Bernard Bolzano
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Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a
Bohemian A Bohemian () is a resident of Bohemia Bohemia ( ; cs, Čechy ; ; hsb, Čěska; szl, Czechy) is the westernmost and largest historical region of the Czech lands in the present-day Czech Republic. Bohemia can also refer to a wider area cons ...
mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

mathematician
,
logician Logic is an interdisciplinary field which studies truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to re ...

logician
,
philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, ...

philosopher
,
theologian Theology is the systematic study of the nature of the divine Divinity or the divine are things that are either related to, devoted to, or proceeding from a deity.
and
Catholic priest The priesthood is the office of the ministers of religion, who have been commissioned ("ordained") with the Holy orders of the Catholic Church. Technically, Bishop in the Catholic Church, bishops and Deacon in the Catholic Church, deacons are pries ...
of Italian extraction, also known for his
liberal Liberal or liberalism may refer to: Politics *a supporter of liberalism, a political and moral philosophy **Liberalism by country *an adherent of a Liberal Party Arts, entertainment and media *''El Liberal'', a Spanish newspaper published betw ...
views. Bolzano wrote in
German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of Germany, see also German nationality law * German language The German la ...

German
, his native language. For the most part, his work came to prominence posthumously.


Family

Bolzano was the son of two pious
Catholics The Catholic Church, also known as the Roman Catholic Church, is the largest Christian church, with 1.3 billion baptised Baptism (from the Greek language, Greek noun βάπτισμα ''báptisma'') is a Christians, Christian ri ...

Catholics
. His father, Bernard Pompeius Bolzano, was an Italian who had moved to
Prague Prague ( ; cs, Praha ; german: Prag, ; la, Praga) is the capital and largest city A city is a large human settlement In geography, statistics and archaeology, a settlement, locality or populated place is a community in which people l ...

Prague
, where he married Maria Cecilia Maurer who came from Prague's German-speaking family Maurer. Only two of their twelve children lived to adulthood.


Career

Bolzano entered the University of Prague in 1796 and studied
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
,
philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical reality Reality is the sum or aggregate of all that is real or existen ...

philosophy
and
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

physics
. Starting in 1800, he also began studying
theology Theology is the systematic study of the nature of the divine Divinity or the divine are things that are either related to, devoted to, or proceeding from a deity.
, becoming a
Catholic The Catholic Church, also known as the Roman Catholic Church, is the List of Christian denominations by number of members, largest Christian church, with 1.3 billion baptised Catholics Catholic Church by country, worldwide . As the wo ...

Catholic
priest A priest is a religious leader authorized to perform the Sacred rite, sacred rituals of a religion, especially as a mediatory agent between humans and one or more deity, deities. They also have the authority or power to administer religious ...

priest
in 1804. He was appointed to the new chair of
philosophy of religion Philosophy of religion is "the philosophical examination of the central themes and concepts involved in religious traditions". Philosophical discussions on such topics date from ancient times, and appear in the earliest known s concerning philo ...
at Prague University in 1805. He proved to be a popular lecturer not only in religion but also in philosophy, and he was elected Dean of the Philosophical Faculty in 1818. Bolzano alienated many faculty and church leaders with his teachings of the social waste of militarism and the needlessness of 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. His political convictions, which he was inclined to share with others with some frequency, eventually proved to be too
liberal Liberal or liberalism may refer to: Politics *a supporter of liberalism, a political and moral philosophy **Liberalism by country *an adherent of a Liberal Party Arts, entertainment and media *''El Liberal'', a Spanish newspaper published betw ...
for the
Austrian Austrian may refer to: * Austrians, someone from Austria or of Austrian descent ** Someone who is considered an Austrian citizen, see Austrian nationality law * Something associated with the country Austria, for example: ** Austria-Hungary ** Austr ...
authorities. On December 24, 1819, he was removed from his professorship (upon his refusal to recant his beliefs) and was
exile To be in exile means to be forced away from one's home (i.e. , , , , , or even ) and unable to return. People (or corporations and even ) may be in exile for legal or other reasons. In , ''exsilium'' denoted both voluntary exile and banishme ...

