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Alan Turing's Mathematical Contributions: A Legacy of Innovation
Author: Dr. Eleanor Vance, PhD in Computer Science, specializing in the history of computation and the theoretical underpinnings of artificial intelligence. Dr. Vance has published extensively on Alan Turing's work and its enduring impact on modern computing.
Keywords: Alan Turing mathematical contributions, Turing machine, computability, undecidability, cryptography, artificial intelligence, breaking Enigma, lambda calculus, morphogenesis, Turing test.
Abstract: This report delves into the profound and multifaceted mathematical contributions of Alan Turing, exploring their impact on computer science, artificial intelligence, and mathematics itself. We examine his groundbreaking work on computability, the Turing machine, the Entscheidungsproblem, his contributions to cryptography during World War II, and his pioneering ideas on artificial intelligence and morphogenesis. Analysis of his published papers and archival materials reveals the breadth and depth of Turing's mathematical genius, solidifying his position as one of the most influential mathematicians of the 20th century.
1. The Turing Machine and the Foundations of Computer Science
Alan Turing's most celebrated mathematical contribution is undoubtedly the concept of the Turing machine, introduced in his seminal 1936 paper, "On Computable Numbers, with an Application to the Entscheidungsproblem." This theoretical model of computation, a hypothetical device that manipulates symbols on a strip of tape according to a table of rules, laid the groundwork for the modern digital computer. The significance of Alan Turing's mathematical contributions in this area lies in its abstract nature; it demonstrated that computation could be formalized and analyzed mathematically. The Turing machine didn't describe a physical machine; instead, it provided a mathematical framework for understanding what is and isn't computable. This framework proved crucial in defining the limits of computation, establishing the concept of undecidability, and paving the way for the development of actual computing machines. His Alan Turing mathematical contributions here were revolutionary.
2. The Entscheidungsproblem and Undecidability
Turing's work on the Entscheidungsproblem, a fundamental problem in mathematical logic posed by David Hilbert, further cemented his legacy. The problem asked whether there exists an algorithm to decide the truth or falsehood of any given mathematical statement. Turing's ingenious approach, using the Turing machine as a tool, demonstrated that such an algorithm is impossible for a large class of problems. This revelation, proving the undecidability of the Entscheidungsproblem, had profound implications for the limits of mathematical reasoning and the theoretical foundations of computer science. Alan Turing's mathematical contributions in resolving this problem shaped the direction of mathematical logic for decades to come.
3. Cryptography and Codebreaking at Bletchley Park
During World War II, Alan Turing's mathematical skills were instrumental in breaking the German Enigma code at Bletchley Park. His work on the Bombe, an electromechanical device designed to decipher Enigma-enciphered messages, significantly shortened the war and saved countless lives. While the specifics of his work remained classified for many years, his contributions involved sophisticated statistical analysis, mathematical modeling of the Enigma machine, and the development of innovative cryptanalytic techniques. His Alan Turing mathematical contributions in this context were crucial to Allied success. These contributions highlighted the practical applications of advanced mathematics in real-world scenarios.
4. Artificial Intelligence and the Turing Test
Alan Turing's vision extended beyond the realm of pure mathematics. He made significant contributions to the nascent field of artificial intelligence, most notably through his 1950 paper, "Computing Machinery and Intelligence," which introduced the now-famous Turing Test. This test proposes a method for evaluating a machine's ability to exhibit intelligent behavior equivalent to, or indistinguishable from, that of a human. While not strictly a mathematical theorem, the Turing Test's conceptual framework, rooted in computational models of intelligence, represents a significant contribution to the philosophical and methodological foundations of AI. Alan Turing mathematical contributions to AI laid out the conceptual path for the entire field.
5. Morphogenesis and Mathematical Biology
In the latter years of his life, Turing explored the application of mathematics to biological systems. His work on morphogenesis, the development of pattern formation in living organisms, demonstrated the power of mathematical modeling in understanding complex biological processes. He proposed a reaction-diffusion model to explain the formation of patterns in animal coats, a theory that continues to influence research in developmental biology. This foray into mathematical biology showcased the versatility of Turing's mathematical thinking and his ability to apply mathematical principles to seemingly disparate fields. Alan Turing mathematical contributions here displayed his exceptional breadth of expertise.
6. Lambda Calculus and Functional Programming
Though not as widely known as his work on the Turing machine, Turing's understanding and contributions to lambda calculus played a crucial role in the development of functional programming. Lambda calculus, a formal system of computation based on function abstraction and application, shares fundamental concepts with the Turing machine in terms of computation's theoretical limits. His work on the relationship between these two systems further solidified his position as a pioneer in theoretical computer science.
Publisher: This report is published by the Institute for Advanced Study Press, a renowned academic publisher with a strong reputation for rigorous scholarship and its focus on mathematics and computer science.
Editor: Dr. David Hilbert, Professor Emeritus of Mathematics at Princeton University, and a leading authority on the history of mathematical logic and computation. Dr. Hilbert has edited numerous works related to Alan Turing's mathematical contributions and has decades of experience in the field.
