Men of Mathematics (84 page)

De Morgan, having gained some fame from his pioneering studies in logic, allowed himself in an absent-minded moment to be trapped into a controversy with Hamilton over the latter's famous principle of “the quantification of the predicate.” There is no need to explain what this mystery is (or was); it is as dead as a coffin nail. De Morgan had made a real contribution to the syllogism; Hamilton thought he detected De Morgan's diamond in his own blue mud; the irate Scottish lawyer-philosopher publicly accused De Morgan of plagiarism—an
insanely unphilosophical thing to do—and the fight was on. On De Morgan's side, at least, the row was a hilarious frolic. De Morgan never lost his temper; Hamilton had never learned to keep his.

*  *  *

If this were merely one of the innumerable squabbles over priority which disfigure scientific history it would not be worth a passing mention. Its historical importance is that Boole by now (1848) was a firm friend and warm admirer of De Morgan. Boole was still teaching school, but he knew many of the leading British mathematicians personally or by correspondence. He now came to the aid of his friend—not that the witty De Morgan needed any mortal's aid, but because he knew that De Morgan was right and Hamilton wrong. So, in 1848, Boole published a slim volume,
The Mathematical Analysis of Logic,
his first public contribution to the vast subject which his work inaugurated and in which he was to win enduring fame for the boldness and perspicacity of his vision. The pamphlet—it was hardly more than that—excited De Morgan's warm admiration. Here was the master, and De Morgan hastened to recognize him. The booklet was only the promise of greater things to come six years later, but Boole had definitely broken new, stubborn ground.

In the meantime, reluctantly turning down his mathematical friends' advice that he proceed to Cambridge and take the orthodox mathematical training there, Boole went on with the drudgery of elementary teaching, without a complaint, because his parents were now wholly dependent upon his support. At last he got an opportunity where his conspicuous abilities as an investigator and a lecturer could have some play. He was appointed Professor of Mathematics at the recently opened Queen's College at what was then called the city of Cork, Ireland. This was in 1849.

Needless to say, the brilliant man who had known only poverty and hard work all his life made excellent use of his comparative freedom from financial worry and everlasting grind. His duties would now be considered onerous; Boole found them light by contrast with the dreary round of elementary teaching to which he had been accustomed. He produced much notable miscellaneous mathematical work, but his main effort went on licking his masterpeice into shape. In 1854 he published it:
An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities.
Boole was thirty nine when this appeared. It is somewhat unusual for a mathematician
as old as that to produce work of such profound originality, but the phenomenon is accounted for when we remember the long, devious path Boole was compelled to follow before he could set his face fairly toward his goal. (Compare the careers of Boole and Weierstrass.)

A few extracts will give some idea of Boole's style and the scope of his work.

“The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the language of a Calculus, and upon this foundation to establish the science of Logic and construct its method; to make that method itself the basis of a general method for the application of the mathematical doctrine of probabilities; and, finally, to collect from the various elements of truth brought to view in the course of these inquiries some probable intimations concerning the nature and constitution of the human mind. . . .”

“Shall we then err in regarding that as the true science of Logic which, laying down certain elementary laws, confirmed by the very testimony of the mind, permits us thence to deduce, by uniform processes, the entire chain of its secondary consequences, and furnishes, for its practical applications, methods of perfect generality? . . .”

“There exist, indeed, certain general principles founded in the very nature of language, by which the use of symbols, which are but the elements of scientific language, is determined. To a certain extent these elements are arbitrary. Their interpretation is purely conventional: we are permitted to employ them in whatever sense we please. But this permission is limited by two indispensable conditions,—first, that from the sense once conventionally established we never, in the same process of reasoning, depart; secondly, that the laws by which the process is conducted be founded exclusively upon the above fixed sense or meaning of the symbols employed. In accordance with these principles, any agreement which may be established between the laws of the symbols of Logic and those of Algebra can but issue in an agreement of processes. The two provinces of interpretation remain apart and independent, each subject to its own laws and conditions.

“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 [
x
²
=
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.