It Began with Babbage (15 page)

So a relay has two “states”: ON and OFF (in present-centered language, a
bistable device
). Suppose the two states, instead of being termed ON and OFF are called TRUE and FALSE. Let us call these two states TRUE and FALSE
logical
or
truth values
. In that case, we may also say that if a current flows through a circuit, its value is TRUE; if no current can flow through, its value is FALSE.

So now, a series connection of two relay switches, X and Y, behaves according to the following statements:

If
X
= TRUE and
Y
= TRUE,
then
X
and
Y
= TRUE;
otherwise,
X
and
Y
= FALSE.

If
X
= FALSE and
Y
= FALSE,
then
X
or
Y
= FALSE;
otherwise,
X
or
Y
= TRUE.

Now, if a relay switch is ON, its
complementary
state is OFF and vice versa. Let us call the complement of a state by the symbol
not
. In which case, we also have a behavior pattern for a relay X:

If
X
= TRUE,
then not
(
X
) = FALSE;
otherwise, not
(
X
) = TRUE.

All this is no mere esoteric conceit. During the mid 19th century, at about the time Babbage was still fretting over his Analytical Engine, another Englishman quite unlike him in social origin named George Boole (1819–1864), born into an impecunious working-class family in Lincoln, England—possessed of the most modest of ordinary school education, almost entirely self-taught in Greek, Latin, German, Italian, French, mathematics, logic, and philosophy—would contribute many papers to mathematical journals and write several books on mathematics and logic. Among them was
The Mathematical Analysis of Logic
(1847) that, according to one historian of mathematics, probably earned him a professorship of mathematics in Queen's College in Cork, Ireland,
in 1849.
²⁸
However, it was a later book, the loftily titled
An Investigation of the Laws of Thought
(1854), that granted Boole a lasting place in the annals of modern thought.

What Boole did was to establish the basis of a new algebra of logic. The variables of this algebra could only have the two truth values—TRUE and FALSE—and were thus called
logical variables
or
symbolic variables
. The rules of this algebra—its “calculus”—allowed one to deduce the truth or falsehood of logical expressions or propositions built up by combining logical variables with the logical operations
and
,
or
, and
not.

Boole died at age 49 in Cork of an illness brought on by getting a drenching in the rain, but not before receiving much recognition, including an honorary LLD degree from the University of Dublin and a medal from the Royal Society. More than a century later, a crater on the moon was named after him.

Boole's
Laws of Thought
had consequences; the laws would later be systematized by other eminent mathematicians, including Augustus de Morgan, Lovelace's one-time tutor; American mathematician–philosopher Charles Sanders Peirce (1839–1914); and British economist and logician William Stanley Jevons (1835–1882). The result came to be called
Boolean algebra
and the development in mathematical logic of the
propositional calculus
for making formal inferences about logical propositions.

In 1937, Boolean algebra formed the logical foundation of the design of circuits composed of switches, such as relays, later called the discipline of
switching circuit design
. Thus, a Victorian professor of mathematics laid one of the cornerstones in the making of a science of computing. If Babbage's ghost hovers continually over this story, Boole's spirit does as well, making his appearance 75 years after his death—and, unlike Babbage, Boole contributed directly to the logical basis of computer design.

In 1937, Claude Shannon (1916–2001), a 21-year-old graduate student, submitted a thesis for a master's degree in electrical engineering at MIT. The thesis was titled
A Symbolic Analysis of Relay and Switching Circuits
.
²⁹
In this work, Shannon drew an analogy between the basic idea in propositional calculus (or, equivalently, in Boolean algebra) and the basic behavior of relay circuits along the lines shown in
Table 5.1
(where + does
not
mean the arithmetic operation of add).

In 1938, Shannon published a paper, based on his thesis, in the
Transactions of the American Institute of Electrical Engineers
. The timing of this publication turned out to be important for Shannon's reputation. In the Soviet Union, a Russian logician and engineering theorist named Victor Shestakov (1907–1987) had proposed a theory of switching circuits based on Boolean algebra in 1935. However, he did not publish his work (in Russian) until 1941.

