Mathematical proof
In mathematics, a proof is an inferential argument for a mathematical statement. In the argument, other established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions. Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning. A proof must demonstrate that a statement is always true, rather than enumerate many confirmatory cases. An unproved proposition, believed to be true is known as a conjecture. Proofs employ logic but include some amount of natural language which admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory; the distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, so-called folk mathematics.
The philosophy of mathematics is concerned with the role of language and logic in proofs, mathematics as a language. The word "proof" comes from the Latin probare meaning "to test". Related modern words are the English "probe", "probation", "probability", the Spanish probar, Italian provare, the German probieren; the early use of "probity" was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, which outweighed empirical testimony. Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof, it is that the idea of demonstrating a conclusion first arose in connection with geometry, which meant the same as "land measurement". The development of mathematical proof is the product of ancient Greek mathematics, one of the greatest achievements thereof. Thales and Hippocrates of Chios proved some theorems in geometry. Eudoxus and Theaetetus formulated did not prove them.
Aristotle said definitions should describe the concept being defined in terms of other concepts known. Mathematical proofs were revolutionized by Euclid, who introduced the axiomatic method still in use today, starting with undefined terms and axioms, used these to prove theorems using deductive logic, his book, the Elements, was read by anyone, considered educated in the West until the middle of the 20th century. In addition to theorems of geometry, such as the Pythagorean theorem, the Elements covers number theory, including a proof that the square root of two is irrational and that there are infinitely many prime numbers. Further advances took place in medieval Islamic mathematics. While earlier Greek proofs were geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers as he considered multiplication, etc. for "lines."
He used this method to provide a proof of the existence of irrational numbers. An inductive proof for arithmetic sequences was introduced in the Al-Fakhri by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal's triangle. Alhazen developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate. Modern proof theory treats proofs as inductively defined data structures. There is no longer an assumption; as practiced, a proof is expressed in natural language and is a rigorous argument intended to convince the audience of the truth of a statement. The standard of rigor has varied throughout history. A proof can be presented differently depending on the intended audience. In order to gain acceptance, a proof has to meet communal statements of rigor; the concept of a proof is formalized in the field of mathematical logic. A formal proof is written in a formal language instead of a natural language. A formal proof is defined as sequence of formulas in a formal language, in which each formula is a logical consequence of preceding formulas.
Having a definition of formal proof makes the concept of proof amenable to study. Indeed, the field of proof theory studies formal proofs and their properties, for example, the property that a statement has a formal proof. An application of proof theory is to show; the definition of a formal proof is intended to capture the concept of proofs as written in the practice of mathematics. The soundness of this definition amounts to the belief that a published proof can, in principle, be converted into a formal proof. However, outside the field of automated proof assistants, this is done in practice. A classic question in philosophy a
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician. He created set theory. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, proved that the real numbers are more numerous than the natural numbers. In fact, Cantor's method of proof of this theorem implies the existence of an "infinity of infinities", he defined their arithmetic. Cantor's work is of a fact he was well aware of. Cantor's theory of transfinite numbers was regarded as so counter-intuitive – shocking – that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. Cantor, a devout Lutheran, believed; some Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God – on one occasion equating the theory of transfinite numbers with pantheism – a proposition that Cantor vigorously rejected.
The objections to Cantor's work were fierce: Leopold Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth". Kronecker objected to Cantor's proofs that the algebraic numbers are countable, that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory", which he dismissed as "utter nonsense", "laughable" and "wrong". Cantor's recurring bouts of depression from 1884 to the end of his life have been blamed on the hostile attitude of many of his contemporaries, though some have explained these episodes as probable manifestations of a bipolar disorder; the harsh criticism has been matched by accolades. In 1904, the Royal Society awarded Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics.
David Hilbert defended it from its critics by declaring, "No one shall expel us from the paradise that Cantor has created." Georg Cantor was born in 1845 in the western merchant colony of Saint Petersburg and brought up in the city until he was eleven. Georg, the oldest of six children, was regarded as an outstanding violinist, his grandfather Franz Böhm was a well-known soloist in a Russian imperial orchestra. Cantor's father had been a member of the Saint Petersburg stock exchange. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt. In 1862, Cantor entered the Swiss Federal Polytechnic. After receiving a substantial inheritance upon his father's death in June 1863, Cantor shifted his studies to the University of Berlin, attending lectures by Leopold Kronecker, Karl Weierstrass and Ernst Kummer, he spent the summer of 1866 at the University of Göttingen and a center for mathematical research. Cantor was a good student, he received his doctorate degree in 1867. Cantor submitted his dissertation on number theory at the University of Berlin in 1867.