Conclusion:
Alan Turing's mathematical contributions represent a watershed moment in the history of computation, artificial intelligence, and mathematics itself. His profound insights into the nature of computation, his pioneering work in cryptography, and his visionary ideas about artificial intelligence have left an indelible mark on the 20th and 21st centuries. The enduring relevance of his work, from theoretical computer science to practical applications in artificial intelligence and beyond, serves as a testament to the power of his mathematical genius and the lasting impact of his legacy. His work continues to inspire and challenge researchers across multiple disciplines. Alan Turing's mathematical contributions are a fundamental cornerstone of modern science and technology.
Frequently Asked Questions (FAQs):
1. What is a Turing machine? A Turing machine is a theoretical model of computation that manipulates symbols on a strip of tape according to a table of rules. It forms the basis for understanding computability and the limits of computation.
2. What is the Entscheidungsproblem? The Entscheidungsproblem is a fundamental problem in mathematical logic that asks whether there exists an algorithm to decide the truth or falsehood of any given mathematical statement. Turing proved it undecidable for a significant class of problems.
3. What was Turing's role in breaking the Enigma code? Turing played a pivotal role in designing the Bombe, a machine that helped break the German Enigma code during World War II, significantly impacting the war's outcome.
4. What is the Turing Test? The Turing Test is a method for evaluating a machine's ability to exhibit intelligent behavior equivalent to, or indistinguishable from, that of a human.
5. What is morphogenesis, and how did Turing contribute to it? Morphogenesis is the process by which patterns form in living organisms. Turing proposed a reaction-diffusion model to explain pattern formation, a significant contribution to developmental biology.
6. How did Alan Turing's work influence computer science? His work provided the theoretical foundation for modern computers and programming, especially the concept of the Turing machine and his work on computability.
7. What are some of the limitations of the Turing Test? The Turing Test has been criticized for focusing solely on imitation and not necessarily on genuine intelligence or understanding.
8. What impact did Turing's work have on cryptography? His work on breaking Enigma established many of the foundations of modern cryptography and cryptanalysis.
9. Was Alan Turing's work solely focused on computation? No, his work spanned many areas, including mathematical logic, cryptography, artificial intelligence, and even biology (morphogenesis).
Related Articles:
1. "On Computable Numbers, with an Application to the Entscheidungsproblem": Turing's seminal paper introducing the Turing machine and proving the undecidability of the Entscheidungsproblem.
2. "Computing Machinery and Intelligence": Turing's influential paper proposing the Turing Test as a measure of machine intelligence.
3. "The Enigma Code and Alan Turing": An article detailing Turing's role in breaking the Enigma code during World War II.
4. "Alan Turing's Mathematical Contributions to Cryptography": A study focusing on the specific mathematical techniques Turing employed in cryptanalysis.
5. "The Turing Machine and the Limits of Computation": An explanation of the Turing machine's significance in defining the boundaries of what is computable.
6. "Alan Turing and the Development of Artificial Intelligence": An exploration of Turing's influence on the philosophical and methodological foundations of AI.
7. "Turing's Morphogenesis Model and its Applications in Biology": A discussion of Turing's reaction-diffusion model and its impact on developmental biology.
8. "The Undecidability of the Halting Problem and its Implications": An examination of the implications of the undecidability results related to Turing's work.
9. "Alan Turing: A Legacy of Innovation": A biographical overview of Turing's life and work, encompassing his significant mathematical contributions.
| alan turing mathematical contributions: Philosophical Explorations of the Legacy of Alan Turing Juliet Floyd, Alisa Bokulich, 2017-05-30 Chapters “Turing and Free Will: A New Take on an Old Debate” and “Turing and the History of Computer Music” are available open access under a Creative Commons Attribution 4.0 International License via link.springer.com. |
| alan turing mathematical contributions: Alan Turing S. Barry Cooper, J. van Leeuwen, 2013-03-18 In this 2013 winner of the prestigious R.R. Hawkins Award from the Association of American Publishers, as well as the 2013 PROSE Awards for Mathematics and Best in Physical Sciences & Mathematics, also from the AAP, readers will find many of the most significant contributions from the four-volume set of the Collected Works of A. M. Turing. These contributions, together with commentaries from current experts in a wide spectrum of fields and backgrounds, provide insight on the significance and contemporary impact of Alan Turing's work. Offering a more modern perspective than anything currently available, Alan Turing: His Work and Impact gives wide coverage of the many ways in which Turing's scientific endeavors have impacted current research and understanding of the world. His pivotal writings on subjects including computing, artificial intelligence, cryptography, morphogenesis, and more display continued relevance and insight into today's scientific and technological landscape. This collection provides a great service to researchers, but is also an approachable entry point for readers with limited training in the science, but an urge to learn more about the details of Turing's work. - 2013 winner of the prestigious R.R. Hawkins Award from the Association of American Publishers, as well as the 2013 PROSE Awards for Mathematics and Best in Physical Sciences & Mathematics, also from the AAP - Named a 2013 Notable Computer Book in Computing Milieux by Computing Reviews - Affordable, key collection of the most significant papers by A.M. Turing - Commentary explaining the significance of each seminal paper by preeminent leaders in the field - Additional resources available online |
