Is God a Mathematician? (24 page)

Laws of Thought

George Boole (figure 47) was born on November 2, 1815, in the industrial town of Lincoln, England. His father, John Boole, a shoemaker in Lincoln, showed great interest in mathematics and was skilled in the construction of a variety of optical instruments. Boole’s mother, Mary Ann Joyce, was a lady’s maid. With the father’s heart not quite
in his formal business, the family was not well off. George attended a commercial school until age seven, and then a primary school, where his teacher was one John Walter Reeves. As a boy, Boole was mainly interested in Latin, in which he received instruction from a bookseller, and in Greek, which he learned by himself. At age fourteen he managed to translate a poem by the first century BC Greek poet Meleager. George’s proud father published the translation in the
Lincoln Herald
—an act that provoked an article expressing disbelief from a local schoolmaster. The poverty at home forced George Boole to start working as an assistant teacher at age sixteen. During the following years, he devoted his free time to the study of French, Italian, and German. The knowledge of these modern languages proved useful, since it allowed him to turn his attention to the great works of mathematicians such as Sylvestre Lacroix, Laplace, Lagrange, Jacobi, and others. Even then, however, he was still unable to take regular courses in mathematics, and he continued to study on his own, while at the same time helping to support his parents and siblings through his teaching job. Nevertheless, the mathematical talents of this auto-didact started to show, and he began publishing articles in the
Cambridge Mathematical Journal.

Figure 47

In 1842, Boole started to correspond regularly with De Morgan, to whom he was sending his mathematical papers for comments. Because of his growing reputation as an original mathematician, and backed by a strong recommendation from De Morgan, Boole was offered the position of professor of mathematics at Queen’s College, in Cork, Ireland, in 1849. He continued to teach there for the rest of his life. In 1855 Boole married Mary Everest (after whose uncle, the surveyor George Everest, the mountain was named), who was seventeen years his junior, and the couple had five daughters. Boole died prematurely at age forty-nine. On a cold winter day in 1864, he got drenched on his way to the college, but he insisted on giving his lectures even though his clothes were soaking wet. At home, his wife may have worsened his condition by pouring buckets of water onto the bed, following a superstition that the cure should somehow replicate the cause of the illness. Boole developed pneumonia and died on December 8, 1864. Bertrand Russell did not hide his admiration
for this self-taught individual: “Pure mathematics was discovered by Boole, in a work which he called
The Laws of Thought
(1854)…His book was in fact concerned with formal logic, and this is the same thing as mathematics.” Remarkably for that time, both Mary Boole (1832–1916) and each of the five Boole daughters achieved considerable fame in fields ranging from education to chemistry.

Boole published
The Mathematical Analysis of Logic
in 1847 and
The Laws of Thought
in 1854 (the full title of the latter read:
An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities
). These were genuine masterworks—the first to take the parallelism between logical and arithmetic operations a giant step forward. Boole literally transformed logic into a type of algebra (which came to be called
Boolean algebra
) and extended the analysis of logic even to probabilistic reasoning. In Boole’s words:

Boole’s calculus could be interpreted either as applying to relations among
classes
(collections of objects or members) or within the logic of
propositions
. For instance, if
x
and
y
were classes, then a relation such as
x y
meant that the two classes had precisely the same members, even if the classes were defined differently. As an example, if all the children in a certain school are shorter than seven feet, then the two classes defined as
x
“all the children in the school” and
y
“all the children in the school that are shorter than seven feet” are
equal. If
x
and
y
represented propositions, then
x y
meant that the two propositions were equivalent (that one was true if and only if the other was also true). For example, the propositions
x
“John Barrymore was Ethel Barrymore’s brother” and
y
“Ethel Barrymore was John Barrymore’s sister” are equal. The symbol “
x
·
y
” represented the common part of the two classes
x
and
y
(those members belonging to both
x
and
y
), or the
conjunction
of the propositions
x
and
y
(i.e., “
x and y
”). For instance, if
x
was the class of all village idiots and
y
was the class of all things with black hair, then
x
·
y
was the class of all black-haired village idiots. For propositions
x
and
y
, the conjunction
x
·
y
(or the word “and”) meant that both propositions had to hold. For example, when the Motor Vehicle Administration says that “you must pass a peripheral vision test and a driving test,” this means that both requirements must be met. For Boole, for two classes having no members in common, the symbol “
x y
” represented the class consisting of both the members of
x
and the members of
y
. In the case of propositions, “
x y
” corresponded to “either
x
or
y
but not both.” For instance, if
x
is the proposition “pegs are square” and
y
is “pegs are round,” then
x y
is “pegs are either square or round.” Similarly, “
x y
” represented the class of those members of
x
that were not members of
y,
or the proposition “
x
but not
y.
” Boole denoted the universal class (containing all possible members under discussion) by 1 and the empty or null class (having no members whatsoever) by 0. Note that the null class (or set) is definitely not the same as the number 0—the latter is simply the number of members in the null class. Note also that the null class is not the same as nothing, because a class with nothing in it is still a class. For instance, if all the newspapers in Albania are written in Albanian, then the class of all Albanian-language newspapers in Albania would be denoted by 1 in Boole’s notation, while the class of all Spanish-language newspapers in Albania would be denoted by 0. For propositions, 1 represented the standard
true
(e.g., humans are mortal) and 0 the standard
false
(e.g., humans are immortal) propositions, respectively.

