마틴 데이비스 (2023) 유니버셜 컴퓨터

• 3 판. 번역. 구입함.

논리 무한 급수

리딩 스텝 : 1 은?! 여기 모든 정보를 체계화 하고 싶다. 그러면

소개

마침내 그들의 꿈은 현실이 됐다 어제와 오늘, 미래를 가로지르며 컴퓨터라는 성을 쌓은 300 년에 걸친 여정

컴퓨터와 수학이 어디선가 연결된다고 누구나 어렴풋이 예상은 할 것이다. 하지만 초기 컴퓨터의 명령어들은 너무나 간단하여 수학과 어떻게 연결되는지 알기 어렵다. 이 책은 그런 수학적 발견이 탄생하기 전으로 돌아가서, 현대 컴퓨터의 근간을 이루는 아이디어와 그들이 처했던 삶의 배경을 보여 준다.

17 세기부터 20 세기까지 약 300 년에 걸친 눈부신 혁신자들의 삶은 각자 달랐지만, 그들은 모두 인간이 생각하는 방식의 근원을 찾고자 했다. 각각의 공헌은 촘촘하게 지식의 기반을 만들었고 범용 디지털 컴퓨터를 가능케 했다. 오늘날 컴퓨터 기술이 눈부신 속도로 발전하고 사람들은 공학 기술의 놀라운 성취에 감탄하지만, 이 모든 걸 가능케 한 사람들은 쉽게 간과하곤 한다. 이 책은 그들에 대한 이야기다.

주요 인물 약력

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Gottfried Wilhelm Leibniz (1 July 1646 [O.S. 21 June] – 14 November 1716)

Gottfried Wilhelm Leibniz (1 July 1646 [O.S. 21 June] – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who invented calculus in addition to many other branches of mathematics and statistics. Leibniz has been called the “last universal genius” due to his knowledge and skills in different fields and because such people became less common during the Industrial Revolution and spread of specialized labor after his lifetime. He is a prominent figure in both the history of philosophy and the history of mathematics. He wrote works on philosophy, theology, ethics, politics, law, history, philology, games, music, and other studies. Leibniz also made major contributions to physics and technology, and anticipated notions that surfaced much later in probability theory, biology, medicine, geology, psychology, linguistics and computer science. In addition, he contributed to the field of library science by devising a cataloguing system whilst working at the Herzog August Library in Wolfenbüttel, Germany, that would have served as a guide for many of Europe’s largest libraries. Leibniz’s contributions to a wide range of subjects were scattered in various learned journals, in tens of thousands of letters and in unpublished manuscripts. He wrote in several languages, primarily in Latin, French and German.As a philosopher, he was a leading representative of 17th-century rationalism and idealism. As a mathematician, his major achievement was the development of the main ideas of differential and integral calculus, independently of Isaac Newton’s contemporaneous developments. Mathematicians have consistently favored Leibniz’s notation as the conventional and more exact expression of calculus.In the 20th century, Leibniz’s notions of the law of continuity and transcendental law of homogeneity found a consistent mathematical formulation by means of non-standard analysis. He was also a pioneer in the field of mechanical calculators. While working on adding automatic multiplication and division to Pascal’s calculator, he was the first to describe a pinwheel calculator in 1685 and invented the Leibniz wheel, later used in the arithmometer, the first mass-produced mechanical calculator. In philosophy and theology, Leibniz is most noted for his optimism, i.e. his conclusion that our world is, in a qualified sense, the best possible world that God could have created, a view sometimes lampooned by other thinkers, such as Voltaire in his satirical novella Candide. Leibniz, along with René Descartes and Baruch Spinoza, was one of the three influential early modern rationalists. His philosophy also assimilates elements of the scholastic tradition, notably the assumption that some substantive knowledge of reality can be achieved by reasoning from first principles or prior definitions. The work of Leibniz anticipated modern logic and still influences contemporary analytic philosophy, such as its adopted use of the term “possible world” to define modal notions.

고트프리트 빌헬름 라이프니츠(1646 년 7 월 1 일[양력 6 월 21 일] ~ 1716 년 11 월 14 일)는 수학자, 철학자, 과학자, 외교관으로 활동한 독일의 다원 수학자로 미적분학을 비롯한 여러 수학 및 통계 분야를 발명한 인물입니다. 라이프니츠는 여러 분야에 걸친 지식과 기술, 그리고 그의 생애 이후 산업혁명과 전문 노동의 확산으로 인해 그런 사람이 흔하지 않게 되었기 때문에 ‘마지막 보편적 천재’로 불려왔습니다. 그는 철학사와 수학사 모두에서 저명한 인물입니다. 그는 철학, 신학, 윤리, 정치, 법, 역사, 문헌학, 게임, 음악 및 기타 연구에 관한 작품을 썼습니다. 라이프니츠는 물리학 및 기술 분야에도 큰 공헌을 했으며, 확률 이론, 생물학, 의학, 지질학, 심리학, 언어학, 컴퓨터 과학 분야에서 훨씬 후에 등장한 개념들을 예견했습니다. 또한 독일 볼펜뷔텔의 헤르초크 아우구스트 도서관에서 일하면서 유럽 최대 도서관의 지침이 된 목록 시스템을 고안하여 도서관 과학 분야에도 기여했습니다. 라이프니츠는 다양한 주제에 대해 다양한 학술지, 수만 통의 편지, 미발표 원고에 걸쳐 많은 공헌을 남겼습니다. 그는 주로 라틴어, 프랑스어, 독일어 등 여러 언어로 글을 썼으며 철학자로서 17 세기 합리주의와 이상주의를 대표하는 인물입니다. 수학자로서 그의 주요 업적은 아이작 뉴턴의 동시대 발전과는 별개로 미분과 적분 미적분의 주요 개념을 발전시킨 것입니다. 수학자들은 미적분학의 전통적이고 보다 정확한 표현으로 라이프니츠의 표기법을 일관되게 선호해 왔으며, 20 세기에는 라이프니츠의 연속의 법칙과 초월적 동질성의 법칙 개념이 비표준 분석을 통해 일관된 수학적 공식을 발견했습니다. 그는 또한 기계식 계산기 분야의 선구자이기도 했습니다. 파스칼의 계산기에 자동 곱셈과 나눗셈을 추가하는 작업을 하던 중 1685 년 바람개비 계산기를 최초로 설명했고, 이후 최초의 대량 생산 기계식 계산기인 아르키메트리에 사용된 라이프니츠 휠을 발명했습니다. 철학과 신학에서 라이프니츠는 낙관주의, 즉 우리가 사는 세계가 신이 창조할 수 있는 최상의 세계라는 결론으로 가장 유명하며, 이러한 견해는 볼테르의 풍자 소설 『캉디드』에서 다른 사상가들의 조롱을 받기도 했습니다. 라이프니츠는 르네 데카르트, 바룩 스피노자와 함께 영향력 있는 초기 근대 합리주의자 세 명 중 한 명입니다. 그의 철학은 또한 스콜라 철학 전통의 요소, 특히 첫 번째 원리 또는 사전 정의로부터 추론함으로써 현실에 대한 실체적 지식을 얻을 수 있다는 가정을 받아들였습니다. 라이프니츠의 연구는 현대 논리학을 예상했으며, 모달 개념을 정의하기 위해 “가능한 세계"라는 용어를 채택하는 등 현대 분석 철학에 여전히 영향을 미치고 있습니다.

Preface to Third Edition

For this edition, I have had the opportunity to write about deep learning algorithms and the astonishing AlphaGo program that defeats human masters of the ancient game of Go. I tried to disentangle what is known about the relationship between Georg Cantor and Leopold Kronecker from the widespread myths about their relationship. In addition, I have detailed the drama of ideas in Princeton during the 1930s involving Alonzo Church, his student Stephen Kleene, and Kurt Gödel, before Alan Turing’s arrival.

이번 호에서는 딥러닝 알고리즘과 고대 바둑의 인간 고수들을 물리친 놀라운 알파고 프로그램에 대해 글을 쓸 기회를 가졌습니다. 저는 게오르그 캔터와 레오폴드 크로네커의 관계에 대해 널리 퍼져 있는 신화에서 그들의 관계에 대해 알려진 내용을 풀어내려고 노력했습니다. 또한 앨런 튜링이 등장하기 전인 1930 년대 프린스턴에서 알론조 처치, 그의 제자 스티븐 클라인, 커트 괴델과 관련된 아이디어의 드라마를 자세히 설명했습니다.

I am grateful to Susan Dickie, Harold Edwards, Dana Scott and François Treves who helped me in various ways. I’m particularly grateful to Thore Graepelof DeepMind for patiently explaining AlphaGo to me. Finally, I want to thank Sarfraz Khan, an editor who understood what I was trying to do and shared my enthusiasm for the project.

다양한 방법으로 저를 도와준 Susan Dickie, Harold Edwards, Dana Scott, François Treves 에게 감사드립니다. 특히 인내심을 갖고 알파고에 대해 설명해준 딥마인드의 토르 그레펠로프에게 감사드립니다. 마지막으로 제가 하고자 하는 일을 이해하고 프로젝트에 대한 열정을 공유해 준 편집자 사프라즈 칸에게 감사의 말을 전하고 싶습니다.

Martin Davis

Berkeley, 2017

Preface to Second Edition

Alan Turing was born on June 23, 1912. The year 2012, the hundredth anniversary of his birth, is providing the occasion for events and publications reflecting on and celebrating his achievements. I am delighted that this updated version of my book will be part of the excitement, and I am very grateful to Klaus Peters and Alice Peters for their help in making it happen. For this edition, I’ve tidied up some loose ends and brought a few things up to date, even commenting on IBM’s achievement in fielding its Watson computer as a successful contestant on the popular television quiz program Jeopardy.

The “universality” of the computers with which we interact today is evident in the myriad unrelated tasks for which the computers that are on our desks or our laps are used, but also in the hidden computers that are embedded in many other devices. If our cameras are computers with lenses and our telephones are computers with microphones and earphones, it could almost be said that the hybrid automobile I drive is a computer with four wheels.

Today it is widely recognized that this universality is an application of a fundamental insight from an article by Turing published in a mathematical journal in 1936. When I began researching these matters in the 1980s, much controversy surrounded credit for the first “stored program” electronic computers, but Turing’s name was never mentioned. The argument was whether the credit was due to the mathematician John von Neumann or to the engineers John Presper Eckert and John Mauchly. David Leavitt kindly suggested that I am responsible for the recognition of Turing’s role. While an article I wrote may have had some influence, probably the publication of some of Turing’s previously unavailable work from the 1940s together with the availability of information about his secret work towards decrypting enemy military communications during the Second World War was more important ¹.

This is a book of stories about seven remarkable people, their ideas and discoveries, and their fascinating lives. They investigated the why and how of logical reasoning. They forced the exhilaration and pitfalls of trying to come to grips with the infinite. Their heroic efforts to buttress the claims of rationality encountered unforeseen obstacles. Finally, Alan Turing’s radically new understanding of the nature of algorithmic processes and its potential to make a single “all-purpose” machine that could be programmed to carry out almost any process was a by-product of this tumultuous development. I had great fun writing this book, and I hope that you will have fun reading it.

Berkeley, June 30, 2011

Preface

This book is about the underlying concepts on which our modern computers are based and about the people who developed these concepts. In the spring of 1951, shortly after completing my doctorate in mathematical logic at Princeton University where Alan Turing worked a decade earlier, I was teaching a course at the University of Illinois based on his ideas. A young mathematician who attended my lectures called my attention to a pair of machines being constructed across the street from my classroom that he insisted were embodiments of Turing’s conception. It was not long before I found myself writing software for early computers. My professional career spanning half a century revolved around this relationship between the abstract logical concepts underlying modern computers and their physical realization.

As computers evolved from the room-filling behemoths of the 1950s to the small powerful machines of today performing a bewildering variety of tasks, their underlying logic has remained the same. These logical concepts developed from the work of a number of gifted thinkers over centuries. In this book I tell the stories of the lives of these people and explain some of their thoughts. The stories are fascinating. My hope is that readers will not only enjoy them, but will come away with a better sense of what goes on inside their computers and with enhanced respect for the value of abstract thought.

In developing this book I benefited from help of various kinds. The John Simon Guggenheim Memorial Foundation provided welcome financial support during the early stages of the studies that led to this book. Patricia Blanchette, Michael Friedman, Andrew Hodges, Lothar Kreiser, and Benson Mates generously shared their expert knowledge with me. Tony Sale kindly acted as my guide to Bletchley Park where Turing played an important part in the decoding of secret German military communications during World War II. Eloise Segal, who alas did not live to see the book completed, was a devoted reader who helped me avoid expository pitfalls. My wife, Virginia, stubbornly refused to let me be obscure. Sherman Stein read the manuscript carefully and suggested many improvements while saving me from a number of errors. I benefited from help with translations by Egon Börger, William Craig, Michael Richter, Alexis Manaster Ramer, Wilfried Sieg, and Francois Treves. Other readers who provided useful comments were Harold Davis, Nathan Davis, Jack Feldman, Meyer Garber, Dick and Peggy Kuhns, and Alberto Policriti. My editor, Ed Barber at W.W. Norton, generously shared his knowledge of English prose style and is responsible for many improvements. Harold Rabinowitz introduced me to my agent, Alex Hoyt, who has been unfailingly helpful. Of course this long list of names is meant only to express gratitude and not to absolve myself of responsibility for the book’s shortcomings. I would be grateful for comments or corrections from readers sent to me at: [[mailto:davis@eipye.com][davis@eipye.com.

Martin Davis

Berkeley, January 2, 2000

Introduction

If it should turn out that the basic logics of a machine designed for the numerical solution of differential equations coincide with the logics of a machine intended to make bills for a department store, I would regard this as the most amazing coincidence I have ever encountered.

—Howard Aiken 1956 ²

Let us now return to the analogy of the theoretical computing machines … It can be shown that a single special machine of that type can be made to do the work of all. It could in fact be made to work as a model of any other machine. The special machine may be called the universal machine …

—Alan Turing 1947³

In the fall of 1945, as the ENIAC, a gigantic calculating engine containing thousands of vacuum tubes, neared completion at the Moore School of Electrical Engineering in Philadelphia, a group of experts met regularly to discuss the design of its proposed successor, the EDVAC. As the weeks went by, the meetings grew increasingly acrimonious, with the experts finding themselves divided into two groups they dubbed the “engineers” and the “logicians.” John Presper Eckert, leader of the “engineers,” was justly proud of his accomplishment with the ENIAC. It had been thought impossible for 15,000 hot vacuum tubes to work together long enough without any of them failing, for anything useful to be accomplished. Nevertheless, by using careful conservative design principles, Eckert had succeeded brilliantly in accomplishing this feat. Things came to a head when, much to Eckert’s displeasure, the group’s leading “logician,” the eminent mathematician John von Neumann, circulated, under his own name, a draft report on the proposed EDVAC that, paying little attention to engineering details, set forth the fundamental logical computer design known to this day as the von Neumann architecture.