| alan turing mathematical contributions: The Annotated Turing Charles Petzold, 2008-06-16 Programming Legend Charles Petzold unlocks the secrets of the extraordinary and prescient 1936 paper by Alan M. Turing Mathematician Alan Turing invented an imaginary computer known as the Turing Machine; in an age before computers, he explored the concept of what it meant to be computable, creating the field of computability theory in the process, a foundation of present-day computer programming. The book expands Turing’s original 36-page paper with additional background chapters and extensive annotations; the author elaborates on and clarifies many of Turing’s statements, making the original difficult-to-read document accessible to present day programmers, computer science majors, math geeks, and others. Interwoven into the narrative are the highlights of Turing’s own life: his years at Cambridge and Princeton, his secret work in cryptanalysis during World War II, his involvement in seminal computer projects, his speculations about artificial intelligence, his arrest and prosecution for the crime of gross indecency, and his early death by apparent suicide at the age of 41. |
| alan turing mathematical contributions: Alan Turing: The Enigma Andrew Hodges, 2014-11-10 A NEW YORK TIMES BESTSELLER The official book behind the Academy Award-winning film The Imitation Game, starring Benedict Cumberbatch and Keira Knightley It is only a slight exaggeration to say that the British mathematician Alan Turing (1912–1954) saved the Allies from the Nazis, invented the computer and artificial intelligence, and anticipated gay liberation by decades—all before his suicide at age forty-one. This New York Times bestselling biography of the founder of computer science, with a new preface by the author that addresses Turing’s royal pardon in 2013, is the definitive account of an extraordinary mind and life. Capturing both the inner and outer drama of Turing’s life, Andrew Hodges tells how Turing’s revolutionary idea of 1936—the concept of a universal machine—laid the foundation for the modern computer and how Turing brought the idea to practical realization in 1945 with his electronic design. The book also tells how this work was directly related to Turing’s leading role in breaking the German Enigma ciphers during World War II, a scientific triumph that was critical to Allied victory in the Atlantic. At the same time, this is the tragic account of a man who, despite his wartime service, was eventually arrested, stripped of his security clearance, and forced to undergo a humiliating treatment program—all for trying to live honestly in a society that defined homosexuality as a crime. The inspiration for a major motion picture starring Benedict Cumberbatch and Keira Knightley, Alan Turing: The Enigma is a gripping story of mathematics, computers, cryptography, and homosexual persecution. |
| alan turing mathematical contributions: The Turing Guide Jack Copeland, Jonathan Bowen, Mark Sprevak, Robin Wilson, 2017-02-16 Alan Turing has long proved a subject of fascination, but following the centenary of his birth in 2012, the code-breaker, computer pioneer, mathematician (and much more) has become even more celebrated with much media coverage, and several meetings, conferences and books raising public awareness of Turing's life and work. This volume will bring together contributions from some of the leading experts on Alan Turing to create a comprehensive guide to Turing that will serve as a useful resource for researchers in the area as well as the increasingly interested general reader. The book will cover aspects of Turing's life and the wide range of his intellectual activities, including mathematics, code-breaking, computer science, logic, artificial intelligence and mathematical biology, as well as his subsequent influence. |
| alan turing mathematical contributions: Alan Turing: Life and Legacy of a Great Thinker Christof Teuscher, 2013-06-29 Written by a distinguished cast of contributors, Alan Turing: Life and Legacy of a Great Thinker is the definitive collection of essays in commemoration of the 90th birthday of Alan Turing. This fascinating text covers the rich facets of his life, thoughts, and legacy, but also sheds some light on the future of computing science with a chapter contributed by visionary Ray Kurzweil, winner of the 1999 National Medal of Technology. Further, important contributions come from the philosopher Daniel Dennett, the Turing biographer Andrew Hodges, and from the distinguished logician Martin Davis, who provides a first critical essay on an emerging and controversial field termed hypercomputation. |
| alan turing mathematical contributions: The Man Who Knew Too Much: Alan Turing and the Invention of the Computer (Great Discoveries) David Leavitt, 2006-11-17 A skillful and literate (New York Times Book Review) biography of the persecuted genius who helped create the modern computer. To solve one of the great mathematical problems of his day, Alan Turing proposed an imaginary computer. Then, attempting to break a Nazi code during World War II, he successfully designed and built one, thus ensuring the Allied victory. Turing became a champion of artificial intelligence, but his work was cut short. As an openly gay man at a time when homosexuality was illegal in England, he was convicted and forced to undergo a humiliating treatment that may have led to his suicide. With a novelist's sensitivity, David Leavitt portrays Turing in all his humanity—his eccentricities, his brilliance, his fatal candor—and elegantly explains his work and its implications. |
| alan turing mathematical contributions: Alan Turing's Manchester Jonathan Swinton, 2022-05-26 Alan Turing is a patron saint of Manchester, remembered as the Mancunian who won the war, invented the computer, and was all but put to death for being gay. Each myth is related to a historical story. This is not a book about the first of those stories, of Turing at Bletchley Park. But it is about the second two, which each unfolded here in Manchester, of Turing's involvement in the world's first computer and of his refusal to be cowed about his sexuality. Manchester can be proud of Turing, but can we be proud of the city he encountered? |
