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- Mathematician Bernhard Placidus Johann Nepomuk Bolzano, 1781
- Writer Guido Karl Anton List, 1848
- 21st President of the United States Chester Alan Arthur, 1829
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Bernhard Placidus Johann Nepomuk Bolzano (October 5, 1781 – December 18, 1848), Bernard Bolzano in English, was a Bohemian mathematician, logician, philosopher, theologian, Catholic priest and antimilitarist of German mother tongue.
Bolzano was the son of two pious Catholics. 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.
Bolzano entered the University of Prague 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 exiled 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 European journals. In 1842 he moved back to Prague, where he died in 1848.
Bolzano's posthumously published work Paradoxien des Unendlichen (The Paradoxes of the Infinite) 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.
In his 1837 Wissenschaftslehre Bolzano attempted to provide logical foundations for all sciences, building on abstractions like part-relation, abstract objects, 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. In the Wissenschaftslehre, Bolzano is mainly concerned with three realms:
(1) The realm of language, consisting of words and sentences. (2) The realm of thought, consisting of subjective ideas and judgements. (3) The realm of logic, consisting of 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. 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 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 - 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.) 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. 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
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. 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 that
obtains between true propositions, when one of the propositions is not
only deducible from, but also explained by the other. 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. 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
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
analysis 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 infinitesimals,
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. Due
to the fact that Bolzano's most important work, the Wissenschaftslehre,
could not be published during his lifetime, 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 and analytic
philosophy.