by E. T. Bell
“Now the actual investigations of the following pages exhibit Logic, in its practical aspect, as a system of processes carried on by the aid of symbols having a definite interpretation, and subject to laws founded upon that interpretation alone. But at the same time they exhibit those laws as identical in form with the laws of the general symbols of Algebra, with this single addition, viz., that the symbols of Logic are further subject to a special law [x2 = x in the algebra of logic, which can be interpreted, among other ways, as “the class of all those things common to a class x and itself is merely the class x”], to which the symbols of quantity, as such, are not subject.” (That is, in common algebra, it is not true that every x is equal to its square, whereas in the Boolean algebra of logic, this is true.)
This program is carried out in detail in the book. Boole reduced logic to an extremely easy and simple type of algebra. “Reasoning” upon appropriate material becomes in this algebra a matter of elementary manipulations of formulas far simpler than most of those handled in a second year of school algebra. Thus logic itself was brought under the sway of mathematics.
Since Boole’s pioneering work his great invention has been modified, improved, generalized, and extended in many directions. Today symbolic or mathematical logic is indispensable in any serious attempt to understand the nature of mathematics and the state of its foundations on which the whole colossal superstructure rests. The intricacy and delicacy of the difficulties explored by the symbolic reasoning would, it is safe to say, defy human reason if only the old, pre-Boole methods of verbal logical arguments were at our disposal. The daring originality of Boole’s whole project needs no signpost. It is a landmark in itself.
Since 1899, when Hilbert published his classic on the foundations of geometry, much attention has been given to the postulational formulation of the several branches of mathematics. This movement goes back as far as Euclid, but for some strange reason—possibly because the techniques invented by Descartes, Newton, Leibniz, Euler, Gauss, and others gave mathematicians plenty to do in developing their subject freely and somewhat uncritically—the Euclidean method was for long neglected in everything but geometry. We have already seen that the British school applied the method to algebra in the first half of the nineteenth century. Their successes seem to have made no very great impression on the work of their contemporaries and immediate successors, and it was only with the work of Hilbert that the postulational method came to be recognized as the clearest and most rigorous approach to any mathematical discipline.
Today this tendency to abstraction, in which the symbols and rules of operation in a particular subject are emptied of all meaning and discussed from a purely formal point of view, is all the rage, rather to the neglect of applications (practical or mathematical) which some say are the ultimate human justification for any scientific activity. Nevertheless the abstract method does give insights which looser attacks do not, and in particular the true simplicity of Boole’s algebra of logic is most easily seen thus.
Accordingly we shall state the postulates for Boolean algebra (the algebra of logic) and, having done so, see that they can indeed be given an interpretation consistent with classical logic. The following set of postulates is taken from a paper by E. V. Huntington, in the Transactions of the American Mathematical Society, (vol. 35, 1933, pp. 274 −304). The whole paper is easily understandable by anyone who has had a week of algebra, and may be found in most large public libraries. As Huntington points out, this first set of his which we transcribe is not as elegant as some of his others. But as its interpretation in terms of class inclusion as in formal logic is more immediate than the like for the others, it is to be preferred here.
The set of postulates is expressed in terms of K, +, X, where K is a class of undefined (wholly arbitrary, without any assigned meaning or properties beyond those given in the postulates) elements a, b, c, . . . , and a + b and a × b (written also simply as ab) are the results of two undefined binary operations, +, × (“binary,” because each of +, × operates on two elements of K). There are ten postulates, I a-VI:
“I a. If a and b are in the class K, then a + b is in the class K.
“I b. If a and b are in the class K, then ab is in the class K.
“II a. There is an element Z such that a + Z = a for every element a.
“II b. There is an element U such that a U = a for every element a.
“III a. a + b = b + a.
“III b. ab = ba.
“IV a. a + bc = (a + b)(a + c).
“IV b. a(b + c) = ab + ac.
“V. For every element a there is an element a′ such that a + a′ = U and aa′ = Z.
“VI. There are at least two distinct elements in the class K.”
