Alessandro Padoa was an Italian mathematician and logician, a contributor to the school of Giuseppe Peano. He is remembered for a method for deciding whether, given some formal theory, a new primitive notion is independent of the other primitive notions. There is an analogous problem in axiomatic theories, namely deciding whether a given axiom is independent of the other axioms; the following description of Padoa's career is included in a biography of Peano: He attended secondary school in Venice, engineering school in Padua, the University of Turin, from which he received a degree in mathematics in 1895. Although he was never a student of Peano, he was an ardent disciple and, from 1896 on, a collaborator and friend, he taught in secondary schools in Pinerolo, Cagliari, at the Technical Institute in Genoa. He held positions at the Normal School in Aquila and the Naval School in Genoa, beginning in 1898, he gave a series of lectures at the Universities of Brussels, Berne, Padua and Geneva, he gave papers at congresses of philosophy and mathematics in Paris, Livorno, Parma and Bologna.
In 1934 he was awarded the ministerial prize in mathematics by the Accademia dei Lincei. The congresses in Paris in 1900 were notable. Padoa's addresses at these congresses have been well remembered for their clear and unconfused exposition of the modern axiomatic method in mathematics. In fact, he is said to be "the first … to get all the ideas concerning defined and undefined concepts straight". At the International Congress of Philosophy Padoa spoke on "Logical Introduction to Any Deductive Theory", he says during the period of elaboration of any deductive theory we choose the ideas to be represented by the undefined symbols and the facts to be stated by the unproved propositions. The system of ideas that we have chosen is one interpretation of the system of undefined symbols, and since the propositions, from the deductive point of view, do not state facts, but conditions, we cannot consider them genuine postulates. Padoa went on to say:...what is necessary to the logical development of a deductive theory is not the empirical knowledge of the properties of things, but the formal knowledge of relations between symbols.
Padoa spoke at the 1900 International Congress of Mathematicians with his title "A New System of Definitions for Euclidean Geometry". At the outset he discusses the various selections of primitive notions in geometry at the time: The meaning of any of the symbols that one encounters in geometry must be presupposed, just as one presupposes that of the symbols which appear in pure logic; as there is an arbitrariness in the choice of the undefined symbols, it is necessary to describe the chosen system. We cite only three geometers who are concerned with this question and who have successively reduced the number of undefined symbols, through them it is possible to define all the other symbols. First, Moritz Pasch was able to define all the other symbols through the following four: 1. Point 2. Segment 3. Plane 4. is superimposable upon Then, Giuseppe Peano was able in 1889 to define plane through point and segment. In 1894 he replaced is superimposable upon with motion in the system of undefined symbols, thus reducing the system to symbols: 1.
Point 2. Segment 3. Motion Finally, in 1899 Mario Pieri was able to define segment through motion. All the symbols that one encounters in Euclidean geometry can be defined in terms of only two of them, namely 1. Point 2. MotionPadoa completed his address by suggesting and demonstrating his own development of geometric concepts. In particular, he showed. A. Padoa "Logical introduction to any deductive theory" in Jean van Heijenoort, 1967. A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press: 118–23. A. Padoa "Un Nouveau Système de Définitions pour la Géométrie Euclidienne", Proceedings of the International Congress of Mathematicians, tome 2, pages 353–63. Secondary: Ivor Grattan-Guinness The Search for Mathematical Roots 1870–1940. Princeton Uni. Press. H. C. Kennedy Peano and Works of Giuseppe Peano, D. Reidel ISBN 90-277-1067-8. Suppes, Patrick Introduction to Logic, Dover. Discusses "Padoa's method." Smith, James T. Methods of Geometry, John Wiley & Sons, ISBN 0-471-25183-6 O'Connor, John J..
