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108-463: Friedrich Ludwig Gottlob Frege ( / ˈ f r eɪ ɡ ə / ; German: [ˈɡɔtloːp ˈfreːɡə] ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena , and is understood by many to be the father of analytic philosophy , concentrating on the philosophy of language , logic , and mathematics . Though he

216-503: A {\displaystyle a} , reflects the function in the y {\displaystyle y} -axis when it is negative. The k {\displaystyle k} and h {\displaystyle h} values introduce translations, h {\displaystyle h} , vertical, and k {\displaystyle k} horizontal. Positive h {\displaystyle h} and k {\displaystyle k} values mean

324-436: A , b , c ) {\displaystyle \mathbf {n} =(a,b,c)} be a nonzero vector. The plane determined by this point and vector consists of those points P {\displaystyle P} , with position vector r {\displaystyle \mathbf {r} } , such that the vector drawn from P 0 {\displaystyle P_{0}} to P {\displaystyle P}

432-410: A x + b y + c z + d = 0 ,  where  d = − ( a x 0 + b y 0 + c z 0 ) . {\displaystyle ax+by+cz+d=0,{\text{ where }}d=-(ax_{0}+by_{0}+cz_{0}).} Conversely, it is easily shown that if a , b , c and d are constants and a , b , and c are not all zero, then

540-415: A dot product , not scalar multiplication.) Expanded this becomes a ( x − x 0 ) + b ( y − y 0 ) + c ( z − z 0 ) = 0 , {\displaystyle a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0,} which is the point-normal form of the equation of a plane. This is just a linear equation :

648-435: A Geometrical Representation of Imaginary Forms in a Plane"), in which he aimed to solve such fundamental problems in geometry as the mathematical interpretation of projective geometry 's infinitely distant (imaginary) points. Frege married Margarete Katharina Sophia Anna Lieseberg (15 February 1856 – 25 June 1904) on 14 March 1887. The couple had at least two children, who unfortunately died when young. Years later they adopted

756-686: A complex variable, applications of physics, selected divisions of mechanics, and mechanics of solids. Abbe was more than a teacher to Frege: he was a trusted friend, and, as director of the optical manufacturer Carl Zeiss AG, he was in a position to advance Frege's career. After Frege's graduation, they came into closer correspondence. His other notable university teachers were Christian Philipp Karl Snell (1806–86; subjects: use of infinitesimal analysis in geometry, analytic geometry of planes , analytical mechanics, optics, physical foundations of mechanics); Hermann Karl Julius Traugott Schaeffer (1824–1900; analytic geometry, applied physics, algebraic analysis, on

864-667: A descendant of Philipp Melanchthon and her father was Johann Heinrich Siegfried Bialloblotzky, a descendant of a Polish noble family who left Poland in the 17th century. Frege was a Lutheran. In childhood, Frege encountered philosophies that would guide his future scientific career. For example, his father wrote a textbook on the German language for children aged 9–13, entitled Hülfsbuch zum Unterrichte in der deutschen Sprache für Kinder von 9 bis 13 Jahren (2nd ed., Wismar 1850; 3rd ed., Wismar and Ludwigslust: Hinstorff, 1862) (Help book for teaching German to children from 9 to 13 years old),

972-497: A foundation for calculus in Europe. Initially the work was not well received, due, in part, to the many gaps in arguments and complicated equations. Only after the translation into Latin and the addition of commentary by van Schooten in 1649 (and further work thereafter) did Descartes's masterpiece receive due recognition. Pierre de Fermat also pioneered the development of analytic geometry. Although not published in his lifetime,

1080-628: A manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (the normal vector ) to indicate its "inclination". Specifically, let r 0 {\displaystyle \mathbf {r} _{0}} be the position vector of some point P 0 = ( x 0 , y 0 , z 0 ) {\displaystyle P_{0}=(x_{0},y_{0},z_{0})} , and let n = (

1188-562: A manuscript form of Ad locos planos et solidos isagoge (Introduction to Plane and Solid Loci) was circulating in Paris in 1637, just prior to the publication of Descartes' Discourse . Clearly written and well received, the Introduction also laid the groundwork for analytical geometry. The key difference between Fermat's and Descartes' treatments is a matter of viewpoint: Fermat always started with an algebraic equation and then described

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1296-543: A method that had a strong resemblance to the use of coordinates and it has sometimes been maintained that he had introduced analytic geometry. Apollonius of Perga , in On Determinate Section , dealt with problems in a manner that may be called an analytic geometry of one dimension; with the question of finding points on a line that were in a ratio to the others. Apollonius in the Conics further developed

1404-402: A method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of Descartes by some 1800 years. His application of reference lines, a diameter and a tangent is essentially no different from our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to

1512-408: A multiple of one equation to the other equation so that one of the variables is eliminated. For our current example, if we subtract the first equation from the second we get ( x − 1 ) 2 − x 2 = 0 {\displaystyle (x-1)^{2}-x^{2}=0} . The y 2 {\displaystyle y^{2}} in the first equation

