Abraham Robinson

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

Leonhard Euler

Leonhard Euler was a Swiss mathematician, astronomer and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while making pioneering contributions to several branches such as topology and analytic number theory. He introduced much of the modern mathematical terminology and notation for mathematical analysis, such as the notion of a mathematical function, he is known for his work in mechanics, fluid dynamics, optics and music theory. Euler was one of the most eminent mathematicians of the 18th century and is held to be one of the greatest in history, he is widely considered to be the most prolific mathematician of all time. His collected works fill more than anybody in the field, he spent most of his adult life in Saint Petersburg, in Berlin the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, Marguerite née Brucker, a pastor's daughter.

He had two younger sisters: Anna Maria and Maria Magdalena, a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, where Euler spent most of his childhood. Paul Euler was a friend of the Bernoulli family. Euler's formal education started in Basel. In 1720, aged thirteen, he enrolled at the University of Basel, in 1723, he received a Master of Philosophy with a dissertation that compared the philosophies of Descartes and Newton. During that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who discovered his new pupil's incredible talent for mathematics. At that time Euler's main studies included theology and Hebrew at his father's urging in order to become a pastor, but Bernoulli convinced his father that Leonhard was destined to become a great mathematician. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono. At that time, he was unsuccessfully attempting to obtain a position at the University of Basel.

In 1727, he first entered the Paris Academy Prize Problem competition. Pierre Bouguer, who became known as "the father of naval architecture", won and Euler took second place. Euler won this annual prize twelve times. Around this time Johann Bernoulli's two sons and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. On 31 July 1726, Nicolaus died of appendicitis after spending less than a year in Russia, when Daniel assumed his brother's position in the mathematics/physics division, he recommended that the post in physiology that he had vacated be filled by his friend Euler. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727, he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration.

Euler settled into life in Saint Petersburg. He took on an additional job as a medic in the Russian Navy; the Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia and to close the scientific gap with Western Europe. As a result, it was made attractive to foreign scholars like Euler; the academy possessed ample financial resources and a comprehensive library drawn from the private libraries of Peter himself and of the nobility. Few students were enrolled in the academy in order to lessen the faculty's teaching burden, the academy emphasized research and offered to its faculty both the time and the freedom to pursue scientific questions; the Academy's benefactress, Catherine I, who had continued the progressive policies of her late husband, died on the day of Euler's arrival. The Russian nobility gained power upon the ascension of the twelve-year-old Peter II; the nobility was suspicious of the academy's foreign scientists, thus cut funding and caused other difficulties for Euler and his colleagues.

Conditions improved after the death of Peter II, Euler swiftly rose through the ranks in the academy and was made a professor of physics in 1731. Two years Daniel Bernoulli, fed up with the censorship and hostility he faced at Saint Petersburg, left for Basel. Euler succeeded him as the head of the mathematics department. On 7 January 1734, he married Katharina Gsell, a daughter of Georg Gsell, a painter from the Academy Gymnasium; the young couple bought a house by the Neva River. Of their thirteen children, only five survived childhood. Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on 19 June 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia, he lived for 25 years in Berlin. In Berlin, he published the two works for which he would become most renowned: the Introductio in analysin infinitorum, a text on functions published in 1748, the Institutiones calculi differentialis, published in 1755 on differential calculus.

In 1755

Natural number

In mathematics, the natural numbers are those used for counting and ordering. In common mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers"; the natural numbers can, at times, appear as a convenient set of codes. Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers; the natural numbers are a basis from which many other number sets may be built by extension: the integers, by including the neutral element 0 and an additive inverse for each nonzero natural number n. These chains of extensions make the natural numbers canonically embedded in the other number systems.

Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, to contrast the discreteness of counting to the continuity of measurement, established by the real numbers; the most primitive method of representing a natural number is to put down a mark for each object. A set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set; the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers; the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, all the powers of 10 up to over 1 million.

A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, 6 ones. The Babylonians had a place-value system based on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context. A much advance was the development of the idea that 0 can be considered as a number, with its own numeral; the use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number. The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica; the use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus, beginning with Dionysius Exiguus in 525, without being denoted by a numeral; the first systematic study of numbers as abstractions is credited to the Greek philosophers Pythagoras and Archimedes.

Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes not as a number at all. Independent studies occurred at around the same time in India and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man". In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not natural but a consequence of definitions. Two classes of such formal definitions were constructed. Set-theoretical definitions of natural numbers were initiated by Frege and he defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox.

