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132-475: Bolzano wrote in German , his native language. For the most part, his work came to prominence posthumously. Bolzano was the son of two pious Catholics . His father, Bernard Pompeius Bolzano, was an Italian who had moved to 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. When he was ten years old, Bolzano entered

264-415: 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

396-660: 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

528-460: A de facto official language of Namibia after the end of German colonial rule alongside English and Afrikaans , and had de jure co-official status from 1984 until its independence from South Africa in 1990. However, the Namibian government perceived Afrikaans and German as symbols of apartheid and colonialism, and decided English would be the sole official language upon independence, stating that it

660-402: A second language , and 75–100   million as a foreign language . This would imply the existence of approximately 175–220   million German speakers worldwide. German sociolinguist Ulrich Ammon estimated a number of 289 million German foreign language speakers without clarifying the criteria by which he classified a speaker. As of 2012 , about 90   million people, or 16% of

792-587: A "commonly used" language and the Pan South African Language Board is obligated to promote and ensure respect for it. Cameroon was also a colony of the German Empire from the same period (1884 to 1916). However, German was replaced by French and English, the languages of the two successor colonial powers, after its loss in World War I . Nevertheless, since the 21st century, German has become

924-512: 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. 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

1056-467: 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 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

1188-647: A convergent subsequence. This reasoning may be applied until we obtain a countable set K n {\displaystyle K_{n}} for which ( x n j ) {\displaystyle (x_{nj})} converges for j = 1 , 2 , … , n {\displaystyle j=1,2,\dots ,n} . Hence, ( x n ) n ∈ K n {\displaystyle (x_{n})_{n\in K_{n}}} converges and therefore since ( x n ) {\displaystyle (x_{n})}

1320-491: 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

1452-405: 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

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1584-1121: A countable set K ⊆ I {\displaystyle K\subseteq I} where I {\displaystyle I} is the index set of the sequence such that ( x n ) n ∈ K {\displaystyle (x_{n})_{n\in K}} converges. Let ( x n ) {\displaystyle (x_{n})} be any bounded sequence in R n {\displaystyle \mathbb {R} ^{n}} and denote its index set by I {\displaystyle I} . The sequence ( x n ) {\displaystyle (x_{n})} may be expressed as an n-tuple of sequences in R 1 {\displaystyle \mathbb {R} ^{1}} such that x n = ( x n 1 , x n 2 , … , x n n ) {\displaystyle x_{n}=(x_{n1},x_{n2},\dots ,x_{nn})} where ( x n j ) {\displaystyle (x_{nj})}

1716-408: A finite-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . The theorem states that each infinite bounded sequence in R n {\displaystyle \mathbb {R} ^{n}} has a convergent subsequence . An equivalent formulation is that a subset of R n {\displaystyle \mathbb {R} ^{n}}

1848-802: 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 an extension of his earlier thoughts in

1980-589: 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 parts and wholes . 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 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

2112-556: A greater need for regularity in written conventions. While the major changes of the MHG period were socio-cultural, High German was still undergoing significant linguistic changes in syntax, phonetics, and morphology as well (e.g. diphthongization of certain vowel sounds: hus (OHG & MHG "house") → haus (regionally in later MHG)→ Haus (NHG), and weakening of unstressed short vowels to schwa [ə]: taga (OHG "days")→ tage (MHG)). A great wealth of texts survives from

2244-452: 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

2376-490: 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

2508-448: 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

2640-439: A monotone subsequence. Now suppose one has a bounded sequence in R 1 {\displaystyle \mathbb {R} ^{1}} ; by the lemma proven above there exists a monotone subsequence, likewise also bounded. It follows from the monotone convergence theorem that this subsequence converges. The general case ( R n {\displaystyle \mathbb {R} ^{n}} ) can be reduced to

2772-453: A native tongue today, mostly descendants of German colonial settlers . The period of German colonialism in Namibia also led to the evolution of a Standard German-based pidgin language called " Namibian Black German ", which became a second language for parts of the indigenous population. Although it is nearly extinct today, some older Namibians still have some knowledge of it. German remained

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2904-679: A popular foreign language among pupils and students, with 300,000 people learning or speaking German in Cameroon in 2010 and over 230,000 in 2020. Today Cameroon is one of the African countries outside Namibia with the highest number of people learning German. In the United States, German is the fifth most spoken language in terms of native and second language speakers after English, Spanish , French , and Chinese (with figures for Cantonese and Mandarin combined), with over 1 million total speakers. In