After teaching in a Berlin girls' school, Cantor took up a position at the University of Halle, where he spent his entire career. He was awarded the requisite habilitation for his thesis on number theory, which he presented in 1869 upon his appointment at Halle University. In 1874, Cantor married Vally Guttmann, they had six children, the last born in 1886. Cantor was able to support a family despite modest academic pay, thanks to his inheritance from his father. During his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met two years earlier while on Swiss holiday. Cantor was promoted to extraordinary professor in 1872 and made full professor in 1879. To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor desired a chair at a more prestigious university, in particular at Berlin, at that time the leading German university. However, his work encountered too much opposition for. Kronecker, who headed mathematics at Berlin until his death in 1891, became uncomfortable with the prospect of having Cantor as a colleague, perceiving him as a "corrupter of youth" for teaching his ideas to a younger generation of mathematicians.
Worse yet, Kronecker, a well-established figure within the mathematical community and Cantor's former professor, disagreed fundamentally with the thrust of Cantor's work since he intentionally delayed the publication of Cantor's first major publication in 1874. Kronecker, now seen as one of the founders of the constructive viewpoint in mathematics, disliked much of Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties. Whenever Cantor applied for a post in Berlin, he was declined, it involved Kronecker, so Cantor came to believe that Kronecker's stance would make it impossible for him to leave Halle. In 1881, Cantor's Halle colleague Eduard Heine died. Halle acc
David Hilbert
David Hilbert was a German mathematician and one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, foundations of mathematics. Hilbert warmly defended Georg Cantor's set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems that set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic, as well as for being among the first to distinguish between mathematics and metamathematics.
Hilbert, the first of two children of Otto and Maria Therese Hilbert, was born in the Province of Prussia, Kingdom of Prussia, either in Königsberg or in Wehlau near Königsberg where his father worked at the time of his birth. In late 1872, Hilbert entered the Friedrichskolleg Gymnasium. Upon graduation, in autumn 1880, Hilbert enrolled at the University of Königsberg, the "Albertina". In early 1882, Hermann Minkowski, returned to Königsberg and entered the university. Hilbert developed a lifelong friendship with the gifted Minkowski. In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius. An intense and fruitful scientific exchange among the three began, Minkowski and Hilbert would exercise a reciprocal influence over each other at various times in their scientific careers. Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen. Hilbert remained at the University of Königsberg as a Privatdozent from 1886 to 1895.
In 1895, as a result of intervention on his behalf by Felix Klein, he obtained the position of Professor of Mathematics at the University of Göttingen. During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world, he remained there for the rest of his life. Among Hilbert's students were Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, Carl Gustav Hempel. John von Neumann was his assistant. At the University of Göttingen, Hilbert was surrounded by a social circle of some of the most important mathematicians of the 20th century, such as Emmy Noether and Alonzo Church. Among his 69 Ph. D. students in Göttingen were many who became famous mathematicians, including: Otto Blumenthal, Felix Bernstein, Hermann Weyl, Richard Courant, Erich Hecke, Hugo Steinhaus, Wilhelm Ackermann. Between 1902 and 1939 Hilbert was editor of the Mathematische Annalen, the leading mathematical journal of the time. "Good, he did not have enough imagination to become a mathematician".
Around 1925, Hilbert developed pernicious anemia, a then-untreatable vitamin deficiency whose primary symptom is exhaustion. Those forced out included Hermann Weyl, Emmy Noether and Edmund Landau. One who had to leave Germany, Paul Bernays, had collaborated with Hilbert in mathematical logic, co-authored with him the important book Grundlagen der Mathematik; this was a sequel to the Hilbert-Ackermann book Principles of Mathematical Logic from 1928. Hermann Weyl's successor was Helmut Hasse. About a year Hilbert attended a banquet and was seated next to the new Minister of Education, Bernhard Rust. Rust asked whether "the Mathematical Institute suffered so much because of the departure of the Jews". Hilbert replied, "Suffered? It doesn't exist any longer, does it!" By the time Hilbert died in 1943, the Nazis had nearly restaffed the university, as many of the former faculty had either been Jewish or married to Jews. Hilbert's funeral was attended by fewer than a dozen people, only two of whom were fellow academics, among them Arnold Sommerfeld, a theoretical physicist and a native of Königsberg.