exile
d to the
countryside A rural landscape in Lappeenranta, South Karelia, Finland. 15 July 2000.">South_Karelia.html" ;"title="Lappeenranta, South Karelia">Lappeenranta, South Karelia, Finland. 15 July 2000. In general, a rural area or a countryside is a geographic ...

countryside
and then devoted his energies to his writings on social, religious, philosophical, and mathematical matters. Although forbidden to
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publish
in
mainstreamThe mainstream is the prevalent current thought In their most common sense, the terms thought and thinking refer to conscious cognitive processes that can happen independently of sensory stimulation. Their most paradigmatic forms are judging, rea ...
as a condition of his exile, Bolzano continued to develop his ideas and publish them either on his own or in obscure
Eastern Europe Eastern Europe is the eastern region of . There is no consistent definition of the precise area it covers, partly because the term has a wide range of , geographical, ethnic, cultural, and connotations. , located in Eastern Europe, is both the ...

Eastern Europe
an journals. In 1842 he moved back to Prague, where he died in 1848.


Mathematical work

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. To this end, he was one of the earliest mathematicians to begin instilling
rigor Rigour (British English British English (BrE) is the standard dialect of the English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval England, wh ...
into
mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical ...
with his three chief mathematical works ''Beyträge zu einer begründeteren Darstellung der Mathematik'' (1810), ''Der binomische Lehrsatz'' (1816) and ''Rein analytischer Beweis'' (1817). These 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 Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematics, mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university withou ...

Karl Weierstrass
. To the foundations of
mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical ...
he contributed the introduction of a fully rigorous ε–δ definition of a mathematical limit. Bolzano was the first to recognize the greatest lower bound property of the real numbers. Like several others of his day, he was skeptical of the possibility of
Gottfried Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the "#1666–1676, 1666–1676" section. ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist, and diplomat. He is a promin ...
's
infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...
s, that had been the earliest putative foundation for
differential calculus In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. 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 also gave the first purely analytic proof of the
fundamental theorem of algebra The fundamental theorem of algebra states that every non- constant single-variable polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, s ...
, which had originally been proven by
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes referr ...

Gauss
from geometrical considerations. He also gave the first purely
analytic proofIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
of the
intermediate value theorem In mathematical analysis, the intermediate value theorem states that if ''f'' is a continuous function whose domain contains the interval 'a'', ''b'' then it takes on any given value between ''f''(''a'') and ''f''(''b'') at some point ...

intermediate value theorem
(also known as Bolzano's theorem). Today he is mostly remembered for the
Bolzano–Weierstrass theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, which
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematics, mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university withou ...

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.


Philosophical work

Bolzano's posthumously published work '' Paradoxien des Unendlichen (The Paradoxes of the Infinite)'' (1851) was greatly admired by many of the eminent
logician Logic is an interdisciplinary field which studies truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to re ...

logician
s who came after him, including
Charles Sanders Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, app ...

Charles Sanders Peirce
,
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...
, and
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory In algebra, ring theory is the study of ring (mathematics), rings ...
. Bolzano's main claim to fame, however, is his 1837 ''Wissenschaftslehre'' (''Theory of Science''), a work in four volumes that covered not only
philosophy of science Philosophy of science is a branch of philosophy concerned with the foundations, methodology, methods, and implications of science. The central questions of this study concern Demarcation problem, what qualifies as science, the reliability of sc ...
in the modern sense but also logic,
epistemology Epistemology (; ) is the concerned with . Epistemologists study the nature, origin, and scope of knowledge, epistemic , the of , and various related issues. Epistemology is considered a major subfield of philosophy, along with other major ...

epistemology
and scientific pedagogy. The logical theory that Bolzano developed in this work has come to be acknowledged as ground-breaking. Other works are a four-volume ''Lehrbuch der Religionswissenschaft'' (''Textbook of the Science of Religion'') and the metaphysical work ''Athanasia'', a defense of the immortality of the soul. Bolzano also did valuable work in mathematics, which remained virtually unknown until Otto Stolz rediscovered many of his lost journal articles and republished them in 1881.