1620-464: A single linear equation, so they are frequently described by parametric equations : x = x 0 + a t {\displaystyle x=x_{0}+at} y = y 0 + b t {\displaystyle y=y_{0}+bt} z = z 0 + c t {\displaystyle z=z_{0}+ct} where: In the Cartesian coordinate system ,

1728-431: A son, Alfred. Little else is known about Frege's family life, however. Though his education and early mathematical work focused primarily on geometry, Frege's work soon turned to logic. His Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens [ Concept-Script: A Formal Language for Pure Thought Modeled on that of Arithmetic ], Halle a/S: Verlag von Louis Nebert, 1879 marked

1836-404: A systematic study of space curves and surfaces. In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates. Similarly, Euclidean space is given coordinates where every point has three coordinates. The value of the coordinates depends on the choice of the initial point of origin. There are a variety of coordinate systems used, but

1944-634: A turning point in the history of logic. The Begriffsschrift broke new ground, including a rigorous treatment of the ideas of functions and variables . Frege's goal was to show that mathematics grows out of logic , and in so doing, he devised techniques that separated him from the Aristotelian syllogistic but took him rather close to Stoic propositional logic. In effect, Frege invented axiomatic predicate logic , in large part thanks to his invention of quantified variables , which eventually became ubiquitous in mathematics and logic, and which solved

2052-600: Is a 2 -dimensional surface in 3-dimensional space defined as the locus of zeros of a quadratic polynomial . In coordinates x 1 , x 2 , x 3 , the general quadric is defined by the algebraic equation ∑ i , j = 1 3 x i Q i j x j + ∑ i = 1 3 P i x i + R = 0. {\displaystyle \sum _{i,j=1}^{3}x_{i}Q_{ij}x_{j}+\sum _{i=1}^{3}P_{i}x_{i}+R=0.} Quadric surfaces include ellipsoids (including

2160-742: Is a Function?" (1904) Logical Investigations (1918–1923). Frege intended that the following three papers be published together in a book titled Logische Untersuchungen ( Logical Investigations ). Though the German book never appeared, the papers were published together in Logische Untersuchungen , ed. G. Patzig, Vandenhoeck & Ruprecht, 1966, and English translations appeared together in Logical Investigations , ed. Peter Geach, Blackwell, 1975. Philosophy Logic and mathematics Historical context University of Jena The University of Jena , officially

2268-676: Is a number?" or "What objects do number-words ('one', 'two', etc.) refer to?" But in pursuing these matters, he eventually found himself analysing and explaining what meaning is, and thus came to several conclusions that proved highly consequential for the subsequent course of analytic philosophy and the philosophy of language. Frege's 1892 paper, " On Sense and Reference " ("Über Sinn und Bedeutung"), introduced his influential distinction between sense ("Sinn") and reference ("Bedeutung", which has also been translated as "meaning", or "denotation"). While conventional accounts of meaning took expressions to have just one feature (reference), Frege introduced

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2376-477: Is a relation in the x y {\displaystyle xy} plane. For example, x 2 + y 2 − 1 = 0 {\displaystyle x^{2}+y^{2}-1=0} is the relation that describes the unit circle. For two geometric objects P and Q represented by the relations P ( x , y ) {\displaystyle P(x,y)} and Q ( x , y ) {\displaystyle Q(x,y)}

2484-477: Is a university association with the Martin Luther University of Halle-Wittenberg and the University of Leipzig . The aim is firstly to give the students the opportunity to visit with relatively few problems at the partner universities and events in order to broaden the range of subjects and topics. Currently e. g. has joined a cooperation in teaching in the field of bioinformatics. In addition,

2592-612: Is an unanalysable whole, and the functional expression "the King of the United Kingdom", which contains the significant parts "the King of ξ" and "United Kingdom", have the same referent , namely, the person best known as King Charles III . But the sense of the word " United Kingdom " is a part of the sense of the latter expression, but no part of the sense of the "full name" of King Charles. These distinctions were disputed by Bertrand Russell, especially in his paper " On Denoting ";

2700-463: Is concerned with defining and representing geometric shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom . The Greek mathematician Menaechmus solved problems and proved theorems by using

2808-484: Is defined by the formula d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 , {\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}},} which can be viewed as a version of the Pythagorean theorem . Similarly, the angle that a line makes with the horizontal can be defined by

2916-592: Is easy to define the relation of membership of a set or extension in Frege's system; Russell then drew attention to "the set of things x that are such that x is not a member of x ". The system of the Grundgesetze entails that the set thus characterised both is and is not a member of itself, and is thus inconsistent. Frege wrote a hasty, last-minute Appendix to Vol. 2, deriving the contradiction and proposing to eliminate it by modifying Basic Law V. Frege opened

3024-694: Is false. ( 0 , 0 ) {\displaystyle (0,0)} is not in P {\displaystyle P} so it is not in the intersection. The intersection of P {\displaystyle P} and Q {\displaystyle Q} can be found by solving the simultaneous equations: x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} ( x − 1 ) 2 + y 2 = 1. {\displaystyle (x-1)^{2}+y^{2}=1.} Traditional methods for finding intersections include substitution and elimination. Substitution: Solve