News of his death only became known to the wider world six months. The epitaph on his tombstone in Göttingen consists of the famous lines he spoke at the conclusion of his retirement address to the Society of German Scientists and Physicians on 8 September 1930; the words were given in response to the Latin maxim: "Ignoramus et ignorabimus" or "We do not know, we shall not know": Wir müssen wissen. Wir werden wissen. In English: We mus
Emmy Noether
Amalie Emmy Noether was a German mathematician who made important contributions to abstract algebra and theoretical physics. She invariably used the name "Emmy Noether" in her life and publications, she was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed the theories of rings and algebras. In physics, Noether's theorem explains the connection between conservation laws. Noether was born to a Jewish family in the Franconian town of Erlangen, she planned to teach French and English after passing the required examinations, but instead studied mathematics at the University of Erlangen, where her father lectured. After completing her dissertation in 1907 under the supervision of Paul Gordan, she worked at the Mathematical Institute of Erlangen without pay for seven years. At the time, women were excluded from academic positions. In 1915, she was invited by David Hilbert and Felix Klein to join the mathematics department at the University of Göttingen, a world-renowned center of mathematical research.
The philosophical faculty objected and she spent four years lecturing under Hilbert's name. Her habilitation was approved in 1919. Noether remained a leading member of the Göttingen mathematics department until 1933. In 1924, Dutch mathematician B. L. van der Waerden joined her circle and soon became the leading expositor of Noether's ideas: Her work was the foundation for the second volume of his influential 1931 textbook, Moderne Algebra. By the time of her plenary address at the 1932 International Congress of Mathematicians in Zürich, her algebraic acumen was recognized around the world; the following year, Germany's Nazi government dismissed Jews from university positions, Noether moved to the United States to take up a position at Bryn Mawr College in Pennsylvania. In 1935 she underwent surgery for an ovarian cyst and, despite signs of a recovery, died four days at the age of 53. Noether's mathematical work has been divided into three "epochs". In the first, she made contributions to the theories of algebraic invariants and number fields.
Her work on differential invariants in the calculus of variations, Noether's theorem, has been called "one of the most important mathematical theorems proved in guiding the development of modern physics". In the second epoch, she began work that "changed the face of algebra". In her classic 1921 paper Idealtheorie in Ringbereichen Noether developed the theory of ideals in commutative rings into a tool with wide-ranging applications, she made elegant use of the ascending chain condition, objects satisfying it are named Noetherian in her honor. In the third epoch, she published works on noncommutative algebras and hypercomplex numbers and united the representation theory of groups with the theory of modules and ideals. In addition to her own publications, Noether was generous with her ideas and is credited with several lines of research published by other mathematicians in fields far removed from her main work, such as algebraic topology. Emmy's father, Max Noether, was descended from a family of wholesale traders in Germany.
At age 14, he had been paralyzed by polio. He regained mobility. Self-taught, he was awarded a doctorate from the University of Heidelberg in 1868. After teaching there for seven years, he took a position in the Bavarian city of Erlangen, where he met and married Ida Amalia Kaufmann, the daughter of a prosperous merchant. Max Noether's mathematical contributions were to algebraic geometry following in the footsteps of Alfred Clebsch, his best known results are the Brill -- AF+BG theorem. Emmy Noether was born on 23 March 1882, the first of four children, her first name was "Amalie", after her mother and paternal grandmother, but she began using her middle name at a young age. As a girl, Noether was well liked, she did not stand out academically although she was known for being friendly. She was talked with a minor lisp during childhood. A family friend recounted a story years about young Noether solving a brain teaser at a children's party, showing logical acumen at that early age, she was taught to cook and clean, as were most girls of the time, she took piano lessons.