3036-454: 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 .) Every proposition in itself

3168-421: 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

3300-639: 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

3432-475: A result, the surviving texts are written in highly disparate regional dialects and exhibit significant Latin influence, particularly in vocabulary. At this point monasteries, where most written works were produced, were dominated by Latin, and German saw only occasional use in official and ecclesiastical writing. While there is no complete agreement over the dates of the Middle High German (MHG) period, it

3564-478: 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

3696-481: A stricter relation of ' grounding ' ( 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: I. Abstract objective meaning: Truth signifies an attribute that may apply to

3828-479: A subsequence converging to an element of A {\displaystyle A} – i.e., the subsets that are sequentially compact in the subspace topology  – are precisely the closed and bounded subsets. This form of the theorem makes especially clear the analogy to the Heine–Borel theorem , which asserts that a subset of R n {\displaystyle \mathbb {R} ^{n}}

3960-798: A subsequence converging to an element of A {\displaystyle A} . Then A {\displaystyle A} must be bounded, since otherwise the following unbounded sequence { x n } ∈ A {\displaystyle \{x_{n}\}\in A} can be constructed. For every n ∈ N {\displaystyle n\in \mathbb {N} } , define x n {\displaystyle x_{n}} to be any arbitrary point such that | | x n | | ≥ n {\displaystyle ||x_{n}||\geq n} . Then, every subsequence of { x n } {\displaystyle \{x_{n}\}}

4092-460: A textbook, is the Theory of Science ( Wissenschaftslehre ). 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

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4224-493: Is compact if and only if it is closed and bounded. In fact, general topology tells us that a metrizable space is compact if and only if it is sequentially compact, so that the Bolzano–Weierstrass and Heine–Borel theorems are essentially the same. There are different important equilibrium concepts in economics, the proofs of the existence of which often require variations of the Bolzano–Weierstrass theorem. One example

4356-422: Is sequentially compact if and only if A {\displaystyle A} is closed and bounded. Proof: ( sequential compactness implies closed and bounded) Suppose A {\displaystyle A} is a subset of R n {\displaystyle \mathbb {R} ^{n}} with the property that every sequence in A {\displaystyle A} has

4488-549: Is sequentially compact if and only if it is closed and bounded . The theorem is sometimes called the sequential compactness theorem . The Bolzano–Weierstrass theorem is named after mathematicians Bernard Bolzano and Karl Weierstrass . It was actually first proved by Bolzano in 1817 as a lemma in the proof of the intermediate value theorem . Some fifty years later the result was identified as significant in its own right, and proved again by Weierstrass. It has since become an essential theorem of analysis . First we prove

4620-393: Is sequentially compact if every sequence { x n } {\displaystyle \{x_{n}\}} in A {\displaystyle A} has a convergent subsequence converging to an element of A {\displaystyle A} . Theorem: A ⊆ R n {\displaystyle A\subseteq \mathbb {R} ^{n}}

4752-688: Is a West Germanic language in the Indo-European language family , mainly spoken in Western and Central Europe . It is the most spoken native language within the European Union . It is the most widely spoken and official (or co-official) language in Germany , Austria , Switzerland , Liechtenstein , and the Italian autonomous province of South Tyrol . It is also an official language of Luxembourg , Belgium and

4884-443: Is a limit point of A {\displaystyle A} and A {\displaystyle A} is a closed set , x {\displaystyle x} must be an element of A {\displaystyle A} . Thus the subsets A {\displaystyle A} of R n {\displaystyle \mathbb {R} ^{n}} for which every sequence in A has

5016-580: Is a recognized minority language in the following countries: In France, the High German varieties of Alsatian and Moselle Franconian are identified as " regional languages ", but the European Charter for Regional or Minority Languages of 1998 has not yet been ratified by the government. Namibia also was a colony of the German Empire, from 1884 to 1915. About 30,000 people still speak German as

5148-446: Is a sequence for j = 1 , 2 , … , n {\displaystyle j=1,2,\dots ,n} . Since ( x n ) {\displaystyle (x_{n})} is bounded, ( x n j ) {\displaystyle (x_{nj})} is also bounded for j = 1 , 2 , … , n {\displaystyle j=1,2,\dots ,n} . It follows then by

