Finding Algebra After a Series of False Starts and Dead Ends

One of the items on my "to do" list is yellow weed digger. Our crab apple tree died last year. I chopped it down this spring. As it turns out, as many as 50 or more shoots are springing up where crab apples rotted and left their seeds in the yard. The yellow weed digger is an implement for uprooting and discarding these shoots. 

Because of the draught this past six weeks the ground had become hard, almost like concrete. Now that we've had a few days of rain I can make time to uproot those shoots in order to have a more normal-looking yard. 

Ideas are a little like all these new shoots. They seem to pop up on their own. Though they must come from somewhere it's hard to explain where or why. Being an avid reader with wide interests may be a contributing factor. My subconscious is constantly being fertilized. 

Here is a list of ideas that have been germinating this summer, producing a myriad quantity of false starts and incomplete or undeveloped blog posts. 

Why Punctuality Matters

Where Did Algebra Come From?

Capers 

The EV Dilemma 

Jurassic Park

The PC Police and 1984 (Thoughts about freedom)

A Strategy for Standing Out from the Crowd

The Future of Education

The Board Game Market

* * * 

The word Algebra comes from an Arabic word, Al-Jabr, which means the reunion of broken parts.

Dylan has written more than once about the brokenness in our world, most directly in the song Everything Is Broken, which appeared on Oh Mercy. What kind of algebra would it take to reunite our multitude of broken parts, broken lives, broken dreams? I do not believe politicians can solve this problem. To achieve a more satisfying solution we need to look for something more transcendent involving faith, hope and love. 

James D Nickel and The Dance of Number (Part 1)

In 1990 James D. Nickel's Mathematics: Is God Silent? was published by Ross House Books, then updated and expanded in 2001. For more than three decades he has been an educator, author and public speaker. He has also served many years as an IT professional. His demeanor is best characterized by the words joy and laughter, not entirely how you'd picture of math man. He's no clown, though. He has an acute mind and a lifelong devotion to sharing his understanding of math with students, fellow teachers and readers. Later this summer his new book The Dance of Number will be published. The title was so intriguing I had to reach out and ask Nickel to elaborate as to what it was all about.

EN: Why is it essential that children understand mathematics?
JDN: The principles of number and space are imbedded in created reality, the way the universe works and the way we think. It is the beauty and power of this reality that should be the primary motivation for studying and understanding mathematics, but in most cases it is not. Since utilitarianism governs most of math instruction (K-12), there is a tendency to focus on dictating rules without the requisite understanding, but it is in understanding why a principle works that a student is (1) introduced to the beauty of mathematics and (2) learns to master its unique symbolic language. And, in understanding the laws of mathematics, one becomes comfortable in the world of God’s making and how man has developed it. We don’t trump utility with beauty because both go together. They are two sides of the same coin. Mathematics is a unique tool of wonder.

EN: What is “good number sense” and why is it important?
JDN: Number sense is a “feel” for numbers. Number sense comes from understanding why, not just following a set of rules. As a high school teacher of mathematics, I have had a front row seat sampling a good number of incoming students. I have observed far too many students who “can’t do the work” simply because they have not mastered arithmetic. When it comes to dividing by a fraction, many cannot remember what to “invert,” a symptom of a deeper conceptual issue, “What does it mean to divide a number by a fraction and what will my answer approximately be?” I have also noted that very few students possess what I call the joy of “number sense”or the pulse of numerical patterns … how they work, how they interact, and how they reveal rational order.

EN: Your book speaks of a “revolution” in the way math is taught. In what ways do we teach arithmetic differently in the U.S. from the way it’s taught in India and the Far East?

JDN: The fundamental paradigm that I use for teaching addition, subtraction, multiplication, and division is based upon the “left to right” methods based upon a book by Edward Stoddard entitled Speed Mathematics Simplified (New York: Dover Publications, [1962, 1965] 1994). Unlike most “speed math” books, rather than providing a series of disjointed “tricks,” Stoddard teaches a streamlined and consistent method of speed arithmetic, based upon the Japanese abacus called the Soroban and its use of complements (1) , that integrates the same principles throughout. I have also incorporated some revisions to Stoddard’s methods using some of the techniques taught in the schools of India, commonly called Vedic mathematics.

EN: What difference does it make?
JDN: The connections and techniques I explore are sound, sensible, interesting, fascinating, and enjoyable. The ease of this method increases speed in computation, guarantees better accuracy, and engenders immediate estimates. For years, I asked myself why, for example, Japanese students excel in numeracy and Americans lope behind at a far distance. It is because of the Soroban and how it uses number complements, and left to right computation, to instill an astounding mastery of number sense.

EN: What is the essence of algebra and what are some of the benefits understanding algebra brings?
JDN: Algebra is the language of mathematics and mathematics is the language of science. You will never understand the nature and structure of mathematics, or the nature and structure of the physical world, unless you master the language of Algebra. As a language, Algebra has its own syntax that requires one to be conversant with it. If these symbols are not mastered, the symphony of mathematics is played in silence and you will never appreciate its profound beauty. In its history, Algebra, as a language, has resolutely pointed to a given order, a specificity of order, in the universe. Naturalistic mathematicians and scientists have done their best to avoid the implications of such order; i.e., the revelation of consistent rationality in the universe points to an ultimate ground of rationality beyond it. The beauty, symmetry, and wonder revealed in the harmonious dance of number is a portal through which one, who has eyes to see, can catch a faint glimpse, an infinitesimal glimmer, of the beauty and wonder of the Author and Sustainer of number.


(1) In the Hindu-Arabic Base Ten Positional System, the complement of a single digit number is the number you must add to it to get the sum of 10. The complement of 1, therefore, is 9, the complement of 2 is 8, the complement of 3 is 7, etc.

