1902 Encyclopedia > George Boole

George Boole
English mathematician and logician

GEORGE BOOLE, one of the most original logicians and mathematicians whom England has produced, was born in Lincoln on the 2d of November 1815. His father was a tradesman of limited means, but of studious char-acter and active mind. Being especially interested in mathematical science the father gave his son early instruc-tion in the rudiments of the science he was so greatly to advance; but it is remarkable that the extraordinary mathematical powers of George Boole did not manifest them-selves in early life. The classical languages formed at first the favourite subject of his studies. Not until the age of seventeen years did he attack the higher mathematics, and his progress was much retarded by the want of efficient help.

When about sixteen years of age he became assistant-master in a private school at Doncaster, and he maintained himself to the end of his life in one grade or other of the scholastic profession. Few distinguished men, indeed, have had a less eventful life. Almost the only changes which can be called events are his successful establishment of a school at Lincoln, its removal to Waddington, his appoint-ment in 1849 as professor of mathematics in the Queen's College at Cork, and his marriage in 1855 to Miss Mary Everest.

To the public Boole was known only as the author of numerous abstruse papers on mathematical topics, and of three or four distinct publications which have become standard works. His earliest published paper was one upon the " Theory of Analytical Transformations," printed in the Cambridge Mathematical Journal for 1839, and it led to a friendship between Boole and D. F. Gregory, the editor of the journal, which lasted until the premature death of the latter in 1844. A long list of Boole's memoirs and detached papers, both on logical and mathematical topics, will be found in the Catalogue of Scientific Memoirs published by the Royal Society, and in the supplementary volume on Differential Equations, edited by Mr Todhunter. To the Cambridge Mathematical Journal and its successor, the Cambridge and Dublin Mathematical Journal, Boole con-tributed in all twenty-two articles. In the third and fourth series of the Philosophical Magazine will be found sixteen papers. The Royal Society printed six important memoirs in' the Philosophical Transactions, and a few other memoirs are to be found in the Transactions of the Royal Society of Edinburgh and of the Royal Irish Academy, in the Bulletin de l'Académie de St Pétersbourg for 1862 (under the name G. Boldt, vol. iv. pp. 198-215), and in Or elle's Journal. To these lists should be added a paper on the mathematical basis of logic, published in the Mechanic's Magazine for 1848. The works of Boole are thus contained in about fifty scattered articles and a few separate publications.

Only two systematic treatises on mathematical subjects were completed by Boole during his lifetime. The well-known Treatise on Differential Equations appeared in 1859, and was followed, the next year, by a Treatise on the Calculus of Finite Differences, designed to serve as a sequel to the former work. These treatises have become the standard text-books on the important branches of mathe-matics in question, and Boole, in composing them, seems to have combined elementary exposition with the profound investigation of the philosophy of the subject in a manner hardly admitting of improvement. To a certain extent these works embody the more important discoveries of their author. In the 16th and 17th chapters of the Differential Equations we find, for instance, a lucid account of the general symbolic method, the bold and skilful employment of which led to Boole's chief discoveries, and of a general method in analysis, originally described in his famous memoir printed in the Philosophical Transactions for 1844. Boole was one of the most eminent of those who perceived that the symbols of operation could be separated from those of quantity and treated as distinct objects of calculation. His principal characteristic was perfect confidence in any result obtained by the treatment of symbols in accordance with their primary laws and conditions, and an almost unrivalled skill and power in tracing out these results.

During the last few years of his life Boole was constantly engaged in extending his researches with the object of pro-ducing a second edition of his Differential Equations much more complete than the first edition; and part of his last vacation was spent in arduous study in the libraries of the Royal Society and the British Museum for the purpose of acquiring a complete knowledge of the less accessible original memoirs on the subject. It must be always a matter of regret that this new edition was never completed. Even the manuscripts left at his death were so incomplete that Mr Todhunter, into whose hands they were put, found it impossible to use them in the publication of a second edition of the original treatise, and wisely printed them, in 1865, in a supplementary volume.

