Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most to objects that are relevant to mathematics; the language of set theory can be used to define nearly all mathematical objects. The modern study of set theory was initiated by Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, such as Russell's paradox, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with or without the axiom of choice, are the best-known. Set theory is employed as a foundational system for mathematics in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
Mathematical topics emerge and evolve through interactions among many researchers. Set theory, was founded by a single paper in 1874 by Georg Cantor: "On a Property of the Collection of All Real Algebraic Numbers". Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity. Notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1870–1874 and was motivated by Cantor's work in real analysis. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking and culminated in Cantor's 1874 paper. Cantor's work polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. Cantorian set theory became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, the "infinity of infinities" resulting from the power set operation.
This utility of set theory led to the article "Mengenlehre" contributed in 1898 by Arthur Schoenflies to Klein's encyclopedia. The next wave of excitement in set theory came around 1900, when it was discovered that some interpretations of Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes. Bertrand Russell and Ernst Zermelo independently found the simplest and best known paradox, now called Russell's paradox: consider "the set of all sets that are not members of themselves", which leads to a contradiction since it must be a member of itself and not a member of itself. In 1899 Cantor had himself posed the question "What is the cardinal number of the set of all sets?", obtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics. In 1906 English readers gained the book Theory of Sets of Points by husband and wife William Henry Young and Grace Chisholm Young, published by Cambridge University Press.
The momentum of set theory was such. The work of Zermelo in 1908 and the work of Abraham Fraenkel and Thoralf Skolem in 1922 resulted in the set of axioms ZFC, which became the most used set of axioms for set theory; the work of analysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory, which has since become woven into the fabric of modern mathematics. Set theory is used as a foundational system, although in some areas—such as algebraic geometry and algebraic topology—category theory is thought to be a preferred foundation. Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member of A, the notation o. Since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the subset relation called set inclusion. If all the members of set A are members of set B A is a subset of B, denoted A ⊆ B. For example, is a subset of, so is but is not; as insinuated from this definition, a set is a subset of itself.
For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined. A is called a proper subset of B if and only if A is a subset of B, but A is not equal to B. Note that 1, 2, 3 are members of the set but are not subsets of it. Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. The: Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both; the union of and is the set. Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B; the intersection of and is the set. Set difference of U and A, denoted U \ A, is the set of all members of U that are not members of A; the set difference \ is, conversely, the set difference \ is. When A is a subset of U, the set difference U \ A is called the complement of A in U. In this case, if the choice of U is clear from the context, the notation Ac is sometimes used instead of U \ A if U is a universal set as in the study of Venn diagrams.
Symmetric difference of sets A and B, denoted A △ B or A ⊖ B, is
Rudolf von Bitter Rucker is an American mathematician, computer scientist, science fiction author, one of the founders of the cyberpunk literary movement. The author of both fiction and non-fiction, he is best known for the novels in the Ware Tetralogy, the first two of which both won Philip K. Dick Awards; until its closure in 2014 he edited. Rucker was raised in Louisville, Kentucky, he is the great-great-great-grandson of Georg Wilhelm Friedrich Hegel. Rucker attended St. Xavier High School before earning a B. A. in mathematics from Swarthmore College and M. S. and Ph. D. degrees in mathematics from Rutgers University. Rucker taught mathematics at the State University of New York at Geneseo from 1972 to 1978. Although he was liked by his students and "published a book and several papers," several colleagues took umbrage at his long hair and convivial relationships with English and philosophy professors amid looming budget shortfalls. Thanks to a grant from the Alexander von Humboldt Foundation, Rucker taught at the Ruprecht Karl University of Heidelberg from 1978 to 1980.
He taught at Randolph-Macon Women's College in Lynchburg, Virginia from 1980 to 1982, before trying his hand as a full-time author for four years. Inspired by an interview with Stephen Wolfram, Rucker became a computer science professor at San José State University in 1986, from which he retired as professor emeritus in 2004. A mathematician with philosophical interests, he has written The Fourth Dimension and Infinity and the Mind. Princeton University Press published new editions of Infinity and the Mind in 1995 and in 2005, both with new prefaces; as his "own alternative to cyberpunk," Rucker developed a writing style. Transrealism, as outlined in his 1983 essay "The Transrealist Manifesto", is science fiction based on the author's own life and immediate perceptions, mixed with fantastic elements that symbolize psychological change. Many of Rucker's novels and short stories apply these ideas. One example of Rucker's transreal works is Saucer Wisdom, a novel in which the main character is abducted by aliens.