3132-420: Is part of the body of Persian mathematics that was eventually transmitted to Europe. Because of his thoroughgoing geometrical approach to algebraic equations, Khayyam can be considered a precursor to Descartes in the invention of analytic geometry. Analytic geometry was independently invented by René Descartes and Pierre de Fermat , although Descartes is sometimes given sole credit. Cartesian geometry ,

3240-513: Is perpendicular to n {\displaystyle \mathbf {n} } . Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be described as the set of all points r {\displaystyle \mathbf {r} } such that n ⋅ ( r − r 0 ) = 0. {\displaystyle \mathbf {n} \cdot (\mathbf {r} -\mathbf {r} _{0})=0.} (The dot here means

3348-634: Is recognized in several university ranking systems. As per the QS World University Rankings for 2024, the university is ranked 461st in the world and 26th nationally. In the Times Higher Education World University Rankings of 2024, it is placed at 201-250th globally and 22-24th within the country. The Academic Ranking of World Universities (ARWU) for 2022 places it within the 401-500 bracket globally, and between 25th and 31st in

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3456-918: Is represented by an ordered triple of coordinates ( x ,  y ,  z ). In polar coordinates , every point of the plane is represented by its distance r from the origin and its angle θ , with θ normally measured counterclockwise from the positive x -axis. Using this notation, points are typically written as an ordered pair ( r , θ ). One may transform back and forth between two-dimensional Cartesian and polar coordinates by using these formulae: x = r cos ⁡ θ , y = r sin ⁡ θ ; r = x 2 + y 2 , θ = arctan ⁡ ( y / x ) . {\displaystyle x=r\,\cos \theta ,\,y=r\,\sin \theta ;\,r={\sqrt {x^{2}+y^{2}}},\,\theta =\arctan(y/x).} This system may be generalized to three-dimensional space through

3564-793: Is subtracted from the y 2 {\displaystyle y^{2}} in the second equation leaving no y {\displaystyle y} term. The variable y {\displaystyle y} has been eliminated. We then solve the remaining equation for x {\displaystyle x} , in the same way as in the substitution method: x 2 − 2 x + 1 − x 2 = 0 {\displaystyle x^{2}-2x+1-x^{2}=0} − 2 x = − 1 {\displaystyle -2x=-1} x = 1 / 2. {\displaystyle x=1/2.} We then place this value of x {\displaystyle x} in either of

3672-420: Is the angle between A and B . Transformations are applied to a parent function to turn it into a new function with similar characteristics. The graph of R ( x , y ) {\displaystyle R(x,y)} is changed by standard transformations as follows: There are other standard transformation not typically studied in elementary analytic geometry because the transformations change

3780-400: Is the equation for any circle centered at the origin (0, 0) with a radius of r. Lines in a Cartesian plane , or more generally, in affine coordinates , can be described algebraically by linear equations. In two dimensions, the equation for non-vertical lines is often given in the slope-intercept form : y = m x + b {\displaystyle y=mx+b} where: In

3888-572: Is the foundation of most modern fields of geometry, including algebraic , differential , discrete and computational geometry . Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and circles, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space. As taught in school books, analytic geometry can be explained more simply: it

3996-696: The Begriffsschrift and work in the foundations of mathematics . His book the Foundations of Arithmetic is the seminal text of the logicist project, and is cited by Michael Dummett as where to pinpoint the linguistic turn . His philosophical papers " On Sense and Reference " and " The Thought " are also widely cited. The former argues for two different types of meaning and descriptivism . In Foundations and "The Thought", Frege argues for Platonism against psychologism or formalism , concerning numbers and propositions respectively. Frege

4104-627: The Friedrich Schiller University Jena ( German : Friedrich-Schiller-Universität Jena , abbreviated FSU , shortened form Uni Jena ), is a public research university located in Jena , Thuringia , Germany . The university was established in 1558 and is counted among the ten oldest universities in Germany . It is affiliated with six Nobel Prize winners, most recently in 2000 when Jena graduate Herbert Kroemer won

4212-628: The University of Jena . Frege matriculated at the University of Jena in the spring of 1869 as a citizen of the North German Confederation . In the four semesters of his studies he attended approximately twenty courses of lectures, most of them on mathematics and physics. His most important teacher was Ernst Karl Abbe (1840–1905; physicist, mathematician, and inventor). Abbe gave lectures on theory of gravity, galvanism and electrodynamics, complex analysis theory of functions of

4320-528: The graph of a quadratic equation in two variables is always a conic section – though it may be degenerate, and all conic sections arise in this way. The equation will be of the form A x 2 + B x y + C y 2 + D x + E y + F = 0  with  A , B , C  not all zero. {\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0{\text{ with }}A,B,C{\text{ not all zero.}}} As scaling all six constants yields

4428-452: The linguistic turn in philosophy. His contributions to the philosophy of language include: As a philosopher of mathematics, Frege attacked the psychologistic appeal to mental explanations of the content of judgment of the meaning of sentences. His original purpose was very far from answering general questions about meaning; instead, he devised his logic to explore the foundations of arithmetic, undertaking to answer questions such as "What