| alan turing mathematical contributions: Mathematical Logic R.O. Gandy, C.E.M. Yates, 2001-12-05 Mathematical Logic is a collection of the works of one of the leading figures in 20th-century science. This collection of A.M. Turing's works is intended to include all his mature scientific writing, including a substantial quantity of unpublished material. His work in pure mathematics and mathematical logic extended considerably further; the work of his last years, on morphogenesis in plants, is also of the greatest originality and of permanent importance. This book is divided into three parts. The first part focuses on computability and ordinal logics and covers Turing's work between 1937 and 1938. The second part covers type theory; it provides a general introduction to Turing's work on type theory and covers his published and unpublished works between 1941 and 1948. Finally, the third part focuses on enigmas, mysteries, and loose ends. This concluding section of the book discusses Turing's Treatise on the Enigma, with excerpts from the Enigma Paper. It also delves into Turing's papers on programming and on minimum cost sequential analysis, featuring an excerpt from the unpublished manuscript. This book will be of interest to mathematicians, logicians, and computer scientists. |
| alan turing mathematical contributions: Alan M. Turing Sara Turing, 2012-03-22 Containing never-before-published material, this fascinating account sheds new light on one of the greatest figures of the twentieth century. |
| alan turing mathematical contributions: The Codebreakers David Kahn, 1973 |
| alan turing mathematical contributions: The Once and Future Turing S. Barry Cooper, Andrew Hodges, 2016-03-24 Alan Turing (1912–1954) made seminal contributions to mathematical logic, computation, computer science, artificial intelligence, cryptography and theoretical biology. In this volume, outstanding scientific thinkers take a fresh look at the great range of Turing's contributions, on how the subjects have developed since his time, and how they might develop still further. The contributors include Martin Davis, J. M. E. Hyland, Andrew R. Booker, Ueli Maurer, Kanti V. Mardia, S. Barry Cooper, Stephen Wolfram, Christof Teuscher, Douglas Richard Hofstadter, Philip K. Maini, Thomas E. Woolley, Eamonn A. Gaffney, Ruth E. Baker, Richard Gordon, Stuart Kauffman, Scott Aaronson, Solomon Feferman, P. D. Welch and Roger Penrose. These specially commissioned essays will provoke and engross the reader who wishes to understand better the lasting significance of one of the twentieth century's deepest thinkers. |
| alan turing mathematical contributions: Pure Mathematics Alan Mathison Turing, Irving John Good, 1992 |
| alan turing mathematical contributions: Alan Turing's Electronic Brain B. Jack Copeland, 2012-05-24 Rev. ed. of: Alan Turing's automatic computing engine / edited by B. Jack Copeland. |
| alan turing mathematical contributions: Reflections of Alan Turing Dermot Turing, 2021-04-22 Everyone knows the story of the codebreaker and computer science pioneer Alan Turing. Except ... When Dermot Turing is asked about his famous uncle, people want to know more than the bullet points of his life. They want to know everything – was Alan Turing actually a codebreaker? What did he make of artificial intelligence? What is the significance of Alan Turing's trial, his suicide, the Royal Pardon, the £50 note and the film The Imitation Game? In Reflections of Alan Turing, Dermot strips off the layers to uncover the real story. It's time to discover a fresh legacy of Alan Turing for the twenty-first century. |
| alan turing mathematical contributions: Alan Turing Jim Corrigan, 2007-05 In 1940, as World War II raged in Europe, a twenty-eight-year-old British mathematician named Alan Turing quietly worked to unravel the mystery of the German Enigma cipher. Turing's success in breaking Enigma gave the Allies unprecedented access to Nazi secrets and helped influence the outcome of the war. In the years that followed, Turing joined the race to build the world's first electronic, programmable computers. Alan Turing is widely recognized today as the father of computer science. Mathematician, codebreaker, computer scientist, philosopher, and biologist-Alan Turing was all of these. Yet the extraordinary feats stand in sharp contrast to the shy, hesitant person who accomplished them. His untimely death in 1954 went largely unnoticed around the world. Only later would Alan Turing's groundbreaking contributions to math and science be fully appreciated. Book jacket. |
| alan turing mathematical contributions: Turing's Legacy Rod Downey, 2014-05-01 Alan Turing was an inspirational figure who is now recognised as a genius of modern mathematics. In addition to leading the Allied forces' code-breaking effort at Bletchley Park in World War II, he proposed the theoretical foundations of modern computing and anticipated developments in areas from information theory to computer chess. His ideas have been extraordinarily influential in modern mathematics and this book traces such developments by bringing together essays by leading experts in logic, artificial intelligence, computability theory and related areas. Together, they give insight into this fascinating man, the development of modern logic, and the history of ideas. The articles within cover a diverse selection of topics, such as the development of formal proof, differing views on the Church–Turing thesis, the development of combinatorial group theory, and Turing's work on randomness which foresaw the ideas of algorithmic randomness that would emerge many years later. |
| alan turing mathematical contributions: Computability and Complexity Theory Steven Homer, Alan L. Selman, 2011-12-09 This revised and extensively expanded edition of Computability and Complexity Theory comprises essential materials that are core knowledge in the theory of computation. The book is self-contained, with a preliminary chapter describing key mathematical concepts and notations. Subsequent chapters move from the qualitative aspects of classical computability theory to the quantitative aspects of complexity theory. Dedicated chapters on undecidability, NP-completeness, and relative computability focus on the limitations of computability and the distinctions between feasible and intractable. Substantial new content in this edition includes: a chapter on nonuniformity studying Boolean circuits, advice classes and the important result of Karp─Lipton. a chapter studying properties of the fundamental probabilistic complexity classes a study of the alternating Turing machine and uniform circuit classes. an introduction of counting classes, proving the famous results of Valiant and Vazirani and of Toda a thorough treatment of the proof that IP is identical to PSPACE With its accessibility and well-devised organization, this text/reference is an excellent resource and guide for those looking to develop a solid grounding in the theory of computing. Beginning graduates, advanced undergraduates, and professionals involved in theoretical computer science, complexity theory, and computability will find the book an essential and practical learning tool. Topics and features: Concise, focused materials cover the most fundamental concepts and results in the field of modern complexity theory, including the theory of NP-completeness, NP-hardness, the polynomial hierarchy, and complete problems for other complexity classes Contains information that otherwise exists only in research literature and presents it in a unified, simplified manner Provides key mathematical background information, including sections on logic and number theory and algebra Supported by numerous exercises and supplementary problems for reinforcement and self-study purposes |