Rucker and his publisher marketed tongue in cheek, as non-fiction. His earliest transreal novel, White Light, was written during his time at Heidelberg; this transreal novel is based on his experiences at SUNY Geneseo. Rucker uses his novels to explore scientific or mathematical ideas, his novels put forward a mystical philosophy that Rucker has summarized in an essay titled, with only a bit of irony, "The Central Teachings of Mysticism". His non-fiction book, The Lifebox, the Seashell, the Soul: What Gnarly Computation Taught Me About Ultimate Reality, the Meaning Of Life, How To Be Happy summarizes the various philosophies he's believed over the years and ends with the tentative conclusion that we might profitably view the world as made of computations, with the final remark, "perhaps this universe is perfect." Rucker was the roommate of Kenneth Turan during his freshman year at Swarthmore College. In 1967, Rucker married Sylvia Rucker. Together they have three children. On July 1, 2008, Rucker suffered a cerebral hemorrhage.
Thinking he may not be around much longer, this prompted him to write Nested Scrolls, his autobiography. Rucker resided in New Jersey during his graduate studies at Rutgers University; the Ware TetralogySoftware Wetware Freeware Realware Transreal novels White Light Spacetime Donuts The Sex Sphere The Secret of Life The Hacker and the Ants Saucer Wisdom novel marketed as non-fiction The Big Aha Other Novels Master of Space and Time The Hollow Earth Spaceland As Above, So Below: A Novel of Peter Bruegel Frek and the Elixir Mathematicians in Love Postsingular Hylozoic Jim and the Flims Turing and Burroughs Return to the Hollow Earth CollectionsThe Fifty-Seventh Franz Kafka Transreal! includes some non-fiction essays Gnarl!, complete short stories Mad Professor Complete Stories Transreal Cyberpunk, with Bruce Sterling Stories Geometry and the Fourth Dimension, Speculations on the Fourth Dimension: Selected Writings of Charles H. Hinton, Dover, ISBN 0-486-23916-0 Infinity and the Mind The Fourth Dimension: Toward a Geometry of Higher Reality Mind Tools All the Visions, memoir Seek!, collected essays Software Engineering and Computer Games, textbook The Lifebox, the Seashell, the Soul: What Gnarly Computation Taught Me about Ultimate Reality, the Meaning of Life, how to be Happy Nested Scrolls - autobiography Collected Essays Journals 1990-2014 Mathenauts: Tales of Mathematical Wonder, Arbor House Semiotext SF, Autonomedia Spinrad, Norman.
"Genre versus literature". On Books. Asimov's Science Fiction. 37: 182–191. CS1 maint: Date format Review of Turing & Burroughs; as actor-speaker in Manual of Evasion LX94, a 1994 film by Edgar Pêra The Rudy Rucker website
In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set. Writing A = means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A, for example, are subsets of A. Sets can themselves be elements. For example, consider the set B =; the elements of B are not 1, 2, 3, 4. Rather, there are only three elements of B, namely the numbers 1 and 2, the set; the elements of a set can be anything. For example, C =, is the set whose elements are the colors red and blue; the relation "is an element of" called set membership, is denoted by the symbol " ∈ ". Writing x ∈ A means that "x is an element of A". Equivalent expressions are "x is a member of A", "x belongs to A", "x is in A" and "x lies in A"; the expressions "A includes x" and "A contains x" are used to mean set membership, however some authors use them to mean instead "x is a subset of A". Logician George Boolos urged that "contains" be used for membership only and "includes" for the subset relation only.
For the relation ∈, the converse relation ∈T may be written A ∋ x, meaning "A contains x". The negation of set membership is denoted by the symbol "∉". Writing x ∉ A means that "x is not an element of A"; the symbol ∈ was first used by Giuseppe Peano 1889 in his work Arithmetices principia, nova methodo exposita. Here he wrote on page X: Signum ∈ significat est. Ita a ∈ b legitur a est quoddam b. So a ∈ b is read; every relation R: U → V is subject to two involutions: complementation R → R ¯ and conversion RT: V → U. The relation ∈ has for its domain a universal set U, has the power set P for its codomain or range; the complementary relation ∈ ¯ = ∉ expresses the opposite of ∈. An element x ∈ U may have x ∉ A, in which case x ∈ U \ A, the complement of A in U; the converse relation ∈ T = ∋ swaps the domain and range with ∈. For any A in P, A ∋ x is true when x ∈ A; the number of elements in a particular set is a property known as cardinality. In the above examples the cardinality of the set A is 4, while the cardinality of either of the sets B and C is 3.