''Wissenschaftslehre (Theory of Science)''

In his 1837 ''Wissenschaftslehre'' Bolzano attempted to provide logical foundations for all sciences, building on abstractions like part-relation,
abstract object In metaphysics, the distinction between abstract and concrete refers to a divide between two types of entities. Many philosophers hold that this difference has fundamental metaphysical significance. Examples of concrete objects include Plant, plant ...
s, attributes, sentence-shapes, ideas and propositions in themselves, sums and sets, collections, substances, adherences, subjective ideas, judgments, and sentence-occurrences. These attempts were basically an extension of his earlier thoughts in the philosophy of mathematics, for example his 1810 ''Beiträge'' where he emphasized the distinction between the objective relationship between
logical consequence Logical consequence (also entailment) is a fundamental concept Concepts are defined as abstract ideas or general notions that occur in the mind, in speech, or in thought. They are understood to be the fundamental building blocks of thoughts ...
s 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.


Introduction to ''Wissenschaftslehre''

Bolzano begins his work by explaining what he means by ''theory of science'', and the relation between our knowledge, truths and sciences. Human knowledge, he states, is made of all truths (or true propositions) that men know or have known. This is, however, only a very small fraction of all the truths that exist, although still too much for one human being to comprehend. Therefore, our knowledge is divided into more accessible parts. Such a collection of truths is what Bolzano calls a science (''Wissenschaft''). It is important to note that not all true propositions of a science have to be known to men; hence, this is how we can make discoveries in a science. To better understand and comprehend the truths of a science, men have created textbooks (''Lehrbuch''), which of course contain only the true propositions of the science known to men. But how to know where to divide our knowledge, that is, which truths belong together? Bolzano explains that we will ultimately know this through some reflection, but that the resulting rules of how to divide our knowledge into sciences will be a science in itself. This science, that tells us which truths belong together and should be explained in a textbook, is the ''Theory of Science'' (''Wissenschaftslehre'').


Metaphysics

In the ''Wissenschaftslehre'', Bolzano is mainly concerned with three realms: (1) The realm of language, consisting in words and sentences.
(2) The realm of thought, consisting in subjective ideas and judgements.
(3) The realm of logic, consisting in objective ideas (or ideas in themselves) and propositions in themselves. Bolzano devotes a great part of the ''Wissenschaftslehre'' to an explanation of these realms and their relations. Two distinctions play a prominent role in his system. Firstly, the distinction between parts and wholes. For instance, words are parts of sentences, subjective ideas are parts of judgments, objective ideas are parts of propositions in themselves. Secondly, all objects divide into those that
exist eXist-db (or eXist for short) is an open source software Open-source software (OSS) is a type of computer software Software is a collection of Instruction (computer science), instructions and data (computing), data that tell a compu ...

exist
, which means that they are causally connected and located in time and/or space, and those that do not exist. Bolzano's original claim is that the logical realm is populated by objects of the latter kind.


''Satz an Sich'' (proposition in itself)

''Satz an Sich'' is a basic notion in Bolzano's ''Wissenschaftslehre''. It is introduced at the very beginning, in section 19. Bolzano first introduces the notions of
proposition In logic and linguistics, a proposition is the meaning of a declarative sentence (linguistics), sentence. In philosophy, "Meaning (philosophy), meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same mea ...
(spoken or written or thought or in itself) and
idea In common usage and in philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philoso ...