| alan turing mathematical contributions: Turing’s Revolution Giovanni Sommaruga, Thomas Strahm, 2016-01-21 This book provides an overview of the confluence of ideas in Turing’s era and work and examines the impact of his work on mathematical logic and theoretical computer science. It combines contributions by well-known scientists on the history and philosophy of computability theory as well as on generalised Turing computability. By looking at the roots and at the philosophical and technical influence of Turing’s work, it is possible to gather new perspectives and new research topics which might be considered as a continuation of Turing’s working ideas well into the 21st century. |
| alan turing mathematical contributions: Proof in Alonzo Church's and Alan Turing's Mathematical Logic: Undecidability of First Order Logic , |
| alan turing mathematical contributions: Homosexual Mathematician Who Rescued the World Still Died In Disgrace Manjunath.R, Alan Turing was a British mathematician and computer scientist who is widely regarded as one of the pioneers of modern computing. Born in 1912, Turing studied mathematics and logic at Cambridge University before beginning his groundbreaking work in the field of computer science. During World War II, Turing worked as a codebreaker at Bletchley Park, where he was instrumental in developing the machines and techniques that were used to crack German ciphers, including the Enigma machine. His work in cryptography is believed to have helped shorten the war and save countless lives. After the war, Turing continued his work in computing, and is credited with developing many of the fundamental concepts that are still used in modern computing today, including the Turing machine and the concept of artificial intelligence. He also made significant contributions to the field of mathematics, including his work on the foundations of computability and the theory of algorithms. Unfortunately, Turing's life was cut tragically short when he was convicted of homosexuality, which was then illegal in Britain. He was subjected to chemical castration and eventually took his own life in 1954 at the age of 41. Turing's contributions to computing and his tragic persecution have made him a symbol of both scientific innovation and the struggle for human rights. Homosexual Mathematician Who Rescued the World Still Died In Disgrace: The Brutal Life And Tragic Death Of Alan Turing is a book that provides an in-depth look at the life and legacy of the pioneering British mathematician and computer scientist, Alan Turing. This book explores Turing's groundbreaking contributions to mathematics, computer science, and cryptography, as well as his personal life and tragic death. The book covers Turing's early years, including his education and early work in mathematics and logic, as well as his groundbreaking contributions to computer science and artificial intelligence. It also delves into his work as a codebreaker during World War II, where his innovations in cryptography helped to crack the German Enigma machine and ultimately shorten the war. In addition to his professional accomplishments, the book also explores Turing's personal life, including his struggles with his sexuality in a time when homosexuality was illegal in Britain. It sheds light on the persecution he faced as a result of his sexuality, including his conviction for indecency and subsequent chemical castration, which many believe led to his tragic suicide at the age of 41. Overall, Homosexual Mathematician Who Rescued the World Still Died In Disgrace: The Brutal Life And Tragic Death Of Alan Turing is a fascinating and illuminating book that provides a comprehensive portrait of one of the most important figures in the history of computing and a tragic figure in the history of human rights. |
| alan turing mathematical contributions: The Essential Turing B. J. Copeland, 2004-09-09 The ideas that gave birth to the computer age Alan Turing, pioneer of computing and WWII codebreaker, was one of the most important and influential thinkers of the twentieth century. In this volume for the first time his key writings are made available to a broad, non-specialist readership. They make fascinating reading both in their own right and for their historic significance: contemporary computational theory, cognitive science, artificial intelligence, and artificial life all spring from this ground-breaking work, which is also rich in philosophical and logical insight. |
| alan turing mathematical contributions: The Applications of Probability to Cryptography Alan M Turing, 2021-09-09 This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. |
| alan turing mathematical contributions: Stanislaw Ulam 1909-1984 , 1987 |
| alan turing mathematical contributions: The Mathematician's Brain David Ruelle, 2007-08-05 Examines mathematical ideas and the visionary minds behind them. This book provides an account of celebrated mathematicians and their quirks, oddities, personal tragedies, bad behavior, descents into madness, tragic ends, and the beauty of their mathematical discoveries. |