5280-421: 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

5412-401: 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

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5544-630: Is a subsequence with the following indices n 1 < n 2 < n 3 < ⋯ < n j < … {\displaystyle n_{1}<n_{2}<n_{3}<\dots <n_{j}<\dots } and the following terms x n 1 ≥ x n 2 ≥ x n 3 ≥ ⋯ ≥ x n j ≥ … {\displaystyle x_{n_{1}}\geq x_{n_{2}}\geq x_{n_{3}}\geq \dots \geq x_{n_{j}}\geq \dots } . So,

5676-596: Is also notable for its broad spectrum of dialects , with many varieties existing in Europe and other parts of the world. Some of these non-standard varieties have become recognized and protected by regional or national governments. Since 2004, heads of state of the German-speaking countries have met every year, and the Council for German Orthography has been the main international body regulating German orthography . German

5808-546: Is an Indo-European language that belongs to the West Germanic group of the Germanic languages . The Germanic languages are traditionally subdivided into three branches: North Germanic , East Germanic , and West Germanic . The first of these branches survives in modern Danish , Swedish , Norwegian , Faroese , and Icelandic , all of which are descended from Old Norse . The East Germanic languages are now extinct, and Gothic

5940-397: Is an accumulation point of ( x n ) {\displaystyle (x_{n})} . Thus, there is a subsequence of ( x n ) {\displaystyle (x_{n})} that converges to x {\displaystyle x} . Definition: A set A ⊆ R n {\displaystyle A\subseteq \mathbb {R} ^{n}}

6072-657: Is an interval I N {\displaystyle I_{N}} that is a subset of U {\displaystyle U} . Because I N {\displaystyle I_{N}} contains by construction infinitely many members of ( x n ) {\displaystyle (x_{n})} and I N ⊆ U {\displaystyle I_{N}\subseteq U} , also U {\displaystyle U} contains infinitely many members of ( x n ) {\displaystyle (x_{n})} . This proves that x {\displaystyle x}

6204-439: 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

6336-517: Is bounded, any sequence { x n } ∈ A {\displaystyle \{x_{n}\}\in A} is also bounded. From the Bolzano-Weierstrass theorem , { x n } {\displaystyle \{x_{n}\}} contains a subsequence converging to some point x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} . Since x {\displaystyle x}

6468-472: Is called the "German Sprachraum ". German is the official language of the following countries: German is a co-official language of the following countries: Although expulsions and (forced) assimilation after the two World wars greatly diminished them, minority communities of mostly bilingual German native speakers exist in areas both adjacent to and detached from the Sprachraum. Within Europe, German

6600-430: Is complicated by the existence of several varieties whose status as separate "languages" or "dialects" is disputed for political and linguistic reasons, including quantitatively strong varieties like certain forms of Alemannic and Low German . With the inclusion or exclusion of certain varieties, it is estimated that approximately 90–95 million people speak German as a first language , 10–25   million speak it as

6732-734: 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

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6864-399: 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

6996-450: Is either non-decreasing or non-increasing ). Proof : Let us call a positive integer-valued index n {\displaystyle n} of a sequence a "peak" of the sequence when x m ≤ x n {\displaystyle x_{m}\leq x_{n}} for every m > n {\displaystyle m>n} . Suppose first that the sequence has infinitely many peaks, which means there

7128-520: Is generally seen as ending when the 1346–53 Black Death decimated Europe's population. Modern High German begins with the Early New High German (ENHG) period, which Wilhelm Scherer dates 1350–1650, terminating with the end of the Thirty Years' War . This period saw the further displacement of Latin by German as the primary language of courtly proceedings and, increasingly, of literature in

7260-595: Is generally seen as lasting from 1050 to 1350. This was a period of significant expansion of the geographical territory occupied by Germanic tribes, and consequently of the number of German speakers. Whereas during the Old High German period the Germanic tribes extended only as far east as the Elbe and Saale rivers, the MHG period saw a number of these tribes expanding beyond this eastern boundary into Slavic territory (known as

7392-433: Is in each interval I n {\displaystyle I_{n}} . Now we show, that x {\displaystyle x} is an accumulation point of ( x n ) {\displaystyle (x_{n})} . Take a neighbourhood U {\displaystyle U} of x {\displaystyle x} . Because the length of the intervals converges to zero, there