An infinite set is a set with an infinite number of elements, while a finite set is a set with a finite number of elements. The above examples are examples of finite sets. An example of an infinite set is the set of positive integers. Using the sets defined above, namely A =, B = and C =: 2 ∈ A ∈ B 3,4 ∉ B is a member of B Yellow ∉ C The cardinality of D = is finite and equal to 5; the cardinality of P = is infinite. Halmos, Paul R. Naive Set Theory, Undergraduate Texts in Mathematics, NY: Springer-Verlag, ISBN 0-387-90092-6 - "Naive" means that it is not axiomatized, not that it is silly or easy. Jech, Thomas, "Set Theory", Stanford Encyclopedia of Philosophy Suppes, Axiomatic Set Theory, NY: Dover Publications, Inc. ISBN 0-486-61630-4 - Both the notion of set, membership or element-hood, the axiom of extension, the axiom of separation, the union axiom are needed for a more thorough understanding of "set element". Weisstein, Eric W. "Element". MathWorld
Julius Wilhelm Richard Dedekind was a German mathematician who made important contributions to abstract algebra, axiomatic foundation for the natural numbers, algebraic number theory and the definition of the real numbers. Dedekind's father was Julius Levin Ulrich Dedekind, an administrator of Collegium Carolinum in Braunschweig. Dedekind had three older siblings; as an adult, he never used the names Julius Wilhelm. He was born, lived most of his life, died in Braunschweig, he first attended the Collegium Carolinum in 1848 before transferring to the University of Göttingen in 1850. There, Dedekind was taught number theory by professor Moritz Stern. Gauss was still teaching, although at an elementary level, Dedekind became his last student. Dedekind received his doctorate in 1852, for a thesis titled Über die Theorie der Eulerschen Integrale; this thesis did not display the talent evident by Dedekind's subsequent publications. At that time, the University of Berlin, not Göttingen, was the main facility for mathematical research in Germany.
Thus Dedekind went to Berlin for two years of study, where he and Bernhard Riemann were contemporaries. Dedekind returned to Göttingen to teach as a Privatdozent, giving courses on probability and geometry, he studied for a while with Peter Gustav Lejeune Dirichlet, they became good friends. Because of lingering weaknesses in his mathematical knowledge, he studied elliptic and abelian functions, yet he was the first at Göttingen to lecture concerning Galois theory. About this time, he became one of the first people to understand the importance of the notion of groups for algebra and arithmetic. In 1858, he began teaching at the Polytechnic school in Zürich; when the Collegium Carolinum was upgraded to a Technische Hochschule in 1862, Dedekind returned to his native Braunschweig, where he spent the rest of his life, teaching at the Institute. He retired in 1894, but continued to publish, he never married. Dedekind was elected to the Academies of Berlin and Rome, to the French Academy of Sciences.
He received honorary doctorates from the universities of Oslo and Braunschweig. While teaching calculus for the first time at the Polytechnic school, Dedekind developed the notion now known as a Dedekind cut, now a standard definition of the real numbers; the idea of a cut is that an irrational number divides the rational numbers into two classes, with all the numbers of one class being greater than all the numbers of the other class. For example, the square root of 2 defines all the nonnegative numbers whose squares are less than 2 and the negative numbers into the lesser class, the positive numbers whose squares are greater than 2 into the greater class; every location on the number line continuum contains an irrational number. Thus there are gaps, or discontinuities. Dedekind published his thoughts on irrational numbers and Dedekind cuts in his pamphlet "Stetigkeit und irrationale Zahlen". Dedekind's theorem states that if there existed a one-to-one correspondence between two sets the two sets were "similar".
He invoked similarity to give the first precise definition of an infinite set: a set is infinite when it is "similar to a proper part of itself," in modern terminology, is equinumerous to one of its proper subsets. Thus the set N of natural numbers can be shown to be similar to the subset of N whose members are the squares of every member of N,: N 1 2 3 4 5 6 7 8 9 10... ↓ N2 1 4 9 16 25 36 49 64 81 100... Dedekind edited the collected works of Lejeune Dirichlet and Riemann. Dedekind's study of Lejeune Dirichlet's work led him to his study of algebraic number fields and ideals. In 1863, he published Lejeune Dirichlet's lectures on number theory as Vorlesungen über Zahlentheorie about which it has been written that: Although the book is assuredly based on Dirichlet's lectures, although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was written by Dedekind, for the most part after Dirichlet's death; the 1879 and 1894 editions of the Vorlesungen included supplements introducing the notion of an ideal, fundamental to ring theory.