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4536-450: The problem of multiple generality . Previous logic had dealt with the logical constants and , or , if... then... , not , and some and all , but iterations of these operations, especially "some" and "all", were little understood: even the distinction between a sentence like "every boy loves some girl" and "some girl is loved by every boy" could be represented only very artificially, whereas Frege's formalism had no difficulty expressing

4644-429: The sphere ), paraboloids , hyperboloids , cylinders , cones , and planes . In analytic geometry, geometric notions such as distance and angle measure are defined using formulas . These definitions are designed to be consistent with the underlying Euclidean geometry . For example, using Cartesian coordinates on the plane, the distance between two points ( x 1 ,  y 1 ) and ( x 2 ,  y 2 )

4752-408: The xy -plane makes with respect to the horizontal axis, and the angle φ that it makes with respect to the z -axis. The names of the angles are often reversed in physics. In analytic geometry, any equation involving the coordinates specifies a subset of the plane, namely the solution set for the equation, or locus . For example, the equation y  =  x corresponds to the set of all

4860-416: The 1st and 3rd or 2nd and 4th quadrant. In general, if y = f ( x ) {\displaystyle y=f(x)} , then it can be transformed into y = a f ( b ( x − k ) ) + h {\displaystyle y=af(b(x-k))+h} . In the new transformed function, a {\displaystyle a} is the factor that vertically stretches

4968-456: The 20th century the university was promoted through cooperation with Carl Zeiss (company) and thereby enabling it to increase the student population as a mass university. In 1905 the university had 1,100 students and 112 university teachers, so this figure has since been almost twenty-fold. The Friedrich-Schiller University is the only comprehensive university in Thuringia. Since 1995, there

5076-554: The Appendix with the exceptionally honest comment: "Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr. Bertrand Russell, just when the printing of this volume was nearing its completion." (This letter and Frege's reply are translated in Jean van Heijenoort 1967.) Frege's proposed remedy

5184-636: The Bachstraße received total or significant physical damage. Completely destroyed were the Botanical Garden, the psychological and the physiological institute and three chemical Institutes. An important event for the National Socialist period was the investigation of the pediatrician Yusuf Ibrahim . A Senate Commission noted the participation of the physician to the "euthanasia" murders of physically or mentally disabled children. In

5292-594: The Emperor Ferdinand I , the university was established on 2 February 1558. The university, jointly maintained by the Saxon Duchies derived from the partitioning of John Frederick's duchy, was thus named Ducal Pan-Saxon University ( German : Herzoglich Sächsische Gesamtuniversität ) or Salana (after the river Saale ). Prior to the 20th century, university enrollment peaked in the 18th century. The university's reputation reached its zenith under

5400-634: The Enlightenment were developed as research institutions. 2014 the "Center of Advanced Research" (ZAF) was established. Jena University is one of the founder of The German Centre for Integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig, that was founded in 2013. It is a research centre of the German Research Foundation (DFG). Friedrich Schiller University is the only German university with chairs for either gravitational theory or Caucasus Studies. The University of Jena

5508-568: The Goethe Memorial at the Botanical Garden are reminders of the two towering geniuses of Jena . Both buildings are also open to the public. Oriental Collections/ Papyrus Collection Archaeological Collections Mineralogy & Geology History of Sciences Medicine Analytic geometry Analytic geometry is used in physics and engineering , and also in aviation , rocketry , space science , and spaceflight . It

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5616-526: The Nobel Prize for physics. It was renamed after the poet Friedrich Schiller who was teaching as professor of philosophy when Jena attracted some of the most influential minds at the turn of the 19th century. With Karl Leonhard Reinhold , Johann Gottlieb Fichte , G. W. F. Hegel , F. W. J. Schelling and Friedrich Schlegel on its teaching staff, the university was at the centre of the emergence of German idealism and early Romanticism . As of 2014 ,

5724-621: The alternative term used for analytic geometry, is named after Descartes. Descartes made significant progress with the methods in an essay titled La Géométrie (Geometry) , one of the three accompanying essays (appendices) published in 1637 together with his Discourse on the Method for Rightly Directing One's Reason and Searching for Truth in the Sciences , commonly referred to as Discourse on Method . La Geometrie , written in his native French tongue, and its philosophical principles, provided

5832-491: The angle between two vectors is given by the dot product . The dot product of two Euclidean vectors A and B is defined by A ⋅ B = d e f ‖ A ‖ ‖ B ‖ cos ⁡ θ , {\displaystyle \mathbf {A} \cdot \mathbf {B} {\stackrel {\mathrm {def} }{=}}\left\|\mathbf {A} \right\|\left\|\mathbf {B} \right\|\cos \theta ,} where θ

5940-419: The auspices of Duke Charles Augustus , Goethe 's patron (1787–1806), when Gottlieb Fichte , G. W. F. Hegel , Friedrich Schelling , Friedrich von Schlegel and Friedrich Schiller were on its teaching staff. Founded as a home for the new religious opinions of the sixteenth century, it has since been one of the most politically radical universities in Germany. Jena was noted among other German universities at