idea
(spoken or written or thought or in itself). "The grass is green" is a proposition (''Satz''): in this connection of words, something is said or asserted. "Grass", however, is only an idea (''Vorstellung''). Something is represented by it, but it does not assert anything. Bolzano's notion of proposition is fairly broad: "A rectangle is round" is a proposition — even though it is false by virtue of self-
contradiction In traditional logicIn philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philo ...

contradiction
— because it is composed in an intelligible manner out of intelligible parts. Bolzano does not give a complete definition of a ''Satz an Sich'' (i.e. proposition in itself) but he gives us just enough information to understand what he means by it. A proposition in itself (i) has no existence (that is: it has no position in time or place), (ii) is either true or false, independent of anyone knowing or thinking that it is true or false, and (iii) is what is 'grasped' by thinking beings. So a written sentence ('Socrates has wisdom') grasps a proposition in itself, namely the proposition ocrates has wisdom The written sentence does have existence (it has a certain location at a certain time, say it is on your computer screen at this very moment) and expresses the proposition in itself which is in the realm of in itself (i.e. ''an sich''). (Bolzano's use of the term ''an sich'' differs greatly from that of Immanuel Kant, Kant; for Kant's use of the term see noumenon, ''an sich''.) Every proposition in itself is composed out of ideas in themselves (for simplicity, we will use ''proposition'' to mean "proposition in itself" and ''idea'' to refer to an objective idea or idea in itself. Ideas are negatively defined as those parts of a proposition that are themselves not propositions. A proposition consists of at least three ideas, namely: a subject idea, a predicate idea and the copula (i.e. 'has', or another form of ''to have''). (Though there are propositions which contain propositions, but we won't take them into consideration right now.) Bolzano identifies certain types of ideas. There are simple ideas that have no parts (as an example Bolzano uses [something]), but there are also complex ideas that consist of other ideas (Bolzano uses the example of [nothing], which consists of the ideas [not] and [something]). Complex ideas can have the same content (i.e. the same parts) without being the same — because their components are differently connected. The idea [A black pen with blue ink] is different from the idea [A blue pen with black ink] though the parts of both ideas are the same.


Ideas and objects

It is important to understand that an idea does not need to have an object. Bolzano uses ''object'' to denote something that is represented by an idea. An idea that has an object, represents that object. But an idea that does not have an object represents nothing. (Don't get confused here by terminology: an objectless idea is an idea without a representation.) Consider, for further explanation, an example used by Bolzano. The idea [a round square], does not have an object, because the object that ought to be represented is self-contrary. A different example is the idea [nothing] which certainly does not have an object. However, the proposition [the idea of a round square has complexity] has as its subject-idea [the idea of a round square]. This subject-idea does have an object, namely the idea [a round square]. But, that idea does not have an object. Besides objectless ideas, there are ideas that have only one object, e.g. the idea [the first man on the moon] represents only one object. Bolzano calls these ideas 'singular ideas'. Obviously there are also ideas that have many objects (e.g. [the citizens of Amsterdam]) and even infinitely many objects (e.g. [a prime number]).


Sensation and simple ideas

Bolzano has a complex theory of how we are able to sense things. He explains sensation by means of the term intuition, in German called ''Anschauung''. An intuition is a simple idea, it has only one object (''Einzelvorstellung''), but besides that, it is also unique (Bolzano needs this to explain sensation). Intuitions (''Anschauungen'') are objective ideas, they belong to the ''an sich'' realm, which means that they don't have existence. As said, Bolzano's argumentation for intuitions is by an explanation of sensation. What happens when you sense a real existing object, for instance a rose, is this: the different aspects of the rose, like its scent and its color, cause in you a change. That change means that before and after sensing the rose, your mind is in a different state. So sensation is in fact a change in your mental state. How is this related to objects and ideas? Bolzano explains that this change, in your mind, is essentially a simple idea (''Vorstellung''), like, ‘this smell’ (of this particular rose). This idea represents; it has as its object the change. Besides being simple, this change must also be unique. This is because literally you can't have the same experience twice, nor can two people, who smell the same rose at the same time, have exactly the same experience of that smell (although they will be quite alike). So each single sensation causes a single (new) unique and simple idea with a particular change as its object. Now, this idea in your mind is a subjective idea, meaning that it is in you at a particular time. It has existence. But this subjective idea must correspond to, or has as a content, an objective idea. This is where Bolzano brings in intuitions (''Anschauungen''); they are the simple, unique and objective ideas that correspond to our subjective ideas of changes caused by sensation. So for each single possible sensation, there is a corresponding objective idea. Schematically the whole process is like this: whenever you smell a rose, its scent causes a change in you. This change is the object of your subjective idea of that particular smell. That subjective idea corresponds to the intuition or ''Anschauung''.