The Principles of Mathematics
The Principles of Mathematics is a 1903 book by Bertrand Russell, in which the author presented his famous paradox and argued his thesis that mathematics and logic are identical. The book has become a classic reference, it reported on developments by Giuseppe Peano, Mario Pieri, Richard Dedekind, Georg Cantor, others. In 1905 Louis Couturat published a partial French translation. In 1937 Russell prepared a new introduction saying, "Such interest as the book now possesses is historical, consists in the fact that it represents a certain stage in the development of its subject." Further editions were printed in 1938, 1951, 1996, 2009. The Principles of Mathematics consists of 59 chapters divided into seven parts: indefinables in mathematics, quantity, order and continuity, space and motion. In chapter one, "Definition of Pure Mathematics", Russell asserts that: There is an anticipation of relativity physics in the final part as the last three chapters consider Newton's laws of motion and relative motion, Hertz's dynamics.
However, Russell rejects what he calls "the relational theory", says on page 489: For us, since absolute space and time have been admitted, there is no need to avoid absolute motion, indeed no possibility of doing so. In his review, G. H. Hardy says "Mr. Russell is a firm believer in absolute position in space and time, a view as much out of fashion nowadays that Chapter will be read with peculiar interest." Reviews were prepared by G. E. Moore and Charles Sanders Peirce, but Moore's was never published and that of Peirce was brief and somewhat dismissive, he indicated that he thought it unoriginal, saying that the book "can hardly be called literature" and "Whoever wishes a convenient introduction to the remarkable researches into the logic of mathematics that have been made during the last sixty years will do well to take up this book."G. H. Hardy wrote a favorable review expecting the book to appeal more to philosophers than mathematicians, but he says: n spite of its five hundred pages.
Many chapters dealing with important questions are compressed into five or six pages, in some places in the most avowedly controversial parts, the argument is too condensed to follow. And the philosopher who attempts to read the book will be puzzled by the constant presupposition of a whole philosophical system utterly unlike any of those accepted. In 1904 another review appeared in Bulletin of the American Mathematical Society written by Edwin Bidwell Wilson, he says "The delicacy of the question is such that the greatest mathematicians and philosophers of to-day have made what seem to be substantial slips of judgement and have shown on occasions an astounding ignorance of the essence of the problem which they were discussing.... All too it has been the result of a wholly unpardonable disregard of the work accomplished by others." Wilson recounts the developments of Peano that Russell reports, takes the occasion to correct Henri Poincaré who had ascribed them to David Hilbert. In praise of Russell, Wilson says "Surely the present work is a monument to patience and thoroughness."
In 1938 the book was re-issued with a new preface by Russell. This preface was interpreted as a retreat from the realism of the first edition and a turn toward nominalist philosophy of symbolic logic. James Feibleman, an admirer of the book, thought Russell’s new preface went too far into nominalism so he wrote a rebuttal to this introduction. Feibleman says, "It is the first comprehensive treatise on symbolic logic to be written in English. In 1959 Russell wrote My Philosophical Development, in which he recalled the impetus to write the Principles: It was at the International Congress of Philosophy in Paris in the year 1900 that I became aware of the importance of logical reform for the philosophy of mathematics.... I was impressed by the fact that, in every discussion, showed more precision and more logical rigour than was shown by anybody else.... It was. Recalling the book after his work, he provides this evaluation: The Principles of Mathematics, which I finished on 23 May 1902, turned out to be a crude and rather immature draft of the subsequent work, from which, however, it differed in containing controversy with other philosophies of mathematics.