6048-555: The belief that it would be best if the Jews of Germany would "get lost, or better would like to disappear from Germany." Some interpretations have been written about that time. The diary contains a critique of universal suffrage and socialism. Frege had friendly relations with Jews in real life: among his students was Gershom Scholem , who greatly valued his teaching, and it was he who encouraged Ludwig Wittgenstein to leave for England in order to study with Bertrand Russell . The 1924 diary

6156-638: The circle with radius 1 and center ( 1 , 0 ) : Q = { ( x , y ) | ( x − 1 ) 2 + y 2 = 1 } {\displaystyle (1,0):Q=\{(x,y)|(x-1)^{2}+y^{2}=1\}} . The intersection of these two circles is the collection of points which make both equations true. Does the point ( 0 , 0 ) {\displaystyle (0,0)} make both equations true? Using ( 0 , 0 ) {\displaystyle (0,0)} for ( x , y ) {\displaystyle (x,y)} ,

6264-530: The controversy has continued into the present, fueled especially by Saul Kripke 's famous lectures " Naming and Necessity ". Frege's published philosophical writings were of a very technical nature and divorced from practical issues, so much so that Frege scholar Dummett expressed his "shock to discover, while reading Frege's diary, that his hero was an anti-Semite." After the German Revolution of 1918–19 his political opinions became more radical. In

6372-572: The cooperation provides the university management the opportunity to share experiences with their regular meetings and initiate common projects. So z. B. went from the successful bid to the German Centre for Integrative Biodiversity Research (iDiv) from the university network. The co-operation continues at other levels: for example in a joint mentoring program for female postdocs or in the central German archives network. And last but not least, there are common sports activities. Since October 2014,

6480-407: The coordinate system was superimposed upon a given curve a posteriori instead of a priori . That is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation. The 11th-century Persian mathematician Omar Khayyam saw a strong relationship between geometry and algebra and

6588-518: The different readings of "every boy loves some girl who loves some boy who loves some girl" and similar sentences, in complete parallel with his treatment of, say, "every boy is foolish". A frequently noted example is that Aristotle's logic is unable to represent mathematical statements like Euclid's theorem , a fundamental statement of number theory that there are an infinite number of prime numbers . Frege's "conceptual notation", however, can represent such inferences. The analysis of logical concepts and

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6696-461: The dominant approach to mathematical logic was still that of George Boole (1815–64) and his intellectual descendants, especially Ernst Schröder (1841–1902). Frege's logical ideas nevertheless spread through the writings of his student Rudolf Carnap (1891–1970) and other admirers, particularly Bertrand Russell and Ludwig Wittgenstein (1889–1951). Frege is one of the founders of analytic philosophy , whose work on logic and language gave rise to

6804-404: The equation for Q {\displaystyle Q} becomes ( 0 − 1 ) 2 + 0 2 = 1 {\displaystyle (0-1)^{2}+0^{2}=1} or ( − 1 ) 2 = 1 {\displaystyle (-1)^{2}=1} which is true, so ( 0 , 0 ) {\displaystyle (0,0)} is in

6912-564: The first equation for y {\displaystyle y} in terms of x {\displaystyle x} and then substitute the expression for y {\displaystyle y} into the second equation: x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} y 2 = 1 − x 2 . {\displaystyle y^{2}=1-x^{2}.} We then substitute this value for y 2 {\displaystyle y^{2}} into

7020-407: The first section of which dealt with the structure and logic of language . Frege studied at Große Stadtschule Wismar  [ de ] and graduated in 1869. Teacher of mathematics and natural science Gustav Adolf Leo Sachse (1843–1909), who was also a poet, played an important role in determining Frege's future scientific career, encouraging him to continue his studies at his own alma mater

7128-679: The formula θ = arctan ⁡ ( m ) , {\displaystyle \theta =\arctan(m),} where m is the slope of the line. In three dimensions, distance is given by the generalization of the Pythagorean theorem: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 + ( z 2 − z 1 ) 2 , {\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}},} while

7236-400: The function g ( x ) if and only if ∀ x [ f ( x ) = g ( x )]. The crucial case of the law may be formulated in modern notation as follows. Let { x | Fx } denote the extension of the predicate Fx , that is, the set of all Fs, and similarly for Gx . Then Basic Law V says that the predicates Fx and Gx have the same extension if and only if ∀x[ Fx ↔ Gx ]. The set of Fs is the same as

7344-441: The function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative a {\displaystyle a} values, the function is reflected in the x {\displaystyle x} -axis. The b {\displaystyle b} value compresses the graph of the function horizontally if greater than 1 and stretches the function horizontally if less than 1, and like

7452-391: The function is translated to the positive end of its axis and negative meaning translation towards the negative end. Transformations can be applied to any geometric equation whether or not the equation represents a function. Transformations can be considered as individual transactions or in combinations. Suppose that R ( x , y ) {\displaystyle R(x,y)}

7560-402: The geometric curve that satisfied it, whereas Descartes started with geometric curves and produced their equations as one of several properties of the curves. As a consequence of this approach, Descartes had to deal with more complicated equations and he had to develop the methods to work with polynomial equations of higher degree. It was Leonhard Euler who first applied the coordinate method in