Although an engineering tour de force, the ENIAC was a logical mess. It was von Neumann’s expertise as a logician and what he had learned from the English logician Alan Turing that enabled him to understand the fundamental fact that a computing machine is a logic machine. In its circuits are embodied the distilled insights of a remarkable collection of logicians, developed over centuries. Nowadays, when computer technology is advancing with such breathtaking rapidity, as we admire the truly remarkable accomplishments of the engineers, it is all too easy to overlook the logicians whose ideas made it all possible. This book tells their story.

CHAPTER 1 Leibniz’s Dream : 라이프니츠의 꿈

Introduction

미분 적분 표기법 수학 언어 -> 모든 지식 표현

Situated southeast of the German city of Hanover, the ore-rich veins of the Harz mountain region had been mined since the middle of the tenth century. Because the deeper parts tended to fill with water, they could only be mined so long as pumps kept the water at bay. During the seventeenth century water wheels powered these pumps. Unfortunately, this meant that the lucrative mining operations had to shut down during the cold mountain winter season when the streams were frozen.

During the years 1680–1685, the Harz mountain mining managers were in frequent conflict with a most unlikely miner. G. W. Leibniz, then in his middle thirties, was there to introduce windmills as an additional energy source to enable all-season operation of the mines. At this point in his life, Leibniz had already accomplished a lot. Not only had he made major discoveries in mathematics, he had also acquired fame as a jurist, and had written extensively on philosophical and theological issues. He had even undertaken a diplomatic mission to the court of Louis XIV in an attempt to convince the French “Sun King” of the advantages of conducting a military campaign in Egypt (instead of against Holland and German territories).

Some 70 years earlier, Cervantes had written of the misadventures of a melancholy Spaniard with windmills. Unlike Don Quixote, Leibniz was incurably optimistic. To those who reacted bitterly to the widespread misery in the world, Leibniz responded that God, from His omniscient view of all possible worlds, had unerringly created the best that could be constructed, that all the evil elements of our world were balanced by good in an optimal manner.*

But Leibniz’s involvement with the Harz Mountain mining project ultimately proved to be a fiasco. In his optimism, he had not foreseen the natural hostility of the expert mining engineers towards a novice proposing to teach them their trade. Nor had he allowed for the inevitable break-in period a novel piece of machinery requires or for the unreliability of the winds. But his most incredible piece of optimism was with respect to what he had imagined he would be able to accomplish with the proceeds he had expected from the project.

Leibniz had a vision of amazing scope and grandeur. The notation he had developed for the differential and integral calculus, the notation still used today, made it easy to do complicated calculations with little thought. It was as though the notation did the work ⁴.

In Leibniz’s vision, something similar could be done for the whole scope of human knowledge. He dreamt of an encyclopedic compilation, of a universal artificial mathematical language in which each facet of knowledge could be expressed, of calculational rules which would reveal all the logical interrelationships among these propositions. Finally, he dreamed of machines capable of carrying out calculations, freeing the mind for creative thought. Even with his optimism, Leibniz knew that the task of transforming this dream to reality was not something he could accomplish alone. But he did believe that a small number of capable people working together in a scientific academy could accomplish much of the design in a few years. It was to fund such an academy that Leibniz embarked on his Harz Mountain project.

Leibniz’s Wonderful Idea

아리스토텔레스 형이상학

Leibniz was born in Leipzig in 1646 into a Germany divided into something like 1,000 separate, semiautonomous political units, and devastated by almost three decades of war. The Thirty Years War, which didn’t end until 1648, was fought mainly on German soil, although all of the major European powers had participated. Leibniz’s father, a professor of philosophy at the University of Leipzig, died when the child was only six. Over the opposition of his teachers, Leibniz gained access to his father’s library at the age of eight, and soon became a fluent reader of Latin.

라이프니츠는 1646 년 라이프치히에서 태어나 1,000 여 개의 개별적인 반자치 정치 단위로 분열되고 거의 30 년에 걸친 전쟁으로 황폐화된 독일에서 태어났습니다. 1648 년까지 끝나지 않은 30 년 전쟁은 유럽의 주요 강대국들이 모두 참전했지만 주로 독일 땅에서 벌어졌습니다. 라이프치히 대학의 철학과 교수였던 라이프니츠의 아버지는 아이가 겨우 여섯 살이었을 때 세상을 떠났습니다. 라이프니츠는 선생님의 반대를 무릅쓰고 8 세 때 아버지의 도서관을 이용할 수 있게 되었고, 곧 라틴어를 유창하게 읽을 수 있게 되었습니다.

Leibniz, destined to become one of the greatest mathematicians of all time, got his first introduction to mathematical ideas from teachers who had no inkling of the exciting work elsewhere in Europe that was revolutionizing mathematics. In the Germany of that day, even the elementary geometry of Euclid was an advanced subject, studied only at the university level. However, in his early teens, his school teachers did introduce Leibniz to the system of logic that Aristotle had developed two millennia earlier, and this was the subject that aroused his mathematical talent and passion.

Fascinated by the Aristotelian division of concepts into fixed “categories,” Leibniz thought of what he came to call his “wonderful idea”: he would seek a special “alphabet” whose elements represented not sounds, but concepts. A language based on such an alphabet should make it possible to determine by symbolic calculation which sentences written in the language were true and what logical relationships existed among them. Leibniz remained under Aristotle’s spell and held fast to this vision for the rest of his life.

Indeed, for his bachelor’s degree at Leipzig, Leibniz wrote a thesis on Aristotelian metaphysics. His master’s thesis at the same university dealt with the relationship between philosophy and law. Evidently attracted to legal studies, Leibniz obtained a second bachelor’s degree, this time in law, writing a thesis emphasizing the use of systematic logic in dealing with the law. Leibniz’s first real contribution to mathematics developed out of his Habilitationsschrift (in Germany, a kind of second doctoral dissertation) in philosophy also at Leipzig: As a first step towards his “wonderful idea” of an alphabet of concepts, Leibniz foresaw the need to be able to count the various ways of combining such concepts. This led him to a systematic study of the problem of counting complex arrangements of basic elements, first in his Habilitationsschrift and then in his more extensive monograph Dissertatio de Arte Combinatoria.²⁵

Continuing his legal studies, Leibniz presented a dissertation for a doctorate in law at the University of Leipzig. The subject, so typical for Leibniz, was the use of reason to resolve cases in law thought too difficult for resolution by the normal methods. For reasons that are not clear the Leipzig faculty refused to accept the dissertation, so Leibniz presented it instead at the University of Altdorf, near Nuremberg where it was received with acclaim. At the age of 21, his formal education completed, Leibniz faced the common problem of the newly graduated: how to develop a career.

Paris (1672 ~ )

후원자를 찾아 그리고 루이 14 세

Not being interested in a career as a university professor in Germany, Leibniz pursued his only real alternative: to find a wealthy noble patron. Baron Johann von Boineburg, nephew of the Elector of Mainz, was quite willing to play this role. In Mainz, Leibniz worked on a project to update the legal system based on Roman civil law, was appointed a judge at the High Court of Appeal, and tried his hand at diplomatic intrigue. This last included an abortive attempt to influence the selection of a new king for Poland and a mission to the court of Louis XIV.

독일에서 대학교수로서의 경력에는 관심이 없던 라이프니츠는 유일한 대안으로 부유한 귀족 후원자를 찾는 길을 택했습니다. 마인츠 총독의 조카인 요한 폰 바인부르크 남작이 이 역할을 기꺼이 맡아주었습니다. 마인츠에서 라이프니츠는 로마 민법에 기반한 법률 시스템을 업데이트하는 프로젝트에 참여했고, 고등항소법원의 판사로 임명되었으며, 외교적 음모에 손을 댔습니다. 마지막에는 폴란드의 새 왕 선출에 영향을 미치려다 실패한 시도와 루이 14 세의 궁정 사절단 파견도 포함되었습니다.

30 년 전쟁 -> 프랑스 : 100 년 후 나폴레옹 이집트 원정

The Thirty Years War had left France as the “superpower” on the European continent. Mainz, situated on the banks of the Rhine, had known military occupation during the war. So, the burghers of Mainz understood very well the importance of forestalling hostile military action, and therefore, of good relations with France. It was in this context that Boineburg and Leibniz concocted the scheme, already mentioned, to convince Louis XIV and his advisers of the great advantages of making Egypt the object of their military endeavors. The most important historical effect of this proposition—essentially the same proposition that led Napoleon to a military disaster over a century later—was that it brought Leibniz to Paris.

파리 정착 - 사칙 연산 기계 모델 : 라이프니츠의 휠 -> 런던 왕립학회 회원

Leibniz arrived in Paris in 1672 to press the Egyptian scheme and to help untangle some of Boineburg’s financial affairs. Before the end of the year disaster struck: the news came that Boineburg had died of a stroke. Although he continued to perform some services for the Boineburg family, Leibniz was left without a reliable source of income. Nevertheless he managed to remain in Paris for another four extremely productive years that included two brief visits to London.³⁶ On the first visit in 1673, he was unanimously elected to the Royal Society of London based on his model of a calculating machine capable of carrying out the four basic operations of arithmetic. Although Pascal had designed a machine that could add and subtract, Leibniz’s was the first that could multiply and divide as well.* Leibniz’s machine incorporated an ingenious gadget that became known as a “Leibniz wheel.” Calculating machines continued to be built incorporating this device well into the twentieth century. About his machine, Leibniz wrote:

라이프니츠 왈 기계는

And now that we may give final praise to the machine we may say that it will be desirable to all who are engaged in computations which, it is well known, are the managers of financial affairs, the administrators of others’ estates, merchants, surveyors, geographers, navigators, astronomers … But limiting ourselves to scientific uses, the old geometric and astronomic tables could be corrected and new ones constructed by the help of which we could measure all kinds of curves and figures … it will pay to extend as far as possible the major Pythagorean tables; the table of squares, cubes, and other powers; and the tables of combinations, variations, and progressions of all kinds, … Also the astronomers surely will not have to continue to exercise the patience which is required for computation. … For it is unworthy of excellent men to lose hours like slaves in the labor of calculation which could safely be relegated to anyone else if the machine were used.⁴⁷

그리고 이제 우리는 기계에 대한 최종 칭찬을 할 수 있으므로 잘 알려진 바와 같이 재정 업무 관리자, 다른 사람의 재산 관리자, 상인, 측량사, 지리학자, 네비게이터, 천문학 자 등 계산에 종사하는 모든 사람에게 바람직 할 것이라고 말할 수 있습니다 … 그러나 과학적 용도로 제한하면 오래된 기하학적 및 천문학적 표를 수정하고 모든 종류의 곡선과 그림을 측정 할 수있는 도움으로 새로운 표를 만들 수 있습니다 … 주요 피타고라스 표, 사각형, 입방체 및 기타 힘의 표, 모든 종류의 조합, 변형 및 진행 표를 가능한 한 멀리 확장하는 것이 좋습니다 …. 또한 천문학자들은 계산에 필요한 인내심을 계속 발휘할 필요가 없을 것입니다. … 기계를 사용한다면 누구에게나 안전하게 맡길 수 있는 계산의 노동에 노예처럼 시간을 허비하는 것은 훌륭한 인간에게 합당하지 않기 때문이다.⁴

곱셉 나눗셈이 가능하기에

The machine Leibniz was “praising” was limited to ordinary arithmetic. But Leibniz grasped the broader significance of mechanizing calculation. In 1674 he described a machine that could solve algebraic equations. A year later, he wrote comparing logical reasoning to a mechanism, thus pointing to the goal of reducing reasoning to a kind of calculation and of ultimately building a machine capable of carrying out such calculations.⁵⁸

네델란드 하위헌스를 만나다

A crucial event for Leibniz, then 26, was meeting the great Dutch scientist Christiaan Huygens then living in Paris. The 43-year-old Huygens had already invented the pendulum clock and discovered the rings of Saturn. What was perhaps to be his most important contribution, the wave theory of light, was still to come. His conception—that light was fundamentally to be viewed like the waves spreading across a pond when a pebble is tossed into it—directly contradicted the great Newton’s account of light as consisting of a stream of discrete bullet-like particles.* Huygens gave Leibniz a reading list enabling the younger man to quickly overcome his lack of knowledge of current mathematical research. Soon Leibniz was making fundamental contributions.

당시 26 세였던 라이프니츠에게 결정적인 사건은 당시 파리에 살고 있던 네덜란드의 위대한 과학자 크리스티안 후이겐스를 만난 것이었습니다. 당시 43 세였던 후이겐스는 이미 진자 시계를 발명하고 토성의 고리를 발견한 바 있었습니다. 하지만 그의 가장 중요한 업적이라 할 수 있는 빛의 파동 이론은 아직 나오지 않은 상태였습니다. 빛은 근본적으로 조약돌을 연못에 던졌을 때 연못에 퍼지는 물결처럼 보아야 한다는 그의 생각은 빛이 총알 같은 입자의 흐름으로 이루어진다는 위대한 뉴턴의 설명과 정면으로 모순되었습니다.* 후이겐스는 라이프니츠가 최신 수학 연구에 대한 지식 부족을 빠르게 극복할 수 있도록 독서 목록을 제공했습니다. 곧 라이프니츠는 근본적인 기여를 하고 있었습니다.

왜 17 세기에 수학이 발전 했는가?

대수학 표현 및 풀기 기하학이 대수학으로 표현 됨

The explosion of mathematical research in the seventeenth century had been fueled by two crucial developments:

• 1. The technique of dealing with algebraic expressions (what is

generally called “high-school algebra”) had been systematized and became essentially the powerful technique we still use today.

• 2. Descartes and Fermat had shown how, by representing points by pairs

of numbers, geometry could be reduced to algebra.

근사치를 이용한 문제 해결 극한

Various mathematicians were using this new power to solve problems that would not previously have been accessible. Much of this work involved what nowadays are called limit processes. Using limits means solving a problem by using approximations to the required answer that get systematically closer and closer to that answer. The idea was not to be satisfied with any particular approximation, but rather, by “going to the limit,” to obtain an exact solution.

다양한 수학자들이 이 새로운 힘을 이용해 이전에는 접근이 불가능했던 문제를 해결했습니다. 이러한 작업의 대부분은 오늘날 ‘극한 프로세스’라고 불리는 것과 관련이 있습니다. 극한을 사용한다는 것은 필요한 해답에 체계적으로 점점 더 가까워지는 근사치를 사용하여 문제를 해결하는 것을 의미합니다. 특정 근사치에 만족하는 것이 아니라 “극한까지 가서” 정확한 해를 구하는 것이 목표였습니다.