She pursued none of these activities with passion. She had three younger brothers: The eldest, was born in 1883, was awarded a doctorate in chemistry from Erlangen in 1909, but died nine years later. Fritz Noether, born in 1884, is remembered for his academic accomplishments; the youngest, Gustav Robert, was born in 1889. Little is known about his life. Noether showed early proficiency in English. In the spring of 1900, she took the examination for teachers of these languages and received an overall score of sehr gut, her performance qualified her to teach languages at schools reserved for girls, but she chose instead to continue her studies at the University of Erlangen. This was an unconventional decision. One of only two wome
Giuseppe Peano
Giuseppe Peano was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation; the standard axiomatization of the natural numbers is named the Peano axioms in his honor. As part of this effort, he made key contributions to the modern rigorous and systematic treatment of the method of mathematical induction, he spent most of his career teaching mathematics at the University of Turin. He wrote an international auxiliary language, Latino sine flexione, a simplified version of Classical Latin. Most of his books and papers are in Latin sine flexione, others are in Italian. Peano was born and raised on a farm at Spinetta, a hamlet now belonging to Cuneo, Italy, he attended the Liceo classico Cavour in Turin, enrolled at the University of Turin in 1876, graduating in 1880 with high honors, after which the University employed him to assist first Enrico D'Ovidio, Angelo Genocchi, the Chair of calculus.
Due to Genocchi's poor health, Peano took over the teaching of calculus course within two years. His first major work, a textbook on calculus, was credited to Genocchi. A few years Peano published his first book dealing with mathematical logic. Here the modern symbols for the union and intersection of sets appeared for the first time. In 1887, Peano married Carola Crosio, the daughter of the Turin-based painter Luigi Crosio, known for painting the Refugium Peccatorum Madonna. In 1886, he began teaching concurrently at the Royal Military Academy, was promoted to Professor First Class in 1889. In that year he published the Peano axioms, a formal foundation for the collection of natural numbers; the next year, the University of Turin granted him his full professorship. Peano's famous space-filling curve appeared in 1890 as a counterexample, he used it to show. This was an early example of. In 1890 Peano founded the journal Rivista di Matematica, which published its first issue in January 1891. In 1891 Peano started the Formulario Project.
It was to be an "Encyclopedia of Mathematics", containing all known formulae and theorems of mathematical science using a standard notation invented by Peano. In 1897, the first International Congress of Mathematicians was held in Zürich. Peano was a key participant, he started to become occupied with Formulario to the detriment of his other work. In 1898 he presented a note to the Academy about binary numeration and its ability to be used to represent the sounds of languages, he became so frustrated with publishing delays that he purchased a printing press. Paris was the venue for the Second International Congress of Mathematicians in 1900; the conference was preceded by the First International Conference of Philosophy where Peano was a member of the patronage committee. He presented a paper which posed the question of formed definitions in mathematics, i.e. "how do you define a definition?". This became one of Peano's main philosophical interests for the rest of his life. At the conference Peano gave him a copy of Formulario.
Russell was so struck by Peano's innovative logical symbols that he left the conference and returned home to study Peano's text. Peano's students Mario Pieri and Alessandro Padoa had papers presented at the philosophy congress also. For the mathematical congress, Peano did not speak, but Padoa's memorable presentation has been recalled. A resolution calling for the formation of an "international auxiliary language" to facilitate the spread of mathematical ideas, was proposed. By 1901, Peano was at the peak of his mathematical career, he had made advances in the areas of analysis and logic, made many contributions to the teaching of calculus and contributed to the fields of differential equations and vector analysis. Peano played a key role in the axiomatization of mathematics and was a leading pioneer in the development of mathematical logic. Peano had by this stage become involved with the Formulario project and his teaching began to suffer. In fact, he became so determined to teach his new mathematical symbols that the calculus in his course was neglected.
As a result, he was dismissed from the Royal Military Academy but retained his post at Turin University. In 1903 Peano announced his work on an international auxiliary language called Latino sine flexione; this was an important project for him. The idea was to use Latin vocabulary, since this was known, but simplify the grammar as much as possible and remove all irregular and anomalous forms to make it easier to learn. On 3 January 1908, he read a paper to the Academia delle Scienze di Torino in which he started speaking in Latin and, as he described each simplification, introduced it into his speech so that by the end he was talking in his new language; the year 1908 was important for Peano. The fifth and final edition of the Formulario project, titled Formulario mathematico, was published, it contained 4200 formulae and theorems, all stated and most of them proved. The book received little attention. However, it remains a significant contribution to mathematical literature; the comments and examples were written in Latino sine flexione
Number theory
Number theory is a branch of pure mathematics devoted to the study of the integers. German mathematician Carl Friedrich Gauss said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of objects made out of integers or defined as generalizations of the integers. Integers can be considered either as solutions to equations. Questions in number theory are best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may study real numbers in relation to rational numbers, for example, as approximated by the latter; the older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory"; the use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is preferred as an adjective to number-theoretic.