7668-445: The graph of the equation a x + b y + c z + d = 0 , {\displaystyle ax+by+cz+d=0,} is a plane having the vector n = ( a , b , c ) {\displaystyle \mathbf {n} =(a,b,c)} as a normal. This familiar equation for a plane is called the general form of the equation of the plane. In three dimensions, lines can not be described by

7776-555: The intersection is the collection of all points ( x , y ) {\displaystyle (x,y)} which are in both relations. For example, P {\displaystyle P} might be the circle with radius 1 and center ( 0 , 0 ) {\displaystyle (0,0)} : P = { ( x , y ) | x 2 + y 2 = 1 } {\displaystyle P=\{(x,y)|x^{2}+y^{2}=1\}} and Q {\displaystyle Q} might be

7884-735: The last year of his life, at the age of 76, his diary contained political opinions opposing the parliamentary system, democrats, liberals, Catholics, the French and Jews, who he thought ought to be deprived of political rights and, preferably, expelled from Germany. Frege confided "that he had once thought of himself as a liberal and was an admirer of Bismarck ", but then sympathized with General Ludendorff . In an entry dated 5 May 1924 Frege expressed agreement with an article published in Houston Stewart Chamberlain's Deutschlands Erneuerung which praised Adolf Hitler . Frege recorded

7992-448: The machinery of formalization that is essential to Principia Mathematica (3 vols., 1910–13, by Bertrand Russell , 1872–1970, and Alfred North Whitehead , 1861–1947), to Russell's theory of descriptions , to Kurt Gödel 's (1906–78) incompleteness theorems , and to Alfred Tarski 's (1901–83) theory of truth, is ultimately due to Frege. One of Frege's stated purposes was to isolate genuinely logical principles of inference, so that in

8100-613: The meteorological institute; the botanical garden; the seminaries of theology, philology, and education; and the well-equipped clinical, anatomical, and physical institutes. After the end of the Saxon duchies in 1918, and their merger with further principalities into the Free State of Thuringia in 1920, the university was renamed as the Thuringian State University (Thüringische Landesuniversität) in 1921. In 1934

8208-473: The most common are the following: The most common coordinate system to use is the Cartesian coordinate system , where each point has an x -coordinate representing its horizontal position, and a y -coordinate representing its vertical position. These are typically written as an ordered pair ( x ,  y ). This system can also be used for three-dimensional geometry, where every point in Euclidean space

8316-677: The national context. Among the collections which are open to the public are the Jena Phyletisches Museum , an institution which is unique in Europe for illustrating the history of evolution, the Ernst-Haeckel-Memorial Museum , the Mineralogical Collection which traces its roots back to Goethe and the second oldest Botanical Garden of Middle Europe . The Schiller Gardenhouse  [ de ] ( Schillers Gartenhaus ) and

8424-839: The original equations and solve for y {\displaystyle y} : ( 1 / 2 ) 2 + y 2 = 1 {\displaystyle (1/2)^{2}+y^{2}=1} y 2 = 3 / 4 {\displaystyle y^{2}=3/4} y = ± 3 2 . {\displaystyle y={\frac {\pm {\sqrt {3}}}{2}}.} So our intersection has two points: ( 1 / 2 , + 3 2 ) and ( 1 / 2 , − 3 2 ) . {\displaystyle \left(1/2,{\frac {+{\sqrt {3}}}{2}}\right)\;\;{\text{and}}\;\;\left(1/2,{\frac {-{\sqrt {3}}}{2}}\right).} Elimination : Add (or subtract)

8532-854: The original equations and solve for y {\displaystyle y} : ( 1 / 2 ) 2 + y 2 = 1 {\displaystyle (1/2)^{2}+y^{2}=1} y 2 = 3 / 4 {\displaystyle y^{2}=3/4} y = ± 3 2 . {\displaystyle y={\frac {\pm {\sqrt {3}}}{2}}.} So our intersection has two points: ( 1 / 2 , + 3 2 ) and ( 1 / 2 , − 3 2 ) . {\displaystyle \left(1/2,{\frac {+{\sqrt {3}}}{2}}\right)\;\;{\text{and}}\;\;\left(1/2,{\frac {-{\sqrt {3}}}{2}}\right).} For conic sections, as many as 4 points might be in

8640-674: The other equation and proceed to solve for x {\displaystyle x} : ( x − 1 ) 2 + ( 1 − x 2 ) = 1 {\displaystyle (x-1)^{2}+(1-x^{2})=1} x 2 − 2 x + 1 + 1 − x 2 = 1 {\displaystyle x^{2}-2x+1+1-x^{2}=1} − 2 x = − 1 {\displaystyle -2x=-1} x = 1 / 2. {\displaystyle x=1/2.} Next, we place this value of x {\displaystyle x} in either of

8748-419: The pharmacologist Walter Rosenthal is the president of the university; Chancellor is since 2007 the mathematician Klaus Bartholmé. The university is organized in 10 schools: Research at Friedrich Schiller University traditionally focusses on both humanities and sciences. In addition to the faculties the following "Collaborative Research Centers" ( German "Sonderforschungsbereich", short: "SFB") operate at