라이프니츠 무한급수

An example that may help to clarify this concept is one of Leibniz’s own early results, one of which he was quite proud. This was the equation:

\(\frac{\pi}{4}\ = \ 1\ - \ \frac{1}{3}\ + \ \frac{1}{5}\ - \ \frac{1}{7}\ + \ \frac{1}{9}\ - \ \frac{1}{11}\ + \ \cdots\)

\[ \frac{\pi}{4}\ = \ 1\ - \ \frac{1}{3}\ + \ \frac{1}{5}\ - \ \frac{1}{7}\ + \ \frac{1}{9}\ - \ \frac{1}{11}\ + \ \cdots \]

On the left side of the equals sign is the familiar number π that occurs in the formulas for the circumference and the area of a circle.* On the right side is what is called an infinite series; the individual numbers alternately added and subtracted are called the terms of the series. The dots … mean that it continues indefinitely. The full infinite pattern consists of fractions, with 1 as numerator and the successive odd numbers as denominators, being alternately added and subtracted, and is intended to be clear from the finite part shown: after subtracting \(\frac{1}{11}\), add \(\frac{1}{13}\), then subtract \(\frac{1}{15}\), etc. But can one actually perform an infinite number of additions and subtractions? Not really. But, starting at the beginning and breaking off at any point, an approximation to a “true” answer is obtained, and that approximation gets better and better as more terms are included. In fact, the approximation can be made as accurate as one wishes by including enough terms. In the table on page 7, it is shown how this works out for Leibniz’s series. When including 10,000,000 terms, a value is obtained that agrees with the true value of \(\frac{\pi}{4}\), namely 0.7853981634 …, to eight places.^†

Table of approximations to Leibniz’s series

Leibniz’s series is so striking because it connects the number π, and therefore the area of a circle, with the succession of odd numbers in a particularly simple way. It is an example of one kind of problem that could be solved using limit processes, that of finding areas of figures with curved boundaries.

라이프니츠의 급수는 숫자 π, 즉 원의 넓이를 홀수의 연속과 매우 간단한 방식으로 연결하기 때문에 매우 인상적입니다. 이는 극한 과정을 사용하여 해결할 수 있는 문제 중 하나인 곡선 경계를 가진 도형의 넓이를 구하는 문제의 한 예입니다.

미적분학의 발명

원주율 (원의 넓이)를 간단한 홀수 패턴으로 표기 달라지는 속도 처럼 변화의 비율을 구하는 것

Another kind of problem susceptible to attack using limits was finding exact rates of change, such as the varying speed of a moving body. During the last months of 1675, towards the end of his stay in Paris, Leibniz made a number of conceptual and computational breakthroughs in the use of limit processes that, taken together, are called his “invention of the calculus”:

극한을 이용해 공격하기 쉬운 또 다른 종류의 문제는 움직이는 물체의 변화하는 속도와 같은 정확한 변화율을 찾는 것이었습니다. 1675 년 파리 체류 마지막 달에 라이프니츠는 극한 과정을 사용하여 여러 개념적, 계산적 혁신을 이루었으며, 이를 종합하여 “미적분의 발명"이라고 불렀습니다:

• 1. Leibniz saw that the problems of finding areas and calculating rates

of change were paradigmatic, in the sense that many different kinds of problems were reducible to one or the other of these two types.*

다양한 종류의 문제가 이 두 가지 유형 중 하나 또는 다른 유형으로 환원될 수 있다는 점에서 변화의 패러다임이 바뀌었습니다*

• 2. He also perceived that the mathematical operations required in

calculating the solutions to problems of these two types were in fact inverse to each other in much the same sense that the operations of addition and subtraction (or multiplication and division) are inverse to one another. Nowadays these operations are called integration and differentiation, respectively, and the fact that they are inverse is called, in the textbooks, the “fundamental theorem of the calculus.”

이 두 가지 유형의 문제에 대한 해를 계산하는 것은 사실 덧셈과 뺄셈 (또는 곱셈과 나눗셈) 연산이 서로 반대인 것과 거의 같은 의미에서 서로 /반대/였습니다. 오늘날에는 이러한 연산을 각각 /적분/과 /미분/이라고 부르며, 교과서에서는 이러한 연산이 역이라는 사실을 “미적분학의 기본 정리"라고 부릅니다

• 3. Leibniz developed an appropriate symbolism (the notation still in

use today) for these operations, ∫ for integration and d for differentiation.^† Finally he found the mathematical rules needed for carrying out the integrations and differentiations that occurred in practice.

오늘날 사용)를 이러한 연산에 사용하고, ∫는 적분, d 는 미분에 사용합니다.^† 마침내 그는 실제로 발생하는 적분과 미분을 수행하는 데 필요한 수학적 규칙을 발견했습니다.

기호를 선택하고 기호를 조작하는 법칙이 얼마나 중요한가!

Taken together these discoveries transformed the use of limit processes, from an exotic method accessible only to a handful of specialists, into a straightforward technique that could be taught in textbooks to many thousands of people.⁶⁹ Most important for the purposes of this book, his success convinced Leibniz of the critical importance of choosing appropriate symbols and finding the rules governing their manipulation. The symbols ∫ and d did not represent meaningless sounds like the letters of a phonetic alphabet; they stood for concepts and thus provided a model for Leibniz’s boyhood “wonderful idea” of an alphabet representing all fundamental concepts.

이러한 발견을 종합하면, 소수의 전문가만 접근할 수 있는 이색적인 방법이었던 극한 과정의 사용이 교과서에서 수천 명의 사람들에게 가르칠 수 있는 간단한 기법으로 바뀌었습니다.⁶ 이 책의 목적에 가장 중요한 것은 그의 성공을 통해 라이프니츠가 적절한 기호를 선택하고 그 조작을 지배하는 규칙을 찾는 것이 매우 중요하다는 것을 확신하게 되었다는 점입니다. 기호 ∫와 d/는 음성 알파벳의 글자처럼 의미 없는 소리를 나타내는 것이 아니라 개념을 나타내며, 따라서 /모든 기본 개념/지식을 나타내는 알파벳이라는 소년 시절 라이프니츠의 “멋진 아이디어"에 대한 모델을 제공했습니다.

Much has been written about the entirely independent development of the calculus by Newton and by Leibniz, and about the bitter accusations of plagiarism tossed back and forth across the English Channel before the foolishness of such charges was finally understood by all. It is the great superiority of Leibniz’s notation that is significant for our story.⁷¹⁰ A key technique used in integration (called in the textbooks, the method of “substitution”) is virtually automatic in Leibniz’s notation, but relatively complicated in Newton’s. It has even been alleged that slavish devotion to their national hero’s methods caused the English followers of Newton to lag far behind their continental contemporaries in developing the mathematical perspectives that the calculus had uncovered.

뉴턴과 라이프니츠에 의한 미적분학의 완전히 독립적인 발전, 그리고 그러한 혐의의 어리석음이 마침내 모든 사람에게 이해되기 전에 영국 해협을 가로질러 앞뒤로 던져진 표절에 대한 격렬한 비난에 대해 많은 글이 쓰여졌습니다. 이 이야기에서 중요한 것은 라이프니츠 표기법의 위대한 우월성입니다 ^{7} 적분에서 사용되는 핵심 기법(교과서에서는 ‘치환 적분법’이라고 부름)은 라이프니츠 표기법에서는 거의 자동이지만 뉴턴의 표기법에서는 상대적으로 복잡합니다. 심지어 뉴턴의 영국 추종자들은 그들의 국가적 영웅의 방법에 대한 맹목적인 헌신으로 인해 미적분이 발견한 수학적 관점을 발전시키는 데 있어 유럽 동시대인들에 비해 훨씬 뒤처졌다는 주장도 제기되었습니다.

Like so many who have tasted the special quality of life in Paris, Leibniz wanted very much to remain there as long as he could. He attempted to maintain his Mainz connections while continuing to live and work in Paris. But it soon became clear that, so long as he remained in Paris no funds from Mainz would be forthcoming.

파리의 특별한 삶의 질을 맛본 많은 사람들처럼 라이프니츠도 가능한 한 파리에 오래 머물고 싶었습니다. 그는 파리에서 계속 거주하고 일하면서 마인츠의 인맥을 유지하려고 노력했습니다. 그러나 곧 그가 파리에 머무는 한 마인츠에서 지원금을 받을 수 없다는 사실이 분명해졌습니다.

Meanwhile an offer of a position arrived from the Dukedom of Hanover, one of the multitude of principalities of which seventeenth century Germany was composed. Although Duke Johann Friedrich had some genuine interest in intellectual matters, and the offer gave some promise of financial security, Leibniz was not eager to live in Hanover. After delaying as long as he could, Leibniz accepted the offer early in 1675.

한편 17 세기 독일을 구성했던 여러 공국 중 하나인 하노버 공국에서 자리 제안이 들어왔습니다. 요한 프리드리히 공작은 지적인 문제에 관심이 많았고 재정적인 보장도 약속했지만 라이프니츠는 하노버에서 살고 싶지 않았습니다. 최대한 시간을 끌다가 라이프니츠는 1675 년 초에 제안을 수락했습니다.

In his letter of acceptance, he asked for the “freedom to pursue his own studies in arts and sciences for the benefit of mankind.” ^{8}¹¹ In no hurry to leave Paris, he stayed until the fall of 1676, departing only when it became clear that no position in Paris would be forthcoming and that the Duke would accept no further delay. Leibniz was to spend the rest of his life in the service of the Dukes of Hanover.

그는 수락서에서 “인류를 위해 예술과 과학에 대한 자신의 연구를 추구할 수 있는 자유"를 요청했습니다 서둘러 파리를 떠나지 않고 1676 년 가을까지 머물렀고, 파리에서 어떤 자리도 나오지 않고 공작이 더 이상 지체할 수 없다는 것이 분명해졌을 때야 떠났습니다. 라이프니츠는 남은 여생을 하노버 공작을 위해 봉사하는 데 보내게 됩니다.

Hanover 하노버

“인류를 위해 학문과 과학 연구를 할 수 있는 자유” 와 현실

Leibniz apparently understood perfectly well that despite his request for “freedom to pursue his own studies in arts and sciences,” success in his new position would require him to do work that his patron would find useful and practical. He undertook to upgrade the ducal library and proposed various ideas for improving public administration and agriculture. Soon thereafter, he began promoting his ill-fated project to use windmills for improving the Harz Mountain mining operations. In 1680, only a year after the Harz project with Leibniz in charge had finally been approved, his position was suddenly endangered by the duke’s sudden death.

라이프니츠는 “예술과 과학에 대한 자신의 학문을 추구할 자유"를 요구했지만, 새로운 직책에서 성공하려면 후원자가 유용하고 실용적이라고 생각하는 일을 해야 한다는 사실을 잘 알고 있었습니다. 그는 공국의 도서관을 업그레이드하고 공공 행정과 농업을 개선하기 위한 다양한 아이디어를 제안했습니다. 얼마 지나지 않아 그는 하르츠 산의 광산 운영을 개선하기 위해 풍차를 사용하려는 불운한 프로젝트를 추진하기 시작했습니다. 1680 년 라이프니츠가 책임자로 있던 하르츠 프로젝트가 마침내 승인된 지 1 년 만에 공작의 갑작스러운 죽음으로 그의 지위는 갑자기 위태로워졌습니다.

It now became necessary to convince the new duke, Ernst August, to continue to found Leibniz’s position and to support the Harz Mountain project. The new duke was a practical man. Unlike his predecessor, he wasn’t willing to spend much on the library. Leibniz soon learned not to involve Ernst August in scholarly discussions.

이제 새로운 공작 에른스트 아우구스트가 라이프니츠의 지위를 계속 유지하고 하르츠 산 프로젝트를 지원하도록 설득해야 했습니다. 새 공작은 실용적인 사람이었습니다. 전임 공작과는 달리 그는 도서관에 많은 돈을 쓰지 않으려 했습니다. 라이프니츠는 곧 에른스트 아우구스트와 학문적인 이야기를 하지 않는 게 좋겠다는 결론을 얻었다.

가계도 프로젝트

To help cement his position, he offered to write a short history of the duke’s family. When the duke finally closed down the Harz project five years later, Leibniz proposed a more elaborate version: if a few gaps were filled, the family tree could be traced back to the year 600. The duke evidently regarded this as a most appropriate way to employ one of the greatest thinkers of all time. Leibniz was granted a regular salary, a personal secretary, and travel funds for searching out genealogical information. Most likely, the optimistic Leibniz hardly imagined that he would find himself chained to this project for the remaining three decades of his life. Georg Ludwig, who succeeded Ernst August on his death in 1698, was especially adamant in nagging Leibniz to get on with the family history.

자신의 입지를 확고히 하기 위해 그는 공작의 가족에 대한 짧은 역사를 쓰겠다고 제안했습니다. 5 년 후 공작이 마침내 하르츠 프로젝트를 마무리했을 때 라이프니츠는 몇 가지 공백을 메우면 가계도를 600 년으로 거슬러 올라갈 수 있다는 더 정교한 버전을 제안했습니다. 공작은 이것이 역사상 가장 위대한 사상가 중 한 명을 고용하는 데 가장 적합한 방법이라고 생각한 것 같습니다. 라이프니츠는 정기적인 급여와 개인 비서, 족보 정보 검색을 위한 여비 등을 제공받았습니다. 낙천적이었던 라이프니츠는 남은 인생의 30 년 동안 이 프로젝트에 묶여 있을 것이라고는 상상도 하지 못했을 것입니다. 1698 년 에른스트 아우구스트가 사망한 후 그 뒤를 이은 게오르크 루트비히는 특히 라이프니츠에게 가족사 연구를 계속하라고 잔소리를 퍼부었습니다.

여성에 대한 편견 없는 자

• 여성에 대한 편견 없이 대함으로서 제자들을 두게 되었으며 도움을 받았다.
• 여행 자유 지식인과 교류 서신과 편지

If Leibniz had any pupils in Hanover, they were women, for he shared none of the common prejudices concerning the intellectual capabilities of the female sex. Duchess Sophie, the talented wife of Ernst August, and Leibniz had frequent conversations about philosophical matters and carried on an extensive correspondence when Leibniz was away from Hanover. She made sure also that her daughter Sophie Charlotte, who was to become Queen of Prussia, also had the benefit of Leibniz’s teachings. Sophie Charlotte, not content simply to receive Leibniz’s wisdom, energetically raised questions that helped Leibniz to clarify his ideas. As the contemporary Leibniz scholar Benson Mates explains:

라이프니츠는 여성의 지적 능력에 대한 일반적인 편견을 전혀 공유하지 않았기 때문에 하노버에 제자가 있었다면 그들은 모두 여성이었습니다. 에른스트 아우구스트의 재능 있는 아내였던 소피 공작 부인과 라이프니츠는 철학적 문제에 대해 자주 대화를 나누었고 라이프니츠가 하노버를 떠나 있을 때에도 광범위한 서신을 주고받았습니다. 그녀는 프로이센의 왕비가 될 딸 소피 샬롯도 라이프니츠의 가르침을 받을 수 있도록 노력했습니다. 소피 샬롯은 단순히 라이프니츠의 지혜를 받는 것에 만족하지 않고 라이프니츠가 자신의 생각을 명확히 하는 데 도움이 되는 질문을 열정적으로 제기했습니다. 현대 라이프니츠 학자 벤슨 메이트는 이렇게 설명합니다:

For most of Leibniz’s life, these women were his principal advocates at the courts in Hanover and Berlin. Sophie Charlotte’s sudden death in 1705 devastated him; it was such an obvious loss to him that he even received formal expressions of sympathy from the emissaries of foreign governments; and when Duchess Sophie … died in 1714, his ability to obtain support for anything other than continuing the Brunswick history came to an end.⁹¹²

라이프니츠의 생애 대부분 동안 이 여성들은 하노버와 베를린의 법정에서 그의 주요 옹호자였습니다. 1705 년 소피 샬롯의 갑작스러운 죽음은 그에게 큰 충격을 주었고, 외국 정부의 사절단으로부터 공식적인 애도의 뜻을 전달받기도 했으며, 1714 년 소피 공작부인이 사망하자 브룬스윅의 역사를 이어가는 것 외에는 어떤 지원도 받을 수 없게 되었습니다.⁹

— Benson Mates

자유의 여행 그리고 신의 섭리 : 연결된 조화로움

The history project did provide Leibniz with an excuse to travel, and he made use of this freedom to an extent that vexed his noble patrons. Of course Leibniz took full advantage of the possibilities of developing and maintaining scholarly contacts. In Berlin he even was able to found a Society of Science, later institutionalized as an academy. His extensive correspondence continued to span the full variety of his interests.

