscript Bolzano proved the theorem missing from 1817: a sequence of measurable numbers satisfying his criterion converges to a measurable number. Rychlík’s 1962 transcription omitted parts of the text. Some readers (B. van Rootselaar, 1963) judged the theory inconsistent; others (Laugwitz, and the fuller edition by Jan Berg in 1976) showed that with Bolzano’s own revision it yields a complete ordered field, isomorphic to \(\mathbb R\).

Paradoxes of the infinite

For the general reader. Galileo had noticed that there are as many square numbers as whole numbers, though the squares are only some of them, and concluded that “more” and “equal” make no sense for infinite collections. Bolzano disagreed. He held that infinite sets are real objects, studied pairings between them, and found that an infinite set can be matched exactly with a part of itself. He drew a different conclusion from Cantor, and the difference is instructive.

Sets and multitudes

The Paradoxien des Unendlichen was finished in 1848 and published in 1851 by Přihonský in Leipzig. Bolzano defines a set as an aggregate in which the arrangement of the parts does not matter, and an infinite multitude as one larger than every finite multitude. He argues, against a long tradition, that actual infinities exist, for instance the set of all true propositions.

Bolzano concluded that a one-to-one pairing is not enough to establish that two infinite multitudes are equal in size, because the whole should still exceed its part. He judged sizes by additional criteria tied to how the sets are determined. Cantor took the opposite decision in the 1870s and 1880s: equal size simply means a bijection exists. Dedekind, in 1888, turned the paradox into a definition: a set is infinite exactly when it is in bijection with a proper subset. Cantor praised Bolzano’s book while rejecting his conclusion. Modern scholarship reads Bolzano’s mature view as a coherent theory of infinite sums and part–whole measure, closer in spirit to measure theory than to cardinal arithmetic.

Logic in the Wissenschaftslehre

For the general reader. When does a conclusion follow from premises? Bolzano answered with a test: vary the ideas in the sentences and see whether every variation that makes the premises true also makes the conclusion true. A century later Alfred Tarski gave the same answer, and it is now the standard definition of logical consequence.

Propositions in themselves

The four-volume Wissenschaftslehre (1837) separates logic from psychology. Its basic objects are propositions in themselves (Sätze an sich): the content of a sentence, true or false independently of whether anyone thinks it. Their parts are ideas in themselves (Vorstellungen an sich). This anti-psychologism shaped Kazimierz Twardowski and Edmund Husserl and through them the Polish school of logic and phenomenology.

Clause (b) is Tarski’s 1936 definition of logical consequence, with substitution in place of models. Clause (a), the requirement that premises and conclusion be compatible, is Bolzano’s own; it excludes the principle that anything follows from contradictory premises.

Grounding

Deducibility is about truth-preservation. Bolzano also defined Abfolge, the relation between true propositions in which some are the objective reason for another. It is asymmetric: that the thermometer rises follows from the air warming, and the warming also follows from the rising, but only the warming is the ground. The Beyträge of 1810 already demanded that proofs follow the order of grounds, and this is why the 1817 paper rejected geometric proofs of an analytic truth.

Dismissal and the years at Těchobuz, 1819–1841

In December 1819 Emperor Franz I ordered Bolzano removed from his chair; the dismissal took effect in early 1820. The charges were heresy and the political content of his sermons, with their pacifism, social egalitarianism and defence of minorities, in the repressive climate of Metternich’s Austria. He was placed under police supervision and forbidden to teach, preach or publish. From 1821 to 1825 the Church examined his orthodoxy. He refused to recant. Support from the Archbishop of Prague and from friends within the clergy, including Michael Josef Fesl, protected him from worse; Fesl himself suffered for the association.

Bolzano’s health was poor, and from 1823 he spent long periods with Josef and Anna Hoffmann on their estate at Těchobuz in southern Bohemia. From 1830 to 1841 he lived there year-round, with Anna Hoffmann caring for him. These were his most productive years. He wrote the Wissenschaftslehre, the Lehrbuch der Religionswissenschaft (1834), the immortality dialogue Athanasia, and the vast unfinished Größenlehre, his planned reconstruction of mathematics, which contains the Reine Zahlenlehre and the Functionenlehre. Censorship blocked his mathematics in the Habsburg lands until after the emperor’s death in 1835, and his books appeared anonymously or abroad. The Wissenschaftslehre was printed in Sulzbach, in Bavaria.

