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 earlier date.
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
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. Devlin profess.
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
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 impact.
"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
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 mathematics.
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
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
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 processing.
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 false.
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.