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Science

The Isaac Newton of logic

It was 150 years ago that George Boole published his classic The Laws of Thought, in which he outlined concepts that form the underpinnings of the modern high-speed computer. SIOBHAN ROBERTS chronicles the man and his method

BySIOBHAN ROBERTS
Saturday, March 27, 2004 - Page F9

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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 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. 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 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 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 failed.

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 innovations."

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 like rats."

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.



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