Profound and important as were Boole's discoveries in pure mathematics, his writings on logic may be considered as still more original. With the exception of De Morgan, he was probably the first English mathematician since the time of Wallis who had also written upon logic; and his wholly novel views of logical method were due to the same profound confidence in symbolic reasoning to which he had successfully trusted in mathematical investigation. From the preface to his Mathematical Analysis of Logic, printed as a separate tract in 1847, we learn that speculations con-cerning a calculus of reasoning had at different times occupied Boole's thoughts, but it was not till the spring of 1847 that a memorable logical controversy led him to put his ideas into a definite form. Boole afterwards regarded this pamphlet as a hasty and imperfect exposition of his logical system, and he desired that his much larger work, An Investigation, of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities, published in 1854, should alone be considered as containing a mature statement of his views. Nevertheless, there is a charm of originality about his earlier logical work which no competent reader can fail to appreciate, and the intro-duction gives striking evidence of his profound views and wide reading.

It is not easy to give in a few words a correct notion of Boole's logical system, and only those who are conversant with the principles of symbolical reasoning can exactly apprehend his position. He did not regard logic as a branch of mathematics, as the title of his earlier pamphlet might be taken to imply, but he pointed out such a deep analogy between the symbols of algebra and those which can be made, in his opinion, to represent logical forms and syllogisms, that we can hardly help saying that logic is mathematics restricted to the two quantities, 0 and 1. By unity Boole denoted the universe of thinkable objects; literal symbols, such as x, y, z, v, u, &c, were used with the elective meaning attaching to common adjectives and sub-stantives. Thus, if x = horned, and y — sheep, then the successive acts of election represented by x and y, if per-formed on unity, give the whole of the class horned sheep. Boole showed that elective symbols of this kind obey the same primary laws of combination as algebraical symbols, whence it followed that they could be added, subtracted, multiplied, and even divided, almost exactly in the same manner as numbers. Thus, 1 - x would represent the operation of selecting all things in the world except horned things, that is, all not horned things, and (1 — x) (1 — y) would give us all things neither horned nor sheep. By the use of such symbols propositions could be reduced to the form of equations, and the syllogistic conclusion from two premises was obtained by eliminating the middle term according to ordinary algebraic rules.

Still more original and remarkable, however, was that part of his system, fully stated in his Laws of Thought, which formed a general symbolic method of logical infer-ence. Given any propositions involving any number of terms, Boole showed how, by the purely symbolic treat-ment of the premises, to draw any conclusion logically contained in those premises. The second part of the Laws of Thought contained a corresponding attempt to discover a general method in probabilities, which should enable us from the given probabilities of any system of events to determine the consequent probability of any other event logically connected with the given events. Soon after its publication this method was the subject of a con-troversy in the Philosophical Magazine ; but it cannot be said that the exact value of this part of his works has ever been clearly ascertained.

It is often supposed that mathematicians are deficient in judgment and knowledge of other matters. In Boole this was not the case; for though he published little except the mathematical and logical works already mentioned, his acquaintance with general literature was wide and deep. Dante was his favourite poet, and he preferred the Paradiso to the Inferno. The metaphysics of Aristotle, the ethics of Spinoza, the philosophical works of Cicero, and many less celebrated works of a kindred character, were also frequent subjects of study. His reflections upon scientific, philosophical, and religious questions are to be mainly gathered from four addresses upon The Genius of Sir Isaac Newton, The Right Use of Leisure, The Claims of Science, and The Social Aspect of Intellectual Culture, which he delivered and printed at different times.

The personal character of Boole inspired all his friends with the deepest esteem. He was marked by the modesty of true genius, and his life was given to the single-minded pursuit of truth. Though he received a royal medal for his memoir of 1844, and the honorary degree of LL.D. from the University of Dublin, it may be said that he neither sought nor received the ordinary rewards to which his discoveries would entitle him.

On the 8th of December 1864, in the full vigour of his intellectual powers, Boole died of an attack of fever, ending in suffusion on the lungs An excellent sketch of his life and works, by the Rev. B. Harley, F.R.S., to which the present writer is indebted for many particulars, is to be found in the British Quarterly Review for July 1866, No. 87. (w. s. J.)

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