Dedekind defined an ideal as a subset of a set of numbers, composed of algebraic integers that satisfy polynomial equations with integer coefficients. The concept underwent further development in the hands of Hilbert and of Emmy Noether. Ideals generalize Ernst Eduard Kummer's ideal numbers, devised as part of Kummer's 1843 attempt to prove Fermat's Last Theorem. In an 1882 article and Heinrich Martin Weber applied ideals to Riemann surfaces, giving an algebraic proof of the Riemann–Roch theorem. In 1888, he published a short monograph titled Was sind und was sollen die Zahlen?, which included his definition of an infinite set. He proposed an axiomatic foundation for the natural numbers, whose primitive notions were the number one and the successor function; the next year, Giuseppe Peano, citing Dedekind, formulated an equivalent but simpler set of axioms, now the standard ones. Dedekind made other
Abraham Robinson was a mathematician, most known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorporated into modern mathematics. Nearly half of Robinson's papers were in applied mathematics rather than in pure mathematics, he was born to a Jewish family with strong Zionist beliefs, in Waldenburg, now Wałbrzych, in Poland. In 1933, he emigrated to British Mandate of Palestine, where he earned a first degree from the Hebrew University. Robinson was in France when the Nazis invaded during World War II, escaped by train and on foot, being alternately questioned by French soldiers suspicious of his German passport and asked by them to share his map, more detailed than theirs. While in London, he joined the Free French Air Force and contributed to the war effort by teaching himself aerodynamics and becoming an expert on the airfoils used in the wings of fighter planes. After the war, Robinson worked in London and Jerusalem, but ended up at University of California, Los Angeles in 1962.
He became known for his approach of using the methods of mathematical logic to attack problems in analysis and abstract algebra. He "introduced many of the fundamental notions of model theory". Using these methods, he found a way of using formal logic to show that there are self-consistent nonstandard models of the real number system that include infinite and infinitesimal numbers. Others, such as Wilhelmus Luxemburg, showed that the same results could be achieved using ultrafilters, which made Robinson's work more accessible to mathematicians who lacked training in formal logic. Robinson's book Non-standard Analysis was published in 1966. Robinson was interested in the history and philosophy of mathematics, remarked that he wanted to get inside the head of Leibniz, the first mathematician to attempt to articulate the concept of infinitesimal numbers. While at UCLA his colleagues remember him as working hard to accommodate PhD students of all levels of ability by finding them projects of the appropriate difficulty.
He was courted by Yale, after some initial reluctance, he moved there in 1967. In the Spring of 1973 he was a member of the Institute for Advanced Study, he died of pancreatic cancer in 1974. Robinson, Introduction to model theory and to the metamathematics of algebra, Amsterdam: North-Holland, ISBN 978-0-7204-2222-1, MR 0153570 Robinson, Keisler, H. Jerome, ed. Complete theories, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland, ISBN 978-0-7204-0690-0, MR 0472504 Robinson, Keisler, H. Jerome, ed. Selected papers of Abraham Robinson. Vol. I Model theory and algebra, Yale University Press, ISBN 978-0-300-02071-7, MR 0533887 Robinson, Luxemburg, W. A. J.. Selected papers of Abraham Robinson. Vol. II Nonstandard analysis and philosophy, Yale University Press, ISBN 978-0-300-02072-4, MR 0533888 Robinson, Young, A. D. ed. Selected papers of Abraham Robinson. Vol. III Aeronautics, Yale University Press, ISBN 978-0-300-02073-1, MR 0533889 Robinson, Non-standard analysis, Princeton Landmarks in Mathematics, Princeton University Press, ISBN 978-0-691-04490-3, MR 0205854 Influence of non-standard analysis J. W. Dauben Abraham Robinson: The Creation of Nonstandard Analysis, A Personal and Mathematical Odyssey, Princeton University Press ISBN 0-691-03745-0 G. D. Mostow Abraham Robinson 1918 — 1974, Israel Journal of Mathematics 25: 5–14 doi:10.1007/BF02756558 A. D. Young, S. Cochen, Stephan Körner & Peter Roquette "Abraham Robinson", Bulletin of the London Mathematical Society 8: 307–23 MR0409084 Abraham Robinson at the Mathematics Genealogy Project Abraham Robinson — Biographical Memoirs of the National Academy of Sciences
In philosophy, the concept of The Absolute known as The Ultimate, The Wholly Other, The Supreme Being, The Absolute/Ultimate Reality, other names, is the thing, entity, force, presence, principle, etc. that possesses maximal ontological status, existential ranking, existential greatness, or existentiality. In layman's terms, this is the one that is, in one way or another, the greatest, truest, or most real being. There are many conceptions of The Absolute in various fields and subjects, such as philosophy, spiritual traditions and natural science; the nature of these conceptions can range from "merely" encompassing all physical existence, nature, or reality, to being unconditioned existentially, transcending all concepts and types, categories of being. The Absolute is thought of as causing to come into being manifestations that interact with lower or lesser forms of being; this is either done passively, through emanations, or through avatars and incarnations. These existential manifestations, which themselves can possess transcendent attributes, only contain minuscule or infinitesimal portions of the true essence of The Absolute.