Such self-deprecation from the author after half a century of philosophical growth is understandable. On the other hand, Jules Vuillemin wrote in 1968: The Principles inaugurated contemporary philosophy. Other works have lost the title; such is not the case with this one. It is serious, its wealth perseveres. Furthermore, in relation to it, in a deliberate fashion or not, it locates itself again today in the eyes of all those that believe that contemporary science has modified our representation of the universe and through this representation, our relation to ourselves and to others; when W. V. O. Quine penned his autobiography, he wrote: Peano's symbolic notation took Russell by storm in 1900, but Russell’s Principles was still in unrelieved prose. I was baffled by its frequent opacity. In part it was rough going because of the cumbersomeness of ordinary language as compared with the suppleness of a notation devised for these intricate themes. Rereading it years I discovered that it had been rough going because matters were unclear in Russell's own mind in those pi
Foundations of geometry
Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries; these are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. The term axiomatic geometry can be applied to any geometry, developed from an axiom system, but is used to mean Euclidean geometry studied from this point of view; the completeness and independence of general axiomatic systems are important mathematical considerations, but there are issues to do with the teaching of geometry which come into play. Based on ancient Greek methods, an axiomatic system is a formal description of a way to establish the mathematical truth that flows from a fixed set of assumptions. Although applicable to any area of mathematics, geometry is the branch of elementary mathematics in which this method has most extensively been applied.
There are several components of an axiomatic system. Primitives are the most basic ideas, they include objects and relationships. In geometry, the objects are things like points and planes while a fundamental relationship is that of incidence – of one object meeting or joining with another; the terms themselves are undefined. Hilbert once remarked that instead of points and planes one might just as well talk of tables and beer mugs, his point being that the primitive terms are just empty shells, place holders if you will, have no intrinsic properties. Axioms are statements about these primitives. Axioms are assumed true, not proven, they are the building blocks of geometric concepts, since they specify the properties that the primitives have. The laws of logic; the theorems are the logical consequences of the axioms, that is, the statements that can be obtained from the axioms by using the laws of deductive logic. An interpretation of an axiomatic system is some particular way of giving concrete meaning to the primitives of that system.
If this association of meanings makes the axioms of the system true statements the interpretation is called a model of the system. In a model, all the theorems of the system are automatically true statements. In discussing axiomatic systems several properties are focused on: The axioms of an axiomatic system are said to be consistent if no logical contradiction can be derived from them. Except in the simplest systems, consistency is a difficult property to establish in an axiomatic system. On the other hand, if a model exists for the axiomatic system any contradiction derivable in the system is derivable in the model, the axiomatic system is as consistent as any system in which the model belongs; this property is referred to as relative model consistency. An axiom is called independent if it can not be proved or disproved from the other axioms of the axiomatic system. An axiomatic system is said to be independent. If a true statement is a logical consequence of an axiomatic system it will be a true statement in every model of that system.
To prove that an axiom is independent of the remaining axioms of the system, it is sufficient to find two models of the remaining axioms, for which the axiom is a true statement in one and a false statement in the other. Independence is not always a desirable property from a pedagogical viewpoint. An axiomatic system is called complete if every statement expressible in the terms of the system is either provable or has a provable negation. Another way to state this is that no independent statement can be added to a complete axiomatic system, consistent with axioms of that system. An axiomatic system is categorical. A categorical system is complete, but completeness does not imply categoricity. In some situations categoricity is not a desirable property, since categorical axiomatic systems can not be generalized. For instance, the value of the axiomatic system for group theory is that it is not categorical, so proving a result in group theory means that the result is valid in all the different models for group theory and one doesn't have to reprove the result in each of the non-isomorphic models.
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, deducing many other propositions from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system; the Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language. For over two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute metaphysical, sense.
Today, many other geometries which are not Euclidean are known, the first ones having
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Bertrand Arthur William Russell, 3rd Earl Russell, was a British philosopher, mathematician, writer, social critic, political activist, Nobel laureate. At various points in his life, Russell considered himself a liberal, a socialist and a pacifist, although he confessed that his skeptical nature had led him to feel that he had "never been any of these things, in any profound sense." Russell was born in Monmouthshire into one of the most prominent aristocratic families in the United Kingdom. In the early 20th century, Russell led the British "revolt against idealism", he is considered one of the founders of analytic philosophy along with his predecessor Gottlob Frege, colleague G. E. Moore and protégé Ludwig Wittgenstein, he is held to be one of the 20th century's premier logicians. With A. N. Whitehead he wrote Principia Mathematica, an attempt to create a logical basis for mathematics, the quintessential work of classical logic, his philosophical essay "On Denoting" has been considered a "paradigm of philosophy".