7524-559: Is less closely related to languages based on Low Franconian dialects (e.g., Dutch and Afrikaans), Low German or Low Saxon dialects (spoken in northern Germany and southern Denmark ), neither of which underwent the High German consonant shift. As has been noted, the former of these dialect types is Istvaeonic and the latter Ingvaeonic, whereas the High German dialects are all Irminonic; the differences between these languages and standard German are therefore considerable. Also related to German are

7656-527: Is not a peak, since n 1 {\displaystyle n_{1}} comes after the final peak, which implies the existence of n 2 {\displaystyle n_{2}} with n 1 < n 2 {\displaystyle n_{1}<n_{2}} and x n 1 < x n 2 {\displaystyle x_{n_{1}}<x_{n_{2}}} . Again, n 2 {\displaystyle n_{2}} comes after

7788-435: Is one of the major languages of the world . German is the second-most widely spoken Germanic language , after English, both as a first and as a second language . German is also widely taught as a foreign language , especially in continental Europe (where it is the third most taught foreign language after English and French), and in the United States. Overall, German is the fourth most commonly learned second language, and

7920-587: Is one of the three biggest newspapers in Namibia and the only German-language daily in Africa. An estimated 12,000 people speak German or a German variety as a first language in South Africa, mostly originating from different waves of immigration during the 19th and 20th centuries. One of the largest communities consists of the speakers of "Nataler Deutsch", a variety of Low German concentrated in and around Wartburg . The South African constitution identifies German as

8052-401: Is partly derived from Latin and Greek , along with fewer words borrowed from French and Modern English . English, however, is the main source of more recent loanwords . German is a pluricentric language ; the three standardized variants are German , Austrian , and Swiss Standard German . Standard German is sometimes called High German , which refers to its regional origin. German

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8184-445: 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

8316-402: 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

8448-412: 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

8580-412: 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.

8712-435: 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

8844-510: Is the Sachsenspiegel , the first book of laws written in Middle Low German ( c.  1220 ). The abundance and especially the secular character of the literature of the MHG period demonstrate the beginnings of a standardized written form of German, as well as the desire of poets and authors to be understood by individuals on supra-dialectal terms. The Middle High German period

8976-404: Is the existence of a Pareto efficient allocation. An allocation is a matrix of consumption bundles for agents in an economy, and an allocation is Pareto efficient if no change can be made to it that makes no agent worse off and at least one agent better off (here rows of the allocation matrix must be rankable by a preference relation ). The Bolzano–Weierstrass theorem allows one to prove that if

9108-566: Is the only language in this branch which survives in written texts. The West Germanic languages, however, have undergone extensive dialectal subdivision and are now represented in modern languages such as English, German, Dutch , Yiddish , Afrikaans , and others. Within the West Germanic language dialect continuum, the Benrath and Uerdingen lines (running through Düsseldorf - Benrath and Krefeld - Uerdingen , respectively) serve to distinguish

9240-474: Is unbounded and therefore not convergent. Moreover, A {\displaystyle A} must be closed, since any limit point of A {\displaystyle A} , which has a sequence of points in A {\displaystyle A} converging to itself, must also lie in A {\displaystyle A} . Proof: (closed and bounded implies sequential compactness ) Since A {\displaystyle A}

9372-433: Is understood in all areas where German is spoken. Approximate distribution of native German speakers (assuming a rounded total of 95 million) worldwide: As a result of the German diaspora , as well as the popularity of German taught as a foreign language , the geographical distribution of German speakers (or "Germanophones") spans all inhabited continents. However, an exact, global number of native German speakers

9504-673: The Ostsiedlung ). With the increasing wealth and geographic spread of the Germanic groups came greater use of German in the courts of nobles as the standard language of official proceedings and literature. A clear example of this is the mittelhochdeutsche Dichtersprache employed in the Hohenstaufen court in Swabia as a standardized supra-dialectal written language. While these efforts were still regionally bound, German began to be used in place of Latin for certain official purposes, leading to

9636-627: The Alamanni , Bavarian, and Thuringian groups, all belonging to the Elbe Germanic group ( Irminones ), which had settled in what is now southern-central Germany and Austria between the second and sixth centuries, during the great migration. In general, the surviving texts of Old High German (OHG) show a wide range of dialectal diversity with very little written uniformity. The early written tradition of OHG survived mostly through monasteries and scriptoria as local translations of Latin originals; as