The term itself was not in use in ancient or medieval philosophy, but related to the description of God as Actus purus in scholasticism. It was introduced in modern philosophy, notably by Hegel, for "the sum of all being and potential"; the term has since been adopted in perennial philosophy. There are three general ways of conceiving the Absolute; the Absolute might be the first and greatest being, not a being at all but the "ground" of being, or both the ground of being and a being. In conception one the Absolute is the most intelligible reality, it can be known. For example, Georg Wilhelm Friedrich Hegel's Absolute Spirit is the most true reality, it is thinkable and exists in the objective world by comprehending everything, including people and world history. In conception two the Absolute might be conceived of as utterly outside of all other reality and hence unintelligible, it can not be spoken about. Plato's Socrates says that "The Form of the Good" is "beyond being", implying that it is beyond thought and normal categories of existence.
St. John of the Cross says: In conception three the Absolute is seen as transcending duality and distinction; this concept of a fundamental reality that transcends or includes all other reality is associated with divinity. While this conception seems contradictory, it has been influential. One way to understand this third conception is to consider the Tao Te Ching; these opening lines distinguish between two Taos. One is the "eternal Tao" and the other "Tao" seems to exist in space and time; the eternal Tao is beyond existence and cannot be named or understood, while the other Tao exists and can be known. The eternal Tao is infinite; the eternal Tao is formless. The eternal Tao is transcendent; the other "Tao" is an attempt to describe the "eternal Tao" in human terms. He continues: In these lines, he further discusses the difference between the two Taos; the eternal Tao is the origin of Heaven and Earth. The "named" Tao, on the other hand, is able to describe specific phenomenons that exist in space and time, hence it is the mother of myriad of things.
He points out that both the "named" and the "nameless" emerge together from the same eternal Tao. This self-contradictory unity, of course, is said to be the mystery to be understood. One or more of these conceptions of the Absolute can be found in various other perspectives; the following is a list of conceptions of the Absolute. Note that the list is ordered alphabetically, but some of the sublists are ordered by historical precedence: General philosophy — God, Conceptions of God, Deity Abrahamic religions — God in Abrahamic religions Alawites — Allah Bahá'í Faith and Bábism — God in the Bahá'í Faith, Báb, He whom God shall make manifest Christianity — God in Christianity, Jehovah Christian theology — Apophatic theology and Cataphatic theology Catholic theology Scholasticism and Thomas Aquinas: Thomism and Thought of Thomas Aquinas — Actus purus, Actus primus Eastern Orthodox theology — Essence–energies distinction Oriental Orthodoxy — Miaphysitism Protestant theology — Five solae Paul Tillich — God Above God Christian philosophy — God in Christianity Nicolas Malebranche — God Christian mysticism — God in Christianity Kimbanguism — Simon Kimbangu Druze — God Islam — God in Islam, Allah Schools of Islamic theology — God in Islam Islamic philosophy — God in Islam Sufism — Haqiqa, Alam-i-HaHoot Judaism — God in Judaism, Tetragrammaton The Kingdoms of Israel and Judah — Yahweh Jewish philosophy — God in Judaism Jewish mysticism / Kabbalah — Ein Sof Mormonism — God in Mormonism Rastafari — Jah Samaritanism — Yahweh Shabakism — Divine Reality Yazdânism — Hâk / Haq Alevism — Haqq-Muhammad-Ali Ishikism — Haqq-Muhammad-Ali Yarsanism — The Divine Essence Yazidis — Melek Taus Acosmism — Unmanifest Adyghe Habze — Theshxwe Akan religion — Anansi Kokuroku Albanian mythology — Perendi Aldous Huxley's — Ground of Being, see The Perennial Philosophy Ancient Canaanite religion — El Ancient Egyptian religion and Egyptian mythology — Ra and assorted aspects, s
Gottfried Wilhelm Leibniz
Gottfried Wilhelm Leibniz was a prominent German polymath and philosopher in the history of mathematics and the history of philosophy. His most notable accomplishment was conceiving the ideas of differential and integral calculus, independently of Isaac Newton's contemporaneous developments. Mathematical works have always favored Leibniz's notation as the conventional expression of calculus, while Newton's notation became unused, it was only in the 20th century that Leibniz's law of continuity and transcendental law of homogeneity found mathematical implementation. He became one of the most prolific inventors in the field of mechanical calculators. While working on adding automatic multiplication and division to Pascal's calculator, he was the first to describe a pinwheel calculator in 1685 and invented the Leibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator, he refined the binary number system, the foundation of all digital computers. In philosophy, Leibniz is most noted for his optimism, i.e. his conclusion that our universe is, in a restricted sense, the best possible one that God could have created, an idea, lampooned by others such as Voltaire.