8856-471: The philosophical doctrines of the mature Frege have parallels in Lotze; it has been the subject of scholarly debate whether or not there was a direct influence on Frege's views arising from his attending Lotze's lectures. In 1873, Frege attained his doctorate under Ernst Christian Julius Schering, with a dissertation under the title of "Ueber eine geometrische Darstellung der imaginären Gebilde in der Ebene" ("On

8964-448: The plane. This is not always the case: the trivial equation x  =  x specifies the entire plane, and the equation x  +  y  = 0 specifies only the single point (0, 0). In three dimensions, a single equation usually gives a surface , and a curve must be specified as the intersection of two surfaces (see below), or as a system of parametric equations . The equation x  +  y  =  r

9072-408: The points on the plane whose x -coordinate and y -coordinate are equal. These points form a line , and y  =  x is said to be the equation for this line. In general, linear equations involving x and y specify lines, quadratic equations specify conic sections , and more complicated equations describe more complicated figures. Usually, a single equation corresponds to a curve on

9180-435: The proper representation of mathematical proof, one would at no point appeal to "intuition". If there was an intuitive element, it was to be isolated and represented separately as an axiom: from there on, the proof was to be purely logical and without gaps. Having exhibited this possibility, Frege's larger purpose was to defend the view that arithmetic is a branch of logic, a view known as logicism : unlike geometry, arithmetic

9288-477: The relation Q {\displaystyle Q} . On the other hand, still using ( 0 , 0 ) {\displaystyle (0,0)} for ( x , y ) {\displaystyle (x,y)} the equation for P {\displaystyle P} becomes 0 2 + 0 2 = 1 {\displaystyle 0^{2}+0^{2}=1} or 0 = 1 {\displaystyle 0=1} which

9396-455: The same locus of zeros, one can consider conics as points in the five-dimensional projective space P 5 . {\displaystyle \mathbf {P} ^{5}.} The conic sections described by this equation can be classified using the discriminant B 2 − 4 A C . {\displaystyle B^{2}-4AC.} If the conic is non-degenerate, then: A quadric , or quadric surface ,

9504-413: The set of Gs just in case every F is a G and every G is an F. (The case is special because what is here being called the extension of a predicate, or a set, is only one type of "value-range" of a function.) In a famous episode, Bertrand Russell wrote to Frege, just as Vol. 2 of the Grundgesetze was about to go to press in 1903, showing that Russell's paradox could be derived from Frege's Basic Law V. It

9612-549: The shape of objects in ways not usually considered. Skewing is an example of a transformation not usually considered. For more information, consult the Misplaced Pages article on affine transformations . For example, the parent function y = 1 / x {\displaystyle y=1/x} has a horizontal and a vertical asymptote, and occupies the first and third quadrant, and all of its transformed forms have one horizontal and vertical asymptote, and occupies either

9720-411: The tangent and intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations (expressed in words) of curves. However, although Apollonius came close to developing analytic geometry, he did not manage to do so since he did not take into account negative magnitudes and in every case

9828-728: The telegraph and other electronic machines ); and the philosopher Kuno Fischer (1824–1907; Kantian and critical philosophy ). Starting in 1871, Frege continued his studies in Göttingen, the leading university in mathematics in German-speaking territories, where he attended the lectures of Rudolf Friedrich Alfred Clebsch (1833–72; analytic geometry), Ernst Christian Julius Schering (1824–97; function theory), Wilhelm Eduard Weber (1804–91; physical studies, applied physics), Eduard Riecke (1845–1915; theory of electricity), and Hermann Lotze (1817–81; philosophy of religion). Many of

9936-527: The time for allowing students to duel and to have a passion for Freiheit , which were popularly regarded as the necessary characteristics of German student life. The University of Jena has preserved a historical detention room or Karzer with famous caricatures by Swiss painter Martin Disteli . In the latter 19th century, the department of zoology taught evolutionary theory , with Carl Gegenbaur , Ernst Haeckel and others publishing detailed theories at

10044-400: The time of Darwin 's " Origin of Species " (1858). The later fame of Ernst Haeckel eclipsed Darwin in some European countries, as the term "Haeckelism" was more common than Darwinism . In 1905, Jena had 1,100 students enrolled and its teaching staff (including Privatdozenten ) numbered 112. Amongst its numerous auxiliaries then were the library, with 200,000 volumes; the observatory;

10152-758: The transfer of power and won great support among the student body elections in January 1933, achieving 49.3% of the vote, which represents the second best result. Between the Jena connections and the NS students wide-ranging human and ideological connections were recorded. When the Allied air raids to Jena in February and March struck in 1945, the University Library, the university main building and several clinics in

10260-449: The truth-value it takes). By contrast, the sense (or "Sinn") associated with a complete sentence is the thought it expresses. The sense of an expression is said to be the "mode of presentation" of the item referred to, and there can be multiple modes of representation for the same referent. The distinction can be illustrated thus: In their ordinary uses, the name "Charles Philip Arthur George Mountbatten-Windsor", which for logical purposes

10368-404: The university has around 19,000 students enrolled and 375 professors. Its current president, Walter Rosenthal , has held the role since 2014. Elector John Frederick of Saxony first thought of a plan to establish a university at Jena upon Saale in 1547 while he was being held captive by emperor Charles V . The plan was put into motion by his three sons and, after having obtained a charter from

