The Man Who Transposed Human Thought Into Algebra

In this excerpt from The Laws of Thought, Tom Griffiths shares how George Boole developed a mathematical theory of logic.

Walking through a field one day, a 17-year-old schoolteacher named George Boole had a vision. His head was full of abstract mathematics — ideas about how to use algebra to solve complex calculus problems. Suddenly, he was struck with a flash of insight: that thought itself might be expressed in algebraic form.

Boole was born on November 2, 1815, at four o’clock in the afternoon, in Lincoln, England. He was the first child of John Boole, a shoemaker, and his wife Mary Ann. But John was no ordinary shoemaker — he was an enthusiast of science and mathematics, as likely to be making telescopes as shoes. Appropriately, his son George received a quality education, studying the classics as well as mathematics and learning to play the flute and piano. He quickly became fluent in Latin and Greek, and his translations of classical poems were published in the local newspaper when he was 14 (creating some controversy when a reader refused to believe they were the work of a schoolboy). When his father’s business began to fail a couple of years later, George became a teacher at a school in Doncaster. He began to read more deeply about mathematics — in part because it helped him stay within his book-buying budget as it took so long to work through each book.

The vision Boole had when walking through that field was arguably partly due to his idiosyncratic education. He had learned calculus from books by French mathematicians — Lacroix, Lagrange, and Laplace — who were part of a tradition that followed the approach to calculus introduced by Leibniz. This was more abstract and algebraic than the geometric approach taken by Newton that was favored in England, disposing him to think about systems that were richer than mere arithmetic. But Boole wasn’t able to immediately pursue his vision. His teaching job led to the opportunity to become the head of another school closer to his family, and ultimately to found a school of his own in Lincoln at age 19. 

Once his family was on a stronger financial footing, Boole began to publish a remarkable series of papers in mathematics. He used his algebraic perspective on calculus to good effect, publishing papers with titles like “Researches on the Theory of Analytical Transformations, with a Special Application to the Reduction of the General Equation of the Second Order” and “On the Integration of Linear Differential Equations with Constant Coefficients” in the Cambridge Mathematical Journal. Soon the lowly schoolteacher, who had never attended university, was corresponding with leading mathematicians. He then hit upon a bigger idea — a method for solving certain tricky differential equations — which he published as “On a General Method in Analysis” in the prestigious Proceedings of the Royal Society in 1844. The paper was so impressive that the Royal Society awarded Boole the first ever Gold Medal in Mathematics.

These papers secured Boole’s reputation as a mathematician, but working on them also gave him the tools that he needed to pursue what would become his life’s work: developing a mathematical theory of logic.

The laws of thought

In retrospect, it’s easy to see Boole as having been perfectly prepared to take on Leibniz’s challenge of arithmetizing Aristotle — his mathematical adventures had left him deeply steeped in algebra, and his study of classics meant that he was able to read Aristotle in the original Greek. However, he came to the challenge completely independently of Leibniz, only discovering their shared passion later in life. Instead, Boole was spurred to revisit his teenage vision by a debate between his friend the mathematician Augustus De Morgan and the philosopher William Hamilton. Both had arrived at a way of extending logic but disdained the methods of the other: De Morgan wanted a mathematical solution, while Hamilton focused on verbal arguments. At one point, Hamilton even accused De Morgan of plagiarism. Boole saw a way of shortcutting the debate, pursuing a mathematical approach in the spirit of De Morgan that formalized the intuitions of Hamilton.

Boole was consumed by the idea of putting logic on a solid mathematical foundation. As his sister MaryAnn described it:

When he first began to work at The Mathematical Analysis of Logic, his absorption was so extreme it seemed as if his brain was never at rest; by day he moved about like one in a dream, and at night lay awake for hours thinking out what had been before him as a vision all his life and many a time unable to sleep had he risen in the dead of the winter night and, wrapping himself in a cloak, descended to the family sitting room to write down the fast-coming trains of thought that would not be repressed.

