DOCSLIB.ORG

Development of Mathematical Theory of Probability: Historical Review

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Development of Mathematical Theory of Probability: Historical Review Development of Mathematical Theory of Probability: Historical Review In the history of probability theory, we can distinguish the following periods of its development (cf. [34, 46]∗): 1. Prehistory, 2. First period (the seventeenth century–early eighteenth century), 3. Second period (the eighteenth century–early nineteenth century), 4. Third period (second half of the nineteenth century), 5. Fourth period (the twentieth century). Prehistory. Intuitive notions of randomness and the beginning of reflection about chances (in ritual practice, deciding controversies, fortune telling, and so on) ap- peared far back in the past. In the prescientific ages, these phenomena were regarded as incomprehensible for human intelligence and rational investigation. It was only several centuries ago that their understanding and logically formalized study began. Archeological findings tell us about ancient “randomization instruments”, which were used for gambling games. They were made from the ankle bone (latin: astra- galus) of hoofed animals and had four faces on which they could fall. Such dice were definitely used for gambling during the First Dynasty in Egypt (around 3500 BC), and then in ancient Greece and ancient Rome. It is known ([14]) that the Roman Emperors August (63 BC–14 AC) and Claudius (10 BC–54 AC) were passionate dicers. In addition to gambling, which even at that time raised issues of favorable and unfavorable outcomes, similar questions appeared in insurance and commerce. The oldest forms of insurance were contracts for maritime transportation, which were found in Babylonian records of the 4th to 3rd Millennium BC. Afterwards the prac- tice of similar contracts was taken over by the Phoenicians and then came to Greece, Rome, and India. Its traces can be found in early Roman legal codes and in legis- lation of the Byzantine Empire. In connection with life insurance, the Roman jurist Ulpian compiled (220 BC) the first mortality tables. ∗ The citations here refer to the list of References following this section. © Springer Science+Business Media, LLC, part of Springer Nature 2019 313 A. N. Shiryaev, Probability-2, Graduate Texts in Mathematics 95, https://doi.org/10.1007/978-0-387-72208-5 314 Development of Mathematical Theory of Probability: Historical Review In the time of flourishing Italian city-states (Rome, Venice, Genoa, Pisa, Flo- rence), the practice of insurance caused the necessity of statistics and actuarial cal- culations. It is known that the first dated life insurance contract was concluded in Genoa in 1347. The city-states gave rise to the Renaissance (fourteenth to early seventeenth cen- turies), the period of social and cultural upheaval in Western Europe. In the Italian Renaissance, there appeared the first discussions, mostly of a philosophical nature, regarding the “probabilistic” arguments, attributed to Luca Pacioli (1447–1517), Celio Calcagnini (1479–1541), and Niccolo` Fontana Tartaglia (1500–1557) (see [46, 14]). Apparently, one of the first people to mathematically analyse gambling chances was Gerolamo Cardano (1501–1576), who was widely known for inventing the Car- dan gear and solving the cubic equation (although this was apparently solved by Tartaglia, whose solution Cardano published). His manuscript (written around 1525 but not published until 1663) “Liber de ludo aleae” (“Book on Games of Chance”) was more than a kind of practical manual for gamblers. Cardano was first to state the idea of combinations by which one could describe the set of all possible outcomes (in throwing dice of various kinds and numbers). He observed also that for true dice “the ratio of the number of favorable outcomes to the total number of possible outcomes is in good agreement with gambling practice” ([14]). 1. First period (the seventeenth century–early eighteenth century). Many math- ematicians and historians, such as Laplace [44](seealso[64]), related the beginning of the “calculus of probabilities” with correspondence between Blaise Pascal (1623– 1662) and Pierre de Fermat (1601–1665). This correspondence arose from certain questions that Antoine Gombaud (alias Chevalier de Mer´ e,´ a writer and moralist, 1607–1684) asked Fermat. One of the questions was how to divide the stake in an interrupted game. Namely, suppose two gamblers, A and B, agreed to play a certain number of games, say a best-of-five series, but were interrupted by an external cause when A has won 4 games and B has won 3 games. A seemingly natural answer is to divide it in the proportion 2 : 1. Indeed, the game certainly finishes in two steps, of which A may win 1 time, while B has to win both. This apparently implies the proportion 2 : 1. However, A has won 4 games against 3 won by B, so that the proportion 4 : 3 also looks natural. In fact, the correct answer found by Pascal and Fermat was 3 : 1. Another question was: what is more likely, to have at least one 6 in 4 throwings of a