Therefore, this formalism was modified so that a natural number is defined as a particular set, any set that can be put into one-to-one correspondence with that set is said to have that number of elements. The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, further explored by Giuseppe Peano, it is based on an axiomatization of the properties of ordinal numbers: each natural number has a

Mathematics

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

Pierre de Fermat

Pierre de Fermat was a French lawyer at the Parlement of Toulouse, a mathematician, given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, analogous to that of differential calculus unknown, his research into number theory, he made notable contributions to analytic geometry and optics. He is best known for his Fermat's principle for light propagation and his Fermat's Last Theorem in number theory, which he described in a note at the margin of a copy of Diophantus' Arithmetica. Fermat was born in the first decade of the 17th century in Beaumont-de-Lomagne, France—the late 15th-century mansion where Fermat was born is now a museum, he was from Gascony, where his father, Dominique Fermat, was a wealthy leather merchant, served three one-year terms as one of the four consuls of Beaumont-de-Lomagne. His mother was Claire de Long.

Pierre had one brother and two sisters and was certainly brought up in the town of his birth. There is little evidence concerning his school education, but it was at the Collège de Navarre in Montauban, he attended the University of Orléans from 1623 and received a bachelor in civil law in 1626, before moving to Bordeaux. In Bordeaux he began his first serious mathematical researches, in 1629 he gave a copy of his restoration of Apollonius's De Locis Planis to one of the mathematicians there. In Bordeaux he was in contact with Beaugrand and during this time he produced important work on maxima and minima which he gave to Étienne d'Espagnet who shared mathematical interests with Fermat. There he became much influenced by the work of François Viète. In 1630, he bought the office of a councillor at the Parlement de Toulouse, one of the High Courts of Judicature in France, was sworn in by the Grand Chambre in May 1631, he held this office for the rest of his life. Fermat thereby became entitled to change his name from Pierre Fermat to Pierre de Fermat.

Fluent in six languages, Fermat was praised for his written verse in several languages and his advice was eagerly sought regarding the emendation of Greek texts. He communicated most of his work in letters to friends with little or no proof of his theorems. In some of these letters to his friends he explored many of the fundamental ideas of calculus before Newton or Leibniz. Fermat was a trained lawyer making mathematics more of a hobby than a profession, he made important contributions to analytical geometry, number theory and calculus. Secrecy was common in European mathematical circles at the time; this led to priority disputes with contemporaries such as Descartes and Wallis. Anders Hald writes that, "The basis of Fermat's mathematics was the classical Greek treatises combined with Vieta's new algebraic methods." Fermat's pioneering work in analytic geometry was circulated in manuscript form in 1636, predating the publication of Descartes' famous La géométrie, which exploited the work. This manuscript was published posthumously in 1679 in Varia opera mathematica, as Ad Locos Planos et Solidos Isagoge.

In Methodus ad disquirendam maximam et minimam and in De tangentibus linearum curvarum, Fermat developed a method for determining maxima and tangents to various curves, equivalent to differential calculus. In these works, Fermat obtained a technique for finding the centers of gravity of various plane and solid figures, which led to his further work in quadrature. Fermat was the first person known to have evaluated the integral of general power functions. With his method, he was able to reduce this evaluation to the sum of geometric series; the resulting formula was helpful to Newton, Leibniz, when they independently developed the fundamental theorem of calculus. In number theory, Fermat studied Pell's equation, perfect numbers, amicable numbers and what would become Fermat numbers, it was while researching perfect numbers. He invented a factorization method—Fermat's factorization method—as well as the proof technique of infinite descent, which he used to prove Fermat's right triangle theorem which includes as a corollary Fermat's Last Theorem for the case n = 4.

Fermat developed the two-square theorem, the polygonal number theorem, which states that each number is a sum of three triangular numbers, four square numbers, five pentagonal numbers, so on. Although Fermat claimed to have proven all his arithmetic theorems, few records of his proofs have survived. Many mathematicians, including Gauss, doubted several of his claims given the difficulty of some of the problems and the limited mathematical methods available to Fermat, his famous Last Theorem was first discovered by his son in the margin in his father's copy of an edition of Diophantus, included the statement that the margin was too small to include the proof. It seems, it was first proven by Sir Andrew Wiles, using techniques unavailable to Fermat. Although he studied and drew inspiration from Diophantus, Fermat began a different tradition. Diophantus was content to find a single solution to his equations if it were an undesired fractional one. Fermat was interested only in integer solutions to his Diophantine equations, he looked for all po