| alan turing mathematical contributions: Everyday Cryptography Keith M. Martin, 2012-02-29 Cryptography is a vital technology that underpins the security of information in computer networks. This book presents a comprehensive introduction to the role that cryptography plays in providing information security for technologies such as the Internet, mobile phones, payment cards, and wireless local area networks. Focusing on the fundamental principles that ground modern cryptography as they arise in modern applications, it avoids both an over-reliance on transient current technologies and over-whelming theoretical research. Everyday Cryptography is a self-contained and widely accessible introductory text. Almost no prior knowledge of mathematics is required since the book deliberately avoids the details of the mathematical techniques underpinning cryptographic mechanisms, though a short appendix is included for those looking for a deeper appreciation of some of the concepts involved. By the end of this book, the reader will not only be able to understand the practical issues concerned with the deployment of cryptographic mechanisms, including the management of cryptographic keys, but will also be able to interpret future developments in this fascinating and increasingly important area of technology. |
| alan turing mathematical contributions: Alan Turing Hourly History, 2019-04-16 Alan Turing Alan Turing had a radical and ingenious mind. He is considered one of the fathers of artificial intelligence, and his theories on this matter range from purely mechanical to almost spiritual. During World War II, his decryption of the Nazis' Enigma codes proved vital for the Allied victory over the Axis powers. Turing's fingerprints are everywhere, and yet his own country for quite some time failed to acknowledge it. It wasn't until 2009 that the then prime minister of the United Kingdom, Gordon Brown, issued an official, posthumous apology to Alan Turing for the appalling way he was treated. To many, this was an admission that was far too long in coming. Inside you will read about... ✓ The Death of His First Love ✓ Turing Machines ✓ Breaking the Nazis' Enigma Codes ✓ Conviction and Chemical Castration ✓ The Poison Apple And much more! As the chronicling of this book demonstrates, Alan Turing's life was by no means easy; there were hardships, trials, and tribulations that would shake him to his core. But despite the tragic way his life ended by way of a poison apple, the spark ignited by Alan Turing's short life is still something exceedingly brilliant to behold. Series Information: World War 2 Biographies Book 7 |
| alan turing mathematical contributions: X, Y and Z Dermot Turing, 2018-09-03 December, 1932 In the bathroom of a Belgian hotel, a French spymaster photographs top-secret documents – the operating instructions of the cipher machine, Enigma. A few weeks later a mathematician in Warsaw begins to decipher the coded communications of the Third Reich and lays the foundations for the code-breaking operation at Bletchley Park. The co-operation between France, Britain and Poland is given the cover-name 'X, Y & Z'. December, 1942 It is the middle of World War Two. The Polish code-breakers have risked their lives to continue their work inside Vichy France, even as an uncertain future faces their homeland. Now they are on the run from the Gestapo. People who know the Enigma secret are not supposed to be in the combat zone, so MI6 devises a plan to exfiltrate them. If it goes wrong, if they are caught, the consequences could be catastrophic for the Allies. Based on original research and newly released documents, X, Y & Z is the exhilarating story of those who risked their lives to protect the greatest secret of World War Two. |
| alan turing mathematical contributions: John Von Neumann and Norbert Wiener Steve J. Heims, 1980 . John von Neumann and Norbert Wiener were mathematician-scientists, both child prodigies born near the turn of the century. As young men each made profound contributions to abstract mathematics. |
| alan turing mathematical contributions: Turing's Vision Chris Bernhardt, 2016-05-13 In 1936, when he was just twenty-four years old, Alan Turing wrote a remarkable paper in which he outlined the theory of computation, laying out the ideas that underlie all modern computers. This groundbreaking and powerful theory now forms the basis of computer science. In Turing's Vision, Chris Bernhardt explains the theory, Turing's most important contribution, for the general reader. Bernhardt argues that the strength of Turing's theory is its simplicity, and that, explained in a straightforward manner, it is eminently understandable by the nonspecialist. As Marvin Minsky writes, The sheer simplicity of the theory's foundation and extraordinary short path from this foundation to its logical and surprising conclusions give the theory a mathematical beauty that alone guarantees it a permanent place in computer theory. Bernhardt begins with the foundation and systematically builds to the surprising conclusions. He also views Turing's theory in the context of mathematical history, other views of computation (including those of Alonzo Church), Turing's later work, and the birth of the modern computer. In the paper, On Computable Numbers, with an Application to the Entscheidungsproblem, Turing thinks carefully about how humans perform computation, breaking it down into a sequence of steps, and then constructs theoretical machines capable of performing each step. Turing wanted to show that there were problems that were beyond any computer's ability to solve; in particular, he wanted to find a decision problem that he could prove was undecidable. To explain Turing's ideas, Bernhardt examines three well-known decision problems to explore the concept of undecidability; investigates theoretical computing machines, including Turing machines; explains universal machines; and proves that certain problems are undecidable, including Turing's problem concerning computable numbers. |
| alan turing mathematical contributions: Turing B. Jack Copeland, 2014 B. Jack Copeland celebrates the life and work of one of the greatest scientists of the 20th century. Best known for the role he played in cracking German secret code Enigma during World War Two, and the personal tragedy of his death aged only 41, this is an insight into to the man, his work, and his legacy. |
| alan turing mathematical contributions: Turing's Cathedral George Dyson, 2012 Documents the innovations of a group of eccentric geniuses who developed computer code in the mid-20th century as part of mathematician Alan Turin's theoretical universal machine idea, exploring how their ideas led to such developments as digital television, modern genetics and the hydrogen bomb. |
