Bernard Bolzano
Encyclopedia
Bernhard Placidus Johann Nepomuk Bolzano (5 October 1781 – December 18, 1848), Bernard Bolzano in English, was a Bohemia
n mathematician
, logician, philosopher, theologian
, Catholic priest and antimilitarist
of German
mother tongue.
. His father, Bernard Pompeius Bolzano, was born in northern Italy and moved to Prague
, where he married Maria Cecilia Maurer, the (German-speaking) daughter of a Prague merchant. Only two of their twelve children lived to adulthood.
in 1796 and studied mathematics
, philosophy
and physics
. Starting in 1800, he also began studying theology
, becoming a Catholic
priest
in 1804. He was appointed to the then newly created chair of philosophy of religion
in 1805. He proved to be a popular lecturer not just in religion but also in philosophy, and was elected head of the philosophy department 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. Upon his refusal to recant his beliefs, Bolzano was dismissed from the university in 1819. His political convictions (which he was inclined to share with others with some frequency) eventually proved to be too liberal
for the Austrian
authorities. He was exile
d to the countryside and at that point devoted his energies to his writings on social, religious, philosophical, and mathematical matters. Although forbidden to publish in mainstream journals
as a condition of his exile, Bolzano continued to develop his ideas and publish them either on his own or in obscure Eastern Europe
an journals. In 1842 he moved back to Prague, where he died in 1848.
was greatly admired by many of the eminent logicians who came after him, including Charles Sanders Peirce, Georg Cantor
, and Richard Dedekind
. 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
in the modern sense but also logic, 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.
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 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.
To better understand and comprehend the truths of a science, men have created textbooks (Lehrbuch), which of course only contain 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).
(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
, 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.
(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
- 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 [Socrates 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 Kant
; for Kant's use of the term see an sich
.) (Bolzano, on the mathematical method, § 2)
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. (Bolzano, on the mathematical method, §3)
Let's 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 a round square. But, the idea [round square] 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]) (Bolzano, on the mathematical method, §4).
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. (Bolzano, Wissenschaftslehre §72).
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 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 explained
by the other.
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.)
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).
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
.
To the foundations of mathematical analysis
he contributed the introduction of a fully rigorous ε-δ definition of a mathematical limit. Bolzano, like several others of his day, was skeptical of the possibility of Gottfried Leibniz
's infinitesimal
s, that had been the earliest putative foundation for differential calculus
. 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
, which had originally been proven by Gauss
from geometrical considerations. He also gave the first purely analytic proof of the intermediate value theorem
(also known as Bolzano's theorem). Today he is mostly 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 .
and Kazimierz Twardowski
, both students of Franz Brentano
. Through them, Bolzano became a formative influence on both phenomenology and analytic philosophy
.
Bohemia
Bohemia is a historical region in central Europe, occupying the western two-thirds of the traditional Czech Lands. It is located in the contemporary Czech Republic with its capital in Prague...
n mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, logician, philosopher, theologian
Theology
Theology is the systematic and rational study of religion and its influences and of the nature of religious truths, or the learned profession acquired by completing specialized training in religious studies, usually at a university or school of divinity or seminary.-Definition:Augustine of Hippo...
, Catholic priest and antimilitarist
Antimilitarism
Antimilitarism is a doctrine commonly found in the anarchist and, more globally, in the socialist movement, which may both be characterized as internationalist movements. It relies heavily on a critical theory of nationalism and imperialism, and was an explicit goal of the First and Second...
of German
German language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....
mother tongue.
Family
Bolzano was the son of two pious CatholicsRoman Catholic Church
The Catholic Church, also known as the Roman Catholic Church, is the world's largest Christian church, with over a billion members. Led by the Pope, it defines its mission as spreading the gospel of Jesus Christ, administering the sacraments and exercising charity...
. His father, Bernard Pompeius Bolzano, was born in northern Italy and moved to Prague
Prague
Prague is the capital and largest city of the Czech Republic. Situated in the north-west of the country on the Vltava river, the city is home to about 1.3 million people, while its metropolitan area is estimated to have a population of over 2.3 million...
, where he married Maria Cecilia Maurer, the (German-speaking) daughter of a Prague merchant. Only two of their twelve children lived to adulthood.
Career
Bolzano entered the University of PragueCharles University in Prague
Charles University in Prague is the oldest and largest university in the Czech Republic. Founded in 1348, it was the first university in Central Europe and is also considered the earliest German university...
in 1796 and studied mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
and physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
. Starting in 1800, he also began studying theology
Theology
Theology is the systematic and rational study of religion and its influences and of the nature of religious truths, or the learned profession acquired by completing specialized training in religious studies, usually at a university or school of divinity or seminary.-Definition:Augustine of Hippo...