9768-522: The Early Middle Ages . German is an inflected language , with four cases for nouns, pronouns, and adjectives (nominative, accusative, genitive, dative); three genders (masculine, feminine, neuter) and two numbers (singular, plural). It has strong and weak verbs . The majority of its vocabulary derives from the ancient Germanic branch of the Indo-European language family, while a smaller share

9900-417: The European Union 's population, spoke German as their mother tongue, making it the second most widely spoken language on the continent after Russian and the second biggest language in terms of overall speakers (after English), as well as the most spoken native language. The area in central Europe where the majority of the population speaks German as a first language and has German as a (co-)official language

10032-514: The German states . While these states were still part of the Holy Roman Empire , and far from any form of unification, the desire for a cohesive written language that would be understandable across the many German-speaking principalities and kingdoms was stronger than ever. As a spoken language German remained highly fractured throughout this period, with a vast number of often mutually incomprehensible regional dialects being spoken throughout

10164-610: The Gymnasium of the Piarists in Prague, which he attended from 1791 to 1796. 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

10296-724: The Holy Roman Emperor Maximilian I , and the other being Meißner Deutsch , used in the Electorate of Saxony in the Duchy of Saxe-Wittenberg . Alongside these courtly written standards, the invention of the printing press led to the development of a number of printers' languages ( Druckersprachen ) aimed at making printed material readable and understandable across as many diverse dialects of German as possible. The greater ease of production and increased availability of written texts brought about increased standardisation in

10428-644: The Old High German language in several Elder Futhark inscriptions from as early as the sixth century AD (such as the Pforzen buckle ), the Old High German period is generally seen as beginning with the Abrogans (written c.  765–775 ), a Latin-German glossary supplying over 3,000 Old High German words with their Latin equivalents. After the Abrogans , the first coherent works written in Old High German appear in

10560-640: The Sprachraum in Europe. German is used in a wide variety of spheres throughout the country, especially in business, tourism, and public signage, as well as in education, churches (most notably the German-speaking Evangelical Lutheran Church in Namibia (GELK) ), other cultural spheres such as music, and media (such as German language radio programs by the Namibian Broadcasting Corporation ). The Allgemeine Zeitung

10692-536: The Standard German language in its written form, and the Duden Handbook was declared its standard definition. Punctuation and compound spelling (joined or isolated compounds) were not standardized in the process. The Deutsche Bühnensprache ( lit.   ' German stage language ' ) by Theodor Siebs had established conventions for German pronunciation in theatres , three years earlier; however, this

10824-503: The Upper German dialects spoken in the southern German-speaking countries , such as Swiss German ( Alemannic dialects ) and the various Germanic dialects spoken in the French region of Grand Est , such as Alsatian (mainly Alemannic, but also Central–and   Upper Franconian dialects) and Lorraine Franconian (Central Franconian). After these High German dialects, standard German

10956-462: The pagan Germanic tradition. Of particular interest to scholars, however, has been the Hildebrandslied , a secular epic poem telling the tale of an estranged father and son unknowingly meeting each other in battle. Linguistically, this text is highly interesting due to the mixed use of Old Saxon and Old High German dialects in its composition. The written works of this period stem mainly from

11088-559: The Bolzano tradition in Austria." Bolzano's work was rediscovered, however, by Edmund Husserl and Kazimierz Twardowski , 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. German language German (German: Deutsch , pronounced [dɔʏtʃ] )

11220-714: The Empire. Its use indicated that the speaker was a merchant or someone from an urban area, regardless of nationality. Prague (German: Prag ) and Budapest ( Buda , German: Ofen ), to name two examples, were gradually Germanized in the years after their incorporation into the Habsburg domain; others, like Pressburg ( Pozsony , now Bratislava), were originally settled during the Habsburg period and were primarily German at that time. Prague, Budapest, Bratislava, and cities like Zagreb (German: Agram ) or Ljubljana (German: Laibach ), contained significant German minorities. In

