Bernard Bolzano

2.1. Birth and Family

Bernard Bolzano was born in Prague, then part of the Kingdom of Bohemia, on 5 October 1781. His parents, Bernard Pompeius Bolzano and Maria Cecilia Maurer, were both devout Roman Catholics from the merchant class. His father, Bernard Pompeius Bolzano, was an Italian merchant who had relocated to Prague. His mother, Maria Cecilia Maurer, was from a German-speaking family in Prague. Out of their twelve children, only two lived to adulthood. Bolzano was raised in a pious Catholic tradition.

2.2. Education and Ordination

At the age of ten, Bolzano began his formal education at the Gymnasium of the Piarists in Prague, attending from 1791 to 1796. In 1796, he enrolled at the University of Prague, where he pursued studies in mathematics, philosophy, and physics. From 1800 onwards, he also dedicated himself to the study of theology, eventually being ordained as a Catholic priest in 1804. During his university years, Bolzano developed a keen interest in the philosophies of Gottfried Leibniz and Christian Wolff, and later engaged deeply with the works of Immanuel Kant. He earned his doctorate in 1804 with a thesis titled Betrachtungen über einige Gegenstände der Elementargeometrie (Considerations on Some Objects of Elementary Geometry). Around this time, he began to critically examine Kant's concept of the thing-in-itself, gradually articulating an anti-Kantian philosophical stance.

3.1. Professorship at Prague University

In 1805, Bernard Bolzano was appointed to the newly established chair of philosophy of religion at the University of Prague. He quickly gained a reputation as a highly popular lecturer, not only in religious studies but also in philosophy. His engaging teaching style and intellectual depth resonated with students. His academic standing continued to rise, culminating in his election as Dean of the Philosophical Faculty in 1818. During this period, he became a prominent figure within the university, known for his intellectual rigor and his commitment to open inquiry.

3.2. Social and Political Views

Bolzano held strong liberal convictions that frequently put him at odds with the conservative Austrian authorities and the prevailing academic and ecclesiastical establishments. He was a vocal critic of militarism, denouncing it as a social waste, and argued passionately against the needlessness of war. His teachings emphasized that armed conflict between nations was unnecessary and destructive. Instead, he advocated for a comprehensive reform of the educational, social, and economic systems. He believed these reforms should redirect national interests towards peace and foster a more equitable society, reflecting a deeply ingrained social liberal perspective. He even envisioned a communist state, drawing inspiration from Saint-Simonian ideas, which further highlighted his radical social philosophy.

3.3. Conflict with Authorities and Exile

Bolzano's progressive teachings, particularly his critiques of militarism and his calls for radical societal reform, alienated many faculty members and church leaders at the University of Prague. His liberal political convictions, which he frequently shared with his students and colleagues, were deemed too subversive by the Austrian authorities and the Roman Catholic Church. He faced immense pressure to recant his beliefs and cease his liberal teachings. However, Bolzano steadfastly refused to compromise his intellectual and moral integrity.

Consequently, on 24 December 1819, he was summarily removed from his professorship, accused of heresy and involvement in the Czech independence movement. Following his dismissal, he was exiled to the countryside and prohibited from publishing his works in mainstream journals or delivering public lectures on religious topics. Despite these severe restrictions, Bolzano continued to develop his ideas and publish them either anonymously, on his own, or in obscure Eastern European journals. His mother's death in 1821 added to his personal misfortunes, and he faced significant financial hardship. However, in 1823, he met Anna Hoffmann, the wife of a Prague merchant, who became a crucial benefactor. She provided him with essential financial and moral support, enabling him to continue his research as a private scholar until her death in 1842. In 1842, Bolzano returned to Prague, where he lived until his death on 18 December 1848.

4.1. Foundations of Mathematical Analysis

Bolzano was one of the earliest mathematicians to champion the introduction of rigor into mathematical analysis, a field that, during his era, often relied on intuitive ideas like time and motion. His philosophical stance was that such intuitive concepts should be excluded from mathematics to achieve greater precision and certainty. He presented his groundbreaking approach in three principal mathematical works: Beyträge zu einer begründeteren Darstellung der Mathematik (Contributions to a Better Grounded Presentation of Mathematics) published in 1810, Der binomische Lehrsatz (The Binomial Theorem) in 1816, and Rein analytischer Beweis (Purely Analytic Proof) in 1817. These works offered "a sample of a new way of developing analysis," a methodological shift that would only be fully appreciated and adopted some five decades later, notably by Karl Weierstrass, who independently arrived at similar conclusions.

