Bernard Bolzano

Bohemia
1781 – 1848

Bolzano graduated from the University of Prague as an ordained priest in 1805 and was immediately appointed professor of philosophy and religion at the university. Within a matter of years, however, Bolzano alienated many faculty and church leaders with his teachings of the social waste of militarism and war. He urged a total reform of the educational, social, and economic systems that would direct the nation’s interests toward peace rather than toward armed conflict between nations. Upon his refusal to recant his beliefs, Bolzano was dismissed from the university in 1819 and at that point devoted his energies to his writings on social, religious, philosophical, and mathematical matters.

Bolzano made several original contributions to mathematics. His overall philosophical stance was that, contrary to much of the prevailing mathematics of the era, it was better not to introduce intuitive ideas such as time and motion into mathematics. He was one of the earliest mathematicians to begin instilling rigor into mathematical analysis. His works presented a sample of a new way of developing analysis, whose ultimate goal would not be realized until some fifty years later when they came to the attention of Karl Weierstrass.

He introduced a fully rigorous ε–δ definition of a mathematical limit and was the first to recognize the greatest lower bound property of the real numbers. Bolzano’s notion of a limit was similar to the modern one: that a limit, rather than being a relation among infinitesimals, must instead be cast in terms of how the dependent variable approaches a definite quantity as the independent variable approaches some other definite quantity. Bolzano gave the first purely analytic proof of the fundamental theorem of algebra. He also gave the first purely analytic proof of the intermediate value theorem (also known as Bolzano’s theorem). Today he is remembered for the Bolzano–Weierstrass theorem, which Karl Weierstrass developed independently and published years after Bolzano’s first proof and which was initially called the Weierstrass theorem until Bolzano’s earlier work was rediscovered. He provided a more detailed proof for the binomial theorem and suggested the means of distinguishing between finite and infinite classes. He may have influenced Georg Cantor, who later developed set theory. Incidentally, Cantor believed his own ideas on infinite sets were divinely inspired. Cantor praised Bolzano for asserting that the actual infinite exists but criticized him for failing to provide either a concept of infinite number or the concept of “power” based on equipollence.

Bolzano emphasized the distinction between the objective relationship between logical consequences and our subjective recognition of these connections. For Bolzano, it was not enough that we merely have confirmation of natural or mathematical truths, but rather it was the proper role of the sciences (both pure and applied) to seek out justification in terms of the fundamental truths that may or may not appear to be obvious to our intuitions.

This makes understandable what could otherwise be seen as very strange for a theological textbook: Bolzano’s Textbook of the Science of Religion contains a section on mathematical probability theory. His choice of examples and his focus on certain methodological questions in can better be understood if one sees that they are theologically motivated. He used his scientific investigations into the discovery and credibility of testimonies and into the degree of credibility of a proposition with respect to testimonies in favor as well as those opposed as a basis for his theory of the divine revelation. The special treatment of proofs, “which are only to show that the probability of a proposition exceeds a given magnitude” is closely connected with the topic of miracles: Bolzano argued that in order to prove an event E is an unusual event and thus qualifies to be a miracle, one must demonstrate that the intrinsic probability of the assumption that E has not occurred is > ½ and therefore exceeds a certain magnitude.