Tuesday, February 7, 2012

Let's Talk Algebra: Ten Minutes with James D Nickel

"I don't know why I should have to learn Algebra... I'm never likely to go there."
~Billy Connolly


My 2009 interview with James D. Nickel explored themes from his book Mathematics: Is God Silent. This past week I read a paper he recently wrote called Algebra: What's It All About, and I had to ask more questions. He answered elaborately and with some keen insights that I find fascinating.

Ennyman: There is a sense in which our understanding of mathematics of today came about by borrowing concepts from various sources. How did these various influences combine to become modern Algebra?

JDN: The development of algebraic syntax and symbolism has a long and complex history. Solving problems using analytical skills (in reality, methods that can be translated into algebraic symbols) is nearly a cultural universal. For example, the ancient Egyptians, Babylonians, and Chinese developed methods (or algorithms) to measure acreage, keep business accounts, assess taxation, and create calendars (by star gazing). Some of the clay Babylonian tablets have problems that require methods of algebra to solve (like completing the square, a method now used to derive the quadratic formula, a formula that almost every high school algebra student dutifully memorizes as a way to find solutions to a quadratic equation).

Today, Algebra is understood in two contexts: (1) Symbolic Algebra or the syntax, symbols, operations, and order by which to find solutions to a wide variety of equations (what we study in high school and what most students must take in college as part of its liberal arts requirement) and (2) the study of mathematical structures. Math majors are required to take courses called Modern or Abstract Algebra. They explore principal framework concepts like fields, groups, and rings. This branch of Algebra is called “Modern” because it was developed in the 19th century. A good book on all of this is Classical Algebra: Its Nature, Origins, and Uses, by Roger Cooke.

Regarding the development of Symbolic Algebra, we can start with the Classical Greeks (600-300 BC). They did not develop good symbolism (their Algebra was rhetorical) partly because of their preference for Geometry. Hence, they understood both arithmetical and algebraic processes in the context of Geometry (e.g., they classified numbers as square, rectangular, or triangular). Not having a positional number system didn’t help them either. In the second century AD, Diophantus of Alexandria, Egypt, did some original work and developed a syncopated form of Algebra (part rhetoric and part symbolic). An algebraic riddle defines his age: “His boyhood lasted 1/6 of his life; his beard began to grow after 1/12 more; after 1/7 more he married; five years later his son was born and the son lived to half his father’s age; the father died four years after his son; how old was Diophantus when he died?” Because of his work, Diophantus is often called “the Father of Algebra.”

Meanwhile, over in India, we see the Hindu development of the base-10 positional number system. For Algebra to really blossom, you need an optimal way to represent numbers.

In a couple of centuries, Islam became a force in world history. In the conquering mode, the warriors of Mohammed eventually reached India and there they recognized the value of the Hindu number system. In the ninth century, at the “House of Wisdom” in Baghdad, the Persian mathematician al-Khwarizmi wrote a book that connected algebraic methods with the new number system, a system we now know as the Hindu-Arabic positional system. Of course, Algebra (al-jabr) is an Arabic word and al-Khwarizmi used that word as a way of naming a foundational method of solving equations, i.e., adding or subtracting a number from both sides of an equation. At its best, al-Khwarizmi’s Algebra was still syncopated.

We have to wait until the sixteenth century for a full-fledged Symbolic Algebra to be developed. Both the French mathematician François Viète and the Flemish mathematician Simon Stevin played key roles. In the 17th century, René Descartes refined their work and then combined algebra with geometry in terms of visualizing algebraic equations, now understood as functions, graphically (i.e., the now familiar x-y coordinate plane). This marriage of Algebra with Geometry is called Analytical Geometry, Coordinate Geometry, or Cartesian Geometry.

It is important to note that Descartes’ invention was not a “bolt from the blue.” A French contemporary, Pierre de Fermat, actually developed this system before Descartes but Descartes was the first to publish. If Fermat had got to the printer earlier, we might now be studying Fermatian Geometry. And, three hundred years earlier, the French medieval scholastic Nicole Oresme had developed the idea of a mathematical function and a way to graph these functions on a “longitude/latitude” coordinate system (and, he borrowed some of his graphing ideas from Arabic mathematicians). What hindered him was the lack of a truly symbolic Algebra. Oresme published his work and both Descartes and Fermat had access to Oresme’s texts (but, like many scholars of the time, both Descartes and Fermat did not “footnote” their sources).

There is much more to this fascinating history, but I think that is enough for a summary.

EN: In what ways has Algebra been essential to our having successfully landed humans on the moon?

JDN: Without Newtonian mechanics, you could not have a successful moon launch. Newton’s seventeenth century publication Principia (or “Mathematical Principles of Natural Philosophy”) relies on the work of medieval scholastics like Oresme and John Buridan (Newton never recognized these “giants” of thinking and innovation), Galileo, Descartes, and Kepler. Newton used algebraic techniques to solve differential equations and thereby calculated the velocity a rocket must attain to escape the gravitational pull of the earth (before such a rocket could even be constructed!). See my paper on Differential Equations and Escape Velocity for how he did that.

NASA scientists also used Newton’s laws of motion, including his Universal Law of Gravitation (written, conveniently, in Algebraic notation), to determine the trajectories necessary to land the lunar module on the “proverbial dime” (if such a coin had been placed on the Moon beforehand!).

TO BE CONTINUED WEDNESDAY

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