역사 프로젝트는 라이프니츠에게 여행의 구실을 제공했고, 그는 고귀한 후원자들을 괴롭힐 정도로 이 자유를 활용했습니다. 물론 라이프니츠는 학문적 인맥을 개발하고 유지할 수 있는 가능성을 최대한 활용했습니다. 베를린에서 그는 나중에 왕립 학술원이 된 과학 학회를 설립할 수 있었습니다. 그의 광범위한 서신은 그의 모든 관심사에 걸쳐 계속 이어졌습니다.

Leibniz seemed never to tire of explaining that, since God had done as well as was possible in creating the world, there must be a pre-established harmony between what existed and what was possible and that there was a sufficient reason (whether or not we could find it) for every single thing in the world.

라이프니츠는 신이 세상을 창조할 때 계획한 대로 잘 해냈기 때문에 실제 존재하는 것과 가능했던 것 사이에는 /서로 연결된 조화로움/이 있어야 하며, 세상의 모든 사물에는 (우리가 그것을 찾을 수 있든 없든) /충분한 이유/가 있다고 믿었다.

In the realm of diplomacy, Leibniz had two pet projects: one was to reunite the various branches of the Christian church; the other, which actually succeeded, was to obtain for the Dukes of Hanover the succession to the British throne. But when Georg Ludwig actually did become George I of England only two years before Leibniz’s death in 1716, he brusquely rejected his employee’s request for permission to leave the Hanovarian backwater for London with his patron, ordering him to hurry up and finish the family history.

외교 영역에서 2 개의 프로젝트 진행 : 하나는 기독교 교회의 여러 분파를 재결합하는 것이었고, 실제로 성공한 다른 하나는 하노버 공작의 영국 왕위 계승권을 획득하는 것이었습니다. 그러나 라이프니츠가 죽기 불과 2 년 전인 1716 년에 게오르크 루트비히가 실제로 영국의 조지 1 세가 되었을 때, 그는 후원자와 함께 하노버를 떠나 런던 변두리로 가고 싶어했던 라이프니츠의 부탁을 퉁명스럽게 거절했다. 빨리 연대기를 완성하라는 지시만 남겼을 뿐이다.

The Universal Characteristic 범용의 문자 체계

But what of the “wonderful idea” of Leibniz’s youth, his grand dream to find a true alphabet of human thought and the appropriate calculational tools for manipulating these symbols? Although he had resigned himself to the fact that unaided he could never accomplish such a thing, he never lost sight of this goal, thinking and writing about it throughout his life. It was clear that the special characters used in arithmetic and algebra, the symbols used in chemistry and astronomy, and the symbols he introduced for the differential and integral calculus provided a paradigm showing how crucial a truly appropriate symbolism could be.

Leibniz referred to such a system of characters as a characteristic. Unlike the alphabetic symbols which had no meaning, the examples just mentioned were, for him, a real characteristic in which each symbol represented some definite idea in a natural and appropriate way. What was needed, Leibniz maintained, was a universal characteristic, a system of symbols that was not only real, but which also encompassed the full scope of human thought.

In a letter explaining this to the mathematician G. F. A. l’Hôspital, Leibniz wrote: “Part of the secret of” algebra “consists of the characteristic, that is to say of the art of properly using” the symbolic expressions. This care for proper use of symbols was to be the “thread of Ariadne” that would guide the scholar in creating his characteristic.

As the early twentieth century logician and Leibniz scholar Louis Coutu-rat explained:

… it is algebraic notation that incarnates, so to speak, the ideal of the characteristic and which is to serve as a model. It is also the example of algebra that Leibniz cites consistently to show how a system of properly chosen symbols is useful and indeed indispensible for deductive thought.¹⁰

Perhaps the most enthusiastic explanation of his proposed characteristic was in another letter, this one to Jean Galloys with whom Leibniz had extensive correspondence:

I am convinced more and more of the utility and reality of this general science, and I see that very few people have understood its extent….This characteristic consists of a certain script or language … that perfectly represents the relationships between our thoughts. The characters would be quite different from what has been imagined up to now. Because one has forgotten the principle that the characters of this script should serve invention and judgment as in algebra and arithmetic. This script will have great advantages; among others, there is one that seems particularly important to me. This is that it will be impossible to write, using these characters, chimerical notions (chimères) such as suggest themselves to us. An ignoramus will not be able to use it, or, in striving to do so, he himself will become erudite.¹¹

In the letter to Galloys quoted above Leibniz refers to arithmetic as well as algebra as showing the importance of an appropriate symbolism. He had in mind in particular the advantage of the Arabic system of notation that we still use today based on the digits 0 to 9 over previous systems (like the Roman numerals) for ordinary calculation. When Leibniz discovered binary notation, in which any number can be written using only the digits 0 and 1, he was impressed by the simplicity of this system. He believed that it would be useful in laying bare properties of numbers that otherwise would be hidden. Although this belief turned out to be unjustified, this interest on Leibniz’s part is remarkable in the light of the importance of this binary notation in connection with modern computers.

Leibniz saw his grand program as consisting of three major components. First, before the appropriate symbols could be selected, it would be necessary to create a compendium or encyclopedia encompassing the full extent of human knowledge. He maintained that once having accomplished this, it should prove feasible to select the key underlying notions and to provide appropriate symbols for each of them. Finally, the rules of deduction could then be reduced to manipulations of these symbols, via what Leibniz called a calculus ratiocinator, what nowadays might be called a symbolic logic.

To a present-day reader, it is hardly surprising that Leibniz did not feel able to accomplish such a program on his own, especially given the constant pressure he was under to produce the family history that his patron regarded as his principal task. It is difficult to understand how Leibniz could have seriously believed that the universe we inhabit, in all of its complexity, could be reduced to a single symbolic calculus.

We can only hope to begin to comprehend the matter by attempting to see the world through the eyes of Leibniz. For him nothing, absolutely nothing, about the world was in any way undetermined or accidental. Everything was in fact entirely determined by the plan, clear in the mind of God, by means of which He had created the best world that could be created. Hence, for Leibniz, all aspects of the world, natural and supernatural, were connected by links one could hope to discover by rational means. Only from this perspective can we understand how, in a famous passage, Leibniz could write of serious “men of good will” sitting around a table to solve some critical problem. After writing out the problem in Leibniz’s projected language, his “universal characteristic,” it would be time to say “Let us calculate!” Out would come the pens and a solution would be found whose correctness would necessarily be accepted by all.¹²

Leibniz wrote with enthusiasm about the importance of producing the calculus ratiocinator, the algebra of logic, that would presumably be needed to carry out these calculations:

For if praise is given to the men who have determined the number of regular solids—which is of no use, except insofar as it is pleasant to contemplate—and if it is thought to be an exercise worthy of a mathematical genius to have brought to light the more elegant properties of a conchoid or cissoid, or some other figure which rarely has any use, how much better will it be to bring under mathematical laws human reasoning, which is the most excellent and useful thing we have.¹³

Unlike the universal characteristic concerning which Leibniz wrote with such passion and conviction, but produced little in the way of specifics, he did make a number of attempts to produce a calculus ratiocinator. Part of his most polished effort in this direction is shown in the above illustration.¹⁴ A good century and a half ahead of his time, Leibniz proposed an algebra of logic that would specify the rules for manipulating logical concepts in the manner in which ordinary algebra specifies the rules for manipulating numbers. He introduced a special new symbol ⊕ to represent the combining of arbitrary “pluralities of terms.” The idea was something like the combining of two collections of things into a single collection containing all of the items in either one. The plus sign encourages us to think of this operation as being like ordinary addition, but the circle around it warns us that it is not exactly like ordinary addition because it is not numbers being added. Some of his algebraic rules are also to be found in high-school algebra textbooks: to some extent the same rules work for logical concepts as for numbers.

DEFINITION 3. A is in L, or L contains A, is the same as to say that L can be made to coincide with a plurality of terms taken together of which A is one. B ⊕ N = L signifies that B is in L and that B and N together compose or constitute L. The same thing holds for a larger number of terms.

AXIOM 1. B ⊕ N = N ⊕ B.

POSTULATE. Any plurality of terms, as A and B, can be added to compose a single term A ⊕ B.

AXIOM 2. A ⊕ A = A.

PROPOSITION 5. If A is in B and A = C, then C is in B. For in the proposition A is in B the substitution of A for B gives C is in B.

PROPOSITION 6. If C is in B and A = B then C is in A. For in the proposition C is in B the substitution of A for B gives C is in A.

PROPOSITION 7. A is in A. For A is in A ⊕ A (by Definition 3). Therefore (by Proposition 6) A is in A.

……………………………………………

PROPOSITION 20. If A is in M and B is in N, then A ⊕ B is in M ⊕ N.

Sample from one of Leibniz’s logical Calculi

But there’s more to the story. There are also rules that are very different from those for numbers. The most striking rule of this latter kind, one that in a somewhat different context George Boole was to make the cornerstone of his algebra of logic, is Leibniz’s Axiom 2, A ⊕ A = A, which expresses the fact that combining a “plurality of terms” with itself will yield nothing new: evidently combining all the things belonging to a given collection with that same collection of things, will just produce that same collection, all over again. Of course addition of numbers is quite different: 2 + 2 = 4 not 2.

In the next chapter, we will see how George Boole, presumably ignorant of Leibniz’s efforts, produced a serviceable symbolic logic along the lines that Leibniz had pioneered. Boole’s logic subsumed the logic Aristotle had introduced 2000 years earlier, but it was only with the work of Gottlob Frege well into the nineteenth century, that the serious limitations shared by the logical systems of Aristotle and of Boole were really overcome.¹⁵

Despite Leibniz’s voluminous correspondence, we have little idea of what he was like as a person. One biographer claims to see in the few portraits of Leibniz we possess, the image of a tired, unhappy, pessimistic man, in contradiction to his optimistic philosophy.¹⁶ Others have remarked that he liked to give cakes to his neighbors’ children. Apparently, he proposed marriage when he was 50, but thought better of it when the lady hesitated.¹⁷ We have the picture of Leibniz spending long days and often entire nights seated at his desk managing his enormous correspondence with remarkable punctuality, his meals brought to him from an inn by his servants. What is clear is that he was indefatigable in his work.*

It is tempting to indulge in a bit of “what if?” What if Leibniz had not been shackled to his patrons’ family history, and was free to devote more time to his calculus rationcinator? Might he not have accomplished what Boole was only to do so much later? But of course, such speculation is useless. What Leibniz has left us is his dream, but even this dream can fill us with admiration for the power of human speculative thought and can serve as a yardstick for judging later developments.

*Voltaire’s Dr. Pangloss in Voltaire’s Candide was a sendup of this Leibnizian doctrine.

*Blaise Pascal, born on June 19, 1623, at Clermont-Ferrand, France, one of the founders of the mathematical theory of probability, was a prolific mathematician, physicist, and religious philosopher. His calculating machine, designed and built circa 1643, brought him considerable fame. He died in 1662

*Although Huygens’s view came to be generally accepted, the coming of quantum physics in the twentieth century made it clear that both Newton and Huygens had been right; each grasped an essential characteristic of light

*The number \(\frac{\text{π}}{4}\) is in fact the area of a circle whose radius is\(\frac{1}{2}\)

†I used my PC to obtain the table of approximations to \(\frac{\pi}{4}\) from Leibniz’s series. A short Pascal program I wrote for the purpose runs for less than a second on a contemporary PC.

*Thus, finding volumes and centers of gravity are problems of the first kind, and computing accelerations and (in economic theory) marginal elasticity are problems of the second type.

†The symbol for integration ∫ is actually a modified “S” intending to suggest “sum,” and the symbol “/d/” is likewise intended to suggest the idea of “difference.”

*In part, this picture comes from the 1951 biography completed by Professor Kurt Huber in prison while awaiting execution by the Nazis. He had supported the efforts of his students at the University of Munich who formed the “White Rose” underground group and were decapitated for distributing anti-Nazi leaflets. There are today a number of streets in Germany named for him including a Professor Huber Platz at the University of Munich. (I am indebted to Benson Mates for this information about Professor Huber’s heroic role.)

*이 사진은 1951 년 쿠르트 후버 교수가 나치의 처형을 기다리는 동안 감옥에서 완성한 전기에서 일부 발췌한 것입니다. 그는 뮌헨 대학교에서 ‘화이트 로즈’ 지하 조직을 결성하고 반나치 전단지를 배포한 혐의로 참수당한 학생들의 활동을 지원했습니다. 오늘날 독일에는 뮌헨 대학교의 후버 플라츠 교수를 비롯해 그의 이름을 딴 거리가 여러 곳 있습니다. (후버 교수의 영웅적인 역할에 대한 이 정보는 벤슨 메이트스에게 빚을 지고 있습니다.)

CHAPTER 2 Booles Turns Logic into Algebra

Booles Turns Logic into Algebra 논리를 수학으로 바꾼 불

George Boole’s Hard Life 불의 삶

The beautiful and intelligent Princess Caroline von Ensbach, one day to be Queen of England as the wife of George II, met Leibniz in Berlin in 1704 when she was 18. After she went to England with the court, their friendship continued by correspondence. She tried to persuade her father-in-law, then George I of England, to bring Leibniz to England, but as we have seen, the king insisted that Leibniz remain in Germany to complete the Hanoverian family history.

Caroline found herself entangled in the continuing dispute between Leibniz and Newton and his followers, each side accusing the other of plagiarism over the invention of the calculus. She tried to convince Leibniz that the issue was of no great importance, but he was having none of it. Indeed, Leibniz sought her support before the king for his desire to be appointed “Historiographer of England” to match Newton’s position as “Master of the Mint,” asserting that only in this way could the honor of Germany vis a vis England be maintained.

Leibniz wrote Caroline that when Newton held that a grain of sand exerted a gravitational force on the distant sun without any evident means by which such a force could be transmitted, he was in effect calling on miraculous means to explain a natural phenomenon, something he assured her was inadmissible. Caroline tried to get some of Leibniz’s writings translated into English. This effort brought her into contact with Samuel Clarke who had been recommended to her as a possible translator.

Clarke was a philosopher and theologian and also a disciple of Newton. In his Being and Attributes of God, dated 1704, Clarke had developed a proof of the existence of God. Caroline showed him a letter from Leibniz attacking certain of Newton’s ideas and asked him to reply. This initiated a correspondence between the two men that continued until just a few days before Leibniz’s death. Not surprisingly, there was no meeting of minds.

