The Essence of Problem-Solving in Algebra

Essay by TyrfingurUniversity, Bachelor's December 2006

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In the minds of many people, the word algebra is synonymous with headache or something equally unpleasant. The reason for this is quite clear; it seems that the educational establishment has really succeeded in convincing generations of intelligent people that the study of algebra is not only totally worthless, but ultimately meaningless. This is indeed a shame because algebra represents a major advance in the intellectual history of our species. In what follows I will attempt to show this, and furthermore, I intend to show that the basic methodological principle of algebra, which I call, for want of a better word, The Solution Principle, goes far beyond the discipline we call algebra or algebraic theory; it actually determines to a large extent our abilities as problem-solvers.

The Beginnings of Algebra:

The day man began to write is the day man began to do algebra! While this may be a slight exaggeration there are reasons to believe in it.

Where are the historical roots of algebra? In Babylonia, like the roots of writing in general. The people of today, having had the benefit of written language that has evolved for millenia, are prone to take it for granted. One has to stop and think deeply to realize to what extent the written word is an addition to our intellectual powers. The computer interface we call pencil and paper has given us among other things, the whole of modern mathematics, the general theory of relativity, the atomic bomb, the computer and the internet! Don´t deceive yourself dear reader, the power of signs on paper with some rules of manipulating those signs is the greatest extension there is, to the human mind. Let´s look more closely at this first and oldest supercomputer, algebra.

The Axiom of ConsistencyE

In "What is Cantoras Continuum...