11352-718: The Frisian languages— North Frisian (spoken in Nordfriesland ), Saterland Frisian (spoken in Saterland ), and West Frisian (spoken in Friesland )—as well as the Anglic languages of English and Scots. These Anglo-Frisian dialects did not take part in the High German consonant shift, and the Anglic languages also adopted much vocabulary from both Old Norse and the Norman language . The history of

11484-524: The German language begins with the High German consonant shift during the Migration Period , which separated Old High German dialects from Old Saxon . This sound shift involved a drastic change in the pronunciation of both voiced and voiceless stop consonants ( b , d , g , and p , t , k , respectively). The primary effects of the shift were the following below. While there is written evidence of

11616-447: The German states; the invention of the printing press c.  1440 and the publication of Luther's vernacular translation of the Bible in 1534, however, had an immense effect on standardizing German as a supra-dialectal written language. The ENHG period saw the rise of several important cross-regional forms of chancery German, one being gemeine tiutsch , used in the court of

11748-675: The Germanic dialects that were affected by the High German consonant shift (south of Benrath) from those that were not (north of Uerdingen). The various regional dialects spoken south of these lines are grouped as High German dialects, while those spoken to the north comprise the Low German and Low Franconian dialects. As members of the West Germanic language family, High German, Low German, and Low Franconian have been proposed to be further distinguished historically as Irminonic , Ingvaeonic , and Istvaeonic , respectively. This classification indicates their historical descent from dialects spoken by

11880-583: The Infinite) (1851) 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

12012-646: The Irminones (also known as the Elbe group), Ingvaeones (or North Sea Germanic group), and Istvaeones (or Weser–Rhine group). Standard German is based on a combination of Thuringian - Upper Saxon and Upper Franconian dialects, which are Central German and Upper German dialects belonging to the High German dialect group. German is therefore closely related to the other languages based on High German dialects, such as Luxembourgish (based on Central Franconian dialects ) and Yiddish . Also closely related to Standard German are

12144-845: The Italian autonomous region of Friuli-Venezia Giulia , as well as a recognized national language in Namibia . There are also notable German-speaking communities in France ( Alsace ), the Czech Republic ( North Bohemia ), Poland ( Upper Silesia ), Slovakia ( Košice Region , Spiš , and Hauerland ), Denmark ( North Schleswig ), Romania and Hungary ( Sopron ). Overseas, sizeable communities of German-speakers are found in Brazil ( Blumenau and Pomerode ), South Africa ( Kroondal ), Namibia , among others, some communities have decidedly Austrian German or Swiss German characters (e.g. Pozuzo , Peru). German

12276-507: The MHG period. Significantly, these texts include a number of impressive secular works, such as the Nibelungenlied , an epic poem telling the story of the dragon -slayer Siegfried ( c.  thirteenth century ), and the Iwein , an Arthurian verse poem by Hartmann von Aue ( c.  1203 ), lyric poems , and courtly romances such as Parzival and Tristan . Also noteworthy

12408-479: 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 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 infinitesimals , that had been the earliest putative foundation for differential calculus . Bolzano's notion of

12540-431: The case of R 1 {\displaystyle \mathbb {R} ^{1}} . Firstly, we will acknowledge that a sequence ( x n ) {\displaystyle (x_{n})} (in R 1 {\displaystyle \mathbb {R} ^{1}} or R n {\displaystyle \mathbb {R} ^{n}} ) has a convergent subsequence if and only if there exists

12672-418: 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

12804-409: 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): B. There is more than one truth in itself: C. There are infinitely many truths in themselves: A known truth has as its parts ( Bestandteile ) a truth in itself and

12936-430: The development of non-local forms of language and exposed all speakers to forms of German from outside their own area. With Luther's rendering of the Bible in the vernacular, German asserted itself against the dominance of Latin as a legitimate language for courtly, literary, and now ecclesiastical subject-matter. His Bible was ubiquitous in the German states: nearly every household possessed a copy. Nevertheless, even with

13068-452: The dialect so as to make the work as natural and accessible to German speakers as possible. Copies of Luther's Bible featured a long list of glosses for each region, translating words which were unknown in the region into the regional dialect. Luther said the following concerning his translation method: One who would talk German does not ask the Latin how he shall do it; he must ask the mother in