GEORGE BOOLE

From the point of view of our story, the most interesting fact about Samuel Clarke is that almost a century and a half after Leibniz’s death, George Boole would demonstrate the efficacy of his own methods by using Clarke’s proof of the existence of God as an example. In effect, with these methods, Boole succeeded in bringing to life part of Leibniz’s dream. He had reduced Clarke’s complicated deduction to a simple set of equations.¹

In proceeding from the world of Leibniz and the seventeenth century European nobility to that of George Boole, we move forward not only two centuries in time, but also down several layers of social class. George, the first of four children, was born on November 2, 1815, in the town of Lincoln in the eastern part of England, to John and Mary Boole who had been childless for the first nine years of their marriage. John Boole, a cobbler who eked out only a meager living from his trade, had a passion for learning, and especially for scientific instruments. He proudly displayed a telescope he had made in his shop window. Unfortunately, he was not an effective business man and his talented, conscientious son soon found himself carrying the burden of supporting the whole family.²

In June 1830, the citizens of Lincoln were treated to a silly controversy in a local newspaper over the originality of an English translation of one of the poems of the ancient Greek writer Meleager. The translation had appeared in the Lincoln Herald as the work of “G.

1. of Lincoln, aged 14 years,” and one P. W. B. took the trouble to

write accusing G. B. of plagiarism. P. W. B. admitted that he was unable to provide a reference to the source from which he was accusing G. B. of copying, but regarded it as simply beyond belief that the work could have been produced by a 14-year-old. The battle led to an exchange of several letters between G.B. and P. W. B., all duly published in the Herald.

George’s family, who early recognized his ability, were far too poor to furnish him with a proper formal education, and so, with the help of his father, George was mainly self-taught. George studied not only Latin and Greek but also taught himself French and German and was able (much later, of course) to write mathematical research papers in these languages. George Boole never belonged to any particular religious denomination, and found it impossible to believe in the divinity of Christ, but throughout his life he held strong religious convictions. He soon abandoned his original ambition to join the clergy of the Church of England, in part because of his beliefs, but also because of his family’s need for immediate financial help when his father’s business collapsed. George was not yet 16 when he began his career as a teacher.

After two years at a small Methodist school some 40 miles from home he was fired, mainly it seems, owing to complaints about his irreligious behavior: he worked on mathematics on Sundays, and even in chapel! Indeed, it was at this time that Boole’s efforts turned more and more to mathematics. In later years, reminiscing about this period in his life, he explained that having a very limited budget for buying books, he found that mathematics books provided the best value because it took longer to work through them than books on other subjects. He also liked to speak of the inspiration that suddenly came to him during his stay at the Methodist school. While walking across a field, the thought flashed across his mind that it should be possible to express logical relationships in algebraic form. This experience, which a biographer compares to that of Paul on the road to Damascus, was to bear fruit many years later.³

After teaching at the Methodist school, Boole took a position in Liverpool. But after six months of living and teaching there, he felt compelled to leave because of (in the words of his sister), “the spectacle of gross appetites and passions unrestrainedly indulged …” presumably by the school head-master.⁴ His next job, in a village only four miles from home, was also of brief duration. This time, the reason was that, at the age of 19, concerned to put his family’s finances on a sound basis, George Boole had decided to start his own school in his home town, Lincoln. For fifteen years, until accepting a professorship at a newly founded university at Cork, Ireland, Boole managed a successful career as a schoolmaster. His schools (there were three in succession) were the sole support of his parents and his siblings, although eventually his sister Mary Ann and brother William did participate in the work.

Although running a day and boarding school, and teaching numerous classes might be thought to be a full-time job, Boole managed during this period to make the transition from student of mathematics to creative mathematician. In addition, he somehow found time for activities of social improvement. He was a founder and trustee of a Female Penitent’s Home in Lincoln whose purpose was “to provide a temporary home in which, by moral and religious instruction and the formation of industrious habits, females, who have deviated from the paths of virtue, may be restored to a reputable place in society.” Boole’s biographer speaks of prostitutes (who were evidently numerous in Victorian Lincoln) as the “penitent” women who were to be helped by this institution.⁵ More likely, the typical client was a young woman of the servant class who found herself pregnant and abandoned after having been promised marriage by a lover of her own social class.* Some insight into George Boole’s personal attitudes towards sexual matters may perhaps be gleaned from what he said in two of his lectures on non-mathematical subjects. In one, a lecture on education, he warned:

A very large proportion of the extant literature of Greece and Rome … is deeply stained with allusions and all too often with more than allusions to the vices of Heathenism. … But that the innocence of youth can be exposed to the contamination of evil without danger I do not believe.⁶

And a lecture on the proper uses of leisure (given after a successful campaign by the “Lincoln Early Closing Association” to obtain a ten-hour working day) included Boole’s stern words:

If you seek gratification in those pursuits from which virtue turns aside, you do so without excuse.⁷

Boole, following in his father’s footsteps, was also deeply involved with the Lincoln Mechanics’ Institute. These institutes, mainly devoted to afterhours education for artisans and other workers, had sprung up all over Victorian Britain. Boole did committee work for the one in Lincoln, made recommendations for improving the library, gave lectures, and provided teaching on a variety of subjects without remuneration.

Yet somehow, amidst all of this, he found time to study some of the most important English and continental mathematical treatises, and to begin making his own contributions. Much of Boole’s early work bears witness to Leibniz’s belief in the power of appropriate mathematical symbolism, of the manner in which the symbols seem to magically produce correct answers to problems almost unaided. Leibniz had pointed to the example of algebra. In England, as Boole began his own work, it was coming to be realized that the power of algebra comes from the fact that the symbols representing quantities and operations obeyed a small number of basic rules or laws. This implied that this same power could be applied to objects and operations of the most varied kind so long as they obeyed some of these same laws.⁸

In Boole’s early work, he applied algebraic methods to the objects that mathematicians call operators. These “operate” on expressions of ordinary algebra to form new expressions. Boole was particularly interested in differential operators, so called because they involve the differentiation operation of the calculus mentioned in the previous chapter.⁹ These operators were seen to be of particular importance because many fundamental laws of the physical universe take the form of differential equations, that is equations involving differential operators. Boole showed how certain differential equations could be solved by using methods of ordinary algebra applied to differential operators. Engineering and science students typically learn some of these methods in their sophomore or junior year in a course in differential equations.

During his years as a schoolmaster, Boole published a dozen research papers in the Cambridge Mathematical Journal. In addition, he submitted a very long paper to the Philosophical Transactions of the Royal Society. At first the Royal Society was loath to consider a submission from such an outsider, but finally decided to accept it, and later awarded it their Gold Medal.¹⁰ Boole’s method was to introduce a technique and then to apply it to a number of examples. He generally asked for no more in the way of proof that his methods were correct than that his examples worked out.¹¹

At this time, Boole developed professional correspondences and friendships with a number of England’s leading young mathematicians. A quarrel with the Scottish philosopher Sir William Hamilton that his friend Augustus De Morgan had fallen into brought Boole’s thoughts back to his long ago flash of insight—that logical relationships might be expressible as a kind of algebra. Although Hamilton was an erudite scholar in aspects of metaphysics, he seems to have been something of a quarrelsome fool. Out of what can only have been his colossal ignorance of the subject, he published diatribes against mathematics. What had set him off was De Morgan’s publication on logic that Hamilton claimed plagiarized what he thought of as his great discovery in logic, what he called the “quantification of the predicate.” We need waste no time trying to understand this idea or the fierce controversy it generated—it is of importance only because of the stimulus it provided to George Boole.¹²

The classical logic of Aristotle that had so fascinated the young Leibniz involved sentences like:

1.All plants are alive.

2.No hippopotamus is intelligent.

3.Some people speak English.

Boole came to realize that what is significant in logical reasoning about such words as “alive,” “hippopotamus,” or “people” is the class or collection of all individuals described by the word in question: the class of living things, the class of hippopotamuses, the class of people. Moreover, he came to see how this kind of reasoning can be expressed in terms of an algebra of such classes. Boole used letters to represent classes just as letters had previously been used to represent numbers or operators. If the letters x and y stand for two particular classes, then Boole wrote xy for the class of things that are both in x and in y. As Boole himself put it:

… if an adjective, as “good,” is employed as a term of description, let us represent by a letter, as y, all things to which the description “good” is applicable, i.e. “all good things,” or the class “good things.” Let it further be agreed, that by the combination xy shall be represented that class of things to which the names or descriptions represented by x and y are simultaneously applicable. Thus, if x alone stands for “white things,” and y for “sheep,” let xy stand for “white sheep;” and in like manner, if z stand for “horned things,” … let zxy represent “horned white sheep,” …¹³

Boole thought of this operation applied to classes like the operation of multiplication applied to numbers. However, he noticed a crucial difference: If once again y is the class of sheep, what is yy? It must be the class of things that are sheep and are also … sheep. But this is the very same thing as the class of sheep; so yy = y. It is only a small exaggeration to say that Boole based his entire system of logic on the fact that when x stands for a class, the equation xx = x is always true. We will return to this point later.*

George Boole was 32 when his first revolutionary monograph on logic as a form of mathematics was published. His more polished exposition, The Laws of Thought appeared seven years later. These were eventful years in Boole’s life. Boole’s social class and unconventional education had apparently ruled out his chances for an appointment at an English university. Strangely, it was the Irish “problem” that gave Boole an opening.

Among the many bitter complaints in Ireland concerning English rule was the Protestant character of its only university, Trinity College in Dublin. In response it was proposed by the British government to found three new universities to be called “Queen’s Colleges” in Cork, Belfast, and Galway. Remarkably for the time, they would be non-denominational. Despite denunciations by Irish political and religious figures, who demanded institutions of a definitely Catholic character, the plans moved forward. Boole decided to apply for an appointment at one of these universities, and finally three years later, in 1849, he was appointed professor of mathematics at Queen’s College in Cork.

By 1849, Ireland had come through the worst of the disaster of famine and disease brought by the potato blight, a devastating fungus that destroyed most of the potato crops on which the Irish poor depended. Many who did not starve to death were killed by the epidemics of typhus, dysentery, cholera, and relapsing fever to which their weakened immune systems had laid them open. The English rulers, slow to recognize the fungus as the underlying cause of the catastrophe, instead blamed the supposed indolence of the Irish. This social fiction was used to justify the continuing export of food from Ireland while millions went hungry and starved. Between 1845 and 1852, out of eight million Irish, at least a million died and another one and a half million emigrated.¹⁴

Boole had little to say about this: his strong expressions of indignation centered on cruelty to animals. Indeed, his attitude to the Irish people was rather equivocal as emerges from these lines from a sonnet to Ireland that Boole wrote just as the college in Cork was being inaugurated:

Yet thou in wisdom still art young, though old
In misery and tears. Oh that thy store
Of bitter thoughts, which brood upon the past,
Were from thy bosom quite erased and worn.¹⁵

Although Cork was certainly no major intellectual or cultural center, the position provided Boole with the possibility of a life far more appropriate to his stature as one of the great mathematicians of the century than that of a schoolmaster. His father had recently died and, after making suitable provision for his mother, he was finally freed from the burden of being the family provider, and could think of having a personal life.

The mathematics taught at Cork was at a rather low level for a university. The syllabus began with “Fractional and Decimal Arithmetic” and continued with topics taught today in secondary school. Boole’s annual salary was £250 in addition to a direct tuition fee of about £2 per term from each student. Since he had no assistant, he did all the grading of the weekly homework assignments.

Controversy over the Queen’s Colleges continued. Although Cork’s president was the distinguished Catholic scientist Sir Robert Kane, Catholics were certainly under-represented: of the academic staff of 21, only one other was Catholic. The Catholic Church hierarchy went so far as to forbid members of the clergy from participating in the work of the colleges. Some felt that Irish candidates for positions were deliberately passed over for relatively mediocre Englishmen or Scots. Nor did President Kane endear himself to his faculty. His wife had no wish to live in Cork, and so the President tried to run the college from Dublin. This, combined with his arbitrary pugnacious manner, led to one fight after another between the president and the faculty, sterile battles in which Boole usually found himself involved.¹⁶

Mary Everest, Boole’s wife-to-be, later recounted some of her first impressions of the attitudes of some of the residents of Cork towards the man she would marry. One lady’s answer to “What is the professor of mathematics like?” was “Oh he’s like—the sort of man to trust your daughter with.” Another lady explained the absence of her young children by informing Miss Everest that George Boole had taken them for a walk and that she was always happy when he walked with them. To the reply that Boole seemed to be a general favorite, the lady demurred:

He is no favorite of mine, … at least, I don’t enjoy his society. I don’t care to be with such very good people. … he never shows you that he thinks you wicked, but when you are near anyone so pure and holy, you can’t help feeling how shocked he must be at you. He makes me feel very wicked; but I am always at ease when the children are with him; I know they are getting some good.¹⁷

Mary Everest was the daughter of an eccentric clergyman and a niece of Lieutenant-Colonel Sir George Everest, whose name was given to the world’s tallest mountain. She was also a niece of Boole’s friend and colleague, John Ryall, Vice-President and Professor of Greek at Cork; this family connection brought George and Mary together. As a child Mary had displayed an aptitude for mathematics and after George began to tutor her, they grew to be good friends and frequent letter writers. It seems that Boole believed that their 17-year age difference precluded anything more, but five years after their first meeting when Boole was 40, matters came to a head with the death of Mary’s father. As Mary was financially impoverished, George proposed at once, and they were married before the year was out.

Their marriage lasted a mere nine years, for Boole died at the age of only 49, after walking three miles to class in a cold October rainstorm. The ensuing bronchitis soon became pneumonia, and he died two weeks later. Tragically, his death may have been hastened by his wife’s unorthodox medical views—apparently she treated his pneumonia by placing him between cold soaking bed sheets.¹⁸

The marriage had evidently been a very happy one.¹⁹ Mary Boole recalled it “like the remembrance of a sunny dream.” They had five children, all girls. Boole’s widow lived well into the twentieth century, dying at the age of 84 while the First World War raged across the channel. She became attached to various systems of mystical belief and wrote a great deal of nonsense.

Boole’s daughters all had interesting lives. The third daughter, Alicia, possessed a very remarkable geometric ability: she was able to visualize clearly geometric objects in four dimensions. This enabled her to make a number of important mathematical discoveries. However, the youngest daughter Ethel Lilian was the most astonishing. She was only six months old when her father died and she remembered her childhood as one of terrible poverty. Lily, as she was called, became involved with the Russian revolutionary emigres who made London their home during the late years of the nineteenth century. While on a trip to the Russian empire (which at that time included much of Poland) to help her revolutionary friends, she was seen by her future husband, Wilfred Voynich, from his prison cell, as she stared up at the Warsaw Citadel. Voynich recognized her years later after he had made his escape to London. This romantic beginning led to their marriage.

