George Boole (1815-64)

Life
b. 2 Nov. 1815, Lincoln city; son of John Boole, shoemaker and impecunious maker of mathematical instruments; left school at 14 to work as schoolteacher; self-taught in continental languages and independent student of Laplace and Lagrange; contrib. Cambridge Journal of Mathematics; made advances in measurement differential and integral calculus; won Gold Medal of Royal Society for a ‘On a General Method of Analysis’, a paper addressing differentiation from the point of view of the operator; devised Boolean alegbra (x²=x), essential to the digital foundations of computer technology later developed by Claude Shannon (MIT) in 1938;

issued The Mathematical Analysis of Logic (1847); appt. Prof. of Mathematics, Queen’s College, Cork, 1849; hon. LLD, TCD, 1851; involved in conflict over issue of principle with Sir Robert Kane, then President; issued An Investigation of the Laws of Thought (1854); tutor to Mary Everest, then half his age; married, and moved in that year to Cork, on the death of her father, 1855; their children were Mary Ellen, Margaret, Alicia, Lucy and Ethel Ethel Lilian Voynich (b.1864; q.v.) - the author of The Gadfly (1897); elected FRS and Keith Prize of Edinburgh Univ., 1857;

he fostered adult education in Lincoln and Cork; d. of pneumonia, 8 Dec., Blackrock, nr. Cork, having walked in a down-pour to lecture all day at College; bur. Blackrock, Cork; honoured in naming of Boole Library and the Boole Crater on the moon; Mary Everest Boole became renowned as a self-taught mathematician, and returned to England at his death, assuming the post of Librarian at Queen's College, London; George Everest, her uncle, was the eponym of Mt. Everest. WJM

[ For Boole’s conception that grammatical conjunctions not only connection propositions - as widely held - but perform them and express them - see extracts from Investigation of the Laws of Thought [... &c.] - infra. ]

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Works

Proposition III: Given any system of probabilities drawn from recorded instances of unanimity, or of assigned numerical majority in the decisions of a deliberative assembly; required, upon a certain determinate hypothesis, the mean probability of correct judgment for a member of the assembly. (p.305.)

Proposition IV: Given any system of probabilities drawn from recorded instances of unanimity, or of assigned numerical majority in the decisions of a criminal court of justice, required upon hypotheses similar to those of the last proposition, the mean probability c of correct judgment for a member of the court, and the general probability k of guilt in an accused person.

We may collect from the above investigations the following facts and conclusions:

1st. That from the mere records of agreement and disagreement in the opinions of any body of men, no definite numerical conclusions can be drawn respecting either the probability of correct judgment in an individual member of the body, or the merit of the questions submitted to its consideration.

2nd. That such conclusions may be drawn upon various distinct hypotheses, as - 1st, Upon the usual hypothesis of the absolute independence of individual judgments; 2ndly, upon certain definite modifications of that hypothesis warranted by the actual data; 3rdly, upon a distinct principle of solution suggested by the appearance of a common form in the solutions obtained by the modifications above adverted to.

Lastly. That whatever of doubt may attach to the final results, rests not upon the imperfection of the method, which adapts itself equally to all hypotheses, but upon the uncertainty of the hypotheses themselves.

It seems, however, probable that with even the widest limits of hypothesis, consistent with the taking into account of all the data of experience, the deviation of the results obtained would be but slight, and that their mean values might be determined with great confidence by the methods of Prop. iii. Of those methods I should be disposed to give the preference to the first. Such a principle of mean solution having been agreed upon, other considerations seem to indicate that the values of c and k for tribunals and assemblies possessing a definite constitution, and governed in their deliberations by fixed rules, would remain nearly constant, subject, however, to a small secular variation, dependent upon the progress of knowledge and of justice among mankind. There exist at present few, if any, data proper for their determination. (pp.309-10; end chapter.)

1. What I mean by the constitution of a system is the aggregate of those causes and tendencies which produce its observed character, when operating, without interference, under those conditions to which the system is conceived to be adapted. Our judgment of such adaptation must be founded upon a study of the circumstances in which the system attains its freest action, produces its most harmonious results, or fulfils in some other way the apparent design of its construction. There are cases in which we know distinctly the causes upon which the operation of a system depends, as well as its conditions and its end. This is the most perfect kind of knowledge relatively to the subject under consideration. There are also cases in which we know only imperfectly or partially the causes which are at work, but are able, nevertheless, to determine to some extent the laws of their action, and, beyond this, to discover general tendencies, and to infer ulterior purpose. It has thus, I think rightly, been concluded that there is a moral faculty in our nature, not because we can understand the special instruments by which it works, as we connect the organ with the faculty of sight, nor upon the ground that men agree in the adoption of universal rules of conduct; but because while, in some form or other, the sentiment of moral approbation or disapprobation manifests itself in all, it tends, wherever human progress is observable, wherever society is not either stationary or hastening to decay, to attach itself to certain classes of actions, consentaneously, and after a manner indicative both of permanency and of law. Always and everywhere the manifestation of Order affords a presumption, not measurable indeed, but real (XX. 22), of the fulfilment of an end or purpose, and the existence of a ground of orderly causation.