13200-507: The eastern provinces of Banat , Bukovina , and Transylvania (German: Banat, Buchenland, Siebenbürgen ), German was the predominant language not only in the larger towns—like Temeschburg ( Timișoara ), Hermannstadt ( Sibiu ), and Kronstadt ( Brașov )—but also in many smaller localities in the surrounding areas. In 1901, the Second Orthographic Conference ended with a (nearly) complete standardization of

13332-478: 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

13464-398: The final peak if one exists (let N = 0 {\displaystyle N=0} otherwise) and let the first index of a new subsequence ( x n j ) {\displaystyle (x_{n_{j}})} be set to n 1 = N + 1 {\displaystyle n_{1}=N+1} . Then n 1 {\displaystyle n_{1}}

13596-862: The final peak, hence there is an n 3 {\displaystyle n_{3}} where n 2 < n 3 {\displaystyle n_{2}<n_{3}} with x n 2 ≤ x n 3 {\displaystyle x_{n_{2}}\leq x_{n_{3}}} . Repeating this process leads to an infinite non-decreasing subsequence  x n 1 ≤ x n 2 ≤ x n 3 ≤ … {\displaystyle x_{n_{1}}\leq x_{n_{2}}\leq x_{n_{3}}\leq \ldots } , thereby proving that every infinite sequence ( x n ) {\displaystyle (x_{n})} in R 1 {\displaystyle \mathbb {R} ^{1}} has

13728-576: 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. Bolzano's posthumously published work Paradoxien des Unendlichen (The Paradoxes of

13860-476: The fundamental truths that may or may not appear to be obvious to our intuitions. 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. However, this is a very small fraction of all the truths that exist, although still too much for one human being to comprehend. Therefore, our knowledge

13992-435: The home, the children on the streets, the common man in the market-place and note carefully how they talk, then translate accordingly. They will then understand what is said to them because it is German. When Christ says ' ex abundantia cordis os loquitur ,' I would translate, if I followed the papists, aus dem Überflusz des Herzens redet der Mund . But tell me is this talking German? What German understands such stuff? No,

14124-420: 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

14256-434: The infinite sequence ( x n ) {\displaystyle (x_{n})} in R 1 {\displaystyle \mathbb {R} ^{1}} has a monotone (non-increasing) subsequence, which is ( x n j ) {\displaystyle (x_{n_{j}})} . But suppose now that there are only finitely many peaks, let N {\displaystyle N} be

14388-474: The influence of Luther's Bible as an unofficial written standard, a widely accepted standard for written German did not appear until the middle of the eighteenth century. German was the language of commerce and government in the Habsburg Empire , which encompassed a large area of Central and Eastern Europe . Until the mid-nineteenth century, it was essentially the language of townspeople throughout most of

14520-409: The largest concentrations of German speakers are in the states of Rio Grande do Sul (where Riograndenser Hunsrückisch developed), Santa Catarina , and Espírito Santo . Bolzano%E2%80%93Weierstrass theorem In mathematics , specifically in real analysis , the Bolzano–Weierstrass theorem , named after Bernard Bolzano and Karl Weierstrass , is a fundamental result about convergence in

14652-465: The lemma that ( x n 1 ) {\displaystyle (x_{n1})} has a convergent subsequence and hence there exists a countable set K 1 ⊆ I {\displaystyle K_{1}\subseteq I} such that ( x n 1 ) n ∈ K 1 {\displaystyle (x_{n1})_{n\in K_{1}}} converges. For

14784-435: The limit of the interval's length is zero. Also, by the nested intervals theorem , which states that if each I n {\displaystyle I_{n}} is a closed and bounded interval, say with then under the assumption of nesting, the intersection of the I n {\displaystyle I_{n}} is not empty. Thus there is a number x {\displaystyle x} that

14916-402: The mother in the home and the plain man would say, Wesz das Herz voll ist, des gehet der Mund über . Luther's translation of the Bible into High German was also decisive for the German language and its evolution from Early New High German to modern Standard German. The publication of Luther's Bible was a decisive moment in the spread of literacy in early modern Germany , and promoted

15048-630: The ninth century, chief among them being the Muspilli , Merseburg charms , and Hildebrandslied , and other religious texts (the Georgslied , Ludwigslied , Evangelienbuch , and translated hymns and prayers). The Muspilli is a Christian poem written in a Bavarian dialect offering an account of the soul after the Last Judgment , and the Merseburg charms are transcriptions of spells and charms from