With these conventions, Boole was able to formulate a set of axioms defining an algebra of logic. For instance, you can check that using the above definitions, the obviously true proposition “every
thing is either
x
or not
x
” could be written in Boole’s algebra as
x
(1
x
) 1, which also holds in ordinary algebra. Similarly, the statement that the common part between any class and the empty class is an empty class was represented by 0 ·
x
0, which also meant that the conjunction of any proposition with a false proposition is false. For instance, the conjunction “sugar is sweet and humans are immortal” produces a false proposition in spite of the fact that the first part is true. Note that again, this “equality” in Boolean algebra holds true also with normal algebraic numbers.

To show the power of his methods, Boole attempted to use his logical symbols for everything he deemed important. For instance, he even analyzed the arguments of the philosophers Samuel Clarke and Baruch Spinoza for the existence and attributes of God. His conclusion, however, was rather pessimistic: “It is not possible, I think, to rise from the perusal of the arguments of Clarke and Spinoza without a deep conviction of the futility of all endeavors to establish, entirely
a priori,
the existence of an Infinite Being, His attributes, and His relation to the universe.” In spite of the soundness of Boole’s conclusion, apparently not everybody was convinced of the futility of such endeavors, since updated versions of ontological arguments for God’s existence continue to emerge even today.

Overall, Boole managed to mathematically tame the logical connectives
and, or, if…then,
and
not,
which are currently at the very core of computer operations and various switching circuits. Consequently, he is regarded by many as one of the “prophets” who brought about the digital age. Still, due to its pioneering nature, Boole’s algebra was not perfect. First, Boole made his writings somewhat ambiguous and difficult to comprehend by using a notation that was too close to that of ordinary algebra. Second, Boole confused the distinction between propositions (e.g., “Aristotle is mortal”), propositional functions or predicates (e.g., “
x
is mortal”), and quantified statements (e.g., “for all
x, x
is mortal”). Finally, Frege and Russell were later to claim that algebra stems from logic. One could argue, therefore, that it made more sense to construct algebra on the basis of logic rather than the other way around.

There was another aspect of Boole’s work, however, that was
about to become very fruitful. This was the realization of how closely related logic and the concept of
classes
or
sets
were. Recall that Boole’s algebra applied equally well to classes and to logical propositions. Indeed, when all the members of one set
X
are also members of set
Y
(
X
is a
subset
of
Y
), this fact can be expressed as a
logical implication
of the form “if
X
then
Y
.” For instance, the fact that the set of all horses is a subset of the set of all four-legged animals can be rewritten as the logical statement “If
X
is a horse then it is a four-legged animal.”

Boole’s algebra of logic was subsequently expanded and improved upon by a number of researchers, but the person who fully exploited the similarity between sets and logic, and who took the entire concept to a whole new level, was Gottlob Frege (figure 48).

Friedrich Ludwig Gottlob Frege was born at Wismar, Germany, where both his father and his mother were, at different times, the principals at a girls’ high school. He studied mathematics, physics,
chemistry, and philosophy, first at the University of Jena and then for an additional two years at the University of Göttingen. After completing his education, he started lecturing at Jena in 1874, and he continued to teach mathematics there throughout his entire professional career. In spite of a heavy teaching load, Frege managed to publish his first revolutionary work in logic in 1879. The publication was entitled
Concept-Script, A Formal Language for Pure Thought Modeled on that of Arithmetic
(it is commonly known as the
Begriffsschrift
). In this work, Frege developed an original, logical language, which he later amplified in his two-volume
Grundgesetze der Arithmetic
(
Basic Laws of Arithmetic
). Frege’s plan in logic was on one hand very focused, but on the other extraordinarily ambitious. While he primarily concentrated on arithmetic, he wanted to show that even such familiar concepts as the natural numbers, 1, 2, 3,…, could be reduced to logical constructs. Consequently, Frege believed that one could prove all the truths of arithmetic from a few axioms in logic. In other words, according to Frege, even statements such as 1 + 1 = 2 were not
empirical
truths, based on observation, but rather they could be derived from a set of logical axioms. Frege’s
Begriffsschrift
has been so influential that the contemporary logician Willard Van Orman Quine (1908–2000) once wrote: “Logic is an old subject, and since 1879 it has been a great one.”