Logical objectivism
Bolzano's theorem (the first purely analytic proof of the intermediate value theorem)
Bolzano–Weierstrass theorem
(ε, δ)-definition of limit
Least-upper-bound property

Influences

Gottfried Wilhelm Leibniz, Immanuel Kant

Influenced

Carl Stumpf, Edmund Husserl, Kevin Mulligan, Gottlob Frege, Kazimierz Twardowski, Heinrich Scholz, Benno Kerry, Georg Cantor, Franz Brentano, Gilbert Ryle, Jean Cavaillès

Bernard Bolzano
Church	Catholic Church
Ordained	1805

Bernard Bolzano (born Bernardus Placidus Johann Gonzal Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a smart person from Bohemia (now part of the Czech Republic). He was a mathematician, a logician (someone who studies how we reason), a philosopher, a theologian (someone who studies religion), and a Catholic priest. He was also known for his liberal ideas, which meant he believed in freedom and individual rights.

Bolzano wrote in German, his first language. Many of his important ideas became well-known only after he had passed away.

Family
Career and Challenges
Amazing Math Ideas
Thinking About Knowledge
- What is the "Theory of Science"?
- Ideas and Truths
His Impact on Philosophy
His Books
See also

Family

Bernard Bolzano grew up in a very religious Catholic family. His father, Bernard Pompeius Bolzano, was from Italy and moved to Prague. There, he married Maria Cecilia Maurer, who came from a German-speaking family in Prague. Sadly, only two of their twelve children lived to become adults.

Career and Challenges

In 1796, Bolzano started studying at the University of Prague. He focused on mathematics, philosophy, and physics. He really loved the thinking side of mathematics, seeing it as a kind of philosophy.

In 1800, he began studying theology (the study of religion) and became a Catholic priest in 1804. The next year, he was given a new job as a professor of the philosophy of religion at Prague University. He was a very popular teacher, not just in religion but also in philosophy. In 1818, he was even chosen to be the Dean of the Philosophy Department.

However, Bolzano's ideas caused problems with many university and church leaders. He spoke out against militarism (the idea that a country should have a strong military and be ready for war). He believed war was unnecessary and wasteful. He wanted big changes in education, society, and the economy to focus on peace instead of fighting between countries.

His strong beliefs were too "liberal" for the government at the time, which was the Austrian Empire. Because he refused to change his views, he was removed from his teaching job on December 24, 1819. He was sent away from the city to the countryside. There, he spent his time writing about social issues, religion, philosophy, and mathematics.

Even though he was not allowed to publish his work in regular journals, Bolzano kept developing his ideas. He found ways to publish them himself or in smaller journals in Eastern Europe. In 1842, he moved back to Prague, where he passed away in 1848.

Amazing Math Ideas

Bolzano made many important contributions to mathematics. He believed that math should be very precise and not rely on "intuitive" ideas like time or movement. He was one of the first mathematicians to make mathematical analysis (a branch of math) much more rigorous and exact.

His main math books were Beyträge zu einer begründeteren Darstellung der Mathematik (1810), Der binomische Lehrsatz (1816), and Rein analytischer Beweis (1817). These books showed a "new way of developing analysis." His ideas were so advanced that they weren't fully understood until about 50 years later when another mathematician, Karl Weierstrass, discovered them.

Bolzano helped create the modern definition of a mathematical limit. This definition is super important in calculus. He was also the first to understand the greatest lower bound property of real numbers. He proved the fundamental theorem of algebra using only mathematical analysis, which was a big step. He also gave the first purely analytic proof of the intermediate value theorem (also known as Bolzano's theorem).

Today, he is most famous for the Bolzano–Weierstrass theorem. Karl Weierstrass also discovered this theorem on his own later, but Bolzano had proven it first.

Thinking About Knowledge

Bolzano's most famous philosophical work was Paradoxien des Unendlichen (The Paradoxes of the Infinite) (1851), published after he died. Many famous logicians admired this book.

However, his most important work is his 1837 book, Wissenschaftslehre (Theory of Science). This huge work, in four volumes, covered not just the philosophy of science but also logic, epistemology (the study of knowledge), and how to teach science. The way Bolzano thought about logic in this book was truly groundbreaking.

What is the "Theory of Science"?

In his Theory of Science, Bolzano wanted to create a strong logical foundation for all sciences. He thought about abstract things like ideas, propositions (statements), and how they relate to each other. He believed it wasn't enough to just confirm scientific or mathematical truths. Instead, science should aim to find the justification for these truths, even if they don't seem obvious at first.

Bolzano explained that human knowledge is made of all the true statements people know. But this is only a tiny part of all the truths that exist. To make knowledge easier to understand, we divide it into "sciences." A "science" is a collection of truths. Not all truths in a science have to be known by people; this is why we can still discover new things in science.

To organize these truths, people create textbooks. The Theory of Science is a special science that tells us how to group truths together and how they should be explained in a textbook.

Ideas and Truths

Bolzano focused on three main areas in his Theory of Science:

The world of language (words and sentences).
The world of thought (our personal ideas and judgments).
The world of logic (objective ideas and "propositions in themselves").

He made a key distinction between things that exist (are in time and space, and are causally connected) and things that do not exist. He argued that the world of logic is filled with things that do not exist in the same way physical objects do.

A central idea for Bolzano was the "proposition in itself" (Satz an Sich). This is a statement that:

Does not exist in time or space (it's not a spoken or written sentence).
Is either true or false, no matter if anyone knows or thinks it.
Is what we "grasp" when we think or understand something.

For example, the written sentence "The grass is green" exists. But it expresses a "proposition in itself" – the idea [The grass is green], which doesn't exist in a physical way but is either true or false.

Bolzano also talked about "ideas." An idea is a part of a proposition that isn't a proposition itself. Ideas can be simple (like [something]) or complex (like [nothing], which is made of [not] and [something]).

He also explained that an idea doesn't always have an "object." An "object" is what an idea represents. For example, the idea [a round square] doesn't have an object because a round square can't exist. But the idea [the first man on the moon] has one object: Neil Armstrong.

Bolzano also had a detailed theory of "truth." He said that a proposition is true if it expresses something that matches reality. He believed there are infinitely many truths in themselves, meaning there are endless true statements that exist whether we know them or not.

His Impact on Philosophy

Bolzano had a group of friends and students who helped spread his ideas. This group was known as the "Bolzano Circle." At first, his impact on philosophy seemed small.

However, his work was later rediscovered by important philosophers like Edmund Husserl and Kazimierz Twardowski. These thinkers were students of Franz Brentano. Because of them, Bolzano became a very important influence on two major philosophical movements: phenomenology (the study of conscious experience) and analytic philosophy (a style of philosophy that uses logic and clear language).

His Books

Most of Bolzano's writings stayed in manuscript form for a long time, so they weren't widely read at first.

Some of his main works include:

Wissenschaftslehre (Theory of Science), his four-volume work on logic and knowledge.
Paradoxien des Unendlichen (The Paradoxes of the Infinite), which explored ideas about infinity.
Rein analytischer Beweis (Purely analytic proof), an important math paper.

Many of his works have now been translated into English, including:

Theory of Science (several translations available).
The Mathematical Works of Bernard Bolzano.
On the Mathematical Method and Correspondence with Exner.
Selected Writings on Ethics and Politics.

Bernard Bolzano facts for kids

Contents