Logic

According to Bolzano, all propositions are composed out of three (simple or complex) elements: a subject, a predicate and a Copula (linguistics), copula. Instead of the more traditional copulative term 'is', Bolzano prefers 'has'. The reason for this is that 'has', unlike 'is', can connect a concrete term, such as 'Socrates', to an abstract term such as 'baldness'. "Socrates has baldness" is, according to Bolzano, preferable to "Socrates is bald" because the latter form is less basic: 'bald' is itself composed of the elements 'something', 'that', 'has' and 'baldness'. Bolzano also reduces existential propositions to this form: "Socrates exists" would simply become "Socrates has existence (''Dasein'')". A major role in Bolzano's logical theory is played by the notion of ''variations'': various logical relations are defined in terms of the changes in truth value that propositions incur when their non-logical parts are replaced by others. Logically Analytic-synthetic distinction, analytical propositions, for instance, are those in which all the non-logical parts can be replaced without change of truth value. Two propositions are 'compatible' (''verträglich'') with respect to one of their component parts x if there is at least one term that can be inserted that would make both true. A proposition Q is 'deducible' (''ableitbar'') from a proposition P, with respect to certain of their 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 said to be 'logically deducible'. Besides the relation of deducibility, Bolzano also has a stricter relation of 'consequentiality' (''Abfolge''). This is an asymmetric relation that obtains between true propositions, when one of the propositions is not only deducible from, but also explanation, explained by the other.


Truth

Bolzano distinguishes five meanings the words ''true'' and ''truth'' have in common usage, all of which Bolzano takes to be unproblematic. The meanings are listed in order of properness: I. Abstract objective meaning: ''Truth'' signifies an attribute that may apply to a proposition, primarily to a proposition in itself, namely the attribute on the basis of which the proposition expresses something that in reality is as is expressed. Antonyms: ''falsity, falseness, falsehood''. II. Concrete objective meaning: (a) ''Truth'' signifies a proposition that has the attribute ''truth'' in the abstract objective meaning. Antonym: (a) ''falsehood''. III. Subjective meaning: (a) ''Truth'' signifies a correct judgment. Antonym: (a) ''mistake''. IV. Collective meaning: ''Truth'' signifies a body or multiplicity true propositions or judgments (e.g. the biblical truth). V. Improper meaning: ''True'' signifies that some object is in reality what some denomination states it to be. (e.g. the true God). Antonyms: ''false, unreal, illusory''. Bolzano's primary concern is with the concrete objective meaning: with concrete objective truths or truths in themselves. All truths in themselves are a kind of propositions in themselves. They do not exist, i.e. they are not spatiotemporally located as thought and spoken propositions are. However, certain propositions have the attribute of being a truth in itself. Being a thought proposition is not a part of the concept of a truth in itself, notwithstanding the fact that, given God's omniscience, all truths in themselves are also thought truths. The concepts ‘truth in itself’ and ‘thought truth’ are interchangeable, as they apply to the same objects, but they are not identical. Bolzano offers as the correct definition of (abstract objective) truth: a proposition is true if it expresses something that applies to its object. The correct definition of a (concrete objective) truth must thus be: a truth is a proposition that expresses something that applies to its object. This definition applies to truths in themselves, rather than to thought or known truths, as none of the concepts figuring in this definition are subordinate to a concept of something mental or known. Bolzano proves in §§31–32 of his ''Wissenschaftslehre'' three things: A There is at least one truth in itself (concrete objective meaning): :1. There are no true propositions (assumption) :2. 1. is a proposition (obvious) :3. 1. is true (assumed) and false (because of 1.) :4. 1. is self-contradictory (because of 3.) :5. 1. is false (because of 4.) :6. There is at least one true proposition (because of 1. and 5.) B. There is more than one truth in itself: :7. There is only one truth in itself, namely A is B (assumption) :8. A is B is a truth in itself (because of 7.) :9. There are no other truths in themselves apart from A is B (because of 7.) :10. 9. is a true proposition/ a truth in itself (because of 7.) :11. There are two truths in themselves (because of 8. and 10.) :12. There is more than one truth in itself (because of 11.) C. There are infinitely many truths in themselves: :13. There are only n truths in themselves, namely A is B .... Y is Z (assumption) :14. A is B .... Y is Z are n truths in themselves (because of 13.) :15. There are no other truths apart from A is B .... Y is Z (because of 13.) :16. 15. is a true proposition/ a truth in itself (because of 13.) :17. There are n+1 truths in themselves (because of 14. and 16.) :18. Steps 1 to 5 can be repeated for n+1, which results in n+2 truths and so on endlessly (because n is a variable) :19. There are infinitely many truths in themselves (because of 18.)