The first historical find of an arithmetical nature is a fragment of a table: the broken clay tablet Plimpton 322 contains a list of "Pythagorean triples", that is, integers such that a 2 + b 2 = c 2. The triples are too large to have been obtained by brute force; the heading over the first column reads: "The takiltum of the diagonal, subtracted such that the width..." The table's layout suggests that it was constructed by means of what amounts, in modern language, to the identity 2 + 1 = 2, implicit in routine Old Babylonian exercises. If some other method was used, the triples were first constructed and reordered by c / a for actual use as a "table", for example, with a view to applications, it is not known whether there could have been any. It has been suggested instead. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, Babylonian algebra was exceptionally well developed. Late Neoplatonic sources state.
Much earlier sources state that Pythagoras traveled and studied in Egypt. Euclid IX 21–34 is probably Pythagorean. Pythagorean mystics gave great importance to the even; the discovery that 2 is irrational is credited to the early Pythagoreans. By revealing that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history; this forced a distinction between numbers, on the one hand, lengths and proportions, on the other hand. The Pythagorean tradition spoke of so-called polygonal or figurate numbers. While square numbers, cubic numbers, etc. are seen now as more natural than triangular numbers, pentagonal numbers, etc. the study of the sums of triangular and pentagonal numbers would prove fruitful in the early modern period. We know of no arithmetical material in ancient Egyptian or Vedic sources, though there is some algebra in both; the Chinese remainder theorem appears as an exercise in Sunzi Suanjing There is some numerical mysticism in Chinese mathematics, unlike that of the Pythagoreans, it seems to have led nowhere.
Like the Pythagoreans' perfect numbers, magic squares have passed from superstition into recreation. Aside from a few fragments, the mathematics of Classical Greece is known to us either through the reports of contemporary non-m
Homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert; the development of homological algebra was intertwined with the emergence of category theory. By and large, homological algebra is the study of homological functors and the intricate algebraic structures that they entail. One quite useful and ubiquitous concept in mathematics is that of chain complexes, which can be studied through both their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, topological spaces, other'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences. From its origins, homological algebra has played an enormous role in algebraic topology.
Its influence has expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, the theory of partial differential equations. K-theory is an independent discipline which draws upon methods of homological algebra, as does the noncommutative geometry of Alain Connes. Homological algebra began to be studied in its most basic form in the 1800s as a branch of topology, but it wasn't until the 1940s that it became an independent subject with the study of objects such as the ext functor and the tor functor, among others; the notion of chain complex is central in homological algebra. An abstract chain complex is a sequence of abelian groups and group homomorphisms, with the property that the composition of any two consecutive maps is zero: C ∙: ⋯ ⟶ C n + 1 ⟶ d n + 1 C n ⟶ d n C n − 1 ⟶ d n − 1 ⋯, d n ∘ d n + 1 = 0; the elements of Cn are called n-chains and the homomorphisms dn are called the boundary maps or differentials.
The chain groups Cn may be endowed with extra structure. The differentials must preserve the extra structure. For notational convenience, restrict attention to abelian groups; every chain complex defines two further sequences of abelian groups, the cycles Zn = Ker dn and the boundaries Bn = Im dn+1, where Ker d and Im d denote the kernel and the image of d. Since the composition of two consecutive boundary maps is zero, these groups are embedded into each other as B n ⊆ Z n ⊆ C n. Subgroups of abelian groups are automatically normal. A chain complex is called an exact sequence if all its homology groups are zero. Chain complexes arise in abundance in algebra and algebraic topology. For example, if X is a topological space the singular chains Cn are formal linear combinations of continuous maps from the standard n-simplex into X. In all these cases, there are natural differentials dn making Cn into a chain complex, whose homology reflects the structure of the topological space X, the simplicial complex K, or the abelian group A.
In the case of topological spaces, we arrive at the notion of singular homology, which plays a fundamental role in investigating the properties of such spaces, for example, manifolds. On a philosophical level, homological algebra teaches us that certain chain complexes associated with algebraic or geometric objects contain a lot of valuable algebraic information about them, with the homology being only the most available part. On a technical level, homological