Overspill estate

An overspill estate is a housing estate planned and built for the housing of excess population in urban areas, both from the natural increase of population and in order to rehouse people from decaying inner city areas as part of the process of slum clearance. They were created on the outskirts of most large British towns and during most of the 20th century, with new towns being an alternative approach outside London after World War II; the Town Development Act, 1952 encouraged the expansion of neighbouring urban areas rather than the creation of satellite communities. Slum clearance tenants had problems with the move, since it separated them from extended family and friends, needed services were lacking, only the better off workers could afford the extra cost of commuting back to their jobs. Another criticism was. London overspill Urban sprawl Chelmsley Wood Darnhill Gamesley Hattersley St Helier, London Wythenshawe

Mathematical induction

Mathematical induction is a mathematical proof technique. It is used to prove that a property P holds for every natural number n, i.e. for n = 0, 1, 2, 3, so on. Metaphors can be informally used to understand the concept of mathematical induction, such as the metaphor of falling dominoes or climbing a ladder: Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung and that from each rung we can climb up to the next one; the method of induction requires two cases to be proved. The first case, called the base case, proves that the property holds for the number 0; the second case, called the induction step, proves that, if the property holds for one natural number n it holds for the next natural number n + 1. These two steps establish the property P for every natural number n = 0, 1, 2, 3... The base step need not begin with zero, it begins with the number one, it can begin with any natural number, establishing the truth of the property for all natural numbers greater than or equal to the starting number.

The method can be extended to prove statements about more general well-founded structures, such as trees. Mathematical induction in this extended sense is related to recursion. Mathematical induction, in some form, is the foundation of all correctness proofs for computer programs. Although its name may suggest otherwise, mathematical induction should not be misconstrued as a form of inductive reasoning as used in philosophy. Mathematical induction is an inference rule used in formal proofs. Proofs by mathematical induction are, in fact, examples of deductive reasoning. In 370 BC, Plato's Parmenides may have contained an early example of an implicit inductive proof; the earliest implicit traces of mathematical induction may be found in Euclid's proof that the number of primes is infinite and in Bhaskara's "cyclic method". An opposite iterated technique, counting down rather than up, is found in the Sorites paradox, where it was argued that if 1,000,000 grains of sand formed a heap, removing one grain from a heap left it a heap a single grain of sand forms a heap.

An implicit proof by mathematical induction for arithmetic sequences was introduced in the al-Fakhri written by al-Karaji around 1000 AD, who used it to prove the binomial theorem and properties of Pascal's triangle. None of these ancient mathematicians, explicitly stated the induction hypothesis. Another similar case was that of Francesco Maurolico in his Arithmeticorum libri duo, who used the technique to prove that the sum of the first n odd integers is n2; the first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique. Another Frenchman, made ample use of a related principle, indirect proof by infinite descent; the induction hypothesis was employed by the Swiss Jakob Bernoulli, from on it became more or less well known. The modern rigorous and systematic treatment of the principle came only in the 19th century, with George Boole, Augustus de Morgan, Charles Sanders Peirce, Giuseppe Peano, Richard Dedekind; the simplest and most common form of mathematical induction infers that a statement involving a natural number n holds for all values of n.

The proof consists of two steps: The base case: prove that the statement holds for the first natural number n0. N0 = 0 or n0 = 1; the step case or inductive step: prove that if the statement holds for any n ≥ n0, it holds for n+1. In other words, assume the statement holds for some arbitrary natural number n ≥ n0, prove that the statement holds for n + 1; the hypothesis in the inductive step, that the statement holds for some n, is called the induction hypothesis or inductive hypothesis. To prove the inductive step, one assumes the induction hypothesis and uses this assumption, involving n, to prove the statement for n + 1. Whether n = 0 or n = 1 is taken as the standard base case depends on the preferred definition of the natural numbers. In the fields of combinatorics and mathematical logic it is common to consider 0 as a natural number. Mathematical induction can be used to prove that the following statement, P, holds for all natural numbers n. 0 + 1 + 2 + ⋯ + n = n 2. P equal to number n; the proof that P is true for each natural number n proceeds as follows.

Base case: Show that the statement holds for n = 0. P is seen to be true: 0 = 0 ⋅ 2. Inductive step: Show that if P holds also P holds; this can be done. Assume P holds, it must be shown that P holds, that is