Death and rediscovery

In 1841 Bolzano returned to Prague with the Hoffmanns and his brother Johann. Anna Hoffmann died in 1842. He served as president of the Royal Bohemian Society of Sciences in 1842–43 and kept writing. He died in Prague on 18 December 1848, of a respiratory illness his lungs could not withstand. His student František Přihonský published the Paradoxien des Unendlichen in 1851.

Rediscovery

• Analysis: Otto Stolz, in an 1881 paper in Mathematische Annalen, showed that Bolzano had anticipated much of the rigorous analysis of Cauchy and Weierstrass. His name became attached to the intermediate value theorem and to Bolzano–Weierstrass.
• The manuscripts: In 1921 Martin Jašek lectured in Prague on the manuscripts held in Vienna. Rychlík and Jarník analysed the nowhere differentiable function in 1922, and Rychlík edited the Functionenlehre in 1930 and the theory of real numbers in 1962.
• Set theory: Cantor and Dedekind knew the Paradoxien; Cantor discussed it in his Grundlagen of 1883.
• Philosophy: Husserl and Twardowski made Bolzano a founding figure for phenomenology and for the Lwów–Warsaw school, and his definition of consequence anticipates Tarski.
• Edition: the Bernard-Bolzano-Gesamtausgabe, begun in 1969 by Eduard Winter, Jan Berg and others, is still in progress, with more than a hundred volumes planned.

Connections

People linked to Bernard Bolzano across this collection. A name without a link has no page here yet. See Bernard Bolzano in the network of all 337 lives →

Teachers and advisors

• František Josef Gerstner 1799-1804

Students

• František Přihonský

Influenced by

• Abraham Gotthelf Kästner

Influenced

• Alfred Tarski
• Georg Cantor
• Karl Weierstrass
• Richard Dedekind
• Edmund Husserl
• Karel Rychlík
• Kazimierz Twardowski
• Otto Stolz
• Vojtěch Jarník

Colleagues and collaborators

• Augustin-Louis Cauchy

Friends

• Christian Doppler 1837-1848
• Michael Josef Fesl

Also mentioned in this life

• Leonhard Euler
• Galileo Galilei
• Carl Friedrich Gauss

Fields

• real analysis 10 lives
• foundations of mathematics 10 lives
• logic 8 lives
• philosophy 12 lives
• theology 8 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Bernard Placidus Johann Nepomuk Bolzano, MacTutor History of Mathematics.
2. Bernard Bolzano, Wikipedia.
3. E. Morscher, Bernard Bolzano, Stanford Encyclopedia of Philosophy.
4. Karel Rychlík (1885–1968), Dějiny matematiky 22 (2003), on the rediscovery of the manuscripts.

Mathematics and logic

1. S. Russ (ed. and trans.), The Mathematical Works of Bernard Bolzano, Oxford University Press, 2004. Introduction.
2. B. Bolzano, Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege, Prague, 1817.
3. B. Bolzano, Paradoxien des Unendlichen, ed. F. Přihonský, Leipzig, 1851; English translation by D. A. Steele, 1950. See Paradoxes of the Infinite.
4. B. Bolzano, Functionenlehre, ed. K. Rychlík, Prague, 1930.
5. K. Rychlík, Über eine Funktion aus Bolzanos handschriftlichem Nachlasse, Věstník Král. České Spol. Nauk, 1921–22.
6. V. Jarník, O funkci Bolzanově, Časopis pro pěstování matematiky a fysiky 51 (1922).
7. S. Russ and K. Trlifajová, Bolzano’s measurable numbers: are they real?, arXiv:1805.02237.
8. O. Stolz, B. Bolzano’s Bedeutung in der Geschichte der Infinitesimalrechnung, Math. Ann. 18 (1881).
9. B. Bolzano, Theory of Science, trans. P. Rusnock and R. George, Oxford University Press, 2014.