Search Results

This chapter discusses the complex conditions for the emergence of 19th-century symbolic logic. The main scope will be on the mathematical motives leading to the interest in logic; the philosophical context will be dealt with only in passing. The main object of study will be the algebra of logic in its British and German versions. Special emphasis will be laid on the systems of George Boole (1815–1864) and above all of his German follower Ernst Schröder (1841–1902).

This chapter presents brief biographical sketches of George Boole and Claude Shannon. George was born in Lincoln, a town in the north of England, on November 2, 1815. His father John, while simple tradesman (a cobbler), taught George geometry and trigonometry, subjects John had found of great aid in his optical studies. Boole was essentially self-taught, with a formal education that stopped at what today would be a junior in high school. Eventually he became a master mathematician (who succeeded in merging algebra with logic), one held in the highest esteem by talented, highly educated men who had graduated from Cambridge and Oxford. Claude was born on April 30, 1916, in Petoskey, Michigan. He enrolled at the University of Michigan, from which he graduated in 1936 with double bachelor's degrees in mathematics and electrical engineering. It was in a class there that he was introduced to Boole's algebra of logic.

This introductory chapter provides an overview of the life and work of George Boole. Boole is known for developing the system of algebraic logic, which eventually found an unexpected engineering application in the design of switching circuits. He was also an early advocate of the mass distribution of knowledge, using the methods at his disposal in early Victorian times. In the classroom and lecture hall, he interpreted the results of recent discoveries and debates originating among specialists in numerous fields—history, psychology, ethnography, and much else—and communicated them to a broad audience. Less known and therefore less appreciated is Boole's role in the history of the making and organization of knowledge. A better understanding of this feature, may eventually provoke a more thoroughgoing reappraisal of the whole figure.

Chapter 16 tells the story of George Boole, his wife Mary Everest Boole, and their five daughters. Boole was largely self-taught and was appointed the first Professor of Mathematics at Queen’s College Cork, now University College Cork, in 1849. When he died, his wife Mary and young daughters returned to London, where Mary made the acquaintance of James Hinton and his circle of literary friends. Mary developed methods of teaching mathematics that she passed on to her daughters, who all lived remarkable lives. Ethel Lily Boole was a very successful novelist, author of The Gadfly. She married Wilfrid Michael Voynich, famous for discovering the Voynich manuscript. Mary Ellen Boole married Charles Howard Hinton, who would popularize the notion of higher dimensions.

This chapter discusses how Boole cast logic into algebraic form. Boole was interested in symbolic analysis years before he wrote his Laws of Thought. In fact, others before him—in particular, the German mathematician Gottfried Wilhelm Leibniz (1646–1716)—had pursued a similar goal of reducing logic to algebra, but it was Boole who finally succeeded. What Boole described in his books is not exactly what modern users call Boolean algebra, but nevertheless it is from Boole that the modern presentation springs. The chapter first describes the essence of what Boole did using the language of sets. It then discusses Boole's algebra of sets, examples of Boolean analysis, and visualizing Boolean functions.

The difference between the transcendent Coleridgean symbol and the unreliable conventional symbol was of explicit concern in Victorian mathematics, where the former was aligned with Euclidean geometry and the latter with algebra. Rather than trying to bridge this divide, practitioners of modern algebra and the pioneers of symbolic logic made it the founding principle of their work. Regarding the content of claims as a matter of “indifference,” they concerned themselves solely with the formal interrelations of the symbolic systems devised to represent those claims. In its celebration of artificial algorithmic structures, symbolic logician Lewis Carroll’s Sylvie and Bruno dramatizes the power of this new formalist ideal not only to revitalize the moribund field of Aristotelian logic but also to redeem symbolism itself, conceived by Carroll and his mathematical, philosophical, and symbolist contemporaries as a set of harmonious associative networks rather than singular organic correspondences.

This paper argues that historians of philosophy must coin concepts that disclose the near or distant past and create a shared horizon for our philosophical future. Two concepts are introduced: “Newton’s challenge to philosophy” and “philosophic prophecy.” “Newton’s challenge to philosophy” explains that from about 1700 onward, “natural science” is increasingly taken to be authoritative in settling debates within philosophy. “Philosophic prophecy,” comprises the structured ways in which concept formation by philosophers can shape possible futures, including that of philosophy. The second half of the paper offers a fresh narrative about the shared origins of analytical philosophy and analytical history of philosophy in the anti-Spinozistic writings of George Boole and Bertrand Russell. Ernest Nagel is treated as the philosophic prophet of analytical philosophy his views are compared to those of Moritz Schlick.

This chapter explores how algebraic logic began in 1847 when George Boole presented his “calculus of deductive reasoning” in a short book titled The Mathematical Analysis of Logic. His calculus reduced known ways of logical reasoning into the solution of algebraic equations. The known ways of logical reasoning were not just accounted for but were extended to full classical propositional logic. Boole reduced Aristotelian syllogistic reasoning to calculation, which was a wonderful achievement. Encouraged by the success, he wrote a book with the bold title An Investigation of the Laws of Thought (1854). However, his logic was not able to treat relations but just one-place predicates.