Leibniz, along with René Descartes and Baruch Spinoza, was one of the three great 17th-century advocates of rationalism. The work of Leibniz anticipated modern logic and analytic philosophy, but his philosophy looks back to the scholastic tradition, in which conclusions are produced by applying reason to first principles or prior definitions rather than to empirical evidence. Leibniz made major contributions to physics and technology, anticipated notions that surfaced much in philosophy, probability theory, medicine, psychology and computer science, he wrote works on philosophy, law, theology and philology. Leibniz contributed to the field of library science. While serving as overseer of the Wolfenbüttel library in Germany, he devised a cataloging system that would serve as a guide for many of Europe's largest libraries. Leibniz's contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters, in unpublished manuscripts, he wrote in several languages, but in Latin and German.
There is no complete gathering of the writings of Leibniz translated into English. Gottfried Leibniz was born on 1 July 1646, toward the end of the Thirty Years' War, in Leipzig, Saxony, to Friedrich Leibniz and Catharina Schmuck. Friedrich noted in his family journal: 21. Juny am Sontag 1646 Ist mein Sohn Gottfried Wilhelm, post sextam vespertinam 1/4 uff 7 uhr abents zur welt gebohren, im Wassermann. In English: On Sunday 21 June 1646, my son Gottfried Wilhelm is born into the world a quarter before seven in the evening, in Aquarius. Leibniz was baptized on 3 July of that year at Leipzig, his father died when he was six years old, from that point on he was raised by his mother. Leibniz's father had been a Professor of Moral Philosophy at the University of Leipzig, the boy inherited his father's personal library, he was given free access to it from the age of seven. While Leibniz's schoolwork was confined to the study of a small canon of authorities, his father's library enabled him to study a wide variety of advanced philosophical and theological works—ones that he would not have otherwise been able to read until his college years.
Access to his father's library written in Latin led to his proficiency in the Latin language, which he achieved by the age of 12. He composed 300 hexameters of Latin verse, in a single morning, for a special event at school at the age of 13. In April 1661 he enrolled in his father's former university at age 14, completed his bachelor's degree in Philosophy in December 1662, he defended his Disputatio Metaphysica de Principio Individui, which addressed the principle of individuation, on 9 June 1663. Leibniz earned his master's degree in Philosophy on 7 February 1664, he published and defended a dissertation Specimen Quaestionum Philosophicarum ex Jure collectarum, arguing for both a theoretical and a pedagogical relationship between philosophy and law, in December 1664. After one year of legal studies, he was awarded his bachelor's degree in Law on 28 September 1665, his dissertation was titled De conditionibus. In early 1666, at age 19, Leibniz wrote his first book, De Arte Combinatoria, the first part of, his habilitation thesis in Philosophy, which he defended in March 1666.
His next goal was to earn his license and Doctorate in Law, which required three years of study. In 1666, the University of Leipzig turned down Leibniz's doctoral application and refused to grant him a Doctorate in Law, most due to his relative youth. Leibniz subsequently left Leipzig. Leibniz enrolled in the University of Altdorf and submitted a thesis, which he had been working on earlier in Leipzig; the title of his thesis was Disputatio Inauguralis de Casibus Perplexis in Jure. Leibniz earned his license to practice law and his Doctorate in Law in November 1666, he next declined the offer of an academic appointment at Altdorf, saying that "my thoughts were turned in an different direction". As an adult, Leibniz often