dice or to have at least one pair (6, 6) in 24 simultaneous throwings of two dice? In this problem, Pascal and Fermat also gave a correct answer: the former combination is slightly more probable than the latter (1 − (5/6)2 =0.516 against 1 − (35/36)2 =0.491). In solving these problems, Pascal and Fermat (as well as Cardano) applied com- binatorial arguments that became one of the basic tools in “calculus of probabil- ities” for the calculation of various chances. Among these tools, Pascal’s triangle also found its place (although it was known before). In 1657, the book by Christianus Huygens (1629–1695) “De Ratiociniis in Ludo Aleæ” (“On Reasoning in Games of Chance”) appeared, which is regarded as the Development of Mathematical Theory of Probability: Historical Review 315 first systematic presentation of the “calculus of probabilities”. In this book, Huy- gens formulates many basic notions and principles, states the rules of addition and multiplication of probabilities, and discusses the concept of expectation. This book became for long the main textbook in elementary probability theory. A prominent figure of this formative stage of probability theory was Jacob (James, Jacques) Bernoulli (1654–1705), who is credited with introducing the clas- sical concept of the “probability of an event” as the ratio of the number of outcomes favorable for this event to the total number of possible outcomes. The main result of J. Bernoulli, with which his name is associated, is, of course, the law of large numbers, which is fundamental for all applications of probability theory. This law, stated as a limit theorem, is dated from 1713 when Bernoulli’s trea- tise “Ars Conjectandi” (“The Art of Guessing”) was published (posthumously) with involvement of his nephew Nikolaus Bernoulli (see [3]). As was indicated by A. A. Markov in his speech on the occasion of the 200th anniversary of the law of large numbers (see [56]), J. Bernoulli wrote in his letters (of October 3, 1703, and April 20, 1704) that this theorem was known to him “already twelve years ago”. (The very term “law of large numbers” was proposed by Poisson in 1835.) Another member of the Bernoulli family, Daniel Bernoulli (1667–1748), is known for probability theory in connection with discussion regarding the so-called “St. Petersburg paradox”, where he proposed to use the notion of “moral expecta- tion”. The first period of development of probability theory coincided in time with the formation of mathematical natural science. This was the time when the concepts of continuity, infinity, and infinitesimally small quantities prevailed. This was when Is- saac Newton (1642–1727) and Gottfried Wilhelm Leibniz (1646–1716) developed differential and integral calculus. As A. N. Kolmogorov [34] wrote, the problem of that epoch was to “comprehend the extraordinary breadth and flexibility (and om- nipotence, as appeared then) of the mathematical method of study of causality. The idea of a differential equation as a law which determines uniquely the evolution of a system from its present state on took then even more prominent position in the mathematical natural science then now. The probability theory is needed in the mathematical natural science when this deterministic approach based on differen- tial equations fails. But at that time there was no concrete numerical material for application of probability theory.” Nevertheless, it became clear that the description of real data by deterministic models like differential equations was inevitably only a rough approximation. It was also understood that, in the chaos of large masses of unrelated events, there may appear in average certain regularities. This envisaged the fundamental natural- philosophic role of the probability theory, which was revealed by J. Bernoulli’s law of large numbers. It should be noted that J. Bernoulli realized the importance of dealing with in- finite sequences of repeated trials, which was a radically new idea in probabilistic considerations restricted at that time to elementary arithmetic and combinatorial tools. The statement of the question that led to the law of large numbers revealed 316 Development of Mathematical Theory of Probability: Historical Review the difference between the notions of the probability of an event and the frequency of its appearance in a finite number of trials, as well as the possibility of determina- tion of this probability (with certain accuracy) from its frequency in large numbers of trials. 2. Second period (the eighteenth century–early nineteenth century). This period is associated, essentially, with the names of Pierre-Remond´ de Montmort (1678– 1719), Abraham de Moivre (1667–1754), Thomas Bayes (1702–1761), Pierre Si- mon de Laplace (1749–1827), Carl Friedrich Gauss (1777–1855), and Simeon´ De- nis Poisson (1781–1840). While the first period was largely of a philosophical nature, in the second one the analytic methods were developed and perfected, computations became necessary in various applications, and probabilistic and statistical approaches were introduced in the theory of observation errors and shooting theory.