A significant contribution to the foundations of mathematical analysis was his introduction of a fully rigorous ε-δ definition of a mathematical limit. This definition, which precisely quantifies the notion of a limit without recourse to infinitesimals, became a cornerstone of modern analysis. Furthermore, Bolzano was the first to explicitly recognize the least-upper-bound property of the real numbers, a fundamental characteristic that underpins the completeness of the real number system. He was also skeptical of Gottfried Leibniz's infinitesimals, which had served as an early, albeit problematic, foundation for differential calculus. Bolzano's concept of a limit was remarkably similar to the modern understanding: it describes how a dependent variable approaches a definite quantity as an independent variable approaches another definite quantity, rather than being a relation among infinitesimals.

4.2. Key Theorems and Proofs

Bolzano's mathematical achievements include several pivotal theorems and their proofs. He provided the first purely analytic proof of the fundamental theorem of algebra, a theorem originally proven by Carl Friedrich Gauss using geometrical considerations. Bolzano's proof demonstrated that this fundamental result could be established without relying on geometric intuition.

He also delivered the first purely analytic proof of the intermediate value theorem, a result often referred to as Bolzano's theorem. This theorem states that if a continuous function takes values of opposite signs at two points, it must have at least one root between those points. While his work on this theorem was significant, Bolzano is perhaps most widely remembered for the Bolzano-Weierstrass theorem. This theorem, which states that every bounded sequence in real numbers has a convergent subsequence, was independently developed and published by Karl Weierstrass years after Bolzano's initial proof. Consequently, it was initially known as the Weierstrass theorem until Bolzano's earlier contributions were rediscovered, leading to its current dual attribution. In his later years, Bolzano also explored the concept of infinity, contributing to the understanding of the "paradoxes of the infinite."

5.1. Wissenschaftslehre (Theory of Science)

Bolzano's most significant philosophical work is his 1837 four-volume treatise, Wissenschaftslehre (Theory of Science). This monumental work aimed to provide a comprehensive and rigorous logical foundation for all sciences. It encompassed not only the philosophy of science in the modern sense but also extensive discussions on logic, epistemology, and scientific pedagogy. The logical theory developed within this work is widely acknowledged as groundbreaking, distinguishing Bolzano as a pivotal figure in the history of logic.

The Wissenschaftslehre built upon Bolzano's earlier thoughts in the philosophy of mathematics, such as his 1810 Beiträge, where he emphasized the critical distinction between the objective relationship among logical consequences and humanity's subjective recognition of these connections. For Bolzano, it was insufficient for the sciences merely to have "confirmation" of natural or mathematical truths; rather, their proper role was to seek out "justification" in terms of fundamental truths, regardless of whether these truths appeared obvious to human intuition.

Bolzano begins his work by defining what he means by "theory of science" and its relationship to human knowledge and truths. He posits that human knowledge comprises all truths (or true propositions) that individuals know or have known. However, this constitutes only a small fraction of all existing truths, yet it is still too vast for any single human to comprehend fully. Therefore, human knowledge is divided into more accessible parts, each such collection of truths forming what Bolzano calls a science (Wissenschaft). Crucially, he notes that not all true propositions within a science need to be known to humans, which allows for the possibility of discoveries within any given science. To better understand and organize these truths, humans create textbooks (Lehrbuch), which contain only the true propositions of a science known at a given time. The Theory of Science itself is the science that provides the rules for how to divide knowledge into coherent sciences and how these truths should be presented in a textbook.

5.2. Logic and Metaphysics

In his Wissenschaftslehre, Bolzano primarily focuses on three distinct realms, meticulously explaining their nature and interrelations:
1. **The realm of language**: This consists of words and sentences.
2. **The realm of thought**: This comprises subjective ideas and judgments.
3. **The realm of logic**: This is populated by objective ideas (or ideas in themselves) and propositions in themselves.

Two fundamental distinctions are central to Bolzano's system. First, he distinguishes between parts and wholes. For example, words are parts of sentences, subjective ideas are parts of judgments, and objective ideas are parts of propositions in themselves. Second, he categorizes all objects into those that exist-meaning they are causally connected and located in time and/or space-and those that do not exist. Bolzano's original and significant claim is that the logical realm, containing objective ideas and propositions in themselves, is populated exclusively by objects of the latter kind; they are non-existent.