Boole published The Mathematical Analysis of Logic as a short book only weeks after he started working on it. Aristotle’s syllogisms take pride of place — the book lays out an algebraic scheme for specifying the relationships between classes of objects, making it possible to derive new conclusions by checking whether certain equations follow from others, which Boole showed could be used to derive the valid syllogisms. The book was a breakthrough — but Boole also felt it had been written too hastily, and spent the next seven years perfecting his system and finally publishing the much longer book An Investigation of the Laws of Thought.

Boole was clear about his goals in the very first paragraph of his book:

The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolical language of a Calculus, and upon this foundation to establish the science of Logic and construct its method; to make that method itself the basis of a general method for the application of the mathematical doctrine of Probabilities; and, finally, to collect from the various elements of truth brought to view in the course of these inquiries some probable intimations concerning the nature and constitution of the human mind.

His path toward this goal was based on a simple idea: Let a symbol, x, denote a class of things. Now the machinery of algebra can be applied: “Thus if x alone stands for ‘white things,’ and y for ‘sheep,’ let xy stand for ‘white sheep’ …” Writing two algebraic symbols like this is a shorthand for multiplication, so what Boole is really saying is that we can write the class of white sheep as x × y. With this simple move, Boole began to identify the rules for manipulating these symbols. 

First, it is clear that xy = yx, as the class of sheep that are white is the same as the class of white things that are sheep. Second, if x and y are the same class of things, then it is reasonable to assert that xy = x, since white things that are white are just white things. We can then rewrite this as xx = x, or x2 = x, which is the first hint that we are entering unusual territory. It’s certainly not the case that x2= x for all possible values of x in the settings where we typically use algebra.

So when does x2 = x? Well, this expression is true if x = 1 or x = 0, which suggests that 1 and 0 are going to play a special role in this system. In fact, Boole noted that the rules of his algebra of logic followed the rules of our more familiar arithmetic when that arithmetic is restricted to variables that take the value 1 or 0. The symbols 1 and 0 also have clear meanings: 1 denotes the class of all things, so 1x = x, and 0 denotes the class that contains nothing, so 0x = 0.

Boole extended this analysis to another operation — if the classes x and y are distinct (with no members in common, so xy = 0), then x + y is the class of things that are in either x or y. Likewise x − y is the class of things in x but not in y. The operations represented here as multiplication and addition combine together in the familiar way, so x(y + z) = xy + xz.

With that, believe it or not, we’re ready to tackle some Aristotle. 

Aristotle expressed his syllogisms using four different kinds of statements: “Every A is B,” “No A is B,” “Some A is B,” and “Some A is not B.” Boole found algebraic expressions that captured each of these statements. For example, Boole’s system can be used to express “Every A is B” as a = ab (since everything that is A is B, the set of things that are both A and B is the same as the set of things that are A). Proving a syllogism to be valid is just a matter of algebraically manipulating the equations that express its premises to produce something equivalent to its conclusion.

For Aristotle’s argument “Every A is B, Every B is C, Therefore, every A is C,” our first step is translating the premises to a = ab and b = bc. We can then prove the syllogism is valid by replacing the b in the first premise with bc to obtain a = abc, and then replacing the ab with a to get a = ac, which is equivalent to “Every A is C.” Boole exhibited similar proofs for the other syllogisms.

An Investigation of the Laws of Thought went much further than this, developing a similar mathematical system for statements that were not explicitly about classes of objects. Boole’s system certainly had flaws — for example, the logician and economist William Stanley Jevons was unsatisfied with the fact that x + y could only be applied when x and y are distinct, and defined a revised system that addressed this problem. Boole’s proofs also contained errors that took many years to discover. But his work is nonetheless the first meaningful instantiation of what Descartes and Leibniz had dreamed of — a formal system that describes aspects of human thought.

- Author: Tom Griffiths, The Conversation