, becoming a Catholic
Catholic
The word catholic comes from the Greek phrase , meaning "on the whole," "according to the whole" or "in general", and is a combination of the Greek words meaning "about" and meaning "whole"...
priest
Priest
A priest is a person authorized to perform the sacred rites of a religion, especially as a mediatory agent between humans and deities. They also have the authority or power to administer religious rites; in particular, rites of sacrifice to, and propitiation of, a deity or deities...
in 1804. He was appointed to the then newly created chair of philosophy of religion
Philosophy of religion
Philosophy of religion is a branch of philosophy concerned with questions regarding religion, including the nature and existence of God, the examination of religious experience, analysis of religious language and texts, and the relationship of religion and science...
in 1805. He proved to be a popular lecturer not just in religion but also in philosophy, and was elected head of the philosophy department 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. Upon his refusal to recant his beliefs, Bolzano was dismissed from the university in 1819. His political convictions (which he was inclined to share with others with some frequency) eventually proved to be too liberal
Liberalism
Liberalism is the belief in the importance of liberty and equal rights. Liberals espouse a wide array of views depending on their understanding of these principles, but generally, liberals support ideas such as constitutionalism, liberal democracy, free and fair elections, human rights,...
for the Austrian
Austrian Empire
The Austrian Empire was a modern era successor empire, which was centered on what is today's Austria and which officially lasted from 1804 to 1867. It was followed by the Empire of Austria-Hungary, whose proclamation was a diplomatic move that elevated Hungary's status within the Austrian Empire...
authorities. He was exile
Exile
Exile means to be away from one's home , while either being explicitly refused permission to return and/or being threatened with imprisonment or death upon return...
d to the countryside and at that point devoted his energies to his writings on social, religious, philosophical, and mathematical matters. Although forbidden to publish in mainstream journals
Magazine
Magazines, periodicals, glossies or serials are publications, generally published on a regular schedule, containing a variety of articles. They are generally financed by advertising, by a purchase price, by pre-paid magazine subscriptions, or all three...
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
Eastern Europe is the eastern part of Europe. The term has widely disparate geopolitical, geographical, cultural and socioeconomic readings, which makes it highly context-dependent and even volatile, and there are "almost as many definitions of Eastern Europe as there are scholars of the region"...
an journals. In 1842 he moved back to Prague, where he died in 1848.
Works
Bolzano's posthumously published work Paradoxien des Unendlichen (The Paradoxes of the Infinite)The Paradoxes of the Infinite
The Paradoxes of the Infinite, is a mathematical work by Bernard Bolzano on the theory of sets. It was published in 1851, three years after Bolzano's death, by a friend. The work contained many interesting results in set theory...
was greatly admired by many of the eminent logicians who came after him, including Charles Sanders Peirce, Georg Cantor
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...
, and Richard Dedekind
Richard Dedekind
Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...
. 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
The philosophy of science is concerned with the assumptions, foundations, methods and implications of science. It is also concerned with the use and merit of science and sometimes overlaps metaphysics and epistemology by exploring whether scientific results are actually a study of truth...
in the modern sense but also logic, 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
Otto Stolz
Otto Stolz was an Austrian mathematician noted for his work on mathematical analysis and infinitesimals. Born in Hall in Tirol, he studied in Innsbruck from 1860 and in Vienna from 1863, receiving his habilitation there in 1867...
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 objectAbstract object
An abstract object is an object which does not exist at any particular time or place, but rather exists as a type of thing . In philosophy, an important distinction is whether an object is considered abstract or concrete. Abstract objects are sometimes called abstracta An abstract object is an...
s, attributes, sentence-shapes, ideas and propositions in themselves, sums
Mereology
In philosophy and mathematical logic, mereology treats parts and the wholes they form...
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 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
Theory of justification
Theory of justification is a part of epistemology that attempts to understand the justification of propositions and beliefs. Epistemologists are concerned with various epistemic features of belief, which include the ideas of justification, warrant, rationality, and probability...
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 are out there, 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 only contain 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
Mereology
In philosophy and mathematical logic, mereology treats parts and the wholes they form...
. 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
Existence
In common usage, existence is the world we are aware of through our senses, and that persists independently without them. In academic philosophy the word has a more specialized meaning, being contrasted with essence, which specifies different forms of existence as well as different identity...
, 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 (spoken or written or thought or in itself) and ideaIdea
In the most narrow sense, an idea is just whatever is before the mind when one thinks. Very often, ideas are construed as representational images; i.e. images of some object. In other contexts, ideas are taken to be concepts, although abstract concepts do not necessarily appear as images...
(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
Contradiction
In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other...
- 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 [Socrates 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 Kant
KANT
KANT is a computer algebra system for mathematicians interested in algebraic number theory, performing sophisticated computations in algebraic number fields, in global function fields, and in local fields. KASH is the associated command line interface...
; for Kant's use of the term see an sich
Noumenon
The noumenon is a posited object or event that is known without the use of the senses.The term is generally used in contrast with, or in relation to "phenomenon", which refers to anything that appears to, or is an object of, the senses...
.) (Bolzano, on the mathematical method, § 2)
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. (Bolzano, on the mathematical method, §3)
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.)Let's 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 a round square. But, the idea [round square] 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]) (Bolzano, on the mathematical method, §4).
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. (Bolzano, Wissenschaftslehre §72).