Load more

Recommended publications
  • On Popularization of Scientific Education in Italy Between 12Th and 16Th Century
    PROBLEMS OF EDUCATION IN THE 21st CENTURY Volume 57, 2013 90 ON POPULARIZATION OF SCIENTIFIC EDUCATION IN ITALY BETWEEN 12TH AND 16TH CENTURY Raffaele Pisano University of Lille1, France E–mail: [email protected] Paolo Bussotti University of West Bohemia, Czech Republic E–mail: [email protected] Abstract Mathematics education is also a social phenomenon because it is influenced both by the needs of the labour market and by the basic knowledge of mathematics necessary for every person to be able to face some operations indispensable in the social and economic daily life. Therefore the way in which mathe- matics education is framed changes according to modifications of the social environment and know–how. For example, until the end of the 20th century, in the Italian faculties of engineering the teaching of math- ematical analysis was profound: there were two complex examinations in which the theory was as impor- tant as the ability in solving exercises. Now the situation is different. In some universities there is only a proof of mathematical analysis; in others there are two proves, but they are sixth–month and not annual proves. The theoretical requirements have been drastically reduced and the exercises themselves are often far easier than those proposed in the recent past. With some modifications, the situation is similar for the teaching of other modern mathematical disciplines: many operations needing of calculations and math- ematical reasoning are developed by the computers or other intelligent machines and hence an engineer needs less theoretical mathematics than in the past. The problem has historical roots. In this research an analysis of the phenomenon of “scientific education” (teaching geometry, arithmetic, mathematics only) with respect the methods used from the late Middle Ages by “maestri d’abaco” to the Renaissance hu- manists, and with respect to mathematics education nowadays is discussed.
    [Show full text]
  • History of Algebra and Its Implications for Teaching
    Maggio: History of Algebra and its Implications for Teaching History of Algebra and its Implications for Teaching Jaime Maggio Fall 2020 MA 398 Senior Seminar Mentor: Dr.Loth Published by DigitalCommons@SHU, 2021 1 Academic Festival, Event 31 [2021] Abstract Algebra can be described as a branch of mathematics concerned with finding the values of unknown quantities (letters and other general sym- bols) defined by the equations that they satisfy. Algebraic problems have survived in mathematical writings of the Egyptians and Babylonians. The ancient Greeks also contributed to the development of algebraic concepts. In this paper, we will discuss historically famous mathematicians from all over the world along with their key mathematical contributions. Mathe- matical proofs of ancient and modern discoveries will be presented. We will then consider the impacts of incorporating history into the teaching of mathematics courses as an educational technique. 1 https://digitalcommons.sacredheart.edu/acadfest/2021/all/31 2 Maggio: History of Algebra and its Implications for Teaching 1 Introduction In order to understand the way algebra is the way it is today, it is important to understand how it came about starting with its ancient origins. In a mod- ern sense, algebra can be described as a branch of mathematics concerned with finding the values of unknown quantities defined by the equations that they sat- isfy. Algebraic problems have survived in mathematical writings of the Egyp- tians and Babylonians. The ancient Greeks also contributed to the development of algebraic concepts, but these concepts had a heavier focus on geometry [1]. The combination of all of the discoveries of these great mathematicians shaped the way algebra is taught today.