15180-434: 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

15312-462: The pronunciation of the ending -ig as [ɪk] instead of [ɪç]. In Northern Germany, High German was a foreign language to most inhabitants, whose native dialects were subsets of Low German. It was usually encountered only in writing or formal speech; in fact, most of High German was a written language, not identical to any spoken dialect, throughout the German-speaking area until well into the 19th century. However, wider standardization of pronunciation

15444-442: 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

15576-594: The sequence ( x n 2 ) {\displaystyle (x_{n2})} , by applying the lemma once again there exists a countable set K 2 ⊆ K 1 ⊆ I {\displaystyle K_{2}\subseteq K_{1}\subseteq I} such that ( x n 2 ) n ∈ K 2 {\displaystyle (x_{n2})_{n\in K_{2}}} converges and hence ( x n 2 ) {\displaystyle (x_{n2})} has

15708-541: The states of North Dakota and South Dakota , German is the most common language spoken at home after English. As a legacy of significant German immigration to the country , German geographical names can be found throughout the Midwest region , such as New Ulm and Bismarck (North Dakota's state capital), plus many other regions. A number of German varieties have developed in the country and are still spoken today, such as Pennsylvania Dutch and Texas German . In Brazil,

15840-441: 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). Bolzano came to be surrounded by

15972-545: The theorem for R 1 {\displaystyle \mathbb {R} ^{1}} (set of all real numbers ), in which case the ordering on R 1 {\displaystyle \mathbb {R} ^{1}} can be put to good use. Indeed, we have the following result: Lemma : Every infinite sequence ( x n ) {\displaystyle (x_{n})} in R 1 {\displaystyle \mathbb {R} ^{1}} has an infinite monotone subsequence (a subsequence that

16104-496: The third most commonly learned second language in the United States in K-12 education. The language has been influential in the fields of philosophy, theology, science, and technology. It is the second most commonly used language in science and the third most widely used language on websites . The German-speaking countries are ranked fifth in terms of annual publication of new books, with one-tenth of all books (including e-books) in

16236-399: 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

16368-543: The world being published in German. German is most closely related to other West Germanic languages, namely Afrikaans , Dutch , English , the Frisian languages , and Scots . It also contains close similarities in vocabulary to some languages in the North Germanic group , such as Danish , Norwegian , and Swedish . Modern German gradually developed from Old High German , which in turn developed from Proto-Germanic during

16500-579: The written form of German. One of the central events in the development of ENHG was the publication of Luther's translation of the Bible into High German (the New Testament was published in 1522; the Old Testament was published in parts and completed in 1534). Luther based his translation primarily on the Meißner Deutsch of Saxony , spending much time among the population of Saxony researching

16632-429: Was a "neutral" language as there were virtually no English native speakers in Namibia at that time. German, Afrikaans, and several indigenous languages thus became "national languages" by law, identifying them as elements of the cultural heritage of the nation and ensuring that the state acknowledged and supported their presence in the country. Today, Namibia is considered to be the only German-speaking country outside of

16764-530: Was an artificial standard that did not correspond to any traditional spoken dialect. Rather, it was based on the pronunciation of German in Northern Germany, although it was subsequently regarded often as a general prescriptive norm, despite differing pronunciation traditions especially in the Upper-German-speaking regions that still characterise the dialect of the area today – especially

16896-412: Was arbitrary, any bounded sequence in R n {\displaystyle \mathbb {R} ^{n}} has a convergent subsequence. There is also an alternative proof of the Bolzano–Weierstrass theorem using nested intervals . We start with a bounded sequence ( x n ) {\displaystyle (x_{n})} : Because we halve the length of an interval at each step,

17028-467: 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

17160-485: Was established on the basis of public speaking in theatres and the media during the 20th century and documented in pronouncing dictionaries. Official revisions of some of the rules from 1901 were not issued until the controversial German orthography reform of 1996 was made the official standard by governments of all German-speaking countries. Media and written works are now almost all produced in Standard German which

17292-484: Was inclined to share with others with some frequency, eventually proved to be too liberal for the Austrian authorities. On December 24, 1819, he was removed from his professorship (upon his refusal to recant his beliefs) and was exiled to the countryside and then devoted his energies to his writings on social, religious, philosophical, and mathematical matters. Although forbidden to publish in mainstream journals as

17424-438: 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

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