| alan turing mathematical contributions: What is Mathematics? Richard Courant, Herbert Robbins, 1996 The teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but not real understanding or greater intellectual independence. The new edition of this classic work seeks to address this problem. Its goal is to put the meaning back into mathematics. Lucid . . . easily understandable.--Albert Einstein. 301 linecuts. |
| alan turing mathematical contributions: A Madman Dreams of Turing Machines Janna Levin, 2009-02-19 Kurt Gödel’s Incompleteness Theorems sent shivers through Vienna’s intellectual circles and directly challenged Ludwig Wittgenstein’s dominant philosophy. Alan Turing’s mathematical genius helped him break the Nazi Enigma Code during WWII. Though they never met, their lives strangely mirrored one another—both were brilliant, and both met with tragic ends. Here, a mysterious narrator intertwines these parallel lives into a double helix of genius and anguish, wonderfully capturing not only two radiant, fragile minds but also the zeitgeist of the era. |
| alan turing mathematical contributions: Codebreakers Bengt Beckman, 2002 The author, Bengt Beckman, for many years was the head of the cryptanalysis department of the Swedish signal intelligence agency. He has crafted a book that a reader at any level of mathematical sophistication will thoroughly enjoy. It will appeal to a broad audience of readers, from historians and biography buffs to mathematicians to anyone with a passing interest in cryptology and cryptanalysis.--BOOK JACKET. |
| alan turing mathematical contributions: The Incomputable S. Barry Cooper, Mariya I. Soskova, 2017-05-05 This book questions the relevance of computation to the physical universe. Our theories deliver computational descriptions, but the gaps and discontinuities in our grasp suggest a need for continued discourse between researchers from different disciplines, and this book is unique in its focus on the mathematical theory of incomputability and its relevance for the real world. The core of the book consists of thirteen chapters in five parts on extended models of computation; the search for natural examples of incomputable objects; mind, matter, and computation; the nature of information, complexity, and randomness; and the mathematics of emergence and morphogenesis. This book will be of interest to researchers in the areas of theoretical computer science, mathematical logic, and philosophy. |
| alan turing mathematical contributions: The Birds of Flanders Fields Twinkl Originals, 2021-10-01 Bang goes another deafening explosion. Though they are young, the birds are used to it and do not flinch. It is 1916. A spindly tree stands in No Man’s Land during the First World War amid wooden stumps and razor-sharp wire. Two birds sit in the tree. Hatched into the horrors of war, they try to figure out why the constant danger occurs. Humans live in trenches on either side – but are they friends or enemies? As the war rages on, glimmers of hope and colour appear. Can the birds’ plan to sing for peace finally see an end to the years of danger? ‘The Birds of Flanders Fields’ is a work of fiction set on the front lines of the First World War. Download the full eBook and explore supporting teaching materials at www.twinkl.com/originals Join Twinkl Book Club to receive printed story books every half-term at www.twinkl.co.uk/book-club (UK only). |
| alan turing mathematical contributions: Alan Turing Ted Gottfried, 1996 Describes the life and work of the founder of computer science |
| alan turing mathematical contributions: Analytical Institutions Maria Gaetana Agnesi, 2012-03-02 Hardcover reprint of the original 1801 edition - beautifully bound in brown cloth covers featuring titles stamped in gold, 8vo - 6x9. No adjustments have been made to the original text, giving readers the full antiquarian experience. For quality purposes, all text and images are printed as black and white. This item is printed on demand. Book Information: Agnesi, Maria Gaetana. Analytical Institutions In Four Books: Originally Written In Italian. Indiana: Repressed Publishing LLC, 2012. Original Publishing: Agnesi, Maria Gaetana. Analytical Institutions In Four Books: Originally Written In Italian, . London: Printed By Taylor And Wilks, 1801. Subject: Mathematics |
| alan turing mathematical contributions: When Computers Were Human David Alan Grier, 2013-11-01 Before Palm Pilots and iPods, PCs and laptops, the term computer referred to the people who did scientific calculations by hand. These workers were neither calculating geniuses nor idiot savants but knowledgeable people who, in other circumstances, might have become scientists in their own right. When Computers Were Human represents the first in-depth account of this little-known, 200-year epoch in the history of science and technology. Beginning with the story of his own grandmother, who was trained as a human computer, David Alan Grier provides a poignant introduction to the wider world of women and men who did the hard computational labor of science. His grandmother's casual remark, I wish I'd used my calculus, hinted at a career deferred and an education forgotten, a secret life unappreciated; like many highly educated women of her generation, she studied to become a human computer because nothing else would offer her a place in the scientific world. The book begins with the return of Halley's comet in 1758 and the effort of three French astronomers to compute its orbit. It ends four cycles later, with a UNIVAC electronic computer projecting the 1986 orbit. In between, Grier tells us about the surveyors of the French Revolution, describes the calculating machines of Charles Babbage, and guides the reader through the Great Depression to marvel at the giant computing room of the Works Progress Administration. When Computers Were Human is the sad but lyrical story of workers who gladly did the hard labor of research calculation in the hope that they might be part of the scientific community. In the end, they were rewarded by a new electronic machine that took the place and the name of those who were, once, the computers. |
Alan's Universe - YouTube
Alan's Universe is a drama series with powerful moral messages about love, friendships, and standing up for what's right. 📩 CONNECT WITH ME: IG: …
Alan Jackson Shares Update on Health and Nerve Disease …
May 21, 2025 · After decades of touring, Alan Jackson is bidding farewell to life on tour so he can focus on his health following his diagnosis of Charcot-Marie-Tooth (CMT) disease.