Judgments and cognitions

A known truth has as its parts (''Bestandteile'') a truth in itself and a judgment (Bolzano, ''Wissenschaftslehre'' §26). A judgment is a thought which states a true proposition. In judging (at least when the matter of the judgment is a true proposition), the idea of an object is being connected in a certain way with the idea of a characteristic (§ 23). In true judgments, the relation between the idea of the object and the idea of the characteristic is an actual/existent relation (§28). Every judgment has as its matter a proposition, which is either true or false. Every judgment exists, but not "für sich". Judgments, namely, in contrast with propositions in themselves, are dependent on subjective mental activity. Not every mental activity, though, has to be a judgment; recall that all judgments have as matter propositions, and hence all judgments need to be either true or false. Mere presentations or thoughts are examples of mental activities which do not necessarily need to be stated (behaupten), and so are not judgments (§ 34). Judgments that have as its matter true propositions can be called cognitions (§36). Cognitions are also dependent on the subject, and so, opposed to truths in themselves, cognitions do permit degrees; a proposition can be more or less known, but it cannot be more or less true. Every cognition implies necessarily a judgment, but not every judgment is necessarily cognition, because there are also judgments that are not true. Bolzano maintains that there are no such things as false cognitions, only false judgments (§34).


Philosophical legacy

Bolzano came to be surrounded by a circle of friends and pupils who spread his thoughts about (the so-called Bolzano Circle), but the effect of his thought on philosophy initially seemed destined to be slight. His work was rediscovered, however, by Edmund Husserl and Kazimierz Twardowski, both students of Franz Brentano. Through them, Bolzano became a formative influence on both phenomenology (philosophy), phenomenology and analytic philosophy.


Writings

* ''Bolzano: Gesamtausgabe'' (''Bolzano: Collected Works''), critical edition edited by Eduard Winter, , Friedrich Kambartel, Bob van Rootselaar, Stuttgart: Fromman-Holzboog, 1969ff. (103 Volumes available, 28 Volumes in preparation).frommann-holzboog.de
/ref> * ''Wissenschaftslehre'', 4 vols., 2nd rev. ed. by W. Schultz, Leipzig I–II 1929, III 1980, IV 1931; Critical Edition edited by Jan Berg: Bolzano's Gesamtausgabe, vols. 11–14 (1985–2000). * ''Bernard Bolzano's Grundlegung der Logik. Ausgewählte Paragraphen aus der Wissenschaftslehre'', Vols. 1 and 2, with supplementary text summaries, an introduction and indices, edited by F. Kambartel, Hamburg, 1963, 1978². * (''Contributions to a better grounded presentation of mathematics''; and ''The Mathematical Works of Bernard Bolzano'', 2004, pp. 83–137). * (''Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation''; . * Franz Prihonsky (1850), ''Der Neue Anti-Kant'', Bautzen (an assessment of the ''Critique of Pure Reason'' by Bolzano, published posthumously by his friend F. Prihonsky).* (''Paradoxes of the Infinite''; (excerpt)).