    [Show full text]
  • The Reception of Gerolamo Cardano's Liber De Ludo Aleae
    READING IN CONTEXT: THE RECEPTION OF GEROLAMO CARDANO’S LIBER DE LUDO ALEAE A RESEARCH PAPER SUBMITTED TO DR. SLOAN DESPEAUX DEPARTMENT OF MATHEMATICS WESTERN CAROLINA UNIVERSITY BY NATHAN BOWMAN 1 Any person who enjoys gambling games would immensely desire to master the theory of probability and find a fool-proof way of winning at these games. Many have made valiant attempts at creating a successful system. One man, in the 16th century, genuinely believed that he had cracked the code. This man was Gerolamo Cardano. He detailed his discovery in his book Liber de Ludo Aleae, also known as The Book on Games of Chance. This book, a discussion of probability and gambling, was frowned upon due to numerous reasons, both moral and practical. Historians of mathematics have often discussed the work by Pascal, Fermat and Bernoulli; however, it seems that Cardano’s Liber de Ludo Aleae has often been left on the back of the shelf to collect dust. This poses the question of why Cardano’s book has been underappreciated. In this paper, we will seek to understand the reception of Liber de Ludo Aleae by looking deeper into Cardano as a gambler and mathematician, considering how he perceived his own book, and finally, by evaluating contemporary opinions and scholarly critiques. Much of what we know about Cardano comes from his own personal writings.1 He wrote many works including his autobiography entitled Da Propia Vita. In it, Cardano wrote that he was born in Pavia on September 24, 1501. He was born outside of wedlock, a situation usually disdained by the general public of the time.
    [Show full text]
  • Some Laws and Problems of Classical Probability and How Cardano Anticipated Them Prakash Gorroochurn
    Some Laws and Problems of Classical Probability and How Cardano Anticipated Them Prakash Gorroochurn n the history of probability, the sixteen–century became the real father of modern probability physician and mathematician Gerolamo Cardano theory. (1501–1575) was among the first to attempt a However, in spite of Cardano’s several contributions systematic study of the calculus of probabilities. Like I to the field, in none of the problems did Cardano reach those of his contemporaries, Cardano’s studies were the level of mathematical sophistication and maturity primarily driven by games of chance. Concerning his that was later to be evidenced in the hands of his suc- gambling for twenty–five years, he famously said in his cessors. Many of his investigations were either too autobiography entitled The Book of My Life: rudimentary or just erroneous. On the other hand, Pas- … and I do not mean to say only from time to cal’s and Fermat’s work were rigorous and provided the time during those years, but I am ashamed to first impetus for a systematic study of the mathematical say it, everyday. theory of probability. In the words of A.W.F. Edwards: Cardano’s works on probability were published post- … in spite of our increased awareness of the humously in the famous 15–page Liber de Ludo Aleae (The earlier work of Cardano (Ore, 1953) and Galileo Book on Games of Chance) consisting of 32 small chapters. (David, 1962) it is clear that before Pascal and Today, the field of probability of widely believed to have Fermat no more had been achieved than the been fathered by Blaise Pascal (1623–1662) and Pierre enumeration of the fundamental probability set de Fermat (1601–1665), whose famous correspondence in various games with dice or cards.