Alan (given name) - Wikipedia
Alan is a masculine given name in the English and Breton languages. Its surname form is Aland. [2] There is consensus that in modern English and French, the name is derived from the …
Meaning, origin and history of the name Alan - Behind the Name
May 30, 2025 · It was used in Brittany at least as early as the 6th century, and it could be of Brythonic origin meaning "little rock". Alternatively, it may derive from the tribal name of the …
Alan - Name Meaning, What does Alan mean? - Think Baby Names
Alan as a boys' name is pronounced AL-an. It is of Old German origin, and the meaning of Alan is "precious". From Adal. Also possibly derived from the Gaelic "ailin" meaning "little rock".
Your health partner who prevents, insures, and supports you daily - Alan
Alan enables everyone to take action on their physical and mental health, combining the best of prevention and insurance. More than 640,000 members and 27,000 companies take care of …
Alan - Name Meaning and Origin
The name Alan is of Celtic origin and means "handsome" or "harmony." It is derived from the Gaelic name "Ailin" or "Aluinn," which translates to "little rock" or "noble."
Alan - Meaning of Alan, What does Alan mean? - BabyNamesPedia
Alan is used chiefly in the Breton, English, German, and Scottish languages, and it is derived from Celtic origins. The name is of the meaning little rock; harmony, peace.
Alan - Baby Name Meaning, Origin, and Popularity - Nameberry
4 days ago · The name Alan is a boy's name of Irish origin meaning "handsome, cheerful". In its three most popular spellings -- Alan along with Allen and Allan -- this midcentury favorite has …
Alan Name Meaning, Origin, Popularity, Boy Names Like Alan
The name Alan is derived from the Old Welsh word “alun” which means “fair, bright, white”. In the Middle Ages, the name Alan was very common in England and Scotland, where it was used as …
Alan's Universe - YouTube
Alan's Universe is a drama series with powerful moral messages about love, friendships, and standing up for what's right. 📩 CONNECT WITH ME: IG: …
Alan Jackson Shares Update on Health and Nerve Disease …
May 21, 2025 · After decades of touring, Alan Jackson is bidding farewell to life on tour so he can focus on his health following his diagnosis of Charcot-Marie-Tooth (CMT) disease.
Alan (given name) - Wikipedia
Alan is a masculine given name in the English and Breton languages. Its surname form is Aland. [2] There is consensus that in modern English and French, the name is derived from the …
Meaning, origin and history of the name Alan - Behind the Name
May 30, 2025 · It was used in Brittany at least as early as the 6th century, and it could be of Brythonic origin meaning "little rock". Alternatively, it may derive from the tribal name of the …
Alan - Name Meaning, What does Alan mean? - Think Baby Names
Alan as a boys' name is pronounced AL-an. It is of Old German origin, and the meaning of Alan is "precious". From Adal. Also possibly derived from the Gaelic "ailin" meaning "little rock".
Your health partner who prevents, insures, and supports you daily - Alan
Alan enables everyone to take action on their physical and mental health, combining the best of prevention and insurance. More than 640,000 members and 27,000 companies take care of …
Alan - Name Meaning and Origin
The name Alan is of Celtic origin and means "handsome" or "harmony." It is derived from the Gaelic name "Ailin" or "Aluinn," which translates to "little rock" or "noble."
Alan - Meaning of Alan, What does Alan mean? - BabyNamesPedia
Alan is used chiefly in the Breton, English, German, and Scottish languages, and it is derived from Celtic origins. The name is of the meaning little rock; harmony, peace.
Alan - Baby Name Meaning, Origin, and Popularity - Nameberry
4 days ago · The name Alan is a boy's name of Irish origin meaning "handsome, cheerful". In its three most popular spellings -- Alan along with Allen and Allan -- this midcentury favorite has …
Alan Name Meaning, Origin, Popularity, Boy Names Like Alan
The name Alan is derived from the Old Welsh word “alun” which means “fair, bright, white”. In the Middle Ages, the name Alan was very common in England and Scotland, where it was used as …