George Boole and Claude Shannon shared a deep interest in the mathematics of probability. Boole's interest was, of course, not related to the theory of computation—he was a century too early for that—while Shannon's mathematical theory of communication and information processing is replete with probabilistic analyses. There is, nevertheless, an important intersection between what the two men did, which is shown in this chapter. The aim is to provide a flavor of how they reasoned and of the sort of probabilistic problem that caught their attention. Once we have finished with Boole's problem, the reader will see that it uses mathematics that will play a crucial role in answering Shannon's concern about “crummy” relays.

This introductory chapter considers the work of mathematician George Boole (1815–1864), whose book An Investigation of the Laws of Thought (1854) would have a huge impact on humanity. Boole's mathematics, the basis for what is now called Boolean algebra, is the subject of this book. It is also called mathematical logic, and today it is a routine analytical tool of the logic-design engineers who create the electronic circuitry that we now cannot live without, from computers to automobiles to home appliances. Boolean algebra is not traditional or classical Aristotelian logic, a subject generally taught in college by the philosophy department. Boolean algebra, by contrast, is generally in the hands of electrical engineering professors and/or the mathematics faculty.

Boole and Shannon never studied the physics of computation. Obviously Boole simply could not have, as none of the required physics was even known in his day, and Shannon was nearing the end of his career when such considerations were just beginning. And yet, both Boole's algebra and Shannon's information concepts to make many of our calculations. This chapter touches on how fundamental physics—the uncertainty principle from quantum mechanics, and thermodynamics, for example—constrain what is possible, in principle, for the computers of the far future. It argues that while there are indeed finite limitations, present-day technology falls so far short of those limits that there will be good employment for computer technologists for a very long time to come.

This chapter provides an overview of the life of Charles Sander Peirce—philosopher, logician, scientist, and father of American pragmatism. This man, unappreciated in his lifetime, virtually ignored by the academic world of his day, is now recognized as perhaps America's most original philosopher and her greatest logician. Indeed, on the latter score, he is surely one of the logical giants of the nineteenth century, which produced such geniuses as Georg Cantor, Gottlob Frege, George Boole, Augustus De Morgan, Bertrand Russell, and Alfred North Whitehead. Today, more than eighty years after his death, another generation of scholars is beginning to pay him the attention he deserves. The chapter shows the brilliant and tragic career of Peirce. Though he never published a book on philosophy, his articles and drafts fill volumes.

This chapter presents George Boole's lecture on the claims of the pursuit of science, He says that like all other claims with which we are concerned, these must ultimately rest upon some intrinsic excellency or special suitableness of the object. Qualities such as these can alone give to it an enduring title to our regard. He draws attention to the ground of those claims, in the immediate or implied relations of science to human nature; in its relations, namely, as an answer to some of the distinctive wants of the human mind, an exercise to its faculties, a discipline of the character and habits, and an instrument of conquest and dominion over the powers of surrounding Nature.

This chapter presents George Boole's lecture on the social aspect of intellectual culture. He says that within proper limits and under proper conditions, intellectual tastes are not only compatible with social enjoyment, but tend to refine and enlarge that enjoyment. An interest in the progress of the arts and sciences and in the researches of the antiquary and the scholar is calculated not to destroy but to deepen our interest in humanity. He further says that the connection between intellectual discovery and the progressive history of our race gives to every stage of the former a deep human interest. Each new revelation, whether of the laws of the physical universe, of the principles of art, or of the great truths of morals and of politics, is a step not only in the progress of knowledge, but also in the history of our species.

This chapter presents George Boole's lecture on the origin of ancient mythologies. Very few ancient writers agree in the representations on this subject despite living in the country whose opinions they profess to relate. Comparison of the statements transmitted by Greek and Roman writers on the religious opinions of nations with which they were less immediately connected show discrepancies so great that the attempt to reconcile them is a hopeless task. Those who have written much on the subject generally started with some theory of their own to which they have endeavored to bend the facts they met with. The attempt to reduce to some single principle the various and discordant relations of mythology is, however, unlikely to meet with any success judging from the past.

This chapter presents George Boole's lecture on whether the planets are inhabited. He first considers the general conditions to which life here has manifested; second, the adaptations of the earth as the abode of life; and third, the presumptions in favor of such adaptation on the part of the remaining orbs of the planetary system. He says that if it is granted that the Author of nature acts consistently in all his works, that he accomplishes the purposes of his will by special means and adaptations, that his purpose with respect to the earth is that it should be an abode of life, and that the mass and adaptations by which that purpose is accomplished are employed in other worlds, then it appears that other worlds are also intended to be habitations of life.

This chapter presents George Boole's lecture on the right use of leisure. He says that every right involves a responsibility. The greater our freedom from external restrictions, the more we become the rightful subjects of the moral law within us. The less our accountability to man, the greater our accountability to a higher power. Such a thing as irresponsible right has no existence in this world. Even in the formation of opinion, which is of all things the most free from human control, and for which something like irresponsible right has been claimed, we are deeply answerable for the use we make of our reason, our means of information, and our various opportunities of arriving at a correct judgment.