His work has had a considerable influence on mathematics, set theory, artificial intelligence, cognitive science, computer science and philosophy the philosophy of language and metaphysics. Russell was a prominent anti-war activist and he championed anti-imperialism, he advocated preventive nuclear war, before the opportunity provided by the atomic monopoly had passed and "welcomed with enthusiasm" world government. He went to prison for his pacifism during World War I. Russell concluded that war against Adolf Hitler's Nazi Germany was a necessary "lesser of two evils" and criticised Stalinist totalitarianism, attacked the involvement of the United States in the Vietnam War and was an outspoken proponent of nuclear disarmament. In 1950, Russell was awarded the Nobel Prize in Literature "in recognition of his varied and significant writings in which he champions humanitarian ideals and freedom of thought". Bertrand Russell was born on 18 May 1872 at Ravenscroft, Monmouthshire, into an influential and liberal family of the British aristocracy.
His parents and Viscountess Amberley, were radical for their times. Lord Amberley consented to his wife's affair with their children's tutor, the biologist Douglas Spalding. Both were early advocates of birth control at a time. Lord Amberley was an atheist and his atheism was evident when he asked the philosopher John Stuart Mill to act as Russell's secular godfather. Mill died the year after Russell's birth, his paternal grandfather, the Earl Russell, had been asked twice by Queen Victoria to form a government, serving her as Prime Minister in the 1840s and 1860s. The Russells had been prominent in England for several centuries before this, coming to power and the peerage with the rise of the Tudor dynasty, they established themselves as one of the leading British Whig families, participated in every great political event from the Dissolution of the Monasteries in 1536–1540 to the Glorious Revolution in 1688–1689 and the Great Reform Act in 1832. Lady Amberley was Lady Stanley of Alderley. Russell feared the ridicule of his maternal grandmother, one of the campaigners for education of women.
Russell had two siblings: brother Frank, sister Rachel. In June 1874 Russell's mother died followed shortly by Rachel's death. In January 1876, his father died of bronchitis following a long period of depression. Frank and Bertrand were placed in the care of their staunchly Victorian paternal grandparents, who lived at Pembroke Lodge in Richmond Park, his grandfather, former Prime Minister Earl Russell, died in 1878, was remembered by Russell as a kindly old man in a wheelchair. His grandmother, the Countess Russell, was the dominant family figure for the rest of Russell's childhood and youth; the countess was from a Scottish Presbyterian family, petitioned the Court of Chancery to set aside a provision in Amberley's will requiring the children to be raised as agnostics. Despite her religious conservatism, she held progressive views in other areas, her influence on Bertrand Russell's outlook on social justice and standing up for principle remained with him throughout his life, her favourite Bible verse, became his motto.
The atmosphere at Pembroke Lodge was one of frequent prayer, emotional repression, formality. Russell's adolescence was lonely, he contemplated suicide, he remarked in his autobiography that his keenest interests were in religion and mathematics, that only his wish to know more mathematics kept him from suicide. He was educated at home by a series of tutors; when Russell was eleven years old, his brother Frank introduced him to the work of Euclid, which he described in his autobiography as "one of the great events of my life, as dazzling as first love."During these formative years he discovered the works of Percy Bysshe Shelley. Russell wrote: "I spent all my spare time reading him, learning him by heart, knowing no one to whom I could speak of what I thought or felt, I used to reflect how wonderful it would have been to know Shelley, to wonder whether
Intuition is the ability to acquire knowledge without proof, evidence, or conscious reasoning, or without understanding how the knowledge was acquired. Different writers give the word "intuition" a great variety of different meanings, ranging from direct access to unconscious knowledge, unconscious cognition, inner sensing, inner insight to unconscious pattern-recognition and the ability to understand something instinctively, without the need for conscious reasoning; the word intuition comes from the Latin verb intueri translated as "consider" or from the late middle English word intuit, "to contemplate". Both Eastern and Western philosophers have studied the concept in great detail. Philosophy of mind deals with the concept of intuition. In the East intuition is intertwined with religion and spirituality, various meanings exist from different religious texts. In Hinduism various attempts have been made to interpret other esoteric texts. For Sri Aurobindo intuition comes under the realms of knowledge by identity.