It will be readily seen that these postulates are satisfied by the following interpretation: a, b, c, . . . are classes’, a + b is the class of all those things that are in at least one of the classes a, b; ab is the class of all those things that are in both of the classes a, b; Z is the “null class”—the class that has no members; U is the “universal class”—the class that contains all the things in all the classes under discussion. Postulate V then states that given any class a, there is a class a’ consisting of all those things which are not in a. Note that VI implies that U, Z are not the same class.
From such a simple and obvious set of statements it seems rather remarkable that the whole of classical logic can be built up symbolically by means of the easy algebra generated by the postulates. From these postulates a theory of what may be called “logical equations” is developed: problems in logic are translated into such equations, which are then “solved” by the devices of the algebra; the solution is then reinterpreted in terms of the logical data, giving the solution of the original problem. We shall close this description with the symbolic equivalent of “inclusion”—also interpretable, when propositions rather than classes are the elements of K, as “implication.”
“The relation a < b [read, a is included in b] is defined by any one of the following equations
a + b = b, ab = a, a′ + b = U, ab′ = Z.”
To see that these are reasonable, consider for example the second, ab = a. This states that if a is included in b, then everything that is in both a and b is the whole of a.
From the stated postulates the following theorems on inclusion (with thousands of more complicated ones, if desired) can be proved. The specimens selected all agree with our intuitive conception of what “inclusion” means.
(1) a < a.
(2) If a < b and b < c, then a < c.
(3) If a < b and b < a, then a = b.
(4) Z < a (where Z is the element in II a—it is proved to be the only element satisfying II a).
(5) a < U (where U is the element in II b—likewise unique).
(6) a < a + b; and if a < y and b < y, then a + b < y.
(7) ab < a; and if x < a and x < b, then x < ab.
(8) If x < a and x < a’, then x = Z; and if a < y and a’ < y, then y = U.
(9) If a < b’ is false, then there is at least one element x, distinct from Z, such that x < a and x < b.
It may be of interest to observe that in arithmetic and analysis is the symbol for “less than.” Note that if a, b, c, . . . are real numbers, and Z denotes zero, then (2) is satisfied for this interpretation of “<,” and similarly for (4), provided a is positive; but that (1) is not satisfied, nor is the second part of (6)—as we see from 5 < 10, 7 < 10, but 5 + 7 < 10 is false.
The tremendous power and fluent ease of the method can be readily appreciated by seeing what it does in any work on symbolic logic. But, as already emphasized, the importance of this “symbolic reasoning” is in its applicability to subtle questions regarding the foundations of all mathematics which, were it not for this precise method of fixing meanings of “words” or other “symbols” once for all, would probably be unapproachable by ordinary mortals.
Like nearly all novelties, symbolic logic was neglected for many years after its inv
ention. As late as 1910 we find eminent mathematicians scorning it as a “philosophical” curiosity without mathematical significance. The work of Whitehead and Russell in Principia Mathematica (1910-1913) was the first to convince any considerable body of professional mathematicians that symbolic logic might be worth their serious attention. One staunch hater of symbolic logic may be mentioned—Cantor, whose work on the infinite will be noticed in the concluding chapter. By one of those little ironies which make mathematical history such amusing reading for the open-minded, symbolic logic was to play an important part in the drastic criticism of Cantor’s work that caused its author to lose faith in himself and his theory.
Boole did not long survive the production of his masterpiece. The year after its publication, still subconsciously striving for the social respectability that he once thought a knowledge of Greek could confer, he married Mary Everest, niece of the Professor of Greek in Queen’s College. His wife became his devoted disciple. After her husband’s death, Mary Boole applied some of the ideas which she had acquired from him to rationalizing and humanizing the education of young children. In her pamphlet, Boole’s Psychology, Mary Boole records an interesting speculation of Boole’s which readers of The Laws of Thought will recognize as in keeping with the unexpressed but implied personal philosophy in certain sections. Boole told his wife that in 1832, when he was about seventeen, it “flashed upon” him as he was walking across a field that besides the knowledge gained from direct observation, man derives knowledge from some source un-definable and invisible—which Mary Boole calls “the unconscious.” It will be interesting (in a later chapter) to hear Poincaré expressing a similar opinion regarding the genesis of mathematical “inspirations” in the “subconscious mind.” Anyhow, Boole was inspired, if ever a mortal was, when he wrote The Laws of Thought.
