Ancient Greek Mathematics

In these chapters It is important to watch the very rapid development of mathematics from primitive algorithms to a full-fledged theory, that stands today as a model for students and all of science, as well. Even still, the Greek contributions took time. There were false starts and dead ends. There was a point of view, a philosophy that denied research into new directions. As indicated above there are three distinct periods to regard the goals below:

1. The early period. (600+ - 450 BCE)
2. The classical period. (450 - 300 BCE)
3. The Helenistic period. (300 - 400 CE)

Goals

Among the many features you should regard are:

Teaching and Pedagogy.

• Pedagogy. Analyse how you would approach teaching mathematics in the style where there are a very limited number of books.
• How would you teach arithmetic in the Greek alphabetic system?
• View the nature of books. With no examples, no color, no preambles, and no discussion, how it is possible to use them to teach? What is the added responsibility of the teacher? Or have things changed very little with our modern textbooks?

Mathematics.

• The idea of proof.
• The concept of modeling natural phenomena (even the cosmos) on a unified basis.
• The introduction of axioms.
• The formation of numbers as objects transcending counting with individual meanings, and the division of valuations to length and number - a necessary consequence of incommensurables.
• Prime numbers and number theory.
• The earliest concepts of limit. (The Method of Exhaustion.)
• Greek counting and calculation.
• Greek astronomy and the impact of geometry on it.
• Greek notions of infinity.

Goals.

• Pedagogy. Analyse how you would approach teaching mathematics in the style where there are no principles, only practice.
• Can you identify problems that could not be solved by ancient methods, but which are very near to ones that can.
• What were the stimuli for the particular methods and algoithms developed.
• Why were conic sections never considered?
• How are nonlinear equations considered, solved? What do the Egyptians do? What do the Babylonians do?
• What evidence can you discern about general principles.
• What was the relation between the exact and the approximate? Was the distinction clearly understood? Is there similarity with the training of young students today. At what age are these distinctions finally absolute?
• Can you identify sufficient mathematics to handle the needs of commerce? to what level?

References

• A History of Mathematics, 2nd Ed., Carl B. Boyer, Revised by Uta C. Merzbach, Wiley, NY, 1991.
• A History of Mathematics - An Introduction, Victor Katz, Harper Collins, NY, 1993.
• The History of Mathematics - An Introduction, David M. Burton, McGraw-Hill, New York, 1997.
• A History of Mathematics, 5th Ed., Florian A. Cajori, Chelsea, New York, 1991.
• History of Mathematics , 2 volumes, David E. Smith, Dover, New York, 1958.