Lily became famous later as the author of The Gadfly, a novel inspired by her short but passionate love affair with Sidney Riley whose incredible life formed the basis for a television mini-series called Riley: Ace of Spies. With irony piled upon irony, Riley, a fervent anti-communist, was executed in Russia by the Bolsheviks, while his lover’s novel, its true inspiration unknown, became required reading for Russian school children. In 1955 Pravda reported to its Moscow readers that the author of The Gadfly was alive and well in New York, and she received from Russia a royalty check for $15,000. She died five years later at the age of 96.²⁰

George Boole’s Algebra of Logic : 불의 논리 대수학

불의 대수학

합집합

Returning to Boole’s new algebra applied to logic, we recall that if x and y represent two classes, Boole would write xy to stand for the class of those things that belong to both x and y. He intended the notation to suggest an analogy with multiplication in ordinary algebra. In contemporary terminology, xy is called the intersection of x and y.²¹ We also saw that the equation xx = x is always true when x represents a class. This led Boole to ask the question: in ordinary algebra, where x stands for a number, when is the equation xx = x true? The answer is straightforward: the equation is true when x is 0 or 1 and for no other numbers. This led Boole to the principle that the algebra of logic was precisely what ordinary algebra would become if restricted to the two values 0 and 1. However, to make sense of this, it became necessary to reinterpret the symbols 0 and 1 as classes. A clue is provided by the behaviors of 0 and 1, respectively, with respect to ordinary multiplication: 0 times any number is 0; 1 times any number is that very number. In symbols,

논리에 적용된 부울의 새로운 대수로 돌아가서, 엑스/와 /와이/가 두 클래스를 나타내는 경우, 부울은 /엑스/와 /와이 모두에 속하는 것들의 클래스를 나타내기 위해 xy/라고 썼음을 상기해 보겠습니다. 그는 일반 대수학의 곱셈과 유사하게 표기하고자 했습니다. 현대 용어로 /xy/는 /x/와 /y/의 /교집합/이라고 합니다.²¹ 또한 /xx = x/라는 방정식은 /x/가 클래스를 나타낼 때 항상 참이라는 것을 알 수 있었습니다. 이에 불은 다음과 같은 질문을 하게 됩니다: /일반 대수학에서 x 가 숫자를 나타낼 때, /x = x 방정식은 언제 참인가? / 대답은 간단합니다. x/가 0 또는 1 이고 다른 숫자가 없을 때 이 방정식은 참입니다. 이를 통해 불은 논리 대수를 0 과 1 이라는 두 값으로 제한하면 일반 대수가 된다는 원리를 발견했습니다. 하지만 이를 이해하기 위해서는 기호 0 과 1 을 클래스로 재해석할 필요가 있었습니다. 일반 곱셈과 관련하여 각각 0 과 1 의 동작에서 단서를 찾을 수 있습니다: 0 은 /어떤 숫자의 곱은 0 이고, 1 은 어떤 숫자의 곱은 바로 그 숫자입니다. 기호에서,

$$\begin{matrix}
{0x\  = \ 0,} & {1x\  = \ x}
\end{matrix}$$

\[\begin{matrix} {0x\ = \ 0,} & {1x\ = \ x} \end{matrix}\]

Now for classes, 0/x/ will be identical to 0 for every x, if we interpret 0 to be a class to which nothing belongs; in modern terminology, 0 is the empty set. Likewise, 1/x/ will be identical to x for every x, if 1 contains every object under consideration, or, as we may say, 1 is the “universe of discourse.”

이제 클래스의 경우, 0/x/는 모든 /x/에 대해 0 과 동일할 것입니다. 0 을 /아무 것도 속하지 않는 클래스/로 해석한다면; 현대 용어로 0 은 /비어 있는 집합/입니다. 마찬가지로, 1/x/는 모든 /x/에 대해 /x/와 동일할 것입니다. 만약 1 이 /고려 중인 모든 대상을 포함/하거나 우리가 1 을 “담론의 우주"라고 말할 수 있다면 말이죠.

Ordinary algebra deals with addition and subtraction as well as multiplication. Thus, if Boole was to present the algebra of logic as just ordinary algebra with the special rule xx = x, he had to provide an interpretation for + and −. So, if x and y represent two classes, Boole took x + y to represent the class of all things to be found either in x or in y, nowadays called the union of x and y. Thus, to use Boole’s own example, if x is the class of men and y is the class of women, then x + y is the class consisting of all men and women. Also, Boole wrote x – y for the class of things in x that are not in y.²² If x represents the class of all people and y represents the class of all children, then x – y would represent the class of adults. In particular, 1 – x would be the class of things not in x, so that

\(x\ + \ (1\ - \ x)\ = \ 1.\)

Let us see how Boole’s algebra works. Using ordinary algebraic notation, let us write x/² for /xx. So Boole’s basic rule can be written as x/² = /x or x − /x/² =0. Factoring this equation, following the usual rules of algebra,

\(x(1\ - \ x)\ = \ 0.\)

In words: nothing can both belong and fail to belong to a given class x. For Boole, this was an exciting result, helping to convince him that he was on the right track. For as he said, quoting Aristotle’s Metaphysics, this equation expresses precisely:

… that “principle of contradiction” which Aristotle has described as the fundamental axiom of all philosophy. “It is impossible that the same quality should both belong and not belong to the same thing … This is the most certain of all principles … Wherefore they who demonstrate refer to this as an ultimate opinion. For it is by nature the source of all the other axioms …”²³

Boole must have been delighted to obtain confirmation such as every scientist seeks when introducing new and general ideas: to see an important earlier landmark turn out to be a particular application of the new ideas, in this case Aristotle’s principle of contradiction. In fact in Boole’s time, it was common for writers on logic to equate the entire subject with what Aristotle had done so many centuries earlier. As Boole put it, “the science of Logic enjoys an immunity from those conditions of imperfection and of progress to which all other sciences are subject …” The part of logic that Aristotle studied deals with inferences, called syllogisms, of a very special and restricted kind. They are inferences from a pair of propositions called premises to another proposition called the conclusion. The premises and conclusions must be representable by sentences of one of the following four types:*

The following is an example of a valid syllogism:

\(\begin{matrix} {\text{All}\ X\ \text{are}\ Y.} \\ \underline{\text{All}\ Y\ \text{are}\ Z.} \\ {\text{All}\ X\ \text{are}\ Z.} \end{matrix}\)

That this syllogism is valid means that whatever properties are substituted for X, Y, and Z, so long as the given two premises are true, the conclusion will be as well. Here are two instances of this syllogism:

Boole’s algebraic methods can easily be used to demonstrate that this syllogism is valid. To say that everything in X also belongs to Y is the same as to say that there is nothing that belongs to X but not to Y, i.e., X (1 − Y) = 0 or equivalently X = XY. Likewise, the second premise can be written Y = YZ. Using these equations we get

\(X\ = \ XY\ = \ X(YZ)\ = \ (XY)Z\ = \ XZ,\)

the desired conclusion.²⁴

Of course, not every proposed syllogism is valid. An example of an invalid syllogism can be obtained by interchanging the second premise with the conclusion in the previous example:

\(\begin{matrix} {\text{All}\ X\ \text{are}\ Y.} \\ \underline{\text{All}\ X\ \text{are}\ Z.} \\ {\text{All}\ Y\ \text{are}\ Z.} \end{matrix}\)

This time there is no way to use the premises X = XY and X = YZ to obtain the supposed conclusion Y = YZ.

In retrospect, it is difficult to understand the widespread belief that syllogistic reasoning constituted the whole of logic, and Boole was quite scathing in his denunciation of this idea. He pointed out that much ordinary reasoning involves what he calls secondary propositions, that is, propositions that express relations between other propositions. Such reasoning is not syllogistic.

For a simple example of such reasoning, let us listen in on a conversation between Joe and Susan. Joe can’t find his checkbook and Susan is helping him.

Joe looks and (if it’s a good day for logic), the missing checkbook is there. Let us see how Boole’s algebra could be used to analyze Joe and Susan’s reasoning.

In their reasoning, Joe and Susan were dealing with the following propositions (each labeled with a letter):

_L_Joe left his checkbook at the supermarket.

_F_Joe’s checkbook was found at the supermarket.

_W_Joe wrote a check at the restaurant last night.

_P_After writing the check last night, Joe put his check book in his jacket pocket.

_H_Joe hasn’t used his check book since last night.

_S_Joe’s checkbook is still in his jacket pocket.

They used the following pattern:

Like Aristotle’s syllogisms, this pattern forms a valid inference. As with any valid inference, the truth of sentences called conclusions is inferred from the truth of other sentences called premises.

Boole saw that the same algebra that worked for classes would also work for inferences of this kind.²⁵ Boole used an equation like X = 1 to mean that the proposition X is true; likewise he used the equation X = 0 to mean that X is false. Thus, for “Not X,” he could write the equation X = 0. Also, for X & Y he wrote the equation XY = 1. This works because X & Y is true precisely when X and Y are both true, while algebraically, XY = 1 if X = Y = 1, but XY = 0 if either X = 0 or Y = 0 (or both). Finally, the statement “If X then /Y/” can be represented by the equation

\(X(1\ - \ Y)\ = \ 0.\)

To see this, think of this statement as asserting

\(if\ X\ = \ 1\ then\ Y\ = \ 1.\)

But indeed, substituting X = 1 in the proposed equation leads to 1 − Y = 0, that is, to Y = 1.

Using these ideas, Joe and Susan’s premises can be expressed by the equations

\(\begin{array}{rll} {L(1\ - \ F)} & = & {0,} \\ F & = & {0,} \\ {W\ P} & = & {1,} \\ {W\ PH(1\ - \ S)} & = & {0,} \\ H & = & 1. \end{array}\)

Substituting the second equation in the first, we get L = 0, the first desired conclusion. Substituting the third and fifth equations in the fourth, we get 1 − S = 0, that is, S = 1, the other desired conclusion.

Now of course, Joe and Susan had no need for this algebra. But the fact that the kind of reasoning that ordinarily takes place informally and implicitly in ordinary human interactions could be captured by Boole’s algebra encouraged the hope that more complicated reasoning could be captured as well. Mathematics may be thought of as systematically encapsulating highly complex logical inferences. This is part of the reason that mathematics is so useful in science. So an ultimate test of a theory of logic that aims at completeness is whether it encompasses all mathematical reasoning. We will return to this matter in the next chapter.

As a final example of Boole’s methods, we turn to Samuel Clarke’s proof of the existence of God mentioned at the beginning of this chapter. Without trying to follow Clarke’s long complex deduction, it is at least amusing to see how Boole proceeds. We quote a small fragment:²⁶

The premises are:—

1st. Something is.

2nd.If something is, either something always was, or the things that now are have risen out of nothing.

3rd.If something is, either it exists in the necessity of its own nature, or it exists by the will of another being.

4th.If it exists in the necessity of its own nature, something always was.

5th.If it exists by the will of another being, then the hypothesis that the things which now are have risen out of nothing, is false.

We must now express symbolically the above propositions.

Let

Boole then obtains from the premises the equations

\(\begin{array}{rll} {1\ - \ x} & = & {0,} \\ {x\left\{ yz\ + \ (1\ - \ y)\ (1\ - \ z) \right\}} & = & {0,} \\ {x\left\{ pq\ + \ (1\ - \ p)\ (1\ - \ q) \right\}} & = & {0,} \\ {p(1\ - \ y)} & = & {0,} \\ {qz} & = & 0. \end{array}\)

One wonders what Clarke would have made of this reduction of his intricate metaphysical reasoning to manipulations of simple equations. Likely, as a disciple of Newton, he would have been pleased. On the other hand, the pugnacious metaphysician Sir William Hamilton who hated mathematics must have been horrified.

Boole and Leibniz’s Dream

Boole’s system of logic included Aristotle’s and went far beyond it. But it still fell far short of what was needed to fulfill Leibniz’s dream. Consider the following sentence:

All failing students are either stupid or lazy.

One might think of this sentence as

\(\text{All}\ X\ \text{are}\ Y.\)

However, this would require that the class of students being stupid or lazy be treated as a unit and would not permit any reasoning that sought to distinguish those who were failing because of stupidity from those who were failing because of laziness. In the next chapter we’ll see how Gottlob Frege’s system of logic does include reasoning of this subtler kind.

It is quite straightforward to use Boole’s algebra as a system of rules for calculating, and say that, within its limits, it provided the calculus ratiocinator Leibniz had sought. Leibniz’s writings on these matters were in the form of letters and other unpublished documents, and it was only late in the nineteenth century that a serious effort to gather and publish these was undertaken. So, there is no reasonable way that Boole could have been aware of his predecessor’s efforts. Nevertheless it is interesting to compare Boole’s full-blown system with Leibniz’s fragmentary attempts.

Leibniz’s fragment quoted in our first chapter included as its second axiom, A ⊕ A = A. Thus the operation Leibniz considered was to obey Boole’s fundamental rule: xx = x. Moreover, Leibniz proposed to present his logic as a full-fledged deductive system in which all of the rules are deduced from a small set of axioms. This is in accord with modern practice and shows Leibniz, in this respect, to have been ahead of Boole.

George Boole’s great achievement was to demonstrate once and for all that logical deduction could be developed as a branch of mathematics. Although there had been some developments in logic after Aristotle’s pioneering work (notably by the stoics in Hellenistic times and by the twelfth century scholastics in Europe), Boole had found the subject essentially as Aristotle left it two millennia earlier. After Boole, mathematical logic has been under continuous development to the present day.*

*The study (Barret-Ducrocq, 1989) of a similar institution in London recounts many such tales of woe

*Boole’s equation xx = x can be compared to Leibniz’s A ⊕ A = A. In both cases, an operation intended to be applied to pairs of items, when applied to an item and itself, yields that same item as a result.

*Lewis Carroll Carroll (1988, pp. 258–259) tells us that in a “sillygism” one proceeds from two “prim Misses” to a “delusion.”

*An international organization, the Association for Symbolic Logic publishes two quarterly journals and holds regular meetings for the dissemination of new research. European logicians also hold annual meetings. New work on the relationships between logic and computers is presented at the annual international Logic in Computer Science and Computer Science Logic conferences.

CHAPTER 3 Frege: From Breakthrough to Despair

3 장: 논리의 표현을 완성한 프레게

In June 1902 a letter arrived in Jena, a medieval German town, addressed to the 53-year-old Gottlob Frege from the young British philosopher Bertrand Russell. Although Frege believed that he had made important and fundamental discoveries, his work had been almost totally ignored. It must then have been with some pleasure that he read, “I find myself in agreement with you in all essentials … I find in your work discussions, distinctions, and definitions that one seeks in vain in the work of other logicians.” But, the letter continued, “There is just one point where I have encountered a difficulty.” Frege soon realized that this one “difficulty” seemed to lead to the collapse of his life’s work. It cannot have helped that Russell went on to write, “The exact treatment of logic in fundamental questions has remained very much behind; in your works I find the best I know of our time, and therefore I have permitted myself to express my deep respect to you.”