    [Show full text]
  • A Short History of Complex Numbers
    A Short History of Complex Numbers Orlando Merino University of Rhode Island January, 2006 Abstract This is a compilation of historical information from various sources, about the number i = √ 1. The information has been put together for students of Complex Analysis who − are curious about the origins of the subject, since most books on Complex Variables have no historical information (one exception is Visual Complex Analysis, by T. Needham). A fact that is surprising to many (at least to me!) is that complex numbers arose from the need to solve cubic equations, and not (as it is commonly believed) quadratic equations. These notes track the development of complex numbers in history, and give evidence that supports the above statement. 1. Al-Khwarizmi (780-850) in his Algebra has solution to quadratic equations of various types. Solutions agree with is learned today at school, restricted to positive solutions [9] Proofs are geometric based. Sources seem to be greek and hindu mathematics. According to G. J. Toomer, quoted by Van der Waerden, Under the caliph al-Ma’mun (reigned 813-833) al-Khwarizmi became a member of the “House of Wisdom” (Dar al-Hikma), a kind of academy of scientists set up at Baghdad, probably by Caliph Harun al-Rashid, but owing its preeminence to the interest of al-Ma’mun, a great patron of learning and scientific investigation. It was for al-Ma’mun that Al-Khwarizmi composed his astronomical treatise, and his Algebra also is dedicated to that ruler 2. The methods of algebra known to the arabs were introduced in Italy by the Latin transla- tion of the algebra of al-Khwarizmi by Gerard of Cremona (1114-1187), and by the work of Leonardo da Pisa (Fibonacci)(1170-1250).
    [Show full text]
  • The Cardano-Tartaglia Dispute
    The Cardano-Tartaglia Dispute R I C H A R D W. F E L D M A N N Introduction fered severe saber cuts about the face and mouth which caused a speech impediment. Because of As the Middle Ages came to a close, a rebirth of this he was nicknamed Tartaglia, "the stutterer." scientific inquiry occurred. Most large cities had He later wore a long beard to hide the scars, but he universities, but these consisted primarily of lec­ could never overcome the stuttering. turers, as there were few books beyond those of the His education was very meager. In fact he tells, classical authors. Hence the reputation of a uni­ in one of his books, that his mother had accumu­ versity depended upon its ability to provide the lated a small amount of money so that he might be best lecturers. But who was the best lecturer? One tutored by a writing master. The money ran out, so way to settle this was in a public debate with the he says, when the instruction reached the letter K winner gaining prestige and academic positions and his education stopped before he could write while the loser was ignored. his own initials. But being an enterprising youth, Since a scholar had to outperform all challeng­ he stole the master's lecture notes and completed ers to maintain his position, he needed every trick the course on his own. In times of extreme poverty he knew. If someone knew something that no one he even used tombstones in place of writing slates.
    [Show full text]
  • Table of Contents
    Dora Bobory BEING A CHOSEN ONE: SELF-CONSCIOUSNESS AND SELF-FASHIONING IN THE WORKS OF GEROLAMO CARDANO M.A. Thesis in Medieval Studies CEU eTD Collection Central European University Budapest June 2002 Being a Chosen One: Self-Consciousness and Self-Fashioning in the Works of Gerolamo Cardano by Dora Bobory (Hungary) Thesis submitted to the Department of Medieval Studies, Central European University, Budapest, in partial fulfillment of the requirements of the Master of Arts degree in Medieval Studies Accepted in conformance with the standards of the CEU ____________________________________________ Chair, Examination Committee ____________________________________________ External Examiner ____________________________________________ Thesis Supervisor CEU eTD Collection ____________________________________________ Examiner Budapest June 2002 I, the undersigned, Dora Bobory, candidate for the M.A. degree in Medieval Studies declare herewith that the present thesis is exclusively my own work, based on my research and only such external information as properly credited in notes and bibliography. I declare that no unidentified and illegitimate use was made of the work of others, and no part of the thesis infringes on any person’s or institution’s copyright. I also declare that no part of the thesis has been submitted in this form to any other institution of higher education for an academic degree. Budapest, 1 June 2002 __________________________ Signature CEU eTD Collection Table of Contents List of Illustrations i Introduction 1 I. The Sources of Renaissance Self-Consciousness 8 I. 1. Centrality of Man in the Hermetic Texts 8 I. 2. Melancholy, Genius, Divination 10 I. 2. 1. The Ancient Greek Tradition of Melancholy 11 I. 2. 2. The Middle Ages and a Different Attitude to the Same Question 18 I.
