40 2. MATHEMATICAL CULTURES II = 1.5 and | = 1.75. Get upper and lower estimates in this way for all numbers of
turns from 1 to 8. What are the narrowest upper and lower bounds you can place
on the number of pins per turn in this way?
2.6. Suppose that the pins in Plate 4 had been joined by a curve winding in the
opposite direction. How would the numbers of turns of the spiral and the number
of pins joined compare? What change would occur in the slope of the spiral?
2.7. With which of the two groups of people mentioned by Plato do you find
yourself more in sympathy: the "practical" people, who object to being taxed to
support abstract speculation, or the "idealists," who regard abstract speculation as
having value to society?2.8. The division between the practical and the ideal in mathematics finds an
interesting reflection in the interpretation of what is meant by solving an equation.
Everybody agrees that the problem is to find a number satisfying the equation, but
interpretations of "finding a number" differ. Inspired by Greek geometric methods,
the Muslim and European algebraists looked for algorithms to invert the operations
that defined the polynomial whose roots were to be found. Their object was to
generate a sequence of arithmetic operations and root extractions that could be
applied to the coefficients in order to exhibit the roots. The Chinese, in contrast,
looked for numerical processes to approximate the roots with arbitrary accuracy.
What advantages and disadvantages do you see in each of these approaches? What
would be a good synthesis of the two methods?2.9. When a mathematical document such as an early treatise or cuneiform tablet
contains problems whose answers "come out even," should one suspect or conclude
that it was a teaching device—either a set of problems with simplified data to
build students' confidence or a manual for teachers showing how to construct such
problems?2.10. From what is known of the Maya codices, is it likely that they were textbooks
intended for teaching purposes, like many of the cuneiform tablets and the early
treatises from India, China, and Egypt?
2.11. Why was the Chinese encounter with the Jesuits so different from the Maya
encounter with the Franciscans? What differences were there in the two situations,
and what conditions account for these differences? Was it merely a matter of
the degree of zeal that inspired Diego de Landa and Matteo Ricci, or were there
institutional or national differences between the two as well? How much difference
did the relative strength of the Chinese and the Maya make?