CONTINUED TOMORROW

Talking Algebra: James Nickel Interview, Part 2

The questions here were derived from James D. Nickel's essay "Algebra: What's It All About?" which I found exceedingly stimulating.
CONTINUED FROM YESTERDAY

EN: The building of the pyramids preceded the discovery of algebraic concepts by several millennia. How did the Egyptians achieve these feats at such an early point in history?

JDN: A good book on how the Egyptians used mathematics and principles of mechanical advantage to build the pyramids is A History of the Circle: Mathematical Reasoning and the Physical Universe by Ernest Zebrowski.

They also based their construction techniques on principles of trigonometry, long before the invention of the words sine, cosine, tangent, cotangent, etc. For more information, see Eli Maor’s Trigonometric Delights or a short essay I have written based upon one of Maor’s chapters entitled “The Proto-Trigonometry of the Pyramids”.

In summary, Egyptian engineers used the concepts of algebra and trigonometry to help them perform their herculean constructions. The concepts used then are the same as the concepts used now, except the way to express them, in algebraic symbols and trigonometric notation, is different. However, because of the lack of symbolic notation and many other necessary components (including a correct view of the nature of the physical world), the Egyptians could never have developed more intricate laws governing the motion of matter, as Newton was able to do.

EN: You write that algebra “is a way in which knowledge of the patterns of creation can be expressed, developed, and used to the benefit of mankind.” Can you elaborate on this?

JDN: Algebra is all about “order and operations.” The way the physical world works is intelligible and therefore orderly. Hence, we can expect that the methods of Algebra, developed by man made in God’s image, will be in sync with the workings of the physical universe, authored by the same Creator.

This creational correlation between the workings of man’s mind and the workings of the physical universe has much more explanatory power than asserting that natural processes involving chance or probabilistic collocations (i.e., evolution) are responsible for this “link” (pun intended). In the doctrine of Creation, we are confronted with an infinitely wise Creator who, by His transcendence and imminence, holds every aspect of the created reality, invisible and invisible, together by His powerful Word (Genesis 1:1; Hebrews 1:1-3; Colossians 1:15-17; John 1:1-3). Scripture asserts that Christ is the true source of the rationality and order revealed in the workings of the physical universe. In naturalism, all we have are blind and, by necessity, irrational forces. Given this premise, we are certainly fortunate that everything “came together” like it did but we really have no justification for end product, rationality, resulting from a process governed by irrationality.

Finally, given the premises of naturalism, the Scientific Revolution, fifteenth to seventeenth centuries, would never have gotten out of the starting blocks. We live today as technological benefactors of this revolution. As C. S. Lewis has observed about this era of history, “Men became scientists because they expected law in nature and they expected law in nature because they believed in a Creator.”

EN: Why did the Greeks have such a difficult time with the matter of infinity?

JDN: The Greeks were rationalistic. In contrast, the Biblical Christian is rational. The difference is that the Greeks believed that reason is the only way to truth while the Biblical Christian believes that reason is a tool to knowing truth, a tool that is built upon revelation; i.e., man made in God’s image.

When it came to infinity in mathematical processes, the Greeks were often stupefied. They could never really get their rationalistic minds around the “concept of infinity”. They found ways, at times, to transform infinite processes into finite ones (see the methods of Eudoxus and the work of Archimedes), but they were never able to formulate a rational, or logical, justification or explanation of infinity. In essence, the mathematical nature of infinity struck at the heart of their rationalism.

Newton’s work in the development of the Calculus depended upon infinite processes, both in differentiation and integration. By his time, thoughts about infinity had been entertained for centuries. Just refer to the meditations of the medieval scholastics and note the construction of the Gothic Cathedrals, how the transcendence of God’s infinity is built into the architecture, sometimes in a dizzying measure. There was no fear of infinity, neither in the mind of Newton nor in the mind of his contemporaries. After Newton, mathematicians seized the mantle of infinity and eventually, in the nineteenth century, defined infinite mathematical processes in terms of limits, called the “epsilon-delta” definition of a limit. Also in the nineteenth century, Georg Cantor, a man who embraced theological propositions, developed the mathematics of infinity in ways that will stagger anyone’s mind. Cantor’s symbolic understanding of infinity duplicated, in a subtle and nuanced way, the grandeur, minus the architecture, of the Gothic Cathedrals.

EN: In what ways does the Christian belief in a triune God help advance our understanding of algebra in general and the world as a whole?

JDN: Algebra is but one branch of mathematics. It is an important one but all its branches reveal both utility and wonder. The spirit of genuine mathematics, i.e., its methods, concepts, and structure, in contrast with mindless calculations, constitutes one of the finest expressions of the human spirit. The great areas of mathematics, algebra, number theory, combinatorics, real and complex analysis, topology, geometry, trigonometry, etc., have arisen from man’s experience of the world that the infinite, personal, Triune (the ultimate One and the Many), and Sovereign God has created and currently sustains. These branches of mathematics, constructively developed by man made in the image of God, enable man to systematize the given order and coherence (the unity in diversity or the proximate one and the many) of creation mediated to us by the Creator and upholder of all things, the logos and wisdom of God revealed in the person of the Lord Jesus Christ. This systematization not only gives man a tool whereby he can take effective dominion over the creation under God in Christ, but also gives man the experience and enjoyment of a rich intellectual beauty that borders the sublime in its infinitely complex, yet structured mosaic.

Thanks for the questions, Ed!

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Mathematics is beautiful. Visit this website for more of James Nickel's ideas and research.
This link will bring you to his essay on Algebra.
To purchase James Nickel's book “Mathematics: Is God Silent?” visit the Chalcedon book store.

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