The second nature being the action when it seeks to be aware of itself, resulting in humans being aware of their existence or aware of being angry & aware of other emotions. He terms this second nature as knowledge by identity, he finds that at present as the result of evolution the mind has accustomed itself to depend upon certain physiological functioning and their reactions as its normal means of entering into relations with the outer material world. As a result, when we seek to know about the external world the dominant habit is through arriving at truths about things via what our senses convey to us. However, knowledge by identity, which we only give the awareness of human beings' existence, can be extended further to outside of ourselves resulting in intuitive knowledge, he finds this intuitive knowledge was common to older humans and was taken over by reason which organises our perception and actions resulting from Vedic to metaphysical philosophy and to experimental science. He finds that this process, which seems to be decent, is a circle of progress, as a lower faculty is being pushed to take up as much from a higher way of working.
He finds when self-awareness in the mind is applied to one's self and the outer -self, results in luminous self-manifesting identity. Osho believed consciousness of human beings to be in increasing order from basic animal instincts to intelligence and intuition, humans being living in that conscious state moving between these states depending on their affinity, he suggests living in the state of intuition is one of the ultimate aims of humanity. Advaita vedanta takes intuition to be an experience through which one can come in contact with an experience Brahman. Buddhism finds intuition to be a faculty in the mind of immediate knowledge and puts the term intuition beyond the mental process of conscious thinking, as the conscious thought cannot access subconscious information, or render such information into a communicable form. In Zen Buddhism various techniques have been developed to help develop one's intuitive capability, such as koans – the resolving of which leads to states of minor enlightenment.
In parts of Zen Buddhism intuition is deemed a mental state between the Universal mind and one's individual, discriminating mind. In Islam there are various scholars with varied interpretations of intuition, sometimes relating the ability of having intuitive knowledge to prophethood. Siháb al Din-al Suhrawadi, in his book Philosophy Of Illumination, finds that intuition is a knowledge acquired through illumination and is mystical in nature and suggests mystical contemplation on this to bring about correct judgments. While Ibn Sīnā finds the ability of having intuition as a "prophetic capacity" and terms it as a knowledge obtained without intentionally acquiring it, he finds that regular knowledge is based on imitation while intuitive knowledge is based on intellectual certitude. In the West, intuition does not appear as a separate field of study, early mentions and definitions can be traced back to Plato. In his book Republic he tries to define intuition as a fundamental capacity of human reason to comprehend the true nature of reality.
In his works Meno and Phaedo, he describes intuition as a pre-existing knowledge residing in the "soul of eternity", a phenomenon by which one becomes conscious of pre-existing knowledge. He provides an example of mathematical truths, posits that they are not arrived at by reason, he argues that these truths are accessed using a knowledge present in a dormant form and accessible to our intuitive capacity. This concept by Plato is sometimes referred to as anamnesis; the study was continued by his followers. In his book Meditations on First Philosophy, Descartes refers to an intuition as a pre-existing knowledge gained through rational reasoning or discovering truth through contemplation; this definition is referred to as rational intuition. Philosophers, such as Hume, have more ambiguous interpretations of intuition. Hume claims intuition is a recognition of relationships while he states that "the resemblance" "will strike the eye" but goes on to stat
Felix Hausdorff was a German mathematician, considered to be one of the founders of modern topology and who contributed to set theory, descriptive set theory, measure theory, function theory, functional analysis. Life became difficult for Hausdorff and his family after Kristallnacht in 1938; the next year he initiated efforts to emigrate to the United States, but was unable to make arrangements to receive a research fellowship. On 26 January 1942, Felix Hausdorff, along with his wife and his sister-in-law, committed suicide by taking an overdose of veronal, rather than comply with German orders to move to the Endenich camp, there suffer the implications, about which he held no illusions. Hausdorff's father, the Jewish merchant Louis Hausdorff, moved in the autumn of 1870 with his young family to Leipzig and worked over time at various companies, including a linen-and cotton goods factory, he was an educated man and had become a Morenu at the age of 14. There are several treatises from his pen, including a long work on the Aramaic translations of the Bible from the perspective of Talmudic law.