• Satz an Sich (proposition in itself)** is a core concept introduced early in Bolzano's Wissenschaftslehre. He differentiates between a proposition (which can be spoken, written, thought, or "in itself") and an idea (similarly, spoken, written, thought, or "in itself"). For instance, "The grass is green" is a proposition because it asserts something, whereas "Grass" is merely an idea, representing something without asserting. Bolzano's definition of a proposition is broad, including even self-contradictory statements like "A rectangle is round," provided they are intelligibly composed of intelligible parts. A proposition in itself (i) has no existence, meaning it is not located in time or space; (ii) is either true or false independently of anyone knowing or thinking it to be so; and (iii) is what is "grasped" by thinking beings. Thus, a written sentence, such as "Socrates has wisdom," exists in time and space and expresses the proposition in itself [Socrates has wisdom], which resides in the non-existent realm of an sich. Bolzano's use of an sich differs significantly from Immanuel Kant's concept of the noumenon.

Every proposition in itself is composed of ideas in themselves. Ideas are negatively defined as those parts of a proposition that are not themselves propositions. A proposition typically consists of at least three ideas: a subject idea, a predicate idea, and the copula (which Bolzano prefers to be 'has' rather than 'is'). Bolzano identifies simple ideas, which have no parts (e.g., [something]), and complex ideas, which are composed of other ideas (e.g., [nothing], formed from [not] and [something]). Complex ideas can have the same constituent parts but differ in their arrangement, leading to distinct ideas, such as [A black pen with blue ink] versus [A blue pen with black ink].

It is crucial to understand that an idea does not necessarily need to have an object. Bolzano uses "object" to denote something that is represented by an idea. An idea that possesses an object represents that object, while an objectless idea represents nothing. For example, the idea [a round square] has no object because the represented object is self-contradictory. Similarly, the idea [nothing] certainly has no object. However, in the proposition [the idea of a round square has complexity], the subject-idea [the idea of a round square] does have an object, namely the idea [a round square] itself, even though that nested idea is objectless. Besides objectless ideas, there are singular ideas that represent only one object (e.g., [the first man on the moon]), and ideas that represent many objects (e.g., [the citizens of Amsterdam]) or even infinitely many objects (e.g., [a prime number]).

Bolzano developed a complex theory of how humans perceive things, centered on the concept of intuition, or Anschauung in German. An intuition is a simple idea that represents only one object (Einzelvorstellung) and is also unique. Bolzano posits that intuitions are objective ideas belonging to the an sich realm, meaning they do not exist in time or space. His argument for intuitions is rooted in his explanation of sensation. When a person senses a real, existing object, such as a rose, the various aspects of the rose (its scent, color, etc.) cause a change in that person. This change signifies a different mental state before and after sensing the rose. Thus, sensation is fundamentally a change in one's mental state. Bolzano explains that this mental change is essentially a simple idea, like 'this smell' (referring to the specific scent of that particular rose). This idea represents the change itself. This change must be unique because no two experiences are ever precisely identical, nor can two individuals smelling the same rose have the exact same subjective experience, though their experiences will be very similar. Consequently, each distinct sensation generates a unique and simple subjective idea, with a particular change as its object. This subjective idea, existing within the mind at a specific time, corresponds to an objective idea. These objective ideas are what Bolzano terms intuitions (Anschauungen), serving as the simple, unique, and objective counterparts to our subjective ideas of changes caused by sensation. Therefore, for every possible sensation, there exists a corresponding objective idea. Schematically, the process involves the rose's scent causing a change in the observer, which becomes the object of a subjective idea of that specific smell, and this subjective idea corresponds to the intuition or Anschauung.

In Bolzano's logical theory, all propositions are composed of three elements (which can be simple or complex): a subject, a predicate, and a copula. He preferred the copula 'has' over the more traditional 'is', arguing that 'has' can more effectively connect a concrete term, such as 'Socrates', to an abstract term, like 'baldness'. According to Bolzano, "Socrates has baldness" is preferable to "Socrates is bald" because the latter form is less fundamental, as 'bald' itself is composed of the elements 'something', 'that', 'has', and 'baldness'. He also reduced existential propositions to this form; for example, "Socrates exists" would be rendered as "Socrates has existence (Dasein)."

