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Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theology and Catholic priest of Italian extraction, also known for his liberal views.
Bolzano wrote in German language, his native language. For the most part, his work came to prominence posthumously.
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 new chair of philosophy of religion 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 for the Austrian Empire authorities. On December 24, 1819, he was removed from his professorship (upon his refusal to recant his beliefs) and was to the countryside and then devoted his energies to his writings on social, religious, philosophical, and mathematical matters.
Although forbidden to publish in mainstream magazine as a condition of his exile, Bolzano continued to develop his ideas and publish them either on his own or in obscure journals. In 1842 he moved back to Prague, where he died in 1848.
To the foundations of mathematical analysis 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'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.
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).
(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. First, the distinction between mereology. For instance, words are parts of sentences, subjective ideas are parts of judgments, objective ideas are parts of propositions in themselves. Second, all objects divide into those that existence, 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.
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. 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; for Kant's use of the term see noumenon.)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, 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 is different from the idea A though the parts of both ideas are the same.Bolzano, "On the Mathematical Method", §3
Consider, for further explanation, an example used by Bolzano. The idea a, does not have an object, because the object that ought to be represented is self-contradictory. A different example is the idea nothing which certainly does not have an object. However, the proposition the has as its subject-idea the. This subject-idea does have an object, namely the idea a. But, that idea does not have an object.
Besides objectless ideas, there are ideas that have only one object, e.g. the idea the represents only one object. Bolzano calls these ideas 'singular ideas'. Obviously there are also ideas that have many objects (e.g. the) and even infinitely many objects (e.g. a).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 'grounding' ( Abfolge).Stefan Roski, Bolzano's Conception of Grounding, Frankfurt, Klostermann, 2017. This is an asymmetric relation that obtains between true propositions, when one of the propositions is not only deducible from, but also explanation 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:
There is at least one truth in itself (concrete objective meaning):
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).
Alois Höfler (1853–1922), a former student of Franz Brentano and Alexius Meinong, who subsequently become professor of pedagogy at the University of Vienna, created the "missing link between the Vienna Circle and the Bolzano tradition in Austria." Bolzano's work was rediscovered, however, by Edmund HusserlWolfgang Huemer, "Husserl's critique of psychologism and his relation to the Brentano school", in: Arkadiusz Chrudzimski and Wolfgang Huemer (eds.), Phenomenology and Analysis: Essays on Central European Philosophy, Walter de Gruyter, 2004, p. 205. and Kazimierz Twardowski,Maria van der Schaar, Kazimierz Twardowski: A Grammar for Philosophy, Brill, 2015, p. 53; Peter M. Simons, Philosophy and Logic in Central Europe from Bolzano to Tarski: Selected Essays, Springer, 2013, p. 15. both students of Brentano. Through them, Bolzano became a formative influence on both phenomenology and analytic philosophy.
Most of Bolzano's work remained in manuscript form, so it had a very small circulation and little influence on the development of the subject.
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