Logic
According to Bolzano, all propositions are composed out of three (simple or complex) elements: a subject, a predicate and a 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 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
Asymmetric relation
Asymmetric often means, simply: not symmetric. In this sense an asymmetric relation is a binary relation which is not a symmetric relation.That is,\lnot....
that obtains between true propositions, when one of the propositions is not only deducible from, but also explained
Explanation
An explanation is a set of statements constructed to describe a set of facts which clarifies the causes, context, and consequencesof those facts....
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).
Mathematics
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 into mathematical analysisMathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
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 Weierstrass
Karl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....
.
To the foundations of mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
he contributed the introduction of a fully rigorous ε-δ definition of a mathematical limit. Bolzano, like several others of his day, was skeptical of the possibility of Gottfried Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
's infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
s, that had been the earliest putative foundation for differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....
. 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
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
proof of the fundamental theorem of algebra
Fundamental theorem of algebra
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
, which had originally been proven by Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
from geometrical considerations. He also gave the first purely analytic proof of the intermediate value theorem
Intermediate value theorem
In mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value....
(also known as Bolzano's theorem). Today he is mostly remembered for the Bolzano–Weierstrass theorem
Bolzano–Weierstrass theorem
In real analysis, the Bolzano–Weierstrass theorem is a fundamental result about convergence in a finite-dimensional Euclidean space Rn. The theorem states thateach bounded sequence in Rn has a convergent subsequence...
, which Karl Weierstrass
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....
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 legacy
The effect of his thought on philosophy initially seemed destined to be slight; his work was rediscovered, however, by Edmund HusserlEdmund Husserl
Edmund Gustav Albrecht Husserl was a philosopher and mathematician and the founder of the 20th century philosophical school of phenomenology. He broke with the positivist orientation of the science and philosophy of his day, yet he elaborated critiques of historicism and of psychologism in logic...
and Kazimierz Twardowski
Kazimierz Twardowski
Kazimierz Jerzy Skrzypna-Twardowski was a Polish philosopher and logician.-Life:Twardowski's family belonged to the Ogończyk coat-of-arms.Twardowski studied philosophy in Vienna with Franz Brentano and Robert Zimmermann...
, both students of Franz Brentano
Franz Brentano
Franz Clemens Honoratus Hermann Brentano was an influential German philosopher and psychologist whose influence was felt by other such luminaries as Sigmund Freud, Edmund Husserl, Kazimierz Twardowski and Alexius Meinong, who followed and adapted his views.-Life:Brentano was born at Marienberg am...
. Through them, Bolzano became a formative influence on both phenomenology and analytic philosophy
Analytic philosophy
Analytic philosophy is a generic term for a style of philosophy that came to dominate English-speaking countries in the 20th century...
.
Writings
- Gesamtausgabe (Collected Works) Critical edition edited by Eduard Winter, Jan Berg, Friedrich Kambartel, Bob van Rootselaar, Stuttgart:Fromman-Holzboog, 1969 ss. (84 volls. published)
- Wissenschaftslehre, 4 Bde Neudr., 2. verb, A. hrsg. W. Schultz, Leipzig I-II 1929, III 1980, IV 1931; Critical edition edited by Jan Berg: Bolzano's Gesamtausgabe, voll. 11-14 (1985–2000). (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; . (Paradoxes of the Infinite; (excerpt)).
Translations and compilations
- Theory of science, attempt at a detailed and in the main novel exposition of logic with constant attention to earlier authors. (Edited and translated by Rolf George University of California Press, Berkeley and Los Angeles, 1972).
- Theory of science (Edited, with an introduction, by Jan Berg. Translated from the German by Burnham Terrell - D. Reidel Publishing Company, Dordrecht and Boston, 1973).
- The mathematical works of Bernard Bolzano - Edited by Steve Russ - Oxford, Oxford University Press, 2004.
- On the mathematical method and correspondence with Exner - Translated by Paul Rusnock and Rolf George - Amsterdam, Rodopi, 2004.
- Selected Writings on Ethics and Politics - Translated by Paul Rusnock and Rolf George - Amsterdam, Rodopi, 2007.
External links
- Bolzano’s Philosophy of Mathematical Knowledge entry by Sandra Lapointe in the Internet Encyclopedia of PhilosophyInternet Encyclopedia of PhilosophyThe Internet Encyclopedia of Philosophy is a free online encyclopedia on philosophical topics and philosophers founded by James Fieser in 1995. The current general editors are James Fieser and Bradley Dowden...
- Bernard Bolzano's Contributions to Logic and Ontology
- Bernard Bolzano: Editions, Translations, Bibliographical Resources and Selected Texts
- Annotated Bibliography on the Philosophical Work of Bolzano (First Part: A - C)
- Annotated Bibliography on the Philosophical Work of Bolzano (Second Part: D - M)
- Annotated Bibliography on the Philosophical Work of Bolzano (Third Part: N - Z)
- Works in German, in the Internet ArchiveInternet ArchiveThe Internet Archive is a non-profit digital library with the stated mission of "universal access to all knowledge". It offers permanent storage and access to collections of digitized materials, including websites, music, moving images, and nearly 3 million public domain books. The Internet Archive...
- Digitized Bolzano's works