Boole died, honored and with a fast-growing fame, on December 8, 1864, in the fiftieth year of his age. His premature death was due to pneumonia contracted after faithfully keeping a lecture engagement when he was soaked to the skin. He fully realized that he had done great work.
CHAPTER TWENTY FOUR
The Man, Not the Method
HERMITE
Talk with M. Hermite: he never evokes a concrete image; yet you soon perceive that the most abstract entities are for him like living creatures.—HENRI POINCARé
OUTSTANDING UNSOLVED PROBLEMS demand new methods for their solution, while powerful new methods beget new problems to be solved. But, as Poincaré observed, it is the man, not the method, that solves a problem.
Of old problems responsible for new methods in mathematics that of motion and all it implies for mechanics, terrestrial and celestial, may be recalled as one of the principal instigators of the calculus and present attempts to put reasoning about the infinite on a firm basis. An example of new problems suggested by powerful new methods is the swarm which the tensor calculus, popularized to geometers by its successes in relativity, let loose in geometry. And finally, as an illustration of Poincaré’s remark, it was Einstein, and not the method of tensors, that solved the problem of giving a coherent mathematical account of gravitation. All three theses are sustained in the life of Charles Hermite, the leading French mathematician of the second half of the nineteenth century—if we except Hermite’s pupil Poincaré, who belonged partly to our own century.
Charles Hermite, born at Dieuze, Lorraine, France, on December 24, 1822, could hardly have chosen a more propitious era for his birth than the third decade of the nineteenth century. His was just the rare combination of creative genius and the ability to master the best in the work of other men which was demanded in the middle of the century to coordinate the arithmetical creations of Gauss with the discoveries of Abel and Jacobi in elliptic functions, the striking advances of Jacobi in Abelian functions, and the vast theory of algebraic invariants in process of rapid development by the English mathematicians Boole, Cayley, and Sylvester.
Hermite almost lost his life in the French Revolution—although the last head had fallen nearly a quarter of a century before he was born. His paternal grandfather was ruined by the Commune and died in prison; his grandfather’s brother went to the guillotine. Hermite’s father escaped owing to his youth.
If Hermite’s mathematical ability was inherited, it probably came from the side of the father, who had studied engineering. Finding engineering uncongenial, Hermite senior gave it up, and after an equally distasteful start in the salt industry, finally settled down in business as a cloth merchant. This resting place was no doubt chosen by the rolling stone because he had married his employer’s daughter, Madeleine Lallemand, a domineering woman who wore the breeches in her family and ran everything from the business to her husband. She succeeded in building both up to a state of solid bourgeois prosperity. Charles was the sixth of seven children—five sons and two daughters. He was born with a deformity of the right leg which rendered him lame for life—possibly a disguised blessing, as it effectively barred him from any career even remotely connected with the army—and he had to get about with a cane. His deformity never affected the uniform sweetness of his disposition.
Hermite’s earliest education was received from his parents. As the business continued to prosper, the family moved from Dieuze to Nancy when Hermite was six. Presently the growing demands of the business absorbed all the time of the parents and Hermite was sent as a boarder to the lycée at Nancy. This school proving unsatisfactory the prosperous parents decided to give Charles the best and packed him off to Paris. There he studied for a short time at the Lycée Henri IV, moving on at the age of eighteen (1840) to the more famous (or infamous) Louis-le-Grand—the “Alma” Mater of the wretched Galois—to prepare for the Polytechnique.
For a while it looked as if Hermite was to repeat the disaster of his untamable predecessor at Louis-le-Grand. He had the same dislike for rhetoric and the same indifference to the elementary mathematics of the classroom. But the competent lectures on physics fascinated him and won his cordial cooperation in the bilateral process of acquiring an education. Later on, unpestered by pedants, Hermite became a good classicist and the master of a beautifully clear prose.