    [Show full text]
  • History of Algebra
    History of Algebra The term algebra usually denotes various kinds of mathematical ideas and techniques, more or less directly associated with formal manipulation of abstract symbols and/or with finding the solutions of an equation. The notion that in mathematics there is such a sepa- rate sub-discipline, as well as the very use of the term “algebra” to denote it, are them- selves the outcome of historical evolution of ideas. The ideas to be discussed in this article are sometimes put under the same heading due to historical circumstances no less than to any “essential” mathematical reason. Part I. The long way towards the idea of “equation” Simple and natural as the notion of “equation” may appear now, it involves a great amount of mutually interacting, individual mathematical notions, each of which was the outcome of a long and intricate historical process. Not before the work of Viète, in the late sixteenth century, do we actually find a fully consolidated idea of an equation in the sense of a sin- gle mathematical entity comprising two sides on which operations can be simultaneously performed. By performing such operations the equation itself remains unchanged, but we are led to discovering the value of the unknown quantities appearing in it. Three main threads in the process leading to this consolidation deserve special attention here: (1) attempts to deal with problems devoted to finding the values of one or more unknown quantities. In Part I, the word “equation” is used in this context as a short-hand to denote all such problems,
    [Show full text]
  • Digitalcommons@Cedarville
    Cedarville University DigitalCommons@Cedarville Science and Mathematics Faculty Presentations Department of Science and Mathematics Fall 2011 On the utility of i = √-1 Adam J. Hammett Cedarville University, [email protected] Follow this and additional works at: http://digitalcommons.cedarville.edu/ science_and_mathematics_presentations Part of the Mathematics Commons Recommended Citation Hammett, Adam J., "On the utility of i = √-1" (2011). Science and Mathematics Faculty Presentations. 228. http://digitalcommons.cedarville.edu/science_and_mathematics_presentations/228 This Local Presentation is brought to you for free and open access by DigitalCommons@Cedarville, a service of the Centennial Library. It has been accepted for inclusion in Science and Mathematics Faculty Presentations by an authorized administrator of DigitalCommons@Cedarville. For more information, please contact [email protected]. Dr. Adam Hammett Math Department February 22, 2012 What even consider = 1 as a legitimate thing? 1. Solving polynomial equations (like − + 3 = 0 for ) is at the core of issues such as taxation,2 maximizing profit, minimizing cost and − a 4host of other economic factors critical to the advancement of society. shows up quickly when one investigates solutions of polynomial equations rigorously. 2. Incorporation of enables us to fully understand models for physical phenomena such as stress on beams, resonance, fluid flow, electrical currents, transmittance of radio waves and population dynamics among other things. Clearly, each
    [Show full text]
  • Some Probability Theory the Math Department's Undergraduate Course
    Some Probability Theory The math department’s undergraduate course in probability theory is MATH 411, Mathematical Probability. Thefirst person to attempt a systematic theory of probability was the Italian mathematician, physician, astrologer, and gambler Gerolamo Cardano (1501–1576). Cardano is perhaps best known for his study of cubic and quartic algebraic equations, which he solved in his 1545 text Ars Magna (The Great Art)—solutions which required the controversial notion of keeping track of the unlikely expression √ 1. − 1 In 1654, Antoine Gombaud Chevalier de Mere (1607–1684), a French nobleman, writer and professional gambler, brought Blaise Pascal’s attention to the following game of chance: is it worthwhile betting even money that double sixes will turn up at least once in 24 throws of a fair pair of dice? This led to a long correspondence between the French mathemati- cians Blaise Pascal (1623–1662) and Pierre de Fermat (1601–1665), during which the classical theory of probability wasfirst formu- lated. Pascal is on the left. 2 The single most influential work on probability theory was writ- ten by another French mathematician Pierre Simon Laplace (1749– 1827), “Thèorie Analytiques des Probabilitès,” published in 1812. As a starting point for our discussion of probability theory, let’s see if we can answer de Mere’s question: is it worthwhile betting even money that double sixes will turn up at least once in 24 throws of a fair pair of dice? 3 4 5 Roulette The game of American roulette employees a large wheel with 38 slots, 36 labeled non-sequentially with the numbers 1–36, and the two house slots labeled 0 and 00.