Translations and compilations

*
Theory of Science
' (selection edited and translated by Rolf George, Berkeley and Los Angeles: University of California Press, 1972). *

' (selection edited, with an introduction, by Jan Berg. Translated from the German by Burnham Terrell, Dordrecht and Boston: D. Reidel Publishing Company, 1973). * ''Theory of Science'', first complete English translation in four volumes by Rolf George and Paul Rusnock, New York: Oxford University Press, 2014. * ''The Mathematical Works of Bernard Bolzano'', translated and edited by Steve Russ, New York: Oxford University Press, 2004 (re-printed 2006). * ''On the Mathematical Method and Correspondence with Exner'', translated by Rolf George and Paul Rusnock, Amsterdam: Rodopi, 2004. * ''Selected Writings on Ethics and Politics'', translated by Rolf George and Paul Rusnock, Amsterdam: Rodopi, 2007. * Franz Prihonsky, ''The New Anti-Kant'', edited by Sandra Lapointe and Clinton Tolley, New York, Palgrave Macmillan, 2014. * (Translation of ''Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege'' (Prague 1817))


See also

* List of Roman Catholic scientist-clerics


Notes


References

* . * . * . * . Retrieved on 2007-03-05 * Kamila Veverková, Veverková, Kamila, "Kleinere Schriften des deutschen Lehrers und Priester (17901869)." In: ''Homiletisch – Liturgisches Korrespondenzblatt – Neue Folge''. Nr 107, Jg 28/2011, str. 758–782. . * .


Further reading

* (1972), "Von Bolzano zu Meinong: Zur Geschichte des logischen Realismus." In: Rudolf Haller (ed.), ''Jenseits von Sein und Nichtsein: Beiträge zur Meinong-Forschung'', Graz, pp. 69–102.


External links

* * *
Bolzano’s Philosophy of Mathematical Knowledge
entry by Sandra Lapointe in the ''Internet Encyclopedia of Philosophy''
The Philosophy of Bernard Bolzano: Logic and Ontology




* [http://www.ontology.co/biblio/bolzanob-biblio-two.htm Annotated Bibliography on the Philosophical Work of Bolzano (Second Part: D - L)]
Annotated Bibliography on the Philosophical Work of Bolzano (Third Part: M - Z)
* *
Digitized Bolzano's works

Volume 1 of ''Wissenschaftslehre'' in Google Books

Volume 2 of ''Wissenschaftslehre'' in Google Books

Volumes 3–4 of ''Wissenschaftslehre'' in Google Books

Volume 1 of ''Wissenschaftslehre'' in Archive.org
(pages 162 to 243 are missing)
Volume 2 of ''Wissenschaftslehre'' in Archive.org

Volume 4 of ''Wissenschaftslehre'' in Archive.org

Volume 3 of ''Wissenschaftslehre'' in Gallica

Volume 4 of ''Wissenschaftslehre'' in Gallica
{{DEFAULTSORT:Bolzano, Bernard 1781 births 1848 deaths 19th-century philosophers 19th-century mathematicians Catholic clergy scientists 19th-century Czech people Czech writers in German Czech philosophers Czech mathematicians Mathematical logicians Mathematical analysts Ontologists Enlightenment philosophers Charles University alumni Czech pacifists Czech people of Italian descent Mathematicians from Prague Burials at Olšany Cemetery