Those who hate examinations will love Hermite. There is something in the careers of these two most famous alumni of Louis-le-Grand, Galois and Hermite, which might well cause the advocates of examinations as a reliable yardstick for arranging human beings in order of intellectual merit to ask themselves whether they have used their heads or their feet in arriving at their conclusions. It was only by the grace of God and the diplomatic persistence of the devoted and intelligent Professor Richard, who had done his unavailing best fifteen years before to save Galois for science, that Hermite was not tossed out by stupid examiners to rot on the rubbish heap of failure. While still a student at the lycée, Hermite, following in the steps of Galois, supplemented and neglected his elementary lessons by private reading at the library of Sainte-Geneviève, where he found and mastered the memoir of Lagrange on the solution of numerical equations. Saving up his pennies, he bought the French translation of the Disquisitiones Arithmeticae of Gauss and, what is more, mastered it as few before or since have mastered it. By the time he had followed what Gauss had done Hermite was ready to go on. “It was in these two books,” he loved to say in later life, “that I learned Algebra.” Euler and Laplace also instructed him through their works. And yet Hermite’s performance in examinations was, to say the most flattering thing possible of it, mediocre. Mathematical nonentities beat him out of sight.
Mindful of the tragic end of Galois, Richard tried his best to steer Hermite away from original investigation to the less exciting though muddier waters of the competitive examinations for entrance to the École Polytechnique—the filthy ditch in which Galois had drowned himself. Nevertheless the good Richard could not refrain from telling Hermite’s father that Charles was “a young Lagrange.”
The Nouvelles Annales de Mathématiques, a journal devoted to the interests of students in the higher schools
, was founded in 1842. The first volume contains two papers composed by Hermite while he was still a student at Louis-le-Grand. The first is a simple exercise in the analytic geometry of conic sections and betrays no originality. The second, which fills only six and a half pages in Hermite’s collected works, is a horse of quite a different color. Its unassuming title is Considerations on the algebraic solution of the equation of the fifth degree (translation).
“It is known,” the modest mathematician of twenty begins, “that Lagrange made the algebraic solution of the general equation of the fifth degree depend on the determination of a root of a particular equation of the sixth degree, which he calls a reduced equation £ today, a ’resolvent’]. . . . So that, if this resolvent were decomposable into rational factors of the second or third degrees, we should have the solution of the equation of the fifth degree. I shall try to show that such a decomposition is impossible.” Hermite not only succeeded in his attempt—by a beautifully simple argument—but showed also in doing so that he was an algebraist. With but a few slight changes this short paper will do all that is required.
It may seem strange that a young man capable of genuine mathematical reasoning of the caliber shown by Hermite in his paper on the general quintic should find elementary mathematics difficult. But it is not necessary to understand—or even to have heard of—much of classical mathematics as it has evolved in the course of its long history in order to be able to follow or work creatively in the mathematics that has been developed since 1800 and is still of living interest to mathematicians. The geometrical treatment (synthetic) of conic sections of the Greeks, for instance, need not be mastered today by anyone who wishes to follow modern geometry; nor need any geometry at all be learned by one whose tastes are algebraic or arithmetical. To a lesser degree the same is true for analysis, where such geometrical language as is used is of the simplest and is neither necessary nor desirable if up-to-date proofs are the object. As a last example, descriptive geometry, of great use to designing engineers, is of practically no use whatever to a working mathematician. Some quite difficult subjects that are still mathematically alive require only a school education in algebra and a clear head for their comprehension. Such are the theory of finite groups, the mathematical theory of the infinite, and parts of the theory of probabilities and the higher arithmetic. So it is not astonishing that large tracts of what a candidate is required to know for entrance to a technical or scientific school, or even for graduation from the same, are less than worthless for a mathematical career. This accounts for Hermite’s spectacular success as a budding mathematician and his narrow escape from complete disaster as an examinee.
