Wednesday, February 8, 2012

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.

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