10476-500: The university was renamed again, receiving its present name of Friedrich Schiller University . During the 20th century, the cooperation between Zeiss corporation and the university brought new prosperity and attention to Jena, resulting in a dramatic increase in funding and enrollment. During the Third Reich, staunch Nazis moved into leading positions at the university. The racial researcher and SS-Hauptscharführer Karl Astel

10584-662: The university: Participations in DFG-Collaborative Research Centres: In 2006 the research center, Jena Center – History of the 20th century, was founded. In 2007 the graduate school "Jena School for Microbial Communication" (JSMC) was established within the German Universities Excellence Initiative . In 2008 the Center for Molecular Biomedicine (CMB) and the interdisciplinary research center Laboratory of

10692-405: The use of cylindrical or spherical coordinates. In cylindrical coordinates , every point of space is represented by its height z , its radius r from the z -axis and the angle θ its projection on the xy -plane makes with respect to the horizontal axis. In spherical coordinates, every point in space is represented by its distance ρ from the origin, the angle θ its projection on

10800-410: The view that expressions have two different aspects of significance: their sense and their reference. Reference (or "Bedeutung") applied to proper names , where a given expression (say the expression "Tom") simply refers to the entity bearing the name (the person named Tom). Frege also held that propositions had a referential relationship with their truth-value (in other words, a statement "refers" to

10908-751: Was appointed professor in 1933, bypassing traditional qualifications and process; he later became rector of the university in 1939. Also in 1933, many professors had to leave the university as a consequence of the Law for the Restoration of the Professional Civil Service . Student fraternities – in particular the Burschenschaften – were dissolved and incorporated into the Nazi student federation. The Nazi student federation enjoyed before

11016-463: Was born in 1848 in Wismar , Mecklenburg-Schwerin (today part of Mecklenburg-Vorpommern ). His father Carl (Karl) Alexander Frege (1809–1866) was the co-founder and headmaster of a girls' high school until his death. After Carl's death, the school was led by Frege's mother Auguste Wilhelmine Sophie Frege (née Bialloblotzky, 12 January 1815 – 14 October 1898); her mother was Auguste Amalia Maria Ballhorn,

11124-417: Was largely ignored during his lifetime, Giuseppe Peano (1858–1932), Bertrand Russell (1872–1970), and, to some extent, Ludwig Wittgenstein (1889–1951) introduced his work to later generations of philosophers. Frege is widely considered to be the greatest logician since Aristotle , and one of the most profound philosophers of mathematics ever. His contributions include the development of modern logic in

11232-427: Was moving in the right direction when he helped close the gap between numerical and geometric algebra with his geometric solution of the general cubic equations , but the decisive step came later with Descartes. Omar Khayyam is credited with identifying the foundations of algebraic geometry , and his book Treatise on Demonstrations of Problems of Algebra (1070), which laid down the principles of analytic geometry,

11340-467: Was published at his own expense), Frege attempted to derive, by use of his symbolism, all of the laws of arithmetic from axioms he asserted as logical. Most of these axioms were carried over from his Begriffsschrift , though not without some significant changes. The one truly new principle was one he called the Basic Law V : the "value-range" of the function f ( x ) is the same as the "value-range" of

11448-844: Was published posthumously in 1994. Frege was described by his students as a highly introverted person, seldom entering into dialogues with others and mostly facing the blackboard while lecturing. He was, however, known to occasionally show wit and even bitter sarcasm during his classes. Begriffsschrift: eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (1879), Halle an der Saale: Verlag von Louis Nebert ( online version ). Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung über den Begriff der Zahl (1884), Breslau: Verlag von Wilhelm Koebner ( online version ). Grundgesetze der Arithmetik , Band I (1893); Band II (1903), Jena: Verlag Hermann Pohle ( online version) . " Function and Concept " (1891) " On Sense and Reference " (1892) " Concept and Object " (1892) "What

11556-901: Was subsequently shown to imply that there is but one object in the universe of discourse , and hence is worthless (indeed, this would make for a contradiction in Frege's system if he had axiomatized the idea, fundamental to his discussion, that the True and the False are distinct objects; see, for example, Dummett 1973), but recent work has shown that much of the program of the Grundgesetze might be salvaged in other ways: Frege's work in logic had little international attention until 1903 when Russell wrote an appendix to The Principles of Mathematics stating his differences with Frege. The diagrammatic notation that Frege used had no antecedents (and has had no imitators since). Moreover, until Russell and Whitehead's Principia Mathematica (3 vols.) appeared in 1910–13,

11664-509: Was to be shown to have no basis in "intuition", and no need for non-logical axioms. Already in the 1879 Begriffsschrift important preliminary theorems, for example, a generalized form of law of trichotomy , were derived within what Frege understood to be pure logic. This idea was formulated in non-symbolic terms in his The Foundations of Arithmetic ( Die Grundlagen der Arithmetik , 1884). Later, in his Basic Laws of Arithmetic ( Grundgesetze der Arithmetik , vol. 1, 1893; vol. 2, 1903; vol. 2

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