    [Show full text]
  • Mathematicians Timeline
    Rikitar¯oFujisawa Otto Hesse Kunihiko Kodaira Friedrich Shottky Viktor Bunyakovsky Pavel Aleksandrov Hermann Schwarz Mikhail Ostrogradsky Alexey Krylov Heinrich Martin Weber Nikolai Lobachevsky David Hilbert Paul Bachmann Felix Klein Rudolf Lipschitz Gottlob Frege G Perelman Elwin Bruno Christoffel Max Noether Sergei Novikov Heinrich Eduard Heine Paul Bernays Richard Dedekind Yuri Manin Carl Borchardt Ivan Lappo-Danilevskii Georg F B Riemann Emmy Noether Vladimir Arnold Sergey Bernstein Gotthold Eisenstein Edmund Landau Issai Schur Leoplod Kronecker Paul Halmos Hermann Minkowski Hermann von Helmholtz Paul Erd}os Rikitar¯oFujisawa Otto Hesse Kunihiko Kodaira Vladimir Steklov Karl Weierstrass Kurt G¨odel Friedrich Shottky Viktor Bunyakovsky Pavel Aleksandrov Andrei Markov Ernst Eduard Kummer Alexander Grothendieck Hermann Schwarz Mikhail Ostrogradsky Alexey Krylov Sofia Kovalevskya Andrey Kolmogorov Moritz Stern Friedrich Hirzebruch Heinrich Martin Weber Nikolai Lobachevsky David Hilbert Georg Cantor Carl Goldschmidt Ferdinand von Lindemann Paul Bachmann Felix Klein Pafnuti Chebyshev Oscar Zariski Carl Gustav Jacobi F Georg Frobenius Peter Lax Rudolf Lipschitz Gottlob Frege G Perelman Solomon Lefschetz Julius Pl¨ucker Hermann Weyl Elwin Bruno Christoffel Max Noether Sergei Novikov Karl von Staudt Eugene Wigner Martin Ohm Emil Artin Heinrich Eduard Heine Paul Bernays Richard Dedekind Yuri Manin 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000 Carl Borchardt Ivan Lappo-Danilevskii Georg F B Riemann Emmy Noether Vladimir Arnold August Ferdinand
    [Show full text]
  • Blaise Pascal Was a French Mathematician and Philosopher. He Was a Child Prodigy in Mathematics and Wrote an Essay on Conic Sections at the Age of 16
    Brittney Miller (112737326) Word Count: 3035 Original Abstract: Blaise Pascal was a French mathematician and philosopher. He was a child prodigy in mathematics and wrote an essay on conic sections at the age of 16. He made many contributions to mathematics, including creating an early calculator called the “Pascaline” and laying the groundwork for the formulation of calculus through his development of probability theory. Maybe his most famous mathematical contribution is his Treatise on the Arithmetical Triangle, where he explains “Pascal’s Triangle.” Pascal’s Triangle is a triangular system of numbers in which each number is the sum of the numbers above it. It is a pattern that is found in many areas of mathematics. Specifically, the number pattern in Pascal’s Triangle is observed in binomial expansions and combinations. There is, however, some confusion as to who really discovered this number pattern first. Multiple other mathematicians, such as Indian Halayudha, Persian Omar Khayyam, and Chinese Yang Hui, discussed this triangle in their own works throughout history. Original Outline: • Blaise Pascal Biography o Early/Personal Life – born in 1623; child prodigy in mathematics; essay on conic sections (Essai pour les coniques) in 1640; Christian theologian o Contributions to mathematics – built an early calculator (the Pascaline) for his father; developed probability theory; laid groundwork for Leibniz’ formulation of calculus; studied “Pascal’s Triangle” • Pascal’s Triangle o What is it? – triangular system of numbers in which each
    [Show full text]