Frege replied at once to Russell, acknowledging the problem. The second volume of his treatise in which he had applied his logical methods to the foundations of arithmetic was already at the printer, and he hastily added an appendix beginning with “There is nothing worse that can happen to a scientist than to have the foundation collapse just as the work is finished. I have been placed in this position by a letter from Mr. Bertrand Russell …”

프레게는 즉시 러셀에게 문제를 인정하며 답장을 보냈습니다. 그는 자신의 논리적 방법을 산술의 기초에 적용한 논문의 두 번째 권이 이미 인쇄소에 도착해 있었고, “작업이 끝나자마자 기초가 무너지는 것보다 과학자에게 더 나쁜 일은 없다"로 시작하는 부록을 서둘러 추가했습니다. 저는 버트런드 러셀 씨의 편지에 의해 이 자리에 오르게 되었습니다…“로 시작하는 부록을 추가했습니다

Many years later, more than four decades after Frege’s death, Bertrand Russell had occasion to write:

As I think about acts of integrity and grace, I realize that there is nothing in my knowledge to compare with Frege’s dedication to truth. His entire life’s work was on the verge of completion, much of his work had been ignored to the benefit of men infinitely less capable, his second volume was about to be published, and upon finding that his fundamental assumption was in error, he responded with intellectual pleasure clearly submerging any feelings of personal disappointment. It was almost superhuman and a telling indication of that of which men are capable if their dedication is to creative work and knowledge instead of cruder efforts to dominate and be known.¹

Much of the contemporary philosopher Michael Dummett’s work has been inspired by Frege’s ideas. Yet when he wrote about Frege’s integrity, it was in a quite different vein:

There is some irony for me in the fact that the man about whose philosophical views I have devoted, over the years, a great deal of time to thinking, was, at least at the end of his life, a virulent racist, specifically an anti-semite. … [His] diary shows Frege to have been a man of extreme right-wing opinions, bitterly opposed to the parliamentary system, democrats, liberals, Catholics, the French and, above all, Jews, who he thought ought to be deprived of political rights and, preferably, expelled from Germany. I was deeply shocked, because I had revered Frege as an absolutely rational man … ^{2}

Frege’s contributions were of immense importance. He provided the first fully developed system of logic that encompassed all of the deductive reasoning in ordinary mathematics, and his pioneering work using tools of logical analysis to study language provided the basis for major developments in philosophy. Today, under the subject heading “Frege, Gottlob” well over 50 items will be found in a typical university library. He died in 1925 a bitter man, believing that his life’s work had led only to futility, his death ignored by the scholarly community.³

Gottlob Frege was born on November 8, 1848, in Wismar a small German town. His father, a theologian in the Evangelical faith, headed a girls’ high school (where his mother was also employed). Frege was 38 when he married the 35-year-old Margarete Lieseberg who, after 17 years of marriage, died leaving no children behind.

At the request of a clergyman who was a relative on his mother’s side, Frege adopted a five-year-old orphan in 1908. It was this son, Alfred, who brought to light the infamous diary Frege had kept in 1924, a year before his death, the diary that so outraged and disillusioned Michael Dummett. Alfred Frege as part of the German military occupation of Paris was killed in action in June 1944, a little over a week after the Allied landings in Normandy and just two months before the liberation of Paris. The diary had been typed by Alfred from his father’s handwritten manuscript and in 1938, five years after Hitler had seized power, Alfred sent it to the Frege archive maintained by Heinrich Scholz. At that time the sentiments that so outraged Michael Dummett would have seemed unexceptional in Germany. The manuscript and a biography Alfred had written of his father are lost.

Frege was 21 when he entered the university. After two years at Jena he moved to Göttingen University where, three years later, he received a Ph.D. in mathematics. Then, he obtained an appointment as lecturer (“Privatdozent”) at the University of Jena, a position without salary. It seems that Frege was supported at this time by his mother who, on his father’s death, had taken over management of the girls’ school. After five years Frege was appointed Associate Professor (“Ausserordentlichen Professor”) at Jena where he remained until his retirement in 1918. Because his colleagues didn’t really value his work, he was never promoted to a full professorship. His death at Bad Kleinen near Wismar, where his impoverishment had forced him to board with relatives, came little over a year after the final entry in his deplorable diary.

GOTTLOB FREGE

(Institute for Mathematical Logic and Foundational Research, Münster University)

In 1873, the year of Frege’s initial appointment at Jena, Germany, newly united, was in a state of euphoria. The war against the France of Napoleon III had ended in a great victory. Industry was expanding at breakneck speed. Until the death of Kaiser Wilhelm I, his Chancellor, Bismarck, continued his cunning policy of maintaining the security of Germany by means of a carefully nurtured system of alliances. Bismarck and the “old Kaiser” remained heroes to Frege for his entire life. However, Bismarck was a reactionary who saw to it that the emperor maintained total control of military affairs and foreign relations. He regarded democracy as anathema, and pushed legislation outlawing many of the activities of the Social Democratic party.

Soon after Wilhelm II succeeded to the throne, he got rid of Bismarck. The new Kaiser, a vainglorious and insecure man, oversaw a disastrous foreign policy. Repeatedly misjudging the effect of his maneuvers, he managed to so alarm the other European powers that France, Russia, and England formed an alliance against Germany. Faced with the danger of a war on two fronts, against Russia on the east and against France on the west, the German general staff produced the clever, but ultimately disastrous, Schlieffen plan, designed to defeat France quickly before Russia could complete its ponderous mobilization. ^{4}

So when, with German encouragement, the Austrians attacked Serbia in the summer of 1914, in response to the assassination of Archduke Ferdinand, and Russia began mobilization to stress its determination that Austria not be permitted to destroy fellow Slavs, the German generals explained to the Kaiser that the Schlieffen plan calling for a German attack through Belgium had to be implemented at once. The attendant violation of Belgium’s neutrality brought England into this catastrophic war whose consequences cast their shadow on the entire twentieth century. In war things rarely go according to plan, and when the Schlieffen plan attack petered out, the fighting degenerated into a murderous stalemate, slaughtering the best part of a generation of European men in trench warfare. Seemingly unaware that the fighting was going badly, many German academics called for a peace in which Germany would annex much territory, including all of Belgium.

As victory continued to elude the Germans and the English siege took its toll, the military command was put into the hands of General Ludendorff. This capricious gambler (who was later to participate in Hitler’s beer hall “Putsch”) refused to consider a compromise peace until a British breakthrough in the Balkans threatened to roll up the German flank. With defeat staring him in the face, Ludendorff told the Kaiser that an armistice was essential. So ended the war and the German monarchy.

The government that assumed power in the new German republic was Social Democratic, and many Germans (Frege among them) came to accept the story that Germany had been forced into the war against its will, had not been defeated, but had been betrayed by the socialists, and (many were soon adding) the Jews. This was the poisonous atmosphere that ultimately made it possible for Hitler to assume power.

The year 1923 saw the great post-war hyper-inflation in Germany, in part the result of the unrealistic reparations imposed by the Versailles treaty. This financial catastrophe wiped out the values of personal savings, and presumably, of Frege’s pension. It was in this situation that Frege produced his terrible diary. He looked for a great leader to rescue Germany from the lowly position into which it had been thrust. Having held high hopes for Ludendorff to play this role, he was disappointed that he had joined Hitler’s Putsch. He still had hope that General Hindenburg might be the leader, but feared that he was too old; Frege did not live to see Hindenburg hand the keys to the republic to Adolf Hitler.

In his diary entry for April 22, 1924, Frege reminisces about a time when the Jews of his home town were treated in what he thought was an appropriate manner and also manages to disclose his views on the French and their baleful influence:

There was a law at that time that Jews were permitted to stay overnight in Wismar only in the time of certain annual fairs, … I suppose this decree was old. The old Wismarkers must have had experiences with the Jews that had led them to this legislation.

It must have been the Jewish way of doing business together with the Jewish national characteristics that is tied together closely with the way of doing business. … There came universal suffrage, even for Jews. There came the freedom of movement, even for Jews, presents from France. We make it so easy for the French to bless us with gifts. If one had only turned to noble and patriotic Germans … The French had treated us nastily enough indeed before 1813, and nevertheless we have this blind admiration of all things French. … I have only in the last years really learned to comprehend antisemitism. If one wants to make laws against the Jews, one must be able to specify a distinguishing mark by which one can recognize a Jew for certain. I have always seen this as a problem.

The problem, merely theoretical for Frege, of defining Jews with sufficient precision so that one could make laws against them, became quite a practical problem under the Nazis. Ludwig Wittgenstein, thought to be one of the great thinkers of the twentieth century and an admirer and disciple of Frege, would have qualified as a Jew under the Nazi racial code.

Other diary entries rail against the Social Democrats and Catholics:

The Reich suffered from a cancer in 1914, namely Social Democracy. (April 24)

To be sure, I regarded Ultramontanism and its embodiment in the Zentrum as very detrimental for our Reich and nation; nevertheless, the revelations of … Ludendorf in his [recent] article on the efforts and machinations of the ultramontanes give me insights which have most deeply disturbed me.* I implore anybody who does not yet believe in the thoroughly unGerman spirit of the Zentrum to read and reflect on the stated article of His Excellency Luden-dorf … This is the most evil enemy which undermined Bismarck’s Reich. … [The Ultramontanes] will always look to the Pope to get their instructions. (April 26)⁵

Frege’s extreme right-wing ideas were hardly rare in Germany after World War I. Nevertheless, we may wonder whether the diary represents only the thoughts of a bitter (and possibly senile) old man within a year of his death. Alas, there is little doubt that Frege had held right-wing views for some time. Frege’s colleague, Bruno Bauch, a philosophy professor at Jena, founded a right-wing philosophical society (the DLG) during the war, and he edited its journal. Frege was one of the early adherents of the DLG, and published in its journal. Bauch’s writings on the concept of nation insisted that no Jew could really be a German. His group came out in full support of the Nazis when they took power in 1933.⁶

Frege’s Begriffsschrift

It is with a sense of relief that one turns from Frege’s awful views, expressed as his life drew towards its end, to the brilliant contributions he made as a young man. In 1879,* he published a booklet of fewer than 100 pages entitled Begriffsschrift, a hard-to-translate word Frege constructed from the German words Begriff (“concept”) and Schrift (roughly “script” or “mode of writing”). It was subtitled, “a formula language, modeled upon that of arithmetic, for pure thought.” This work has been called “perhaps the most important single work ever written in logic.”⁷

Frege sought a system of logic that included all of the deductive inferences in mathematical practice. Boole took ordinary algebra as his starting point and used the symbols of algebra to represent logical relations. Since Frege intended algebra, like other parts of mathematics, to be built as a superstructure with his logic as a foundation, he regarded it as important to introduce his own special symbols for logical relationships to avoid confusion.

Also, where Boole had thought of propositions that express relations between other propositions as “secondary propositions,” Frege saw that the same relations that connect propositions can also be used to analyze the structure of individual propositions, and he made these relations the basis of his logic. This crucial insight has gained general acceptance and forms the basis of modern logic.

For example, Frege would analyze the statement that “All horses are mammals” using the logical relationship if … then …:

\({if}\ x\ \text{is}\ \text{a}\ \text{horse},\ \ \text{then}\ x\ \text{is}\ \text{a}\ \text{mammal}\text{.}\)

Likewise, he would analyze the statement that “Some horses are pure-bred” using the logical relationship … and …:

\(x\ \text{is}\ \text{a}\ \text{horse},\ \ \text{and}\ x\ \text{is}\ \ \text{pure-bred}\text{.}\)

However, the letter x is used differently in these two examples. In the first example one wants to say that what is asserted is true whatever x might be, that is, for every x. But in the second example what is wanted is only the assertion for some x. In the symbolism in current use, for every is written ∀ and for some is written ∃. So, the two sentences could be written as follows:

\(\begin{matrix} {\text{(}\forall x\text{)}\ \text{(}if\ x\ \ \text{is}\ \ \text{a}\ \text{horse},\ \ \text{then}\ \ x\ \ \text{is}\ \ \text{a}\ \text{mammal)}} \\ {(\exists x)\ (x\ \text{is}\ \text{a}\ \text{horse},\ \ \text{and}\ x\ \text{is}\ \ \text{pure-bred})} \end{matrix}\)

The symbol ∀, an upside-down A, suggests “all” and is called a universal quantifier. Likewise the symbol ∃, a backwards E, is called an existential quantifier, and is intended to suggest “exists.” So this second sentence could be read:

  There exists x such that x is a horse and x is pure-bred.

The logical relation if … then … is usually symbolized ⊃, and the relation … and … is symbolized …. Using these symbols, the sentences become:⁸

\(\begin{matrix} {\text{(}\forall x\text{)}\ \text{(}\ x\ \ \text{is}\ \ \text{a}\ \text{horse}\ \ \supset \ \ x\ \ \text{is}\ \ \text{a}\ \text{mammal)}} \\ {(\exists x)\ (x\ \ \text{is}\ \ \text{a}\ \text{horse}\ \ \land \ x\ \ \text{is}\ \ \text{pure-bred})} \end{matrix}\)

They can be abbreviated as follows:

\(\begin{matrix} {\text{(}\forall x\text{)}\ \text{(}\ \text{horse}\ \ \supset \ \ \ \text{mammal}\ \text{(}x\text{))}} \\ {(\exists x)\ (\ \text{horse}\ \ \land \ \ \ \text{pure-bred}\ \text{(}x\text{)})} \end{matrix}\)

Or more briefly:

\(\begin{matrix} {\text{(}\forall x\text{)}\ \text{(}\ h\text{(}x\text{)}\ \ \supset \ \ \ m\ \text{(}x\text{))}} \\ {(\exists x)\ (\ h\text{(}x\text{)}\ \ \land \ \ \ p\ \text{(}x\text{)})} \end{matrix}\)

Joe and Susan’s effort to use logic in locating Joe’s wallet was used as an example in the previous chapter. In that example, we used letters to abbreviate sentences as follows:

_L_Joe left his checkbook at the supermarket.

_F_Joe’s checkbook was found at the supermarket.

_W_Joe wrote a check at the restaurant last night.

_P_After writing the check last night, Joe put his check book in his jacket pocket.

_H_Joe hasn’t used his check book since last night.

_S_Joe’s checkbook is still in his jacket pocket.

Their reasoning came down to the following pattern:

Using the symbol ¬ to stand for “not,” and the other symbols we’ve introduced, this now becomes

\(\begin{matrix} {L\ \supset \ F} \\ {\neg F} \\ {W\ \land \ P} \\ \underline{\begin{matrix} {W\ \land\ P\ \land\ H\ \supset\ S} \\ H \end{matrix}} \\ {\neg L} \\ S \end{matrix}\)

One final symbol should be mentioned: ∨ standing for … or …. The following table provides a summary of the symbols that have been introduced:

At the end of the previous chapter, the statement that all failing students are either stupid or lazy was exhibited as an example whose logical structure would be missed by Boole’s analysis. In Frege’s logic, it is easy. Writing

\(\begin{array}{lll} {F(x)} & \text{for} & {x\ \ \text{is}\ \ \text{a}\ \text{failing}\ \ \text{student,}} \\ {S(x)} & \text{for} & {x\ \ \text{is}\ \ \text{stupid,}} \\ {L(x)} & \text{for} & {x\ \ \text{is}\ \ \text{lazy,}} \end{array}\)

the sentence can be expressed as

\(\text{(}\forall x\text{)}\ \text{(}\ F\text{(}x\text{)}\ \ \supset \ \ \ S\ \text{(}x\text{)}\ \ \vee \ L\ \text{(}x\text{))}\text{.}\)

By now it should be clear that Frege was not just developing a mathematical treatment of logic, but was creating a new language. In this he was guided by Leibniz’s notion of a universal language that would gain its power from a judicious choice of symbols.⁹ The expressiveness of this language can be gauged from the following examples using L/(/x, y) to stand for x loves y.