2. The particular question of the constitution of the intellect has, it is almost needless to say, attracted the efforts of speculative ingenuity in every age. For it not only addresses itself to that desire of knowledge which the greatest masters of ancient thought believed to be innate in our species, but it adds to the ordinary strength of this motive the inducement of a human and personal interest. A genuine devotion to truth is, indeed, seldom partial in its aims, but while it prompts to expatiate over the fair fields of outward [311] observation, forbids to neglect the study of our own faculties. Even in ages the most devoted to material interests, some portion of the current of thought has been reflected inwards, and the desire to comprehend that by which all else is comprehended has only been baffled in order to be renewed. It is probable that this pertinacity of effort would not have been maintained among sincere inquirers after truth, had the conviction been general that such speculations are hopelessly barren. We may conceive that it has been felt that if something of error and uncertainty, always incidental to a state of partial information, must ever be attached to the results of such inquiries, a residue of positive knowledge may yet remain; that the contradictions which are met with are more often verbal than real; above all, that even probable conclusions derive here an interest and a value from their subject, which render them not unworthy to claim regard beside the more definite and more splendid results of physical science. Such considerations seem to be perfectly legitimate. Insoluble as many of the problems connected with the inquiry into the nature and constitution of the mind must be presumed to be, there are not wanting others upon which a limited but not doubtful knowledge, others upon which the conclusions of a highly probable analogy, are attainable. As the realms of day and night are not strictly conterminous, but are separated by a crepuscular zone, through which the light of the one fades gradually off into the darkness of the other, so it may be said that every region of positive knowledge lies surrounded by a debateable and speculative territory, over which it in some degree extends its influence and its light. Thus there may be questions relating to the constitution of the intellect which, though they do not admit, in the present state of knowledge, of an absolute decision, may receive so much of reflected information as to render their probable solution not difficult; and there may also be questions relating to the nature of science, and even to particular truths and doctrines of science, upon which they who accept the general principles of this work cannot but be led to entertain positive opinions, differing, it may be, from those which are usually received in the present day. [1] In what follows I shall recapitulate some of the more definite conclusions established in the former parts of this treatise, and shall then indicate one or two trains of thought, connected with the general objects above adverted to, which they seem to me calculated to suggest. 3. Among those conclusions, relating to the intellectual constitution, which may be considered as belonging to the realm of positive knowledge, we may reckon the scientific laws of thought and reasoning, which have formed the basis of the general methods of this treatise, together with the principles, Chap, V., by which their application has been determined. The resolution of the domain of thought into two spheres, distinct but coexistent (IV. XI.); the subjection of the [312] intellectual operations within those spheres to a common system of laws (XI.); the general mathematical character of those laws, and their actual expression =(II. III.); the extent of their affinity with the laws of thought in the domain of number, and the point of their divergence therefrom; the dominant character of the two limiting conceptions of universe and eternity among all the subjects of thought with which Logic is concerned; the relation of those conceptions to the fundamental conception of unity in the science of number, — these, with many similar results, are not to be ranked as merely probable or analogical conclusions, but are entitled to be regarded as truths of science. Whether they be termed metaphysical or not, is a matter of indifference. The nature of the evidence upon which they rest, though in kind distinct, is not inferior in value to any which can be adduced in support of the general truths of physical science. [... &c.; italics mine - BS.] (pp.310-13)

1. The following illustration may suffice:- It is maintained by some of the highest modern authorities in grammar that conjunctions connect propositions only. Now, without inquiring directly whether this opinion is sound or not, it is obvious that it cannot consistently be held by any who admit the scientific principles of this treatise; for to such it would seem to involve a denial, either, 1st, of the possibility of performing, or 2ndly, of the possibility of expressing, a mental operation, the laws of which, viewed in both these relations, have been investigated and applied in the present work. (Latham on the English Language; Sir John Stoddart’s Universal Grammar, &c.)