Algebra and Number Theory
In
this chapter, the level of abstraction increases. Mathematicians, still reeling
from the staggering successes in analysis in the 18th century, turn their attention
in the 19th century to fundamental and underlying structural issues and relations.
This is the century when the three ancient problems are resolved — all
in the negative. Mathematicians begin to see the power of symbolism, even to
the point of conceiving of relationships that are muddled by specific examples.
We also will see the beginning of how algebra
can be used to link diverse areas of mathematics and give new structure where
there was none before. This same program was one of the "pearls" of
the 20th century, where it was carried to remarkable depths.
Goals
Among the many features you should regard
are:
Philosophy and Mathematics.
- What are the combination of factors that led to the development of algebra.
- Trace the development, or meaning, of the term algebra throughout the century.
- There are several remarkable applications of algebraic and number theoretic
developments. What are they?
- The players. Are the mathematicians of this day truly professional or are
they of the class of wealthy (or poor) practitioners of their craft dependent
on other employment to earn money.
- Note the power and importance of symbolism at this juncture.
- Consider the circle squaring problem. Could the ancients have resolved this
problem? That is, were they philosophically capable of absorbing the new concepts
needed.
- What are the new pedagogical issues for teaching these new theories that
may differ from an earlier age?
References
- Ivan Niven, Numbers: Irrational
and Rational, Mathematical Association of America, NY, 1963.
- Uspensky, J. V. Theory of Equations. New York: McGraw-Hill, p. 256, 1948.
- A Brief History
of Algebra and Computing: An Eclectic Oxonian View Jonathan P. Bowen,
online.
