Ivars Peterson's MathTrek

January 17, 2000

# Algebra, Philosophy, and Fun

I don't often encounter the words "philosophy" and "fun" right next to the term "algebra." Nowadays, these words don't seem to fit together comfortably. However, the three terms do appear in the title of an engaging little book called Philosophy and Fun of Algebra, written by Mary Everest Boole (1832-1916) and published in 1909.

I discovered the book while browsing the online Cornell University Library Math Book Collection, which consists of 571 volumes that were scanned from originals held by the library. The collection includes a number of historically significant works, by such prominent mathematicians as Jean Bernoulli, René Descartes, G.H. Hardy, and Henri Poincaré, many in French, German, and other languages. It also has a smattering of mathematical curiosities. I have my eye on How to Draw a Straight Line: A Lecture on Linkages by A.B. Kempe and Euclid and His Modern Rivals by Charles L. Dodgson.

Boole's gentle, conversational introduction to algebra is meant for children. It also serves as a window on math education--as it was perceived in some circles about a century ago.

"Arithmetic means dealing logically with facts which we know (about questions of number)," Boole begins. And she immediately launches into a discussion of what "logically" means and the scope and primacy of the laws of logic.

No Parliament can pass a law to make an answer come out right, she insists. ". . .governments have grown wiser by experience and found out that, as far as arithmetic goes, there is no use in ordering people to go contrary to the laws of the Logos [hidden wisdom], because the Logos has the whip hand, and knows its own business, and is master of the situation."

Why bother with algebra if you already know arithmetic? Boole answers: "When people had only arithmetic and not algebra, they found out a surprising amount of things about number and quantities. But there remained problems which they very much needed to solve and could not. They had to guess the answer; and, of course, they usually guessed wrong. And I am inclined to think they disagreed. . . . Probably they quarreled, and got nervous and overstrained and miserable, and said things which hurt the feelings of their friends. . . ."

That impasse led to the birth of x. "Instead of guessing whether we are to call it nine, or seven, or a hundred and twenty, or a thousand and fifty, let us agree to call it x, and let us always remember that x stands for the Unknown," she declares.

In 17 short chapters, Boole presents the basic concepts of algebra, with a variety of examples and snippets of mathematical theory, all seasoned with anecdotal pinches of history and philosophy. She frequently refers to biblical events and characters. One chapter on the question of choosing the proper working hypothesis focuses on Macbeth's tragic mistake in the play by William Shakespeare of failing to distinguish between the real and the imagined.

In the book's final chapters, Boole confronts the perplexities of the square root of -1 and the unlimited vistas of the infinite. As it does elsewhere in the book, her tone verges on the mystical.

"A story is told of a man at Cambridge who was expected to be Senior Wrangler [top mathematics student]; but he got thinking about the square root of minus one as if it were a reality, till he lost his sleep and dreamed that he was the square root of minus one and could not extract himself; and he became so ill that he could not go to his examination at all," Boole recounts.

"Angels, and the square roots of negative quantities, and the other things that have no existence in three dimensions, do not come to us to gossip about themselves; or the place they came from; or where they are going to; or where we are going to in the far future," she continues. "They are messengers from the As-Yet-Unknown; and come to tell us where we are to go next; and the shortest road to get there; and where we ought not to go just at present."

Boole's reference to the "As-Yet-Unknown" is sometimes quoted in present-day discussions of pantheism and other systems of belief.

"When square root of minus one comes to you, behave reasonably about him," Boole advises. "Treat him logically, exactly as if he were six or nine; only always remember to keep well in front of you the fact of your own ignorance. You may never find out any more about him than you know now; but if you treat him sensibly he will tell you plenty of truths about your x's and y's, and other unknown things."

Boole herself is a fascinating character. She was born in England but raised in France, where her father, a minister, sought a cure for a serious, lengthy illness that afflicted him. Her uncle, George Everest, had made the family name famous, leading a surveying party up to the mountain that now bears his name.

At the age of 11, Mary returned with her family to England. She was taken out of school to assist her father, teaching Sunday School and helping with sermons. She did not abandon her studies, however, and used books in her father's library to teach herself calculus. Through another uncle, John Ryall, she met the famous mathematician George Boole (1815-1864). He ended up tutoring her in mathematics, and she helped edit Boole's epochal 1854 book Laws of Thought. Even though Mary was 17 years younger, the two were married not long after her father died.

Tragically, George Boole died of pneumonia in 1864, leaving Mary to take care of five daughters, the youngest only 6 months old. She accepted a job as a librarian at Queen's College, London, which trained women to become governesses. Her real love was teaching, and, when she got the chance, she proved very good at it. Her fascination with the spirit world, however, led to her eventual resignation from the college, when controversy dogged the publication of her book on the message of "psychic science" for mothers and nurses.

Many years later, Mary's 1904 book, The Preparation of the Child for Science, had a lasting effect on the move toward progressive schools in England and the United States during the early part of the 20th century. She followed up that initial success with Algebra, Philosophy, and Fun; Logic Taught by Love: Rhythm in Nature and in Education; and Mathematical Psychology of Gratry and Boole for Medical Students. She often wrote about and encouraged the use of hands-an activities and items that are now often called "manipulatives" as a crucial part of math education.

Mary Everest Boole's eldest daughter married Charles Howard Hinton (1853-1907), who devised methods for visualizing the fourth dimension, invented the word tesseract to describe a hypercube, and wrote a story that is said to have inspired Edwin A. Abbott's Flatland (see A Stranger from Spaceland, Dec. 20, 1999). Another daughter, Alicia, developed an amazing feel for four-dimensional geometry as a child, became a mathematician, and introduced into English the word "polytope" to describe a four- or higher-dimensional convex solid. She later worked with the famed geometer H.S.M. Coxeter.

So, I started with an intriguing book title, and I ended up immersed in four-dimensional geometry!

References:

Bashmakova, I.G., and G.S. Smirnova. 2000. The Beginnings and Evolution of Algebra, trans. by A. Shenitzer. Washington, D.C.: Mathematical Association of America. (Not for children, but an interesting contemporary account of the origins of algebra.)

Boole, G. 1854. An Investigation of the Laws of Thought, on Which are Founded the Mathematical Theories of Logic and Probabilities. London: Walton and Maberley. A biography of George Boole is available at http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Boole.html.

Boole, M.E. 1909. Philosophy and Fun of Algebra. London: C.W. Daniel.

Coxeter, H.S.M. 1973. Regular Polytopes. New York: Dover.

Michalowicz, K.D.A. 1996. Mary Everest Boole (1832-1916): An erstwhile pedagogist for contemporary times. In Vita Mathematica: Historical Research and Integration with Teaching, Ronald Calinger, ed. Washington, D.C.: Mathematical Association of America.

A biography of Mary Everest Boole is available at http://www.agnesscott.edu/lriddle/women/boole.htm.