A central concept in Bolzano's logic is the notion of variations. Various logical relations are defined by observing the changes in truth value that propositions undergo when their non-logical parts are systematically replaced by others. For instance, logically analytical propositions are those in which all non-logical parts can be replaced without altering the truth value. Two propositions are considered 'compatible' (verträglich) with respect to one of their component parts x if there is at least one term that can be inserted to make both propositions true. A proposition Q is 'deducible' (ableitbar) from a proposition P, with respect to certain non-logical parts, if any replacement of those parts that makes P true also makes Q true. If a proposition is deducible from another with respect to all its non-logical parts, it is termed 'logically deducible'. Beyond the relation of deducibility, Bolzano also introduced a stricter relation called 'grounding' (Abfolge). This is an asymmetric relation that holds between true propositions when one proposition is not only deducible from but also explained by the other.

• Truth** is a central theme in Bolzano's philosophy. He distinguishes five meanings of the words "true" and "truth" as commonly used, all of which he considers unproblematic:
  • I. Abstract objective meaning**: "Truth" signifies an attribute that applies to a proposition, primarily a proposition in itself. This attribute indicates that the proposition expresses something that genuinely corresponds to reality as it is expressed. Its antonyms are falsity, falseness, and falsehood.
  • II. Concrete objective meaning**: "Truth" refers to a proposition that possesses the attribute of "truth" in the abstract objective sense. Its antonym is falsehood.
  • III. Subjective meaning**: "Truth" denotes a correct judgment. Its antonym is mistake.
  • IV. Collective meaning**: "Truth" signifies a body or multiplicity of true propositions or judgments (e.g., "the biblical truth").
  • V. Improper meaning**: "True" indicates that an object is in reality what a certain denomination states it to be (e.g., "the true God"). Its antonyms are false, unreal, and illusory.

  Bolzano's primary focus is on the concrete objective meaning of truth, or truths in themselves. All truths in themselves are a type of proposition in themselves. They do not exist in the sense of being spatiotemporally located, unlike thought or spoken propositions. However, certain propositions possess the attribute of being a truth in itself. While, given God's omniscience, all truths in themselves are also thought truths, being a thought proposition is not an inherent part of the concept of a truth in itself. Thus, the concepts 'truth in itself' and 'thought truth' are interchangeable in their application to the same objects, but they are not identical in meaning. Bolzano defines (abstract objective) truth as a proposition being true if it expresses something that applies to its object. Consequently, a (concrete objective) truth is a proposition that expresses something that applies to its object. This definition applies to truths in themselves, as none of the concepts within this definition are subordinate to mental or known concepts.
  
  In sections 31-32 of his Wissenschaftslehre, Bolzano provides proofs for three fundamental assertions about truths in themselves:

  • A. There is at least one truth in itself (concrete objective meaning):**

  1. Assume there are no true propositions.
  2. Statement (1.) is itself a proposition.
  3. Statement (1.) is assumed to be true (from 1.) and simultaneously false (because if there are no true propositions, then 1. itself cannot be true).
  4. Therefore, statement (1.) is self-contradictory.
  5. A self-contradictory proposition is false.
  6. Thus, statement (1.) is false, which implies that its negation is true: there is at least one true proposition.

  • B. There is more than one truth in itself:**

  7. Assume there is only one truth in itself, namely "A is B."
  8. "A is B" is a truth in itself (from 7.).
  9. "There are no other truths in themselves apart from A is B" (from 7.).
  10. Statement (9.) is a true proposition / a truth in itself (because it is a true statement about the assumed state of affairs).
  11. Therefore, there are at least two truths in themselves (from 8. and 10.).
  12. This contradicts the initial assumption (7.), proving that there is more than one truth in itself.

  • C. There are infinitely many truths in themselves:**

  13. Assume there are only n truths in themselves, specifically "A is B, ..., Y is Z."
  14. "A is B, ..., Y is Z" are n truths in themselves (from 13.).
  15. "There are no other truths apart from A is B, ..., Y is Z" (from 13.).
  16. Statement (15.) is a true proposition / a truth in itself (because it is a true statement about the assumed state of affairs).
  17. Therefore, there are n+1 truths in themselves (from 14. and 16.).
  18. Steps 1 to 5 (the proof for at least one truth) can be repeated for n+1 truths, which results in n+2 truths, and so on endlessly, because n is a variable.
  19. Thus, there are infinitely many truths in themselves.
  