Here is one more example:

\(\text{Everyone}\ \ \text{loves}\ \ \text{a}\ \ \text{lover}\text{.}\)

As a first stab we write:

\(\text{(}\forall x\text{)}\ \text{(}\forall y\text{)}\ \text{[}y\ \ \text{is}\ \ \text{a}\ \ \text{lover}\ \ \supset \ \ L\ \text{(}x,y\text{)]}\text{.}\)

Now, if we construe being a lover as simply meaning loving someone, we can replace y is a lover by (∃/z/)/L/(y, z), finally obtaining:

\(\text{(}\forall x\text{)}\ \text{(}\forall y\text{)}\ \text{[(}\exists z\text{)}\ L\ \text{(}y,z\text{)}\ \ \supset \ \ L\ \text{(}x,y\text{)]}\text{.}\)

Frege Invents Formal Syntax

Boole’s logic was simply another branch of mathematics to be developed using ordinary mathematical methods. This of course includes using logical reasoning. But there is something circular about using logic to develop logic. For Frege this was unacceptable. His intention was to show how all mathematics could be based on logic; logic was to provide a foundation for all the rest of mathematics. For this to be at all convincing, Frege had to find some way to develop his logic without using logic in the process.

His solution was to develop his Begriffsschrift as an artificial language with mercilessly precise rules of grammar, or as one says, of syntax. This made it possible to exhibit logical inferences as purely mechanical operations, so-called rules of inference, having reference only to the patterns in which symbols are arranged. It was also the first example of a formal artificial language constructed with a precise syntax. From this point of view, the Begriffsschrift was the ancestor of all computer programming languages in common use today.

The most fundamental of Frege’s rules of inference works like this: if ⋄ and ⋄ are any two sentences of Frege’s Begriffsschrift, then if ⋄ and (⋄ ⊃ △) are both asserted, then one is permitted to also assert the sentence Δ. It is important to notice that to carry out this operation, no understanding of what ⊃ means is required. Of course we can see that the rule cannot lead to error because it only enables one to proceed from ⋄ and (if ⋄ then △) to △. But to actually employ the rule, it is only necessary to match up the individual symbols constituting the sentence ⋄ with symbols in the first part of the longer sentence.¹⁰ In our example of locating Joe’s wallet, we had the premise

\(W\ \land \ P\ \land \ H\ \supset \ S.\)

If we were able to also assert W ∧ P ∧ H, then the rule would enable us to also assert one of the desired conclusions, namely S. Here is how the match-up would go:

\(\begin{array}{l} {W\ \land \ P\ \land \ H\ \supset \ S.} \\ {W\ \land \ P\ \land \ H\ .} \end{array}\)

Frege’s logic has become the standard taught to undergraduate students in logic courses in mathematics, computer science, and philosophy departments.¹¹ It has been the basis for an enormous body of research, and indirectly led Alan Turing to formulate the idea of an all-purpose computer. But this is getting ahead of ourselves.

Frege’s logic was an enormous advance over Boole’s. For the first time an exact system of mathematical logic encompassed, at least in principle, all the reasoning ordinarily used by mathematicians. But in attaining this goal, something was given up. Beginning with some premises in Frege’s logic, Frege’s rules could be applied in an attempt to reach a desired conclusion. But if the attempt failed, Frege provided no means to know whether this was because not enough cleverness or persistence was employed, or whether the desired conclusion simply did not follow from the given premises. This lack meant that Frege’s logic did not fulfill Leibniz’s dream that with the words “Let us calculate,” those knowing the rules of logic would be able to proceed to determine unfailingly whether or not some conclusion follows.

Why Bertrand Russell’s Letter Was So Devastating

If Frege’s logic was such a great achievement, why did Russell’s letter lead Frege to despair? Frege regarded his logic as only a stepping stone towards complete foundation for arithmetic. Although the differential and integral calculus of Leibniz and Newton led to fruitful developments, there were serious problems in justifying some of the steps in the reasoning mathematicians were in the habit of employing. During the nineteenth century these problems were gradually cleared up, ultimately by developing a new and profound theory of the number system of mathematics. However, in the end we still rely on the so-called natural (or counting) numbers:

\(1,\ 2,\ 3,\ \ldots\)

Frege wanted to provide a purely logical theory of the natural numbers and thereby to demonstrate that arithmetic, and indeed all of mathematics including developments stemming from the differential and integral calculus, could be regarded as a branch of logic. This point of view, which came to be called logicism, was also that of Bertrand Russell. Logicism has been explained by the American logician Alonzo Church as maintaining that the relationship between logic and mathematics is that between the elementary and the advanced part of one and the same subject.*

Thus Frege wanted to be able to define the natural numbers in purely logical terms, and then to use his logic to derive their properties. The number 3 for example was to be explained as part of logic. How could this be possible? A natural number is a property of a set, namely, the number of its elements. The number 3 is something that all of the following have in common: the Holy Trinity, the set of horses pulling a troika, the set of leaves on a (normal) clover leaf, the set of letters {a, b, c}. Without saying anything about the number 3, one can see that any two of these sets have the same number of elements. We can simply match them up. Frege’s idea was to identify the number 3 with the collection of all of these sets. That is, the number 3 is just the set of all triples. In general, the number of elements in a given set can be defined to be the collection of all those sets that can be matched one-to-one with the given set.¹²

Frege’s two-volume treatise on the foundations of arithmetic showed how to develop the arithmetic of natural numbers using the logic developed in his Begriffsschrift. Bertrand Russell’s letter of 1902 showed Frege that this entire development was inconsistent, that is, self-contradictory. Frege’s arithmetic, in effect, made use of sets of sets. Russell showed in his letter that reasoning with sets of sets can easily lead to contradictions.

Russell’s “paradox” can be explained as follows: Call a set extraordinary if it is a member of itself; otherwise call it ordinary. How could a set be extraordinary? Russell’s own example of an extraordinary set is: the set of all those things that can be defined in fewer than 19 English words. Since we have just defined this set using only 16 words, it belongs to itself and therefore is extraordinary. Another example is the set of all things that are not sparrows. Whatever this set might be, it is surely not a sparrow. So this set too is extraordinary.

Russell proposed to Frege the set \(\mathcal{E}\) of all ordinary sets. Is \(\mathcal{E}\) ordinary or extraordinary? It must be one or the other. But it seems to be neither. Could \(\mathcal{E}\) be ordinary. If so, since \(\mathcal{E}\) is the class of all ordinary sets, it would belong to itself. But then it would be extraordinary. OK. Then \(\mathcal{E}\) would have to be extraordinary. Therefore, it would not belong to itself, since it is the set of ordinary sets. But that would make it ordinary! Either way one is led to a contradiction!

Russell’s paradox is first cousin to a large number of puzzles that are simply amusing. But when Frege received Russell’s letter, he was not amused. He realized at once that the contradiction could be readily derived within the system he was using to develop arithmetic. Now, a mathematical proof that runs into a contradiction is a demonstration that one of the premises of the argument was false. This principle is used all the time as a useful proof method: to prove a proposition, one shows that its denial leads to a contradiction. But for poor Frege, the contradiction had shown that the very premises on which his system was built were untenable. Frege never recovered from this blow.¹³

Frege and the Philosophy of Language

In 1892 Frege published a paper in a philosophical journal whose title may be translated as On Sense and Denotation.¹⁴ Along with Frege’s logic, it is because of the issues raised in this paper that philosophers have been so interested in his work.

Frege pointed out that different words may be used to denote one specific object although they may have quite different senses or meanings. His famous example uses the phrases “the morning star” and “the evening star.” Their sense is quite different: one is the bright star seen after sunset, the other the one seen before sunrise. But both denote the same planet, Venus. The fact that both phrases refer to the same object is not obvious; it was at one time a real astronomical discovery. Some of Frege’s concerns have to do with substitutivity: Consider the sentence

\(\text{Venus}\ \ \text{is}\ \ \text{the}\ \ \text{morning}\ \ \text{star}\text{.}\)

This is very different from

\(\text{Venus}\ \ \text{is}\ \ \text{Venus}\text{.}\)

This is the case although in fact, one sentence was derived from the other by replacing one phrase by another denoting the same object.

These ideas represent the beginning of a major branch of twentieth century philosophy: the philosophy of language.¹⁵ In addition, some key concepts in contemporary computer science may be said to have their origin in this same essay.¹⁶

Frege and Leibniz’s Dream

Frege thought of his Begriffsschrift as embodying the universal language of logic that Leibniz had called for. Indeed, Frege’s logic can deal with the most diverse subjects. But to Leibniz it would likely have been a disappointment. It fell short of his desires in at least two important respects. Leibniz had imagined a language that was capable not only of logical deduction but that also would automatically include all the truths of science and of philosophy. This naive expectation was only conceivable before the massive development of science in the eighteenth and nineteenth centuries based on careful experiment as well as theorizing.

From the point of view of our story, it is more appropriate to point to a different limitation of Frege’s logic. Leibniz had called for a language that would also be an efficient instrument of calculation, one that would enable logical inferences to be carried out systematically by the direct manipulation of symbols. In fact any but the simplest of deductions are almost unbearably complicated in Frege’s logic. Not only are such deductions tediously long, but Frege’s rules provide no calculational procedures for determining whether some desired conclusion can be deduced from given premises in the logic of his Begriffsschrift.

Because the Begriffsschrift did fully encapsulate the logic used in ordinary mathematics, it became possible for mathematical activity to be investigated by mathematical methods. As we will see, these investigations led to some very remarkable and unexpected developments. The search for a calculational method that could show whether or not a proposed inference in Frege’s logic is correct reached its climax in 1936 with a proof that no such general method exists.

This was bad news for Leibniz’s dream. However, in the process of proving this negative result, Alan Turing discovered something that would have delighted Leibniz: he found that it was possible, in principle, to devise one single “universal” machine that could alone carry out any possible computation.

*The Zentrum party was oriented towards the Catholic Church. Its “Ultramontanism” referred to the influence from “over the mountains,” that is, Rome

*I was invited to present an address at a scientific conference in 1979 commemorating the hundredth anniversary of the Begriffsschrift in which I was to trace its consequences for computer science. This was the beginning of my second career as a historian of science

*It is now generally recognized that, by the use of numerical coordinates, geometry can also be reduced to arithmetic. However, Frege always believed that geometry had to be regarded as separate. I’m indebted to Patricia Blanchette for emphasizing this aspect of Frege’s thought and for other helpful comments on this section.

프레게의 개념 표기법

정규화된 문법을 만든 프레게

버트런드 러셀의 편지가 그리도 절망적이었던 이유

프레게와 언어의 철학

프레게와 라이프니츠의 꿈

4 장: 무한을 탐험한 칸토어

공학자 혹은 수학자

서로 다른 크기의 무한 집합들

무한한 수를 향한 칸토어의 탐구

대각선 논법

우울증과 비극

결정적인 전투?

부록: 칸토어와 크로네커

5 장: 완전한 알고리즘을 꿈꾼 힐베르트

힐베르트의 초기 업적

새로운 한 세기를 향하여

무한을 둘러싼 싸움

메타수학

파국

6 장: 완전한 계산의 꿈을 뒤흔든 괴델

괴델의 박사 학위 논문

결정불가능 명제들

컴퓨터 프로그래머 쿠르트 괴델

쾨니히스베르크 학술회의

사랑과 혼란

1930 년대 프린스턴에서 수학을 둘러싸고 벌어진 일들

빈으로의 회귀

힐베르트의 사상

불가사의한 인물의 슬픈 마지막

부록: 괴델의 불완전 명제

7 장: 범용 컴퓨터를 생각해 낸 튜링

대영 제국의 아이

힐베르트의 결정 문제

튜링이 분석한 계산의 과정

튜링 기계의 동작

칸토어의 대각선 논법을 적용한 튜링

해결할 수 없는 문제

튜링의 범용 기계

프린스턴에서의 앨런 튜링

앨런 튜링의 전쟁

8 장: 최초의 디지털 범용 컴퓨터

누가 컴퓨터를 발명했나?

존 폰 노이만과 무어 공과 대학

앨런 튜링의 에이스(ACE)

에커트, 폰 노이만, 그리고 튜링

감사해야 할 국가가 영웅을 대접한 방식

9 장: 라이프니츠의 꿈을 넘어

엘리자, 왓슨, 그리고 딥 블루

바둑을 두는 컴퓨터

컴퓨터, 두뇌, 마음

Epilogue

We have followed the lives of a group of brilliant innovators spanning three centuries. All of them in one way or another were concerned with the nature of human reason. Their individual contributions added up to the intellectual matrix out of which emerged the all-purpose digital computer. Except for Turing, none of them had any idea that his work might be so applied.

우리는 3 세기에 걸친 뛰어난 혁신가들의 삶을 추적해 왔습니다. 이들은 모두 어떤 식으로든 인간 이성의 본질에 관심을 가졌습니다. 이들의 개별적인 공헌이 더해져 다목적 디지털 컴퓨터가 탄생한 지적 매트릭스가 탄생했습니다. 튜링을 제외하고는 누구도 자신의 연구가 이렇게 응용될 수 있을 것이라고는 생각하지 못했습니다.

Leibniz saw far, but not that far. Boole could hardly have imagined that his algebra of logic would be used to design complex electric circuits. Frege would have been amazed to find equivalents of his logical rules incorporated into computer programs for carrying out deductions. Cantor certainly never anticipated the ramifications of his diagonal method. Hilbert’s program to secure the foundations of mathematics was pointed in a very different direction. And Gödel, living his life of the mind, hardly thought of applications to mechanical devices.

라이프니츠는 멀리 내다봤지만 그렇게 멀리 본 것은 아니었습니다. 불은 자신의 논리 대수가 복잡한 전기 회로를 설계하는 데 사용될 것이라고는 상상도 하지 못했을 것입니다. 프레게는 자신의 논리 규칙과 동등한 규칙이 추론을 수행하는 컴퓨터 프로그램에 통합되어 있다는 사실에 놀랐을 것입니다. 캔터는 자신의 대각선 방법이 가져올 파급력을 전혀 예상하지 못했을 것입니다. 수학의 기초를 다지기 위한 힐베르트의 프로그램은 전혀 다른 방향을 지향했습니다. 그리고 괴델은 정신의 삶을 살면서 기계 장치에 대한 응용은 거의 생각하지 못했습니다.

This story underscores the power of ideas and the futility of predicting where they will lead. The Dukes of Hanover thought they knew what Leibniz should be doing with his time: working on their family history. Too often today, those who provide scientists with the resources necessary for their lives and work try to steer them in directions deemed most likely to provide quick results. This is not only likely to be futile in the short run, but more important, by discouraging investigations with no obvious immediate payoff, it short-changes the future.

이 이야기는 아이디어의 힘과 그것이 어디로 이어질지 예측하는 것의 무익함을 강조합니다. 하노버 공작들은 라이프니츠가 자신의 시간에 무엇을 해야 하는지 알고 있었다고 생각했습니다. 오늘날 과학자들에게 삶과 연구에 필요한 자원을 제공하는 사람들은 종종 빠른 결과를 가져올 가능성이 가장 높다고 여겨지는 방향으로 그들을 이끌려고 합니다. 이는 단기적으로는 쓸데없는 일일 뿐 아니라, 더 중요한 것은 즉각적인 보상이 분명하지 않은 연구를 막음으로써 미래를 단축시킨다는 점입니다.

Related-Notes

• #수학:
• 박정일 (2010) 지식인마을: 추상적 사유의 위대한 힘 - 튜링 괴델

References