Shannon would later co-invent (with Warren Weaver [1894–1978]) a branch of applied mathematics called
information theory
that laid the theoretical foundation for communication. As a scientific eclectic, his work crossed several scientific boundaries. When he died in 2001, Shannon was a much-honored scientist, but his place in this story rests primarily (but not solely, as we will later see) in that 1937 master's thesis.

TABLE
5.1 Correspondence Between Relay Circuit and Propositional Calculus Concepts

It was not only Boolean algebra that he introduced into switching circuit design, but also the fact that the binary states ON and OFF (or open and closed, in Shannon's terminology) could be represented by the truth values TRUE and FALSE from propositional logic, and the binary digits 1 and 0, respectively. Boolean algebra or propositional logic, switching circuit design, and binary arithmetic came together by way of Shannon.

Thus we find George Stibitz, in his 1940 memorandum, explicating the advantages of the binary system in the context of computers—notably, the simplicity of binary addition: 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 10. This “extraordinary” simplicity, Stibitz wrote, made it so much easier to design computing machines composed of relays, for a relay has only two positions: open and closed. The former represents the binary digit 0; the latter, the digit 1.
³⁰

One of the earliest textbooks on the design of arithmetic units for digital computers, published in 1955, described the role Boolean algebra could play in computer design. It is enough for any basic circuit element having just two states (that is, is a bistable device) to be represented by a Boolean variable—not just relays, but other circuit elements that came to be used in later computers, such as vacuum tubes and diodes. Associated with Boolean variables are three fundamental Boolean operations AND (logical product), OR (logical sum), and NOT (complement). Using these operations, Boolean expressions can represent
any
circuit composed of bistable or binary circuit elements.
³¹

For example, a switching network with four inputs
A, B, C, D
and a single output
E
such that 1 is produced on
E
only when both
A
and
B
are 1s or when both
C
and
D
are 1s (
Figure 5.1
) can be expressed by the Boolean expression
E = AB + CD
, where
AB
and
CD
represent
A
AND
B, C
AND
D
, respectively, and the + represents OR.
³²
A very different switching network is shown in
Figure 5.2
, represented by the Boolean expression
Z
= (
A
+
C
)(
A
+
D
)(
B
+
C
)(
B
+
D
). However, using the
laws
of Boolean algebra, one can transform this last expression to the expression
E
=
AB
+
CD
, corresponding to the much simpler and economic circuit of
Figure 5.1
.
³³
So the power of Boolean algebra lay not only in its concise notation, but also as a
calculus
.

We have seen that, ever since Babbage, the desire for automatic computation was stimulated by the felt need for the speedy, accurate, and reliable production of tables of all
sorts. In Babbage's time, these tables were mathematical in nature. Later, after Hollerith, punched-card tabulators were used busily through the first decades of the 20th century in compiling statistical tables and mortality data, and in accounting and other business-related statements.

Beginning in about 1928, Hollerith-style data processing machines began to be used extensively to produce astronomical tables. In particular, a Cambridge-educated New Zealander, Leslie John Comrie (1893–1950), superintendent of the Nautical Almanac Office in England, used the Hollerith machine to compile tables of the moon's position.
³⁴
In the United States, Wallace J. Eckert (1902–1971), an astronomer, teaching in Columbia University, and seriously interested in automating astronomical calculations, established in 1926 a computer laboratory in the university.
³⁵
This invoked the interest of Thomas J. Watson (1874–1956), president of IBM, the undisputed industrial leader in punched-card data processing machines. Watson was persuaded to establish a computational bureau at Columbia. In 1933, financed by IBM, this computer laboratory—for that is what it was—was expanded and named the Thomas J. Watson Astronomical Computing Bureau, a joint enterprise of IBM, the department of astronomy in Columbia, and the American Astronomical Society.
³⁶
Eckert was one of those in the electromechanical era who understood the need for automatic digital computing machines to support the mathematical work of scientists. So did Stibitz at Bell Laboratories, as we have seen. And so did a graduate student in physics in Harvard University named Howard H. Aiken (1900–1973).