  A known truth, according to Bolzano, consists of a truth in itself and a judgment. A judgment is a thought that affirms a true proposition. When judging (at least when the content of the judgment is a true proposition), the idea of an object is connected in a specific way with the idea of a characteristic. In true judgments, the relationship between the idea of the object and the idea of the characteristic is an actual, existing relation. Every judgment has a proposition as its content, which is either true or false. Unlike propositions in themselves, judgments exist but are dependent on subjective mental activity. However, not every mental activity is a judgment; only those that affirm propositions and are therefore either true or false (e.g., mere presentations or thoughts are not judgments). Judgments whose content is a true proposition are called cognitions. Cognitions, being dependent on the subject, permit degrees; a proposition can be more or less known, but it cannot be more or less true. Every cognition necessarily implies a judgment, but not every judgment is necessarily a cognition, as there are also false judgments. Bolzano maintains that there are no such things as false cognitions, only false judgments.

5.3. Paradoxes of the Infinite

Bolzano's posthumously published work, Paradoxien des Unendlichen (Paradoxes of the Infinite), released in 1851, was highly regarded by many eminent logicians who followed him, including Charles Sanders Peirce, Georg Cantor, and Richard Dedekind. In this work, written in his final years (1848), Bolzano ventured to integrate the philosophical concept of infinity into mathematics. Georg Cantor, the founder of set theory, particularly admired Bolzano, referring to him as the "decisive defender" of the concept of the actual infinite. This work proved to be a crucial contribution to the development of mathematical thought on the infinite.

6.1. Liberalism and Pacifism

Bolzano was a staunch advocate of classical liberal views. His deep-seated convictions led him to openly criticize militarism and denounce the needlessness of war, which he considered a significant social waste. These teachings often alienated him from the conservative academic and ecclesiastical authorities of his time. He firmly believed that peace should be the fundamental goal of any society, and that national interests should be directed towards peaceful coexistence rather than armed conflict. His political stance was considered excessively liberal by the Austrian authorities, leading to his eventual dismissal from his professorship.

6.2. Social Reform Advocacy

Driven by his liberal and pacifist ideals, Bolzano proposed comprehensive reforms across various societal systems. He urged a total overhaul of the educational, social, and economic structures. His aim was to reshape these systems in a way that would foster peace and promote social equity, directing the nation's collective interests away from conflict and towards societal well-being. He even envisioned a form of communist state, influenced by Saint-Simonian thought, demonstrating his radical commitment to social transformation and justice.

7.1. Mathematical Legacy

Bolzano's mathematical discoveries and his pioneering approach to introducing rigor into mathematical analysis had a profound, albeit delayed, influence on later mathematicians and the development of modern analysis. While much of his work remained in manuscript form and had limited circulation during his lifetime, it was eventually brought to light. His contributions to the Bolzano-Weierstrass theorem are particularly notable, even though Karl Weierstrass independently developed and published the theorem years after Bolzano's initial proof. The significance of Bolzano's earlier work was rediscovered by Otto Stolz, who republished many of his lost journal articles in 1881, thereby ensuring his rightful place in the history of mathematics. His work on the Paradoxien des Unendlichen also contributed significantly to the development of the concept of the actual infinite, with Georg Cantor, the founder of set theory, explicitly praising Bolzano as its "decisive defender."

7.2. Philosophical Influence

Initially, the impact of Bolzano's philosophical thought appeared limited, largely due to the opposition he faced and the restrictions on his publications. However, he was surrounded by a dedicated group of friends and pupils, collectively known as the Bolzano Circle, who helped disseminate his ideas. His work experienced a significant rediscovery through the efforts of Edmund Husserl and Kazimierz Twardowski, both of whom were students of Franz Brentano. Through their engagement with his philosophy, Bolzano became a formative influence on the development of both phenomenology and analytic philosophy. Edmund Husserl, in his seminal work Logical Investigations, famously lauded Bolzano as "one of the greatest logicians of all time." Today, Bolzano is recognized as a crucial figure in the modern period of logic and mathematics. In recognition of his contributions, the asteroid 2622 Bolzano was named in his honor.