Hausdorff's mother, referred to in various documents as Johanna, came from the Jewish Tietz family. From another branch of this family came Hermann Tietz, founder of the first department store, co-owner of the department store chain called "Hermann Tietz". During the period of Nazi dictatorship the name was "Aryanised" to Hertie. From 1878 to 1887 Felix Hausdorff attended the Nicolai School in Leipzig, a facility that had a reputation as a hotbed of humanistic education, he was an excellent student, class leader for many years and recited self-written Latin or German poems at school celebrations. In his graduation in 1887, he was the only one; the choice of subject was not easy for Hausdorff. Magda Dierkesmann, a guest in the home of Hausdorff as a student in Bonn in the years 1926–1932, reported in 1967 that: His versatile musical talent was so great that only the insistence of his father made him give up his plan to study music and become a composer; the decision was made to study the sciences in high school.
From summer term 1887 to summer semester 1891 Hausdorff studied mathematics and astronomy in his native city of Leipzig, interrupted by one semester in Freiburg and Berlin. The surviving testimony of other students show him as versatile interested young man, who, in addition to the mathematical and astronomical lectures, attended lectures in physics and geography, lectures on philosophy and history of philosophy as well as on issues of language and social sciences. In Leipzig he heard lectures on the history of music from musicologist Paul, his early love of music lasted a lifetime. As a student in Leipzig, he was an admirer and connoisseur of the music of Richard Wagner. In semesters of his studies, Hausdorff was close to Heinrich Bruns. Bruns was professor of director of the observatory at the University of Leipzig. Under him, Hausdorff graduated in 1891 with a work on the theory of astronomical refraction of light in the atmosphere. Two publications on the same subject followed, in 1895 his habilitation followed with a thesis on the absorbance of light in the atmosphere.
These early astronomical works of Hausdorff have—despite their excellent mathematical working through—not gained importance. Firstly, the underlying idea of Bruns has not proved viable. On the other hand, the progress in the direct measurement of atmospheric data has since made the painstaking accuracy of this data from refraction observations unnecessary. In the time between PhD and habilitation Hausdorff completed the yearlong-volunteer military requirement and worked for two years as a human computer at the observatory in Leipzig. With his habilitation, Hausdorff became a lecturer at the University of Leipzig and began an extensive teaching in a variety of mathematical areas. In addition to teaching and research in mathematics, he went with his literary and philosophical inclinations. A man of varied interests, educated sensitive and sophisticated in thinking and experiencing, he frequented in his Leipzig period with a number of famous writers and publishers such as Hermann Conradi, Richard Dehmel, Otto Erich Hartleben, Gustav Kirstein, Max Klinger, Max Reger and Frank Wedekind.
The years 1897 to about 1904 mark the high point of his literary and philosophical creativity, during which time 18 of his 22 pseudonymous works were published, including a book of poetry, a play, an epistemological book and a volume of aphorisms. Hausdorff married Charlotte Goldschmidt in daughter of Jewish doctor Siegismund Goldschmidt, her stepmother was preschool teacher Henriette Goldschmidt. Hausdorff's only child, daughter Lenore, was born in 1900. In December 1901 Hausdorff was appointed as adjunct associate professor at the University of Leipzig; the repeated assertion that Hausdorff got a call from Göttingen and rejected it cannot be verified and is wrong. When applying in Leipzig, Dean Kirchner had been led to positive vote of his colleagues, written by Hei