Year 1804: George Boole's
The Laws of Thought = Modern high-speed computer
By SIOBHAN ROBERTS
Globe and Mail: Saturday, March 27, 2004 - Page
There is nothing more ubiquitous these days than the computer, the thinking
machine that has hardwired itself to our lives.
A quick Google search of "history of the computer" yields the website http://www.computerhistory.org,
which pegs the computer's invention to 1945. That year, John von Neumann, a
Hungarian-born mathematician at Princeton, wrote his "First Draft of a Report of
the EDVAC" (the Electronic Discrete Variable Automatic Computer).
In his report, Von Neumann outlined the architecture of a stored-program digital
computer, an ancestor of most computers in use today. (Also that year, Grace
Hopper, an admiral in the U.S. Navy, recorded the first computer "bug" -- a moth
stuck between the relays of a pre-digital computer.)
But the existence of both the computer and Google can be traced to a much
It was 150 years ago that George Boole published his literary classic The Laws
of Thought, wherein he devised a mathematical language for dealing with mental
machinations of logic. It was a symbolic language of thought -- an algebra of
logic (algebra is the branch of mathematics that uses letters and other general
symbols to represent numbers and quantities in formulas and equations).
In doing so, he provided the raw material needed for the design of the modern
high-speed computer. His concepts, developed over the past century by other
mathematicians but still known as "Boolean algebra," form the underpinnings of
computer hardware, driving the circuits on computer chips. And, at a much higher
level in the brain stem of computers, Boolean algebra operates the software of
search engines such as Google.
"Boole was the first cognitive scientist," says Keith Devlin, executive director
of the Center for the Study of Language and Information at Stanford University.
Dr. Devlin's work attempts to take Boole's concepts -- the mathematics of human
thought -- and apply them to human communication. "I'm trying to take it one
step further and it's damn hard," he says. "Boole was bold and successful, and
that was a mixture of genius and good luck."
How Boolean logic works isn't very difficult, or so the experts such as Dr.
The most basic and tangible example is the machinations of Boolean searches,
which operate on three logical operators: and, or, not.
Algebra gets factored in to this logical equation when Boole designates a
multiplication sign (x) to represent "and," an addition sign (+) to represent
"or," and a subtraction sign (-) to represent "not."
For example, in a Boolean search with the terms "Martin and sponsorship," the
"and" logic collates the search results to retrieve all records with both terms.
"Or" logic collates results to retrieve all the records containing one term, the
other or both. "Not" logic excludes records from your search results.
The same "and" gates and "or" gates drive computer circuitry, with streams of
electrons performing Boole's algebraic operations -- a computer's bits and bytes
operate on the binary system, as does Boole's algebra. He employs the number 1
to represent the universal class of everything (or true) and 0 to represent the
class of nothing (false).
But rather than delving any deeper in Boole's algebra (which now may seem not so
simple; consult the sidebar if you're still curious), it would be logical to
examine instead the historical context in which his invention had such an
"Boole's primary contribution was in showing that logic could be conceived of in
a radically different way," says Jim Van Evra, an associate professor of
philosophy at the University of Waterloo.
As Prof. Van Evra chronicles in an article to be published in the Biographical
Dictionary of Nineteenth Century British Scientists, logic was considered a dead
subject from the 17th to the 19th century. It was criticized as being
superfluous, a device that merely stirred the pot of knowledge already at hand.
In England during the early 19th century, however, that perception began to
change. Logic began to develop into a serious science.
Boole was born in Lincoln, England, in 1815, the eldest son of a poor shoemaker
who also had a passion for mathematics. He was a precocious child. His mother
boasted that young George, 18 months, wandered out of the house and was found in
the centre of town, spelling words for money.
Boole was fluent in Latin and Greek by the time he was 12, and subsequently
self-taught in French, German, Italian and Spanish. He became the sole support
for his family (as a teacher) at the age of 16, when his father's business
Having Cambridge University close at hand, he consulted the elite mathematicians
of the day. They invited him to attend as a student, but he could not afford the
time or money.
"Everything he did was from his own mind. That's why he was such a trailblazer,"
says Desmond Mac-Hale, author of George Boole: His Life and Works and an
associate professor of mathematics at University College, Cork. "Had he gone
down the standard path of schooling, he might not have hit upon such major
Cambridge mathematicians, still keen to encourage Boole, provided him access to
the mathematical library. And he succeeded in publishing several papers in the
Cambridge Mathematical Journal -- one of which, published in 1844, was awarded
the first-ever gold medal from London's Royal Society for a paper in
And though Boole was never offered a position at Cambridge, the university asked
for his well-regarded opinion about whom they should hire when they were seeking
a new professor of mathematics.
In 1849, he became the founding professor of mathematics at Queen's College (now
University College). In 1855, he married Mary Everest (niece of Sir George
Everest, for whom the mountain is named) and they raised five daughters in
Ireland not long after the potato famine.
He was also a very religious man. According to Prof. MacHale, all evidence
points to Boole's faith as Unitarian -- believing in God as one, not the
Trinity, which meshes with the prominent position he gave the number one in his
work. "It's my feeling that his motivation with his logic was religious," he
says. "He believed that the human mind was the greatest of God's creations."
Prof. MacHale also notes that subsequent to The Laws of Thought, Boole undertook
to rewrite the Bible in his mathematical logic. "He was slightly out of touch
with reality," he says. "It was a foolhardy project and it caused him a great
deal of torment because he could never accomplish it."
One anecdote about Boole's life that comes to the mind of Geoffrey Hinton, a
computer-science professor at the University of Toronto and his
great-great-grandson, was the way the mathematician died.
A devoted professor to his detriment, he walked the four miles one day from his
house to the college in a rainstorm. Soaking wet, he lectured all day and
subsequently died of pneumonia.
As Prof. Hinton tells it, "He was killed by homeopathy. His wife wrapped him in
wet sheets, thinking what caused the pneumonia would cure it." (Tangentially,
Prof. Hinton is quick to mention that his other great-great-grandfather was also
famous and ahead of his time -- James Hinton founded the first Victorian sex
cult, advocating woman should have fun while having sex, and profoundly
influenced the work of sexologist and psychologist Havelock Ellis.)
With his PhD in artificial intelligence, it might appear that Prof. Hinton
followed after Boole. But in fact, he says, "I'm entirely on the other side."
The field of artificial intelligence, in its early years circa 1950-60, was
committed to the Boolean idea that symbols effectively represent human
reasoning. Since the eighties, however, artificial intelligence has come to see
human reasoning as not purely logical. Rather, it is more about what is
intuitively plausible. "Boole thought the human brain worked like a pocket
calculator or a standard computer," Prof. Hinton says. "I think we're more like
Despite the fact that he is universally admired, Boole does have his detractors.
"People have their own heroes and they serve their heroes by being critical of
Boole," says John Corcoran, a professor of the history and philosophy of logic
at the University of Buffalo.
"[Gottlob] Frege is the main hero whose worshippers denigrate Boole," he says,
adding that there are five giants of logic: Aristotle, Boole, Frege, Kurt Godel
and Alfred Tarski. "Perhaps a few worshippers of Tarski or Godel will
occasionally take a swipe at Boole in order to show how 'advanced' they are.
Many of the Boole-bashers are people dedicated to proving that new ideas are
always better than old. Many of the Boole worshippers are also people dedicated
to proving that new ideas are always better than old, but they do not realize
how old Boole's ideas really are."
Prof. Corcoran, of course, falls into the class of Boole worshippers. But not
beyond all reason.
"There are major flaws in Boole's work that have come to light over the years.
It's been said that Boolean algebra isn't Boole's algebra -- it's the modern
refinement of Boole's work."
With the advantage of hindsight on the occasion of the sesquicentennial of the
publication of The Laws of Thought, the imperfections in his work go undisputed;
the analogy Boole drew between algebra and logic was not a perfect fit.
Prof. Corcoran addresses one flaw in a paper titled, Boole's Solutions Fallacy.
"Boole did not recognize the difference between the consequences of an equation
and the solution of an equation," he says. "This mistake might seem like a
technicality, but it mars a lot of Boole's thinking."
Nonetheless, Prof. Corcoran chooses to focus on Boole's positive contribution.
"Boole's book is really a classic of literature," he says. "He brought about a
revolutionary paradigm shift that dramatically changed the nature of logic. He
thought he was the Isaac Newton of logic, and he was."
Even Boole, dying at just 49, was well aware that The Laws of Thought would give
him a lasting reputation. In a letter penned while his book was still in
progress, he betrayed what Prof. MacHale calls an uncharacteristic lack of
modesty: "I am now about to set seriously to work upon preparing for the press
an account of my theory of Logic and Probabilities, which in its present state I
look upon as the most valuable, if not the only valuable contribution that I
have made or am likely to make to Science and the thing by which I would desire
if at all to be remembered hereafter."
Siobhan Roberts is a Toronto writer whose biography of geometer Donald Coxeter
will be published by Penguin in 2005
An idiot's guide To
The following is a bit of an idiot's guide to Boolean algebra (for something
more sophisticated, see John Corcoran's introduction to the latest edition of
The Laws of Thought, published by Prometheus Books, 2003).
The gist of George Boole's idea was to reduce logical thought to the mathematics
taught in an elementary algebra class. He showed how the numbers 1 and 0 and the
standard mathematical operations could be hijacked to perform logical reasoning
-- operations such as addition, multiplication and methods for solving equations
formed his symbolic language of thought.
Boole wanted his algebra of thought to include what is called the logic of
classes, which expanded on Aristotle's logic (the famous "All men are mortal"
syllogisms). And he wanted his method to encompass the logic of propositions,
based on logical work originating with the Stoics.
He employed the symbols x, y, z, etc. to denote arbitrary collections of objects
-- the collection of all men, the collection of all documents with the word "Boole,"
and so on -- and with the number 1 representing the set of everything and 0
representing the set of nothing.
He then explained how performing algebra with the symbols corresponded to
performing logical deductions.
In conducting a Boolean search, for example, an "and" operator (or a
multiplication sign -- x) between two words or other values (for example, "pear
and apple") means one is searching for documents containing both of the words,
not just one of them. An "or" operator (an addition sign -- +) between two words
or other values (for example, "pear or apple") means one is searching for
documents containing at least one of the words, not necessarily both.
In computers based on binary operations, Boolean logic is used to describe
electromagnetically charged memory locations or circuit states that are either
charged (1, or true) or not charged (0, or false). The computer can use an "and"
gate or an "or" gate operation to obtain a result that can be used for further
Boole's logic of propositions, similarly, is used to derive the truth-value of a
complicated proposition from the truth-values of simpler propositions.
An example might be the predicament of then-finance minister Paul Martin when
the sponsorship debacle was underfoot: Suppose, for example, we want to
contemplate the proposition that Mr. Martin knew about the scandalous
sponsorship slush fund "and" did nothing about it.
We first assign a value of 1 or 0 to the first proposition: Mr. Martin knew
about the slush fund. That is, we compute the truth-value: 1 for true, or 0 for
Then we assign a value of 1 or 0 to the second proposition: Mr. Martin did
nothing about it: again, 1 for true, 0 for false.
Boolean logic tells us to multiply these two truth-values together to get the
truth-value of the whole, compound proposition. One possibility being, 1 x 1=1 =
True: Mr. Martin knew about the sponsorship slush fund and did nothing about it.
The Prime Minister is saved from culpability for the